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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0658.00011.]
The image of the period mapping \(\phi\) of a simple hypersurface singularity X is shown to be a punctured neighbourhood of 0 when dim X is odd (respectively, its closure is a neighbourhood of 0 when dim X is even). Moreover in the first case (excepting \(X=A_ 1)\) there are arbitrarily many points with the same image under \(\phi\), while in the second case there is a bound for this number in terms of dim X and the Milnor number \(\phi\) (X). period mapping; hypersurface singularity Deformations of complex singularities; vanishing cycles, Singularities of surfaces or higher-dimensional varieties, Complex singularities Image of period mappings for simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The theory of moduli of curves has been extremely successful and part of this success is due to the compactification of the moduli space of smooth projective curves by the moduli space of stable curves. A similar construction is desirable in higher dimensions but unfortunately the methods used for curves do not produce the same results in higher dimensions. In fact, even the definition of what stable should mean is not entirely clear a priori. In order to construct modular compactifications of moduli spaces of higher dimensional canonically polarized varieties one must understand the possible degenerations that would produce this desired compactification that itself is a moduli space of an enlarged class of canonically polarized varieties.
The main purpose of the present article is to discuss the relevant issues that arise in higher dimensions and how these lead us to the definition of stable varieties and stable families. Particular emphasis is placed on understanding the singularities of stable varieties including some recent results. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Minimal model program (Mori theory, extremal rays), Families, moduli of curves (algebraic) Singularities of stable varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a \(\mathbb Q\)-Gorenstein normal variety over a field \(K\) of characteristic zero and \(\mathfrak{a}\subseteq \mathcal O_X\) be an ideal sheaf of \(X\). Let \(f:\widetilde{X}\to X\) be a log resolution of \((X, V(\mathfrak{a}))\), i.e. \(f\) is proper and birational, \(\widetilde{X}\) is nonsingular and \(f^{-1}V(\mathfrak{a})=:F\) is a divisor with simple normal crossing support. Let \(K_{\widetilde{X}| X}\) be the relative canonical divisor of \(f\). The multiplier ideal of \(\mathfrak{a}\) with coefficient \(t\in \mathbb R_{>0}\) ist
\[
\mathcal J(\mathfrak{a}^t)=\mathcal J(t\cdot\mathfrak{a})=f_\ast\mathcal O_{\widetilde{X}}(\lceil K_{\widetilde{X}| X} -t F\rceil)\subseteq \mathcal O_X\;.
\]
Let \(J (X| K)\) be the Jacobian ideal sheaf of \(X\) over \(K\), \(\mathfrak{a}, \mathfrak{b} \subseteq \mathcal O_X\) be two nonzero ideal sheaves on \(X\) and \(s,t\) positive real numbers. It is proved that \(J(X| K)\mathcal J (\mathfrak{a}^t \mathfrak{b}^s)\subseteq\mathcal J(\mathfrak{a}^t\mathcal J(\mathfrak{b}^s))\) and \(\mathcal J((\mathfrak{a}+\mathfrak{b})^t)=\sum\limits_{\lambda+\mu=t}\mathcal J(\mathfrak{a}^\lambda\mathfrak{b}^\mu)\).
As an application, Hochster-Huneke's formula on the growth of symbolic powers of ideals in singular affine algebras is improved:
Let \(R\) be an equidimensional reduced affine algebra over a perfect field \(K\) of positive characteristic or a normal \(\mathbb Q\)-Gorenstein affine domain over a field \(K\) of characteristic zero. Let \(J(R| K)=:J\) be the Jacobian ideal of \(R\) over \(K\) and \(\mathfrak{a}\subseteq R\) be any ideal which is not contained in any minimal prime ideal. Let \(h\) bet he largest analytic spread of \(\mathfrak{a}R_P\) as \(P\) runs through the associated primes of \(\mathfrak{a}\). Then for every integer \(m\geq 0\) and every integer \(n\geq 1\), \(J^n\mathfrak{a}^{(hn+mn)}\subseteq (\mathfrak{a}^{(m+1)})^n\). \(\mathbb Q\)-Gorenstein; Jacobian ideal Ta4 S.~Takagi, Formulas for multiplier ideals on singular varieties, Amer. J. Math. \textbf 128 (2006), no. 6, 1345--1362. Singularities in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Divisors, linear systems, invertible sheaves Formulas for multiplier ideals on singular varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider a complex hypersurface \(V\) given by an algebraic equation in \(k\) unknowns, where the set \(A\subset\mathbb{Z}^k\) of monomial exponents is fixed, and all the coefficients are variable. In other words, we consider a family of hypersurfaces in \((\mathbb{C}\setminus 0)^k\) parametrized by its coefficients \(a =(a_{\alpha})_{\alpha \in A} \in\mathbb{C}^A \). We prove that when \(A\) generates the lattice \(\mathbb{Z}^k\) as a group, then over the set of regular points \(a\) in the \(A\)-discriminantal set, the singular points of \(V\) admit a rational expression in \(a\). Laurent polynomials; general algebraic hypersurfaces; \(A\)-hypersurfaces; \(A\)-discriminantal set; singular point; \(A\)-discriminant; logarithmic Gauss map Singular points of complex algebraic hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert schemes of points on a \(K3\) surface are irreducible holomorphic symplectic manifolds. The cup products in the integral cohomology are studied. A computer algebra program is used to compute the cup products. The source code and an explanation how to use it is given in an appendix.
Let \(S^{[3]}\) be the Hilbert scheme of three points on a projective \(K3\) surface. \(\mathrm{Sym}^kH^2(S^{[3]}, \mathbb{Z})\) can be identified with its image in \(H^{2k}(S^{[3]}, \mathbb{Z})\) under the cup product mapping. As a result of the computations the following theorem is obtained:
\(H^4(S^{[3]}, \mathbb{Z})/\mathrm{Sym}^2H^2(S^{[3]}, \mathbb{Z})\simeq\mathbb{Z}/3\mathbb{Z}\oplus \mathbb{Z}^{23}\)
\(H^6(S^{[3]}, \mathbb{Z})/H^2 (S^{[3]}, \mathbb{Z})\cup H^4(S^{[3]}, \mathbb{Z})\simeq(\mathbb{Z}/3\mathbb{Z})^{23}\). \(K3\) surface; Hilbert scheme of points; integral cohomology; cup product 8. S. Kapfer, Computing cup-products in integral cohomology of Hilbert schemes of points on K3 surfaces, to appear in LMS J. Comput. Math. Families, moduli, classification: algebraic theory, Computational aspects of higher-dimensional varieties, Combinatorial aspects of partitions of integers, Orbifold cohomology Computing cup products in integral cohomology of Hilbert schemes of points on \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let X be a regular scheme, G a reductive group scheme over X. Serre and Grothendieck conjectured that any rationally trivial G-torsor is locally trivial in the Zariski topology of X. We prove this conjecture when \(\dim (X)=2\) and G is quasi-split over X. reductive group scheme; torsor Nisnevich, Y., Rationally trivial principal homogeneous spaces, purity and arithmetic of reductive group schemes over extensions of two-dimensional regular local rings, C. R. Acad. Sci. Paris, Sér. I, Math., 309, 651-655, (1989) Homogeneous spaces and generalizations, Group schemes Rationally trivial principal homogeneous spaces, purity and arithmetic of reductive group schemes over extensions of two-dimensional local regular rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f\) be a real analytic function defined in a neighbourhood of the origin in \(\mathbb{R}^{n}.\) The main theorem of the paper is the resolution of singularities of \(f\) in the following sense: there is a ``partition'' of a neighbourhood of the origin such that for each piece \(N\) of this partition the function \(f,\) after appropriate composition, is a monomial i.e.
\[
f\circ \Psi (x)=c(x)m(x),
\]
where \(c(x)\) is nonvanishing, \(m(x)\) is a monomial and \(\Psi \) is a composition of reflections, translations, invertible monomial maps and quasi-translations (the last maps are:
\[
(x_{1},\ldots ,x_{n})\mapsto (x_{1},\ldots ,x_{j-1},x_{j}-r(x_{1},\ldots ,x_{j-1},x_{j+1},\ldots ,x_{n}),x_{j+1},\ldots ,x_{n}),
\]
where \(r\) is analytic).
As applications the author gives:
1. the criteria for the exponents \(\epsilon _{i}\) for the integrability of \( \int_{O}\prod_{i}f_{i}^{-\epsilon _{i}},\) where \(f_{i}\) are real analytic functions and \(O\) is a neighbourhood of the origin in \(\mathbb{R} ^{n}.\)
2. a proof of the Łojasiewicz inequality\(\;\left| f_{2}\right| \geq C\left| f_{1}\right| ^{\mu },\) \(C,\mu >0,\) on compact sets for any two analytic functions \(f_{1},\) \(f_{2}\) for which \(V(f_{2})\subset V(f_{1}).\) singularity; real analytic function; resolution of singularity; Lojasiewicz inequality Greenblatt M.: A coordinate-dependent local resolution of singularities and applications. J. Funct. Anal. 255(8), 1957--1994 (2008) Local complex singularities, Real algebraic and real-analytic geometry, Global theory and resolution of singularities (algebro-geometric aspects) An elementary coordinate-dependent local resolution of singularities and applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A numerical criterion is given for the vanishing of the first cohomology of an invertible sheaf on a resolution of a normal complex analytic surface singularity. The criterion is sharp for rational surface singularities. Applications are given, including simplified proofs of vanishing results already in the literature. vanishing of the first cohomology; rational surface singularities A. Röhr, A vanishing theorem for line bundles on resolutions of surface singularities, Abh. Math. Semin. Univ. Hambg. 65 (1995), 215--223. Vanishing theorems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) A vanishing theorem for line bundles on resolutions of surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper contains a lot of material and information about maximal Cohen--Macaulay modules \(M\) over the local ring \(A\) of a surface singularity (\(\dim (A)=\text{depth}(M)\)). It starts with basic results and definitions as for instance the depth lemma and the Auslander--Buchsbaum formula, contains Matlis Duality, Grothendieck's Local Duality and a lot of other stuff from commutative algebra related to the study of maximal Cohen--Macaulay modules. Then general properties of maximal Cohen--Macaulay modules over surface singularities are presented as for instance the fact that in case \(A\) is normal \(M\) is maximal Cohen--Macaulay if and only if it is reflexive. It follows a section about maximal Cohen--Macaulay modules over two--dimensional quotient singularities containing the result that a normal surface singularity is a quotient singularity if and only if it has finite Cohen--Macaulay representation type.
The algebraic and the geometric approaches to McKay correspondence for quotient surface singularities as well as its generalization for simply elliptic and cusp singularities are described. A new proof of a result of \textit{R.-O. Buchweitz, G.-M. Greuel} and \textit{F.-O. Schreyer} [Invent. Math. 88, 165--182 (1987; Zbl 0617.14034)] (the surface singularities \(A _{\infty}\) and \(D_{\infty}\) have countable Cohen--Macaulay representation type) is given. At the end one can find some conjectures concerning the Cohen--Macaulay representation type and an example for a \textsc{Singular}--computation. Cohen-Macaulay modules; surface singularity; quotient singularity Burban, I., Drozd, Y. (2008). Maximal Cohen--Macaulay modules over surface singularities, trends in representation theory of algebras and related topics.EMS Ser. Congr. Rep., Eur. Math. Soc.Zürich, pp. 101--166. Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Cohen-Macaulay modules Maximal Cohen-Macaulay modules over surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be the ring of integers of an algebraic number field \(K\), and denote by \(X\) a separated noetherian scheme which is projective, flat, and with one-dimensional fibers over the arithmetic curve \(S = \text{Spec} (A)\). Such a datum, with \(X\) being a two-dimensional arithmetic variety, is called a relative arithmetic curve over \(S\), and \(X\) is referred to as an arithmetic surface over \(A\). If the generic fibre of the structure morphism \(f : X \to S\) is smooth, then \(X\) is called a regular arithmetic surface over \(A\). For regular arithmetic surfaces, there is an intersection theory for divisors and line bundles, which has been established by S. Arakelov in the early 1970s [cf. \textit{S. Yu. Arakelov}, Math. USSR, Izv. 8(1974), 1167-1180 (1976); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1179-1192 (1974; Zbl 0355.14002)]. Arakelov's intersection theory on arithmetic surfaces is based upon ``completing'' \(X\) by adding fibers (Riemann surfaces) over the archimedean places of the number field \(K\), and by enlarging both the divisor class group and the Picard group of \(X\) in a suitable way. The so-called Arakelov-Picard group (of isomorphism classes of ``admissibly metrized line bundles'') of \(X\) is equipped with an intersection pairing \(\langle , \rangle\), whose significance culminates in a Riemann-Roch-type theorem for regular arithmetic surfaces. In his celebrated paper ``Calculus on arithmetic surfaces'' [Ann. Math., II. Ser. 119, 387-424 (1984; Zbl 0559.14005)], \textit{G. Faltings} constructed certain volume forms on the cohomology of a metrized line bundle \(L\) restricted to the fibres over the points at infinity, and these allowed to define the notion of an ``Euler characteristic'' \(\chi (L)\) for an admissibly metrized line bundle \(L\) on \(X\). Falting's Riemann-Roch theorem for regular arithmetic surfaces implies the formula \(\chi (L) = {1 \over 2} \cdot \langle L,L - K_{X/S} \rangle + \chi ({\mathcal O}_X)\), where \(K_{X/S}\) denotes the relative dualizing sheaf on \(X\) over \(S\). Actually, Faltings's Riemann-Roch theorem has a formulation not only in terms of numerical invariants but in terms of a certain metric isomorphism between certain sheaves on \(X\). In the meantime, the arithmetic Riemann-Roch theorem has been generalized to arbitrary projective and flat morphisms \(f : X \to Y\) between regular arithmetic varieties \(X\) and \(Y\). The ``arithmetic Riemann-Roch theorem'', in this general setting, is due to \textit{J. M. Bismut}, \textit{G. Lebeau}, \textit{M. Gillet} and \textit{C. Soulé}, and was obtained around 1990. Detailed accounts on these recent developments are given in the lecture notes by \textit{G. Faltings}: ``Lectures on the arithmetic Riemann-Roch theorem'', Ann. Math. Stud. 127 (1992; Zbl 0744.14016), and in the book of \textit{C. Soulé}, \textit{D. Abramovich}, \textit{J.-F. Burnol} and \textit{J. Kramer}: ``Lectures on Arakelov geometry'', Camb. Stud. Adv. Math. 33 (1992; Zbl 0812.14015).
In the present work, the author turns to the (so far still open) case of non-regular arithmetic varieties. His objects of study are singular arithmetic surfaces \(X\) over \(S\), whose fibers are Cohen-Macaulay curves, and whose relative dualizing sheaf \( K_{X/S}\) is invertible. The text gives a development of a modified Arakelov theory for arithmetic surfaces, which is general enough to handle also a large class of singular surfaces, and which culminates in a Riemann-Roch-type theorem for such arithmetic surfaces.
Basically, the work is divided into two parts.
The first part, chapters 1 through 3, provides a very detailed and self-contained discussion of Deligne's functorial intersection theory for invertible sheaves on families of curves. The author's approach, however, is more direct and concrete, especially tailored to the specific arithmetic situation to come. Bypassing the abstract framework, as developed by \textit{P. Deligne} [in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata 1985, Contemp. Math. 67, 93-117 (1987; Zbl 0629.14008)], and \textit{D. Mumford} and \textit{F. Knudsen} [Math. Scand. 39, 19-55 (1976; Zbl 0343.14008)], he gives a merely ad-hoc treatment which is closer to the construction of admissible metrics on arithmetic line bundles due to \textit{L. Moret-Bailly} [in: Séminaire sur les pinceaux arithmétiques: La conjecture de Mordell, Astérisque 127, 29-87 (1985; Zbl 0588.14028)]. This approach also yields much of the needed duality theory, including the geometry of the dualizing sheaf, the associated duality isomorphisms, and the respective Riemann-Roch isomorphism.
The second part, chapters 4 and 5, is devoted to developing a class of intersection functions on complex curves, as they occur as fibers over the points at infinity in (possibly singular) arithmetic surfaces, and to extending the relative Riemann-Roch isomorphism (of the first part) to the ``completed'' object. The intersection functions constructed here behave analogously to the canonical Green's functions used in the smooth case, but they also show major differences to them. For example, Arakelov's canonical Green's function for regular arithmetic surfaces is unique, whereas the author's functions are parametrized by a finite-dimensional vector space. Also, Arakelov's function is bounded from below, whereas the author's functions are not bounded but, nevertheless, asymptotically nicely behaved at the singularities. Using his intersection functions in combination with his adapted functorial intersection theory on singular relative curves, the author obtains metrized invertible sheaves on particular (see above) singular arithmetic surfaces, which have a well-defined degree and a meaningful Euler characteristic. The general Riemann-Roch isomorphism extended to the ``completed'' (singular) arithmetic surfaces yields then a Riemann-Roch formula, just by taking degrees on both sides of it.
Altogether, the present work is a comprehensive and self-contained treatise which provides not only a generalization of Faltings's Riemann-Roch theorem to certain singular arithmetic surfaces, but also an alternative approach to two-dimensional Arakelov theory from a functorial point of view. arithmetic curve; Arakelov-Picard group; arithmetic Riemann-Roch theorem; non-regular arithmetic varieties; Green's functions; Riemann-Roch isomorphism; Arakelov theory Arithmetic varieties and schemes; Arakelov theory; heights, Singularities of surfaces or higher-dimensional varieties, Curves of arbitrary genus or genus \(\ne 1\) over global fields An arithmetic Riemann-Roch theorem for singular arithmetic surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the review in Zbl 0829.32012. Compact complex \(3\)-folds, \(3\)-folds, Modifications; resolution of singularities (complex-analytic aspects) Smoothing 3-folds with trivial canonical bundle and ordinary double points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth threefold with trivial canonical bundle and disjoint rational curves \(C_1, \dots, C_l\) of type \((-1, 1)\), then one can contact those rational curves to obtain a singular threefold \(X_0\) with \(l\) ordinary double points. Here, by type \((-1, 1)\), we mean that the normal bundle of each \(C_1, \dots, C_l\) is a direct sum of two tautological line bundles on the rational curve. A natural question is whether or not there exists a smoothing of \(X_0\), in other words, there exists a complex fourfold \(\mathcal X\), together with a proper flat map \(\pi : {\mathcal X} \mapsto \Delta\), where \(\Delta\) is the unit disk in \(C\), such that \(\pi^{-1} (0) = X_0\) and \(\pi^{-1} (t) = X_t\) is smooth for \(t\neq 0\). It is shown by R. Friedman that there is an infinitesimal smoothing of \(X_0\) if and only if the fundamental classes \([C_i]\) in \(H^2 (X; \Omega^2_X)\) satisfy a relation \(\sum_i \lambda_i [C_i] = 0\) such that, for every \(i\), \(\lambda_i \neq 0\). This infinitesimal smoothing can be realized by a real smoothing in case the obstruction group \({\mathbf T}^2_{X_0}\) happens to be zero.
The purpose of this paper is to show that the infinitesimal smoothing can always be ralized by a real smoothing. The main result is the following.
Theorem 0.1. Let \(X_0\) be a singular threefold with \(l\) ordinary double points as only singular points \(p_1, \dots, p_l\). Let \(X\) be the small resolution of \(X_0\) by replacing \(p_i\) by a smooth rational curve \(C_i\). Assume that \(X\) is Kähler and has trivial canonical line bundle. Furthermore, we assume that the fundamental classes \([C_i]\) in \(H^2(X, \Omega^2_X)\) satisfy a relation \(\sum_i \lambda_i [C_i] = 0\) such that, for every \(i\), \(\lambda_i \neq 0\). Then \(X_0\) admits a smoothing. smooth threefold; singular threefold; ordinary double points; infinitesimal smoothing; real smoothing; resolution G Tian, Smoothing \(3\)-folds with trivial canonical bundle and ordinary double points (editor S T Yau), Int. Press, Hong Kong (1992) 458 Compact complex \(3\)-folds, \(3\)-folds, Modifications; resolution of singularities (complex-analytic aspects) Smoothing 3-folds with trivial canonical bundle and ordinary double points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0723.00022.]
The author makes a brief historical note on desingularization: there is an interesting survey of the techniques of \textit{Zariski}, \textit{Abhyankar} and \textit{Hironaka} and a clear summary of the classical proofs of desingularization in dimension 1 and 2.
At the end, there is an announcement of a resolution theorem of dimension 3 singularities of the equation: \(T^ p-f(x_ 1,x_ 2,x_ 3)\) where \(p\) is the characteristic. The author gives some hints about this proof: there are 60 different cases which are controlled with numerical characters built with Newton polyhedras.
Unfortunately, the author did not quote the last results of \textit{E. Bierstone} and \textit{P. D. Milman} [cf. Effective methods in algebraic geometry, Proc. Symp., Castiglioncello 1990, Prog. Math. 94, 11-30 (1991; Zbl 0743.14012) and J. Am. Math. Soc. 2, No. 4, 801-836 (1989; Zbl 0685.32007), of the reviewer in Géométrie algébrique et applications, C. R. 2. Conf. int., La Rabida 1984, Vol. I: Géométrie et calcul algébrique, Trav. Cours 22, 1-21 (1987; Zbl 0621.14015)] and of \textit{O. Villamayor} [``Patching local uniformizations'', Ann. Sci. Éc. Norm. Supér., IV. Sér. 25 (1992)]. desingularization; resolution; dimension 3 singularities; Newton polyhedras Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Development of contemporary mathematics On the resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Für ein \(p\)-fach lineares System der Curven \(L_{n\nu} \; n^{\text{ter}}\) Ordnung \(\nu^{ten}\) Geschlechtes mit gegebenen \(i\)-fachen Punkten finden bekanntlich die Gleichungen:
\[
\begin{aligned} \varSigma i^2 -i& =n^2 -3n+2-2\nu\\ \varSigma i^2 +i& =n^2 +3n-2p\end{aligned}
\]
statt. Der Verfasser stellt sich nun die Aufgabe, für Curvenbüschel \((p=1)\) und Netze \((p=2)\) die Fälle zu untersuchen, in denen sämmtliche Curven \(L_{n\nu}\) in Curven niederer Ordnung zerfallen, von denen dann eine variabel, die anderen fest sein werden. Ueser diese Zerfällungen wird eine Reihe merkwürdiger Theoreme hergeleitet,welche sich namentlich auf den folgenden Satz stützen:
Wenn alle Curven eines Büschels \(L_{n\nu}\) unter den genannten Voraussetzungem aus einer variabelen Curve \(C_{s\sigma}\) und einer festen Curve \(C_{m\mu}\) betehen, so muss die erstere durch alle gegebenen vielfachen Punkte des Büschels und zwar durch jeden mit einer mindestens eben so hohen Multiplicität, wie \(C_{m\mu}\) hindurch gehen. Referent hat sich indessen nicht davon überzeugen können, dass der dafür gegebene beweis ausreichend ist , insofern derselbe sich auf die gleich Eingangs erwähnte Bemerkung stützt, dass bei den Curven \(L_{n\nu}\) das Auftreten von weiteren vielfachen Punkten gleich ein Zerfallen derselben bewirkt; ein Umstand, der allgemein nur bei den Curven \(\nu=0\) stattfindet. algebraic plane curves; singularities Pencils, nets, webs in algebraic geometry, Plane and space curves, Singularities of curves, local rings About some singularities in pencils and nets of plane algebraic curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Es sei \(K\) ein Körper, \((X^i_u)\) eine \((m,n)\)-Matrix von Unbestimmten über \(K\) und \(r\) eine ganze Zahl, derart daß \(1\leq r< m\leq n\) ist. \(P\) sei die Lokalisierung von \(K[X^i_u: 1\leq i\leq m,\ 1\leq u\leq n]\) nach dem irrelevanten maximalen Ideal, \(R\) der Restklassenring von \(P\) nach dem von allen \((r+1)\)-Minoren der Matrix \((X^i_u)\) erzeugten Ideal. \(D\) bezeichne den Differentialmodul von \(R\) über \(K\), \(D^*\) sein \(R\)-Dual \(\Hom_R(D,R)\).
In der Arbeit wird zunächst die Tiefe von \(D^*\) berechnet: Bei \(m=n\) ist \(\operatorname{Tiefe} D^*=\dim R-1\), und bei \(m<n\) ist \(D^*\) ein Cohen-Macaulay-Modul. Hieraus ergibt sich das folgende Resultat von \textit{T. Svanes} [``Coherent cohomology on flag manifolds and rigidity'', Ph. D. Thesis (MIT, Cambridge Mass. 1972), 6.8.1]:
Es ist \(\operatorname{Ext}^i_R(D,R)=0\) für \(i=1,\ldots,2(m-r-1)\) im Falle \(m=n\) und für \(i=1,\ldots,m+n-2r-1\) im Falle \(m\neq n\). Dabei sind die angegebenen Grenzen für das Verschwinden von \(\operatorname{Ext}^i_ R(D,R)\) in den Fällen \(m\neq n-1\) scharf.
Bei \(r+1=m=n-1\) gilt \(\operatorname{Ext}^i_R(D,R)=0\) für \(i=1,\ldots,5\), \(\operatorname{Ext}^6_ R(D,R)\neq 0,\) und bei \(r+1<m=n-1\) hat man \(\operatorname{Ext}^i_R(D,R)=0\) für \(i=1,\ldots,2(m- r+1)\), \(\operatorname{Ext}_R^{2(m-r)+3}(D,R)\neq 0\).
Die beschriebenen Ergebnisse benutzen ganz wesentlich die Untersuchungen über Hodge-Algebren von \textit{C. de Concini}, \textit{D. Eisenbud} und \textit{C. Procesi} [Astérisque 91, 87 p. (1982; Zbl 0509.13026)].
Der letzte Abschnitt enthält als Anwendung und in Anlehnung an Resultate von \textit{R. O. Buchweitz} [''Déformations de diagrammes, déploiements et singularités très rigides, liaisons algébriques'' (Thèse, Paris 1981)] eine in gewissem Sinne komplette Beschreibung der Starrheit von \(R\). Hodge algebra; syzygy; determinantal singularity; depth of dual of module of derivations Vetter, U.: Generische determinantielle singularitäten: homologische eigenschaften des derivationenmoduls. Manuscripta math. 45, 161-191 (1984) Determinantal varieties, Morphisms of commutative rings, Singularities in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Local rings and semilocal rings, Polynomial rings and ideals; rings of integer-valued polynomials Generic determinantal singularities: homological properties of the module of derivations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((X,p)\) be a normal complex surface singularity and \(\pi:(M,A)\rightarrow (X,p),\;\pi^{-1}(p)=A\) be a minimal resolution. It is assumed that \((X,p)\) is numerical Gorenstein, i.e., the canonical divisor in \(M\) is numerically equivalent to a cycle supported in \(A\). For any cycle \(B\) in \(A\) let \(Z_B\) be the minimal positive cycle supported in Supp\((B)\) such that \(-Z_B\) is nef on \(B\), i.e., \(-Z_B\cdot C\geq 0\) for all irreducible components \(C\) of \(B\).
Let \(C_t:=\sum_{i=0}^t Z_{B_{i}}\) and \((X_j,p_j)\) be the singularity obtained by contracting the support of \(Z_{B_j}\), where \(B_0,B_1,\dots, B_m\) is the Yau sequence as in [\textit{A. Némethi}, Invent. Math. 137, No.~1, 145--167 (1999; Zbl 0934.32018)]. Let also \( {\mathcal A}_f=\{j\mid 0\leq j\leq m-1, H^0({\mathcal O}_M(-C_{j+1}))\subsetneqq H^0({\mathcal O}_M(-C_j))\}\cup\{m\}\), \({\mathcal A}_g=\{j\mid 0\leq j\leq m, X_j \text{ is Gorenstein }\} \), \(\alpha=\min {\mathcal A}_g\) and \(\beta=\min{\mathcal A}_f\).
The main theorem of this article is the following: Let \(d=-C_\beta^2\). Then one of the following cases hold: 1) \(-Z_{B_\beta}^2\geq 2\), or \(-Z_{B_\beta}^2=1\) and \(\alpha<\beta\). In this case \({\mathcal O}_M(-C_\beta)\) is free and \(\text{ mult}(X,p)=d\geq 2\). If \((X,p)\) is Gorenstein then \(\text{ embdim}(X,p)=\max\{d,3\}\). 2) \(-Z_{B_\beta}^2=1\), \(\alpha=\beta\) and \(C_\beta\) is the maximal ideal cycle; \(\text{ mult}(X,p)=\text{embdim}(X,p)-1=d+1\). 3) \(-Z_{B_\beta}^2=1\), \(\alpha=\beta>0\) and \(C_\beta\) is not the maximal ideal cycle; \(d\geq 2\) and \(\text{mult}(X,p)=\text{embdim}(X,p)-1=d+2\).
The author also proves the following statement: For \(j\geq \alpha\), \(X_j\) is a \(\mathbb Q\)-Gorenstein singularity of index \(\gamma/(j-\alpha,\gamma)\). Thus \(\gamma | (m-\alpha)\) and \({\mathcal A}_g=\{\alpha+i\gamma\mid 0\leq i\leq (m-\alpha)/\gamma\}\). Since \(p_g=\#{\mathcal A}_g\), it follows that \(p_g=(m-\alpha)/\gamma+1\). We also have \(0\leq\beta-\alpha<\gamma\). If \((X,p)\) is Gorenstein, then \(\beta=\gamma-1\).
The above theorems generalize the results of the above cited article to Gorenstein singularities. Gorenstein singularities Okuma, T., Numerical Gorenstein elliptic singularities, Math. Z., 249, 31-62, (2005) Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties Numerical Gorenstein elliptic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth separated scheme over a Noetherian base ring \(\mathbb{K}\). The decomposition we are interested in is an isomorphism
\[
\text{R}{\mathcal H}\text{om}_{{\mathcal O}_{X\times_\mathbb{K} X}}({\mathcal O}_X,{\mathcal O}_X)\cong\bigoplus_q\bigl(\bigwedge_{{\mathcal O}_X}^q{\mathcal T}_{X/\mathbb{K}}\bigr)[-q]
\]
in the derived category \(\text{D}(\text{Mod }{\mathcal O}_{X\times_\mathbb{K} X})\). Here \({\mathcal T}_{X/\mathbb{K}}\) is the relative tangent sheaf. Upon passing to cohomology sheaves such an isomorphism recovers the Hochschild-Kostant-Rosenberg Theorem. If \(\mathbb{K}\) has characteristic 0 there is a decomposition that relies on a particular homomorphism of complexes from poly-tangents to continuous Hochschild cochains. We discuss sheaves of continuous Hochschild cochains on schemes and show why this approach to decomposition fails in positive characteristics. Our main result is a proof of the decomposition valid for any Gorenstein Noetherian ring \(\mathbb{K}\) of finite Krull dimension, regardless of characteristic. The proof is based on properties of minimal injective resolutions. smooth separated schemes; relative tangent sheaves; cohomology sheaves; continuous Hochschild cochains; injective resolutions; Hochschild complexes; derived categories Amnon Yekutieli, Decomposition of the Hochschild complex of a scheme in arbitrary characteristic, Canad. J. Math. 54 (2002), no. 4, 866 -- 896. Amnon Yekutieli, The continuous Hochschild cochain complex of a scheme, Canad. J. Math. 54 (2002), no. 6, 1319 -- 1337. (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Resolutions; derived functors (category-theoretic aspects), Syzygies, resolutions, complexes in associative algebras Decomposition of the Hochschild complex of a scheme in arbitrary characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We check that the Hilbert scheme, \({\mathcal H}_{d,g}\), of smooth and connected curves of degree \(d\) and genus \(g\) in projective three-dimensional space over \(\mathbb{C}\) is smooth provided that \(d\leq 11\). The proof uses essentially our good knowledge of curves lying on cubic surfaces and the possibility to endow a curve having a special normal bundle with a double structure of high arithmetic genus. Then we give some partial results in the case of degree 12. Namely, we obtain that \({\mathcal H}_{12,g}\) is smooth for \(g<15\) except cases \(g=11,12\), for which we were able to establish only that \({\mathcal H}_{12,g}\) is smooth in codimension 1. This shows that (12,15) is the lexicographically first pair \((d,g)\) such that \({\mathcal H}_{d,g}\) is singular in codimension 1. space curves; normal bundle; double structure Sébastien Guffroy, Lissité du schéma de Hilbert en bas degré, J. Algebra 277 (2004), no. 2, 520 -- 532 (French, with English summary). Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special algebraic curves and curves of low genus, Plane and space curves, Singularities in algebraic geometry, Infinitesimal methods in algebraic geometry Smoothness of Hilbert scheme in low degree. (Lissité du schéma de Hilbert en bas degré). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the paper under review, the author continues the study of normal surface singularities \((A,\mathfrak m)\) and \(p_g\)-ideals in \(A\) [\textit{T.\ Okuma}, Math. Z. 249, No. 1, 31--62 (2005; Zbl 1091.32010)], [\textit{T. Okuma} et al, J. Algebra 499, 450--468 (2018; Zbl 1390.14016); Manuscripta Math. 150, No. 3--4, 499--520 (2016; Zbl 1354.13011); Proc. Am. Math. Soc. 145, No. 1, 39--47 (2016; Zbl 1357.13011)].
In this review, let \((A,\mathfrak m,k)\) be a two-dimensional local ring which is a normal surface singularity, i.e., \(A\) is excellent and normal, and contains an algebraically closed field isomorphic to \( k\). The author proves his results using assumption 1.1; this assumption holds if \(k\) has characteristic \(0\).
Let \(f: X\to \text{Spec}(A)\) be a resolution of singularity, \(E:=f^{-1}(\mathfrak m)\) the exceptional locus, and \(E=\bigcup E_i\) the decomposition of \(E\) into its irreducible components. The \textit{geometric genus} \(p_g(A)\) of \(A\) is defined by \(p_g(A)=\ell_A(H^1(X,\mathcal O_X))\); this does not depend on the choice of the resolution \(X\). A rational singularity is characterized by \(p_g(A)=0\) [\textit{M. Artin}, Am. J. Math. 88, 129--136 (1966; Zbl 0142.18602)]. A \textit{cycle} is a divisor on \(X\) supported in \(E\). For a cycle \(B>0\) let \(\chi(\mathcal O_B)\) be the Euler characteristic and set \(\chi(B):=\chi(\mathcal O_B)\). The fundamental cycle on \(\text{Supp}(B)\) is the minimal cycle \(Z_B\) such that \(\text{Supp}(Z_B)=\text{Supp}(B)\) and \(Z_BE_i\leq 0\) for for all \(E_i\) with \(E_i\leq B\) (for a divisor \(D\) on \(X\) let \(DE_i\) be the intersection number). The following conditions are equivalent:
(1) \(\chi (D)\geq0\) for all cycles \(D>0\) and \(\chi(F)=0\) for some cycle \(F>0\);
(2) \(\chi(Z_E)=0\).
A normal surface singularity satisfying these conditions was called elliptic by \textit{P. Wagreich} [Am. J. Math. 92, 419--454 (1970; Zbl 0204.56404)]. Let \((A,\mathfrak m)\) be an elliptic singularity. Then there exits a unique cycle \(E_{\min}\) such that \(\chi(E_{\min})=0\) and \(\chi(D)>0\) for all cycles \(D\) such that \(0<D<E_{\min}\) [\textit{H. B. Laufer}, Am. J. Math. 99, No. 6, 1257--1295 (1977; Zbl 0384.32003)]. The cycle \(E_{\min}\) is called a minimally elliptic cycle. The singularity \((A,\mathfrak m)\) is said to be \textit{minimally elliptic} if the fundamental cycle is minimally elliptic on the minimal resolution.
Let \(I\) be an integrally closed \(\mathfrak m\)-primary ideal in \(A\); then there exists a resolution \(X\to\text{Spec}(A)\) and a cycle \(Z>0\) on \(X\) such that \(I\mathcal O_X=\mathcal O_X(-Z)\). In this case the ideal \(I\) is denoted by \(I_Z\) and \(I\) is said to be represented on \(X\) by \(Z\); clearly \(I_Z=H^0(X,\mathcal O_X(-Z))\). The invariant \(q(I)\) is defined by \(q(I)=\ell_A(H^1(X,\mathcal O_X(-Z)))\); \(q(I)\) does not depend on the choice of the representation of the ideal \(I\) [3].
The ideal \(I\) is called a \(p_g\)-\textit{ideal} if \(q(I)=p_g(A)\), and a cycle \(Z>0\) is called a \(p_g\)-\textit{cycle} if \(\mathcal O_X(-Z)\) is generated by global sections and \(\ell_A(H^1(X,\mathcal O_X(-Z)))=p_g(A)\). \(p_g\)-ideals have many nice properties [Zbl 1354.13011]: Let \(I\), \(I'\) be integrally closed \(\mathfrak m\)-primary ideals. Then \(I\) and \(I'\) are \(p_g\)-ideals iff \(II'\) is a \(p_g\)-ideals. If \(Q\) is a minimal reduction of \(I\), then \(I^2=QI\) (These results are well-known if \(A\) is regular and \(I\), \(I'\) are integrally closed; cf. [\textit{J. Sally} and \textit{C. Huneke}, Integral closure of ideals, ring and modules. Cambridge: Cambridge University Press (2006; Zbl 1117.13001)]. They go back to Zariski. They holds also if \(A\) is rational [\textit{J. Lipman}, Publ. Math. Inst. Hautes Études Sci. 36, 195--279 (1969; Zbl 0181.48903)]). The ideal \(I\) is a \(p_g\)-ideal iff the Rees algebra \(\bigoplus_{n\geq0}I^nt^n\subset A[t]\) is a Cohen-Macaulay normal domain [Zbl 1357.13011]. There is also a criterion for a cycle to be a \(p_g\)-cycle. In [Zbl 1390.14016] it is shown: For \(h\in I\) there exists \(h'\in I\) such that the integral closure of the ideal \((h,h')\) is a \(p_g\)-ideal.
If \(A\) is rational, i.e. \(p_g(A)=0\), then every integrally closed \(\mathfrak m\)-primary ideal is a \(p_g\)-ideal [Zbl 0181.48903]. Conversely, this property characterizes a rational singularity because there always exist integrally closed \(\mathfrak m\)-primary ideals \(I\) with \(q(I)=0\) [3].
Let \(I\) be an \(\mathfrak m\)-primary ideal in \(A\) and \(Q\) a minimal reduction of \(I\). The \textit{normal reduction number} \(\overline r(I)\) of \(I\) is defined by \( \overline r(I)=\min\{r\in\mathbb Z_{\geq0}\mid \overline {I^{n+1}}=Q\overline {I^n}\,\, \text{for all}\,\,\, n\geq r\}\); this number is independent of the choice of the minimal reduction \(Q\). Set \(\overline r(A)=\max\{\overline r(I)\mid I\subset A \,\, \text{integrally closed}\,\, \mathfrak m\text{-primary ideal}\}\). If \(A\) is rational then from results of Lipman and [\textit{S. D. Cutkosky}, Invent. Math. 102, No. 1, 157--177 (1990; Zbl 0718.14025)] it follows that \(\overline r(A)=0\) iff \(A\) is a rational singularity.
Let \(Z>0\) be a cycle on \(X\). The author calls \(\mathcal O_X(-Z)\) \textit{to have no fixed component} if \(H^0(X,\mathcal O_X(-Z))\neq H^0(X,\mathcal O_X(-Z-E_i))\) for every \(E_i\); to such a cycle the author associates a nonnegative integer \(n_0(Z)\) [\,cf.\ Remark 3.5]. Note that \(\mathcal O_X(-Z)\) has no fixed component when \(I\) is represented by \(Z\); then \(n_0(I):=n_0(Z)\) is independent of the representation \(Z\) of \(I\). In Cor.\ 3.9 it is shown that \(\overline r(I)=n_0(I)+1\). From this the author gets one of his main results, Theorem 3.3: If \(A\) is an elliptic singularity, then \(\overline r(A)=2\) .
The invariant \(q\) is a function on the set of integrally closed \(\mathfrak m\)-primary ideals in \(A\). The set \(\text{Im}_A(q)\subset \mathbb Z\) is defined by \(\text{Im}_A(q)=\{q(I)\mid I\subset A\,\,\text{is an integrally closed}\,\, \mathfrak m\text{-primary ideal}\}\). One has \(\text{Im}_A(q)\subset\{0,1,\ldots,p_g(A)\}\); and one has equality if \(A\) is an elliptic singularity by Cor.\ 3.13; the converse does not hold, cf.\ Example 3.15.
In the last part the author is interested in singularities for which the maximal ideal \(\mathfrak m\) is a \(p_g\)-ideal. In Def.\ 4.7 it is defined when \(A\) is a maximal elliptic singularity. Such a singularity is Gorenstein by a result of [\textit{S. S. Yau}, Trans. Am. Math. Soc. 257, No. 2, 269--329 (1980; Zbl 0343.32009]. In Theorem 4.10 the author proves the following result: Assume that \(A\) is not rational. Then \(A\) is Gorenstein and \(\mathfrak m\) is a \(p_g\)-ideal iff \(A\) is a maximally elliptic singularity with \(-Z_E^2=1\) where \(Z_E\) is the fundamental cycle on \(E\). normal surface singularity; \(p_g\)-ideals; elliptic surface singularities Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Cohomology of ideals in elliptic surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study constructive embedded resolutions of irreducible quasi-ordinary singularities in \(\mathbb{C}^3\). A surface germ \((V,p)\subset (\mathbb{C}^3,0)\) is a quasi-ordinary singularity if it admits a finite projection \(\pi:(V,p) \to(\mathbb{C}^2,0)\) such that the discriminant locus (i.e., the plane curve over which \(\pi\) ramifies) has only normal crossings. If \(f\) is such a singularity and is irreducible, it admits a parametrization (analogous to the Puiseux series of an irreducible algebroid plane curve) from which certain pairs of numbers, called the characteristic pairs, can be extracted. They are important in the study of the germ. For instance, explicit resolutions of such a singularity have been studied by \textit{J. Lipman} [in: Singularities, Summer Inst., Arcata 1981, Proc. Symp. Pure Math. 40, Part 2, 161-172 (1983; Zbl 0521.14014)]. In this process, which does not lead to embedded resolutions, an important role is played by the characteristic pairs. Recall that, informally, ``embedded resolution'' means a process where along which the singular variety \(V\) one transforms the ambient space where it is defined, so that eventually the strict transform of \(V\) is non-singular and its union with the exceptional divisor is locally defined by simple, ``nice'' equations. Recently, several (closely related) methods to constructively (or canonically) obtain embedded resolutions of general singularities have been proposed. That is, procedures which involve a finite sequence of blowing-ups and which tell us, at each stage of the resolution process, how to choose the center of the transformation.
More precisely, in this note the authors explicitly study what results from the application of the general canonical process devised by \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No. 2, 207-302 (1997; Zbl 0896.14006)] to an irreducible quasi-ordinary singularity. It turns out that the process depends on the (suitably normalized) characteristic pairs of the singularity only. The description of the algorithm given in the paper is very explicit. The authors apply it to a non-trivial example, and they affirm that the method can be actually implemented by a computer.
The authors plan to apply similar techniques to the problem of simultaneous desingularization of a family of quasi-ordinary singularities in a future work. canonical resolution; quasi-ordinary singularities; characteristic pairs; embedded resolutions C. Ban and L. J. McEwan, Canonical resolution of a quasi-ordinary surface singularity , Canad. J. Math. 52 (2000), 1149--1163. Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Complex surface and hypersurface singularities Canonical resolution of a quasi-ordinary surface singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review studies the Segre classes of subschemes of projective space defined by monomial ideals. The Segre class is shown to be computable by a formula involving the log canonical thresholds of certain extensions of the ideal. The author poses the question to what extent his result might hold for non-monomial schemes.
The proof relies on a result of \textit{J. A. Howald} describing the multiplier ideal of a monomial ideal [Trans. Am. Math. Soc. 353, No. 7, 2665--2671 (2001; Zbl 0979.13026)] and a more recent result of the author [``Segre classes as integrals over polytopes'', to appear in J. Eur. Math. Soc., \url{arXiv:1307.0830}]. log canonical threshold; Segre class; multiplier ideal; Howald's theorem Paolo Aluffi, Log canonical threshold and Segre classes of monomial schemes, Manuscripta Math. 146 (2015), no. 1-2, 1 -- 6. Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Log canonical threshold and Segre classes of monomial schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Recall that for any complete intersection defined by \(n\) forms in \(K[x_1,\dots, x_n]\), where \(K\) is a field, there exists a unique almost revlex ideal with the same Hilbert function. The authors present a new connstruction of this revlex ideal. Quoted from the authors' abstract,
``Properties of the reduction numbers for an almost revlex ideal have an important role in our inductive and constructive proof, which is different from the more general construction given by \textit{K. Pardue} [J. Algebra 324, No. 4, 579--590 (2010; Zbl 1200.13025)].'' The authors obtain, an exact closed formula for the number of the
minimal generators of an almost revlex ideal in terms of the Hilbert function. They find several classes of almost revlex ideals with the Hiberrt function of a complete intersection for an infinite field \(K\), that are singular points in a Hilbert scheme. ``This second result is the outcome of a more general study of lower bounds for the dimension of the tangent space to a Hilbert scheme at stable ideals, in terms of the number of minimal generators.'' almost revlex ideal; reduction number; complete intersection; Hilbert scheme Computational aspects and applications of commutative rings, Parametrization (Chow and Hilbert schemes), Singularities of surfaces or higher-dimensional varieties On almost revlex ideals with Hilbert function of complete intersections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A well known theorem of K. Saito states that an isolated hypersurface singularity admits a \({\mathbb{C}}^*\) action if and only if the defining function belongs to its own Jacobian ideal. This is equivalent to equality of the Milnor and Tyurina numbers, \(\mu\) and \(\tau\) respectively. Here the result (in this form) is generalized to isolated complete intersection surface singularities. More generally, the difference \(\mu\)-\(\tau\) is expressed in terms of other (nonnegative) analytic invariants. The theory is applied to obtain results about the irregularity and about equisingular deformations. The proof depends on choosing a component C of the exceptional divisor which either has positive genus or meets at least three other components, and extending a derivation from C to the whole divisor. quasi-homogeneous Gorenstein surface singularities; equality of the Milnor and Tyurina numbers; isolated complete intersection surface singularities; irregularity; equisingular deformations J. M. Wahl, A characterization of quasihomogeneous Gorenstein surface singularities, Compos. Math., 55 (1985), 269--288. MR799816 (87e:32013) Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects), Complete intersections, Deformations of singularities, Global theory and resolution of singularities (algebro-geometric aspects), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities in algebraic geometry A characterization of quasi-homogeneous Gorenstein surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We discuss Hilbert functions and graded Betti numbers of arithmetically Gorenstein subschemes of projective space, and the recent results obtained by \textit{J. Migliore} and \textit{U. Nagel} [Adv. Math. 180, 1--63 (2003; Zbl 1053.13006)] for arithmetically Gorenstein subschemes with the subspace property, a generalization of the weak Lefschetz property. Artinian-Hilbert function; Stanley-Iarrobino sequence; SI-sequence Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Syzygies, resolutions, complexes and commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Linkage, complete intersections and determinantal ideals Reduced arithmetically Gorenstein schemes with prescribed properties. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Nach Cayley kann jede höhere Singularität einer ebenen algebraischen Curve durch eine gewisse Zahl äquivalenter ``elementarer'' Plücker'scher Singularitäten vertreten werden. Diese Cayley'schen Aequivalenzzahlen haben zunächst nur eine functionentheoretische Bedeutung. So wird z. B. die Zahl der Rückkehrpunkte, die in eine Singularität entfallen, gleich der Multiplicität des Punktes, vermindert um die Zahl seiner cyclischen Gruppen, gesetzt; zu erweisen bleibt dann, dass diese Zahlen in den Plücker'schen Gleichungen als Repräsentanten ebensoviel elementarer Singularitäten auftreten. Doch lag von vornherein die Tendenz nahe, ihnen die geometrische zuzuweisen, dass jede höhere Singularität in eine derselben entsprechende Gruppe elementarer durch Deformation der Curve aufgelöst werden könne, mit anderen Worten, dass jede Curve mit einer höheren Singularität in eine von gleicher Classe, Ordnung und Geschlecht deformirt werden könne deren elementare Singularitäten den `Cayley'schen Formeln entsprechen und durch Aufhebung der Deformation in die gegebenen übergehen.
Mit dieser auch schon von anderer Seite bezeichneten Fragestellung beschäftigt sich die vorliegende Arbeit. Da in der Nähe jeder singulären Stelle die Curve mit beliebiger Annäherung durch ein System rationaler Curven, resp. eine einzige ersetzt werden kann (die Berechtigung dieser oft benutzten Schlussweise für die in Rede stehenden algebraischen Untersuchungen wird sorgfältig erörtert) so wird man die Untersuchung über den Einfluss solcher Deformationen auf die Vorgänge bei rationalen Curven beschränken dürfen. Es zeigt sich nun, dass in der That immer die Deformation so ausgeführt werden kann, dass bei ungeänderter Ordnung, Classe und Geschlecht an Stelle der höheren Singularitäten die äquivalenten elementaren auftreten.
Die Frage nach der Bestimmung der Aequivalenzzahlen läuft demnach darauf hinaus, dieselben für gewisse rationale Curven zu ermitteln.
Der Verfasser führt dieses Problem indessen nicht an den speciellen rationalen Curven aus, welche einen oder mehrere unicursale Zweige vertreten können, sondern an einer allgemeineren Classe, den ``rational ganzen'' Curven. Diese besitzen insofern ein selbständiges Interesse, als bei ihnen algebraische Untersuchungen, die im allgemeinen nicht mehr explicite ausführbar scheinen, eine übersichtliche Gestalt gewinnen; sie sind dadurch ausgezeichnet, dass ihre cartesischen Punkt- und Liniencoordinaten \(x, y\); \(u, v\) sich als ganze rationale Functionen eines Parameters \(\lambda\) in folgender Weise ausdrücken lassen:
\[
\begin{aligned} x & =\int\varrho d\lambda,\quad u =\int\omega d\lambda\\ y & =\int u\varrho d\lambda,\quad v =\int x\omega d\lambda,\end{aligned}
\]
wo \(\varrho\), \(\omega\) rationale ganze Functionen sind, die bis auf einen constanten Factor durch die im Endlichen gelegenen Rückkehrpunkte und Inflexionspunkte bestimmt sind. Die Aequivalenzen für Rückkehrpunkte und Wendetangenten, die an einen singulären Punkt, etwa \(x=0\), \(y=0\), entfallen, sind leicht zu bestimmen; für die Doppelpunkte resp. tangenten wird eine genauere Untersuchung der bezüglichen Doppeldiscriminante nöthig, bei welcher Gelegenheit gezeigt wird, dass diese beiden Discriminanten aus je fünf mit bestimmter Multiplicität auftretenden Factoren bestehen, deren jeder eine leicht zu erkennende geometrische Bedeutung besitzt. Gleichzeitig ergiebt sich auch eine solche für gewisse in die Zahl der Doppelpunkte eingehende characteristische Zahlen, deren Wichtigkeit als ``kritische'' Exponenten bei der functionen-theoretischen Untersuchung der Aequivalenzen zuerst von Herrn H. J. S. Smith hervorgehoben wurde. Als eine interessante Consequenz der in der Abhandlung des Herrn Brill entwickelten Betrachtungen ist noch zu bezeichnen, dass, wie auch eine gegebene Curve in eine ``äquivalente'' deformirt werden mag, eine aus der Zahl der reellen Rückkehr- und Wendepunkte, der isolirten Doppeltangenten und Doppelpunkte zusammengesetzte Zahl, welche als Realitätsindex der Singularität bezeichnet wird, ungeändert bleibt. singularities; rational curves; Cayley's equivalence numbers; Plücker Singularities; Cayley's formulae. Plane and space curves, Rational and birational maps On singularities of plane algebraic curves and a new species of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This very readable article is a collection of results on \({\mathbb F}_1\)-schemes, the principle ones being definitions of flat, unramified and étale morphisms; and (Theorem 4.1) that integral \({\mathbb F}_1\)-schemes of finite type are essentially the same as toric varieties. No previous knowledge of \({\mathbb F}_1\)-schemes is needed by the reader.
The idea of the ``field of one element'' \({\mathbb F}_1\) has been discussed since [\textit{J.~Tits}, Centre Belge Rech. math., Colloque d'Algébre supérieure, Bruxelles du 19 au 22 déc. 1956, 261--289 (1957; Zbl 0084.15902)] noted some properties that such an object would have, in the context of Chevalley groups. Various approaches have been taken to defining \({\mathbb F}_1\) and schemes over it; see \textit{C.~Soulé} [Mosc.\ Math.\ J., 4, 217--244 (2004; Zbl 1103.14003)] and \textit{N.~Durov} [New approach to Arakelov geometry. \url{arXiv:0704.2030} (2007)]. In the author's previous article [in: Number fields and function fields -- two parallel worlds. Boston, MA: Birkhäuser. Progress in Mathematics 239, 87--100 (2005; Zbl 1098.14003)], he showed how one can define the category of schemes over \({\mathbb F}_1\) by replacing the rings occurring in the classical definition with monoids. The resulting schemes satisfy expected properties with respect to Chevalley groups and zeta functions.
This article begins with a short but clear account of the definition of the category of \({\mathbb F}_1\)-schemes, as in [loc. cit.]. In Section~1, a notion of flatness of modules over monoids is defined, and hence flatness of morphisms of \({\mathbb F}_1\)-schemes. Algebraic extensions of monoids are defined in Section~2, and so unramified morphisms in Section~3; this leads to the definition of an étale morphism of \({\mathbb F}_1\)-schemes. It is shown that the corresponding notion of simply connected holds in some important cases, notably that of \({\mathbb P}^1\) over the algebraic closure of \({\mathbb F}_1\).
Section~4 deals with toric varieties. Every toric variety is the lift of an \({\mathbb F}_1\)-scheme; Theorem~4.1 gives a result in the converse direction: ``Let \(X\) be a connected integral \({\mathbb F}_1\)-scheme of finite type. Then every irreducible component of \(X_{\mathbb C}\) is a toric variety. The components of \(X_{\mathbb C}\) are mutually isomorphic as toric varieties.'' Proposition~4.3 then describes explicitly the zeta function of such an \({\mathbb F}_1\)-scheme.
Section~5 gives a lemma concerning valuations on monoids; Section~6 gives an example, attributed to Gabber, showing that cohomology is not defined over \({\mathbb F}_1\) . \(\mathbb{F}_1\)-schemes; field of one element; toric varieties Deitmar, A., \(\mathbb {F}_{1}\)-schemes and toric varieties, Beitr. Algebra Geom., 49, 517-525, (2008) Schemes and morphisms, Toric varieties, Newton polyhedra, Okounkov bodies, Varieties over finite and local fields \(\mathbb F_1\)-schemes and toric varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We generalize Kahn's correspondence between Cohen-Macaulay modules over normal surface singularities over an algebraically closed field and vector bundles over some projective curves to abstract surface singularities, which need not be algebras over a field. As a consequence, we also generalize to the abstract case the Drozd-Greuel criterion for tameness of curve singularities [\textit{Yu. A. Drozd} and \textit{G. M. Greuel}, Compos. Math. 89, No. 3, 315--338 (1993; Zbl 0794.14010)]. V. Gavran, Kahn's correspondence and Cohen-Macaulay modules over abstract surface and curve singularities, Journal of Singularities 4 (2012), 68-73. Cohen-Macaulay modules, Vector bundles on curves and their moduli, Singularities of curves, local rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Kahn's correspondence and Cohen-Macaulay modules over abstract surface and curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a compact connected Riemann surface of genus \(g\), with \(g\geq 2\), and let \(\mathcal{O}_{X}\) denote the sheaf of holomorphic functions on \(X\). Fix positive integers \(r\) and \(d\) and let \(\mathcal{Q}(r,d)\) be the Quot scheme parametrizing all torsion coherent quotients of \(\mathcal{O}^{\oplus r}_{X}\) of degree \(d\). We prove that \(\mathcal{Q}(r,d)\) does not admit a Kähler metric whose holomorphic bisectional curvatures are all nonnegative.
For Part I, cf. [the authors, ibid. 57, No. 4, 1019--1024 (2013; Zbl 1304.14012)]. Biswas, I.; Seshadri, H., On the Kähler structures over quot schemes II, Ill. J. math., 58, 689-695, (2014) Divisors, linear systems, invertible sheaves, Computational aspects of algebraic curves, Positive curvature complex manifolds, Vector bundles on curves and their moduli, Parametrization (Chow and Hilbert schemes) On the Kähler structures over Quot schemes. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Using the theory of dimer models \textit{N. Broomhead} [Dimer models and Calabi-Yau algebras. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1237.14002)] proved that every 3-dimensional Gorenstein affine toric variety \(\mathrm{Spec}R\) admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of \(\mathrm{Spec}R\) using standard toric methods. Our proof does not use dimer models.
For Part I, see [the authors, ibid. 2020, No. 21, 8120--8138 (2020; Zbl 1457.14003)]. toric varieties; tilting bundle; noncommutative resolution Actions of groups on commutative rings; invariant theory, Toric varieties, Newton polyhedra, Okounkov bodies, Modifications; resolution of singularities (complex-analytic aspects) Non-commutative crepant resolutions for some toric singularities. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Several recent papers look to the question of whether, for a given Hilbert function \(H\), there is a unique minimum set of graded Betti numbers or not [e.g. \textit{B. Richert}, J. Algebra 244, No.~1, 236--259 (2001; Zbl 1027.13008)]. Moreover by semicontinuity of the graded Betti numbers different minima correspond to different irreducible components of the postulation Hilbert scheme, Hilb\((H)\), of zero-schemes [see \textit{A. Ragusa} and \textit{G. Zappalá}, Rend. Circ. Matem. di Palermo Serie II, 53, 401--406 (2004; Zbl 1099.13029)]. In the present paper the author gives infinitely many families of reduced zero-schemes of height 3, having at least two irreducible components of Hilb\((H)\) which he separates by the incomparability of the set of graded Betti numbers. Moreover the general element of one of the components has the weak Lefschetz property (WLP) while every zero-scheme of the second component fails to have WLP. Linkage is important in constructing the general elements of the two components.
In [J. Algebra 262, No.~1, 99--126 (2003; Zbl 1018.13001)] \textit{T. Harima}, the author, \textit{U. Nagel} and \textit{J. Watanabe} characterized Artinian quotients having WLP in terms of their Hilbert function \(h\). Comparing \(h\) of this result with the first difference of \(H\) of the classes of reducible Hilbert schemes of the present paper, the existence range is very similar except for the number in the ``flat part'' of \(h\), which may be much more limited. Still, due to this and several other papers [e.g. \textit{A. Iarrobino} and \textit{H. Srinivasan}, J. Pure Appl. Algebra 201, No.~1--3, 62--96 (2005; Zbl 1107.13020) and the reviewer in Trans. Am. Math. Soc. 358, No.~7, 3133--3167 (2006; Zbl 1103.14005)], Hilb\((H)\) seems to be very rich in reducible Hilbert schemes, even in height 3. The stratum of Hilb\((H)\) of fixed graded Betti numbers should have a much better chance of being irreducible, but also this fails by infinitely many families of zero-schemes (of height 4) by the mentioned paper of the reviewer. postulation Hilbert scheme; graded Betti numbers; linkage Migliore, J.: Families of reduced zero-dimensional schemes. Collect. math. 57, No. 2, 173-192 (2006) Syzygies, resolutions, complexes and commutative rings, Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Families of reduced zero-dimensional schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For part I see the preceding review.]
The author introduces a notion of infinite-dimensionality of the Chow group of 0-cycles of degree 0 in the case when the considered surface is singular. This is a generalization of well known definition given by Mumford. There are proved generalizations of Mumford's and Rojtman's theorems for singular surfaces. zero-cycles; infinite-dimensionality of the Chow group; singular surfaces V. Srinivas, Zero cycles on a singular surface. I, J. Reine Angew. Math. 359 (1985), 90 -- 105. With an appendix by S. Bloch. V. Srinivas, Zero cycles on a singular surface. II, J. Reine Angew. Math. 362 (1985), 4 -- 27. Algebraic cycles, Singularities of surfaces or higher-dimensional varieties, Parametrization (Chow and Hilbert schemes) Zero cycles on a singular surface. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this very well-written article, the authors establish several results relating finite schemes and secant varieties over arbitrary fields. Some of these results were previously known only over the complex numbers. The paper is in part expository and contains background material on scheme theory, apolarity theory, Castelnuovo-Mumford regularity, Hilbert schemes, and secant varieties. Let \(\mathbb K\) be a field and \(R\) be a finite scheme over \(\mathbb K\). One of the main objectives is to study the \textit{smoothability} of \(R\) both as an abstract scheme and as an embedded scheme in some algebraic variety \(X\). The condition of smoothability can be easily seen over an algebraically closed field: a finite scheme \(R\) is smoothable if and only if it is a flat limit of distinct points. Theorem 1.1 gives the equivalence between the abstract smoothability and the embedded smoothability in some algebraic variety \(X\), whenever \(X\) is smooth. Moreover, smoothability over \(\mathbb K\) is equivalent to smoothability in the algebraic closure of \(\mathbb K\) (Proposition 1.2). Let \(\mathbb K\) be an algebraically closed field. Let \(X\) be an algebraic variety \(\mathbb K\) and let \(r\) be an integer. Condition \((\star)\) holds if every finite \textit{Gorenstein} subscheme over \(\mathbb K\) of \(X\) of degree at most \(r\) is smoothable in \(X\). One of the main results is Theorem 1.7. This relates the scheme theoretic condition above with the possibility of giving \textit{set-theoretic equations} for secants of sufficiently high Veronese embeddings of \(X\), by determinantal equations from vector bundles on \(X\). If condition \((\star)\) does not hold, then those equations are not enough to cut them. Interestingly, the locus of determinantal equations from vector bundles contain more general loci than secants: the \textit{cactus varieties}. This containment is the ultimate reason for the failure of present methods to give good enough lower bounds on tensor ranks. smoothable; secant varieties; finite Gorenstein scheme; cactus variety; Veronese reembedding; Hilbert scheme Buczyński, J.; Jelisiejew, J., Finite schemes and secant varieties over arbitrary characteristic, Differential Geom. Appl., 55, 13-67, (2017) Determinantal varieties, Local deformation theory, Artin approximation, etc., Parametrization (Chow and Hilbert schemes), Schemes and morphisms, Homogeneous spaces and generalizations Finite schemes and secant varieties over arbitrary characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Here we prove the existence of many zero-dimensional schemes \(Z\subset\mathbb P^n\) with a small number of connected components, all of them defined over \(\mathbb F_q\), and with good postulation (even if \(\text{length}(Z)\gg \sharp(\mathbb P^n(\mathbb F_q)))\). Projective techniques in algebraic geometry, Finite ground fields in algebraic geometry Unreduced zero-dimensional schemes defined over \(\mathbb F_q\) with good postulation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we consider congruences of Hilbert modular forms. Sturm showed that mod \(\ell\) elliptic modular forms of weight \(k\) and level \(\Gamma_1(N)\) are determined by the first \((k/12)[\Gamma_1(1):\Gamma_1(N)] \bmod\ell\) Fourier coefficients. We prove an analogue of Sturm's result for Hilbert modular forms associated to totally real number fields. The proof uses the positivity of ample line bundles on toroidal compactifications of Hilbert modular varieties. Hilbert modular forms and varieties; congruences of modular forms; Sturm's theorem; toroidal and minimal compactifications; intersection numbers Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Fourier coefficients of automorphic forms, Congruences for modular and \(p\)-adic modular forms, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry An analogue of Sturm's theorem for Hilbert modular forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Der Grundkörper ist \(k=\mathbb{C}\), und \({\mathcal H}:=H_{d,g}= \text{Hilb}^P (\mathbb{P}^3_k)\), \(P(T)= dT-g+1\), ist das (volle) Hilbertschema der Raumkurven vom Grad \(d\) und Geschlecht \(g\). Für \(3\leq d\leq 11\) definiert man \(g(d)\) durch die Tabelle
\[
\begin{matrix} d & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11\\ g(d) & -2 & 0 & 1 & 2 & 4 & 6 & 9 & 11 & 15\end{matrix}
\]
und für \(d\geq 12\) durch die Formel \(g(d)={1\over 6}d(d-3)\). Die vorliegende Arbeit bringt zunächst eine Ergänzung und verschiedene Verbesserungen zu früheren Arbeiten des Autors [\textit{G. Gotzmann}, ``Der kombinatorische Teil der ersten Chowgruppe eines Hilbertschemas von Raumkurven'' und ``Der algebraische Teil der ersten Chowgruppe eines Hilbertschemas von Raumkurven'' (Münster 1994; Zbl 0834.14004 und Münster 1997; Zbl 0954.14002)], und man erhält als Zusammenfassung:
Satz I. Wenn \(d\geq 3\) und \(g\leq g(d)\) ist, dann ist
(i) \(\dim_\mathbb{Q} A_1({\mathcal H})\otimes_\mathbb{Z} \mathbb{Q}=3\);
(ii) \(\text{Pic} ({\mathcal H}) \simeq \mathbb{Z}^3\oplus \mathbb{C}^r\), mit \(r=\dim_kH^1 ({\mathcal H},{\mathcal O}_{\mathcal H})\);
(iii) \(\text{NS}({\mathcal H})\simeq\mathbb{Z}^3\).
Der Satz ist eine Folgerung aus etwas genaueren Ergebnissen in Abschnitt 6, wo explizite Basen von \(A_1({\mathcal H})\otimes\mathbb{Q}\) und \(\text{NS}({\mathcal H})\) bestimmt werden. -- Man kann die in Satz I genannten Ergebnisse für \({\mathcal H}\) auf die zugehörige universelle \({\mathcal C}\subset{\mathcal H}\times\mathbb{P}^3\) übertragen, und man erhält:
Satz II. Wenn \(d\geq 3\) und \(g\leq g(d)\) ist, dann ist
(i) \(\dim_\mathbb{Q} A_1({\mathcal C})\otimes_\mathbb{Z} \mathbb{Q}=4\);
(ii) \(\text{Pic}({\mathcal C})\simeq \mathbb{Z}^4\oplus \mathbb{C}^r\), mit \(r=\dim_kH^1 ({\mathcal C},{\mathcal O}_{\mathcal C})\);
(iii) \(\text{NS}({\mathcal C})\simeq\mathbb{Z}^4\). Néron-Severi group; Hilbert scheme; universal curve; Picard group Parametrization (Chow and Hilbert schemes), Plane and space curves The Néron-Severi group of a Hilbert scheme of space curves and the universal curve | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Assume that \((X,\mathcal{O}_X)\) is an arbitrary scheme. The concept of the big (resp. little) finitistic flat dimension \(\mathrm{FFD}(X)\) (resp. \(\mathrm{fFD}(X)\)) of \(X\) will be introduced. It is shown that if \(X\) is affine and any flat quasi-coherent \(\mathcal{O}_X\)-module has finite projective dimension, then finitistic flat dimensions are finite if and only if the finitistic projective dimensions are finite. We will find the minimum requirements for \(\mathrm{FFD}(X)\) (resp. \(\mathrm{fFD}(X)\)) to be finite. Furthermore, if \(R\) is a commutative \(n\)-perfect ring, we prove that \(\mathrm{fPD}(R)<+\infty\) if and only if \(\sup_{\mathfrak{m}\in \mathrm{Max}R}\mathrm{fPD}(R_{\mathfrak{m}})<+\infty\) where \(\mathrm{fPD}(R)\) (resp. \(\mathrm{fPD}(R_{\mathfrak{m}})\)) is the little finitistic projective dimension of \(R\) (resp. \(R_{\mathfrak{m}}\)). quasi-coherent sheaf; flat dimension; finitistic dimension Homological dimension in associative algebras, Sheaves in algebraic geometry On finitistic flat dimension of rings and schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In an earlier paper [J. Am. Math. Soc. 14, 941--1006 (2001; Zbl 1009.14009)] we showed that the Hilbert scheme of points in the plane \(H_n= \text{Hilb}^n(\mathbb{C}^2)\) can be identified with the Hilbert scheme of regular orbits \(\mathbb{C}^{2n}//S_n\). Using our earlier result and a recent result of \textit{T. Bridgeland}, \textit{A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, 535--554 (2001; Zbl 0966.14028)], we prove vanishing theorems for tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish the conjectured character formula for diagonal harmonics of \textit{A. M. Garsia} and the author [J. Algebr. Comb. 191--244 (1996; Zbl 0853.05008)]. In particular we prove that the dimension of the space of diagonal harmonics is \((n+1)^{n-1}\). This is a preliminary report. We state the main results and outline the proofs. Detailed proofs, a more systematic study of the applications, and a fuller exposition will be given in a future publication. M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, http://arxiv.org/math.AG/0201148 . Parametrization (Chow and Hilbert schemes), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Symmetric functions and generalizations, Combinatorial aspects of representation theory Vanishing theorems and character formulas for the Hilbert scheme of points in the plane. (Abreviated version). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities These notes aim at providing a complete and systematic account of some foundational aspects of algebraic supergeometry, namely, the extension to the geometry of superschemes of many classical notions, techniques and results that make up the general backbone of algebraic geometry, most of them originating from Grothendieck's work. In particular, we extend to algebraic supergeometry such notions as projective and proper morphisms, finiteness of the cohomology, vector and projective bundles, cohomology base change, semicontinuity theorems, relative duality, Castelnuovo-Mumford regularity, flattening, Hilbert and Quot schemes, faithfully flat descent, quotient étale relations (notably, Picard schemes), among others. Some results may be found elsewhere, and, in particular, there is some overlap with [51]. However, many techniques and constructions are presented here for the first time, notably, a first development of Grothendieck relative duality for proper morphisms of superschemes, the construction of the Hilbert superscheme in a more general situation than the one already known (which in particular allows one to treat the case of sub-superschemes of supergrassmannians), and a rigorous construction of the Picard superscheme for a locally superprojective morphism of noetherian superschemes with geometrically integral fibres. Moreover, some of the proofs given here are new as well, even when restricted to ordinary schemes. In a final section we construct a period map from an open substack of the moduli of proper and smooth supercurves to the moduli stack of principally polarized abelian schemes. algebraic supergeometry; finiteness of cohomology in algebraic supergeometry; base change and semicontinuity for superschemes; relative Grothendieck duality; Hilbert and Picard superschemes; superperiod maps Supervarieties, Algebraic moduli of abelian varieties, classification, Fine and coarse moduli spaces, Families, moduli of curves (algebraic), String and superstring theories in gravitational theory Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let a representation of a reductive group \(G\) in a linear space \(V\) be given. Let \(X\) be an irreducible \(G\)-invariant subvariety of the projective space \(\mathbb{P}(V)\). The variety \(X\) is said to be spherical if a Borel subgroup \(B\) of the group \(G\) has an open orbit on \(X\). We denote by \(F[X]\) the homogeneous coordinate ring of the variety \(X\); this ring is graded by the degree of polynomials, \(F[X]= \bigoplus^\infty_{n=0} F[X]_n\). We note that, for any classical group \(G\) and any spherical \(G\)-variety \(X\), we can define, in the real space \(\mathbb{R}^{\dim B}\) of dimension \(\dim B\), a polytope \(\Delta(X)\) and a lattice \(\widetilde P\) with the following properties.
Proposition. If the variety \(X\) is normal, then \(\dim F[X]_n=\text{card}\{n\Delta(X) \cap \widetilde P\}\), i.e., the Hilbert polynomial of the variety \(X\) coincides with the Ehrhart polynomial of the polytope \(\Delta(X)\). For every variety \(X\) we have \(\deg X=(\dim X) !\text{ vol} \Delta(X)\), where the volume of the cell of \(\widetilde P\) is normalized to unity. spherical variety; Hilbert polynomial Okounkov A.\ Y., Note on the Hilbert polynomial of a spherical variety, Funct. Anal. Appl. 31 (1997), no. 2, 138-140. Geometric invariant theory, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Note on the Hilbert polynomial of a spherical variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a very interesting paper which gives new insight in Hilbert's theorem. The authors prove:
Theorem 1.1: Suppose \(f(x,y,z)\) is a nonnegative real quartic form which defines a smooth plane curve \(Q= \{(x: y: z)\in\mathbb{P}^2(\mathbb{C}): f(x,y,z)= 0\}\). Then the inequivalent representations of \(f\) as a sum of three squares (of real quadratic forms) -- modulo the real orthogonal group \(O(\mathbb{R}^3)\) -- are in one-to-one correspondence with the eight 2-torsion points in the non-identity component of \(J(\mathbb{R})\), where \(J\) is the Jacobian of \(Q\).
The assumptions imply in particular: \(f\) is irreducible, \(Q(\mathbb{R})= \emptyset\), \(Q\) has genus 3, \(J\) has 63 non-zero complex 2-torsion points. Hilbert's original theorem (i.e. any nonnegative quartic form \(f\) is a sum of three squares in at least one way) follows from the above theorem by continuity arguments.
The proof of Theorem 1.1 proceeds in two steps:
(1) The non-trivial 2-torsion points of \(J(\mathbb{C})\) are in one-to-one correspondence with the equivalence classes -- modulo \(O(\mathbb{C}^3)\) -- of representations of \(f\) as a sum of three squares of complex quadratic forms. A condensed proof using Weil divisors, the Picard group \(\text{Pic}(Q)\) and the Riemann-Roch Theorem is given in the paper. The result itself goes back to A. B. Coble (1929; JFM 55.0808.02), it was rediscovered by C. T. C. Wall (1991; Zbl 0741.14014).
(2) Show that under (1) the non-trivial 2-torsion points of \(J(\mathbb{R})\) correspond to ``signed quadratic representations'' \(f=\pm q^2_1\pm q^2_2\pm q^2_3\) with \(q_i\in\mathbb{R}[x,y,z]\), and the 2-torsion points in the non-identity component of \(J(\mathbb{R})\) correspond to the representations with \(+\) signs. This is the essential new result of the paper. For the proof one needs the following exact sequences
\[
\begin{gathered} 0\to \text{Pic}(Q_r)\to \text{Pic}(Q)@>\partial>> \text{Br}(R)\to \text{Br}(Q_r),\\ 0\geq J(\mathbb{R})^0\to J(\mathbb{R})@>\partial>> \text{Br}(\mathbb{R})\to 0,\end{gathered}
\]
where \(Q_r\) is the curve \(Q\) as a curve over \(\mathbb{R}\) such that \(Q= Q_r\otimes\mathbb{C}\).
The first exact sequence follows from Hochschild-Serre spectral sequence for étale cohomology, the second sequence follows from a theorem of G. Weichold (1883; JFM 15.0431.01) reproved by Geyer (1964). Powers, V.; Reznick, B.; Scheiderer, C.; Sottile, F.: A new approach to Hilbert's theorem on ternary quartics, C. R. Acad. sci. Paris, sér. I 339, 617-620 (2004) Sums of squares and representations by other particular quadratic forms, Forms of degree higher than two, Jacobians, Prym varieties A new approach to Hilbert's theorem on ternary quartics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\) be a noetherian ground ring. The author establishes that an \(R\)-scheme \(X\) of finite type that admits an ample family of line bundles allows a closed embedding into another \(R\)-scheme \(W\) of finite type that admits an ample family of line bundles and is furthermore smooth. This generalizes a result of \textit{H. Brenner} and \textit{S. Schröer} [Pac. J. Math. 208, No. 2, 209--230 (2003; Zbl 1095.14004)], who proved that \(X\) has an ample family if and only if it is isomorphic to a closed subscheme of some multihomogeneous homogeneous spectrum of some finitely generated polynomial ring, endowed with a \(\mathbb{Z}^n\)-grading. Note that such spectra are usually non-separated, unlike the classical case of \(\mathbb{N}\)-gradings.
Recall that \(\mathscr{L}_1,\ldots,\mathscr{L}_n\) is called an ample family is the non-zero loci of global sections in tensor combinations form a basis of the Zariski topology. For \(n=1\) this gives back the notion of an ample sheaf. The generalization
was introduced by \textit{M. Borelli} [Pac. J. Math. 13, 375--388 (1963; Zbl 0123.38102)]. Schemes admitting an ample family are also called divisorial. Examples are the regular noetherian schemes.
A crucial step in the paper is an essentially combinatorial lemma, which gives a sufficient conditions for homogeneous localizations of \(\mathbb{Z}^n\)-graded rings to be polynomial rings. As an application, Zanchetta shows that if \(X\) is divisorial, then each collection \(\mathscr{E}_1,\ldots,\mathscr{E}_n\) of locally free sheaves of finite rank arises as a pull-back of locally free sheaves under a morphism \(f:X\to Y\) to a smooth divisorial scheme. algebraic geometry; \(K\)-theory; toric geometry Schemes and morphisms, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Embedding divisorial schemes into smooth ones | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Using the theory of jet schemes, we give a new approach to the description of a minimal generating sequence of a divisorial valuations on \(\mathbf{A}^2\). For this purpose, we show how to recover the approximate roots of an analytically irreducible plane curve from the equations of its jet schemes. As an application, for a given divisorial valuation \(v\) centered at the origin of \(\mathbf{A}^2\), we construct an algebraic embedding \(\mathbf{A}^2\hookrightarrow\mathbf{A}^N,\,N\geq2\) such that \(v\) is the trace of a monomial valuation on \(\mathbf{A}^N\). We explain how results in this direction give a constructive approach to a conjecture of Teissier on resolution of singularities by one toric morphism. Global theory and resolution of singularities (algebro-geometric aspects), Arcs and motivic integration, Toric varieties, Newton polyhedra, Okounkov bodies Jet schemes and generating sequences of divisorial valuations in dimension two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We compute the generating series for the intersection pairings between the total Chern classes of the tangent bundles of the Hilbert schemes of points on a smooth projective surface and the Chern characters of tautological bundles over these Hilbert schemes. Modulo the lower weight term, we verify \textit{A. Okounkov}'s conjecture [Funct. Anal. Appl. 48, No. 2, 138--144 (2014; Zbl 1327.14026); translation from Funkts. Anal Prilozh. 48, No. 2, 79--87 (2014)] connecting these Hilbert schemes and multiple \(q\)-zeta values. In addition, this conjecture is completely proved when the surface is abelian. We also determine some universal constants in the sense of Boissière and Nieper-Wisskirchen [\textit{S. Boissière}, J. Algebr. Geom. 14, No. 4, 761--787 (2005; Zbl 1120.14002);
\textit{S. Boissière} and \textit{M. A. Nieper-Wisskirchen}, LMS J. Comput. Math. 10, 254--270 (2007; Zbl 1221.14005)] regarding the total Chern classes of the tangent bundles of these Hilbert schemes. The main approach of this article is to use the set-up of Carlsson and Okounkov outlined in
\textit{E. Carlsson}, Vertex operators and moduli spaces of sheaves. Princeton University (PhD thesis) (2008); \textit{E. Carlsson} and \textit{A. Okounkov}, Duke Math. J. 161, No. 9, 1797--1815 (2012; Zbl 1256.14010)] and the structure of the Chern character operators proved in [\textit{W.-P. Li} et al., Int. Math. Res. Not. 2002, No. 27, 1427--1456 (2002; Zbl 1062.14010)]. Parametrization (Chow and Hilbert schemes), Divisors, linear systems, invertible sheaves, Binomial coefficients; factorials; \(q\)-identities, Sheaves in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry On Okounkov's conjecture connecting Hilbert schemes of points and multiple \(q\)-zeta values | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(U\subseteq \mathbb{R}^n\) be a small open neighbourhood of 0 and \(I\subseteq \mathbb{R}\) an open interval containing \([0,1]\). Let \(N_i(x,t)\), \(D_i(x,t)\) be analytic functions in \((x, t) \in U\times J\). Suppose that the weighted initial forms (with respect to positive rational weights \(w_1, \dots, w_n\) for \(x_1, \dots, x_n\) and 0 weight for \(t)\) \(w(D_i)\) of \(D_i\) are positive and for the corresponding weighted degree holds
\[
w\deg \bigl(W (N_i)\bigr) \geq w\deg \bigl(W(D_i) \bigr)+ w_i, \quad i=1, \dots,n.
\]
Then the integration of the vector field \(\delta= \sum {N_i \over D_i} {\partial \over \partial x_i} +{\partial \over \partial t}\) generates an arc-analytic homeomorphism. Furthermore it is proved that for a \(t\)-parametrized analytic family \(F_t(x)= W_d(x,t) +\cdots, W_d\) the weighted initial form of \(F_t(x)\), such that for each \(g\), \(W_d\) admits an isolated singularity at 0, the family is arc-analytically trivial, i.e. there exist a \(t\)-level preserving arc-analytic homeomorphism \(H:(U \times J,0 \times J) \to(U \times J,0 \times J)\) such that \(F\circ H\) is independent of \(t\). singular vector field; arc-analytically trivial family; arc-analytic homeomorphism Kuo, T-C., and Milman, P. D.: On arc-analytic trivialisation of singularities. Real Analytic and Algebraic Singularities. Pitman Res. Notes Math. ser. 381, Longman, Harlow, pp. 38-42 (1998). Real-analytic manifolds, real-analytic spaces, Real-analytic and semi-analytic sets On arc-analytic trivialization of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We realized that there is one exception in Theorem 4.3 of our paper [Adv. Geom. 2, No. 4, 371--389 (2002; Zbl 1054.14052)], namely, the case where the web \(\Delta\) is contained in the tangent space to \(G(3,4)\) at a point. Fania M.L. and Mezzetti E. (2008). Erratum to ''On the Hilbert scheme of Palatini threefolds''. Adv. Geom. 8: 153--154 (to appear) \(3\)-folds, Parametrization (Chow and Hilbert schemes), Low codimension problems in algebraic geometry, Varieties of low degree, Adjunction problems Erratum to ``On the Hilbert scheme of Palatini threefolds'' | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We investigate finite torsors over big opens of spectra of strongly \(F\)-regular germs that do not extend to torsors over the whole spectrum. Let \((R,\mathfrak{m},\mathfrak{k},K)\) be a strongly \(F\)-regular \(\mathfrak{k}\)-germ where \(\mathfrak{k}\) is an algebraically closed field of characteristic \(p>0\). We prove the existence of a finite local cover \(R\subset R^*\) so that \(R^*\) is a strongly \(F\)-regular \(\mathfrak{k}\)-germ and: for all finite algebraic groups \(G/\mathfrak{k}\) with solvable neutral component, every \(G\)-torsor over a big open of \(\mathrm{Spec}R^*\) extends to a \(G\)-torsor everywhere. To achieve this, we obtain a generalized transformation rule for the \(F\)-signature under finite local extensions. Such formula is used to show that the torsion of \(\mathrm{Cl}R\) is bounded by \(1/s(R)\). By taking cones, we conclude that the Picard group of globally \(F\)-regular varieties is torsion-free. Likewise, this shows that canonical covers of \(\mathbb{Q}\)-Gorenstein strongly \(F\)-regular singularities are strongly \(F\)-regular. \(F\)-regularity; \(F\)-signature; finite torsors; local Nori fundamental group-scheme Group schemes, Homotopy theory and fundamental groups in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Hopf algebras and their applications, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Actions of groups on commutative rings; invariant theory Finite torsors over strongly \(F\)-regular singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author treats many inequalities and introduces the following main theorems of analytic irreducible plane curve singularities.
Let \({\mathcal V} =\{(z,y): f(z,y)= z^n+ {\mathcal A} y^\alpha z^p +y^\beta z^q+ y^k=0\}\) and \({\mathcal W} = \{(z,y): g(z,y)= z^n+{\mathcal B} y^\gamma z^s+ y^\delta z^i+ y^k=0\}\) be germs of analytic irreducible subvarieties of a polydisc near the origin in \(\mathbb{C}^2\) with \(n<k\) and \((n,k)=1\) where \({\mathcal A}\) and \({\mathcal B}\) are complex numbers. Assume that \({\mathcal V}\) and \({\mathcal W}\) are topologically equivalent near the origin. Then we denote this relation by \(f\sim g\) for brevity. If \({\mathcal V}\) and \({\mathcal W}\) are analytically equivalent at the origin, then we write \(f\approx g\).
Theorem. Let \(f\sim g \sim z^n+y^k\) at the origin. Assume that \(1\leq q< p\leq n-2\), \(1\leq \alpha< \beta\leq k-2\), \(\alpha+ p<\beta +q\), \(a/(n-p) > \beta/(n-q)\); \(1\leq t< s\leq n-2\), \(1\leq\gamma < \delta\leq k-2\), \(\gamma+ s< \delta+t, \gamma/(n-s)> \delta/(n-t)\). Then \(f\approx g\) if and only if \((\alpha,p) = (\gamma, s)\), \((\beta,q) = (\delta,t)\) and \(a^n= d^k= a^qd^\beta\), \({\mathcal A} a^pd^\alpha = a^n {\mathcal B}\) for some nonzero numbers \(a,d\).
In detail, \(f\approx g\) implies that \(a^N= d^N=1\) and \({\mathcal A}^N ={\mathcal B}^N\) where \({\mathcal N}= n\beta+ kq-nk\).
Theorem. Let \(f=z^n+ {\mathcal A}_1 y^{\alpha_1} z^{p_1} + \cdots + {\mathcal A}_{t-1} y^{\alpha_{t-1}} z^{p_{t-1}} + y^{\alpha_t} z^{p_t} + y^k\) where \(n<k\), \((n,k)=1\), \(n-2 \geq p_1> \cdots >p_t\geq 1\), \(\alpha_t \leq k-2\), \(\alpha_1+ p_1< \cdots <\alpha_t+ p_t, \alpha_1/(n-p_1) > \cdots > \alpha_t/(n-p_t) > k/n\) and each \({\mathcal A}_i = {\mathcal A}_i (z,y)\) is a unit in \(_2{\mathcal O}\) for \(i=1, \dots, t-1\). Let \(g=z^n+ {\mathcal B}_1 y^{\beta_1} z^{q_1} + \cdots + {\mathcal B}_{s-1} y^{\beta_{s-1}} z^{q_{s-1}} + y^{\beta_s} z^{q_s} + y^k\) where exponents \(\{\beta_*, q_*\}\) and coefficients \(\{{\mathcal B}_j\}\) hold the same inequalities as above.
If \(f\approx g\) then \(t=s\), \((\alpha_i,p_i) = (\beta_i,q_i)\) for \(i=1, \dots, t\) and \({\mathcal A}_i (0,0)^{n\alpha_t + kp_t-nk} = {\mathcal B}_i (0,0)^{n\alpha_t + kp_t-nk}\) for \(i=1, \dots, t-1\).
In particular, if \({\mathcal A}_i\) and \({\mathcal B}_i\) are nonzero complex numbers and \(n\alpha_t + kp_t-n k=1\) with the same assumption above, then \(f\approx g\) if and only if \((\alpha_i,p_i) = (\beta_i,q_i)\) for \(i=1, \dots, t=s\) and \({\mathcal A}_i = {\mathcal B}_i\). plane curve singularities; topologically equivalent; analytically equivalent Equisingularity (topological and analytic), Global theory and resolution of singularities (algebro-geometric aspects) Some analytic irreducible plane curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field, and let \(P_ 1,\ldots,P_ s\) be points in general position in \(\mathbb{P}^ 2(k)\) (no three on a line), and let \(m_ 1,\ldots,m_ s\) be positive integers. The paper concerns the Hilbert function of the coordinate ring of the points \(P_ i\) with multiplicities \(m_ i\), respectively. The first main result provides a characterization for the points to lie on an irreducible conic. The second one gives, in case \(m_ i=2\) for all \(i\), a relation between the index of regularity (the degree where the Hilbert function stabilizes to the Hilbert polynomial) and the number of points lying on a conic. points in general position; Hilbert function; multiplicities; index of regularity; Hilbert polynomial; number of points lying on a conic Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Projective techniques in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Enumerative problems (combinatorial problems) in algebraic geometry Some remarks on the Hilbert function of a zero-cycle in \(\mathbb{P}^ 2\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(Z \subset \mathbb{P}^2\) be a zero-dimensional scheme. Fix \(t \in \mathbb N\). In this paper we study the following question: find assumptions on \(Z\) and \(t\) such that \(h^1 (\mathcal{I}_A (t)) < h^1 (\mathcal{I}_Z (t))\) for all \(A\subsetneq Z\) and check if \(t\) does not exist for a certain class of schemes \(Z\). zero-dimensional scheme; plane curve; Hilbert function Projective techniques in algebraic geometry Zero-dimensional schemes in the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f \colon X \to Z\) be a contraction of normal projective varieties defined over an algebraically closed field of characteristic zero, and \((X,B)\) be a klt pair of dimension \(d\) such that \(K_X + B \sim_{\mathbb{R}} 0/Z\). By \textit{Y. Kawamata} [Contemp. Math. 207, 79--88 (1997; Zbl 0901.14004); Am. J. Math. 120, No. 5, 893--899 (1998; Zbl 0919.14003)] and \textit{F. Ambro} [``The adjunction conjecture and its applications'', Preprint, \url{arXiv:math/9903060}],
we may write \[ K_X + B \sim_{\mathbb{R}} f^*(K_Z + B_Z + M_Z), \] where \(B_Z\) is the \textit{discriminant divisor} and \(M_Z\) is the \textit{moduli divisor}, which is determined up to \(\mathbb{R}\)-linear equivalence. Note that \((Z, B_Z + M_Z)\) is a \textit{generalised pair}. From now on, assume that \(X\) is of Fano type over \(Z\). Shokurov conjectured that for any real number \(\varepsilon>0\), if \((X, B)\) is \(\varepsilon\)-lc, then there is a real number \(\delta > 0\) depending on \(d\) and \(\varepsilon\) such that \((Z, B_Z+M_Z)\) is generalised \(\delta\)-lc. A special case of Shokurov's conjecture is McKernan's conjecture. The main result of the paper under review essentially says that Shokurov's conjecture holds in the toric setting after taking an average with the toric boundary divisor (Theorems 1.4 and 1.7). A consequence of the main theorem asserts that if \(f\) is a toric Fano contraction and \(X\) is \(\varepsilon\)-lc, then there is an integer \(m \geq 1\) depending only on \(\varepsilon\) and \(d\) such that the multiplicities of the fibers of \(f\) over codimension one points of \(Z\) are bounded by \(m\) (Corollary 1.5). Another consequence is a new proof of the result of \textit{V. Alexeev} and \textit{A. Borisov} [Proc. Am. Math. Soc. 142, No. 11, 3687--3694 (2014; Zbl 1305.14006)] on McKernan's conjecture for toric morphisms of toric varieties (Corollary 1.6). toric varieties; Shokurov's conjecture; singularities of pairs Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Minimal model program (Mori theory, extremal rays) Singularities on toric fibrations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert-Kunz multiplicity is a numerical invariant introduced by \textit{E. Kunz} [Am. J. Math. 91, 772--784 (1969; Zbl 0188.33702)] for an \(m\)-primary ideal \(I\) of a Noetherian local ring \((A,m)\) of characteristic \(p>0\). It is a real number which is in general difficult to compute, but it is known to be a rational number for some classes of rings. In particular \textit{K. Watanabe} [`Hilbert-Kunz multiplicity of toric rings', Proc. Inst. Nat. Sci., Nihon Univ. 35, 173--177 (2000; Zbl 1172.13309)] proved that the Hilbert-Kunz multiplicity of a toric ring is a rational number. In this paper an explicit formula is given to compute the Hilbert-Kunz multiplicity of two dimensional toric rings. The formula shows that the Hilbert-Kunz multiplicity of two-dimensional non-regular toric rings is at least \(3/2\). Hilbert-Kunz multiplicity; toric rings; characteristic \(p\) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Toric varieties, Newton polyhedra, Okounkov bodies, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure The Hilbert-Kunz multiplicity of two-dimensional toric rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The 14th International Workshop of Real and Complex Singularities took place in São Carlos (São Paulo, Brazil), at the Instituto de Ciências Matemáticas e de Computação of São Paulo University (ICMC-USP) from 24th to 30th of July 2016, and was preceded by a School on Singularity Theory from 17th to 22nd of July. Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Proceedings, conferences, collections, etc. pertaining to manifolds and cell complexes, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to differential geometry, Proceedings, conferences, collections, etc. pertaining to global analysis Real and complex singularities and their applications in geometry and topology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This survey discusses hyperbolicity properties of moduli stacks and generalisations of the Shafarevich Hyperbolicity Conjecture to higher dimensions. It concentrates on methods and results that relate moduli theory with recent progress in higher dimensional birational geometry. hyperbolicity properties of moduli spaces; differential forms on singular spaces; minimal model program [21] Stefan Kebekus, &Differential forms on singular spaces, the minimal program, and hyperbolicity of moduli&#xhttp://arxiv.org/abs/1107.4239 Families, moduli, classification: algebraic theory, Structure of families (Picard-Lefschetz, monodromy, etc.), Fine and coarse moduli spaces, Rational and birational maps, Minimal model program (Mori theory, extremal rays) Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In algebraic geometry, moduli spaces are constructed by forming suitable quotients, the most common of these being projective methods. Grothendieck generalized this to the functor of points, and the main problem considered concerns representable functors, where the scheme representing the functor is the moduli space for families of geometric points. To prove representability, Grothendieck (together with Serre, MacLane et. al.) developed deformation theory and formal geometry as main tools for proving representability, but left this for the projective methods in proving the existence of Hilbert and Picard schemes.
Later on, Artin generalized Grothendieck's existence results, and proved that Hilbert and Picard schemes exists as algebraic spaces. The authors claim that the correct setting is that of algebraic spaces and stacks (not schemes).
Artin gave precise criteria for algebraicity of functors and stacks. These were clarified by Conrad and De Jong using Néron-Popescu desingularization, by \textit{H. Flenner} [Math. Z. 178, 449--473 (1981; Zbl 0453.14002)] using Exal, and by the first author [J. Reine Angew. Math. 722, 137--182 (2017; Zbl 1362.14012)] using coherent functors.
This article uses the ideas of Flenner and the first author to give a new criterion for algebraicity of functors and stacks, giving a supplement to Artin's criteria.
To give a wider application of the criteria, and to simplify the proofs of Artin and Flenner, several new concepts are introduced. Also, a place where Artin's proof for algebraicity of stacks doesn't work in positive characteristic is circumvented by these new techniques.
The main result is stated for \(S\) an excellent scheme. Then a category \(X\) fibred in groupoids over \(\operatorname{Sch}/S,\) is an algebraic stack, locally of finite presentation over \(S\), if and only if it satisfies:
\begin{itemize} \item[(1)] \(X\) is a stack over \(\operatorname{Sch}/S\) with the fppf topology, \item[(2)] \(X\) preserves limits, \item[(3)] \(X\) is weakly effective, \item[(4)] \(X\) is \(\mathbf{Art}^{\text{triv}}\)-homogeneous, \item[(5a)] \(X\) has bounded aumorphisms and deformations, \item[(5b)] \(X\) has constructible automorphisms and deformations, \item[(5c)] \(X\) has Zariski local automorphisms and deformations, \item[(6b)] \(X\) has constructible obstructions, \item[(6c)] \(X\) has Zariski local obstructions. \end{itemize}
In addition, the main result contains special cases where some of the criteria are superfluous or can be replaced by simpler ones.
The main content of the paper is to give the necessary definitions of the nine concepts (criteria) above, and to prove the result.
The authors point out the homogeneity condition (4) above as the most striking difference between their condition and Artin's condition as this condition only involves local artinian schemes and that there is no conditions on étale localization of deformations and obstruction theories. When \(S\) is Jacobson, e.g. of finite type over a field, no compatibility with zariski localization is needed, nor conditions on the compatibility with completions for automorphisms and deformations. A detailed comparison between the results of this article and the results of Artin is given.
The authors single out the following steps in algebraicity proofs up to now: \begin{itemize} \item[(i)] Existence of formally versal deformations, \item[(ii)] algebraization of formally versal deformations, \item[(iii)] openess of formal versality, \item[(iv)] formal versality implies formal smoothness. \end{itemize}
Notice that these steps gives an explicit explanation of the link between deformation theory and the theory of algebraic stacks. The last two steps are the ones where the treatment of Artin, Flenner, Starr and Hall differs from the present. This yields both the criteria themselves and the techniques, and this treatment fills up the main body of the paper.
Step (iv), formal versality implies formal smoothness: This criterion is weaker than Artin's two criteria, except that in positive characteristic, \(X\) needs to be a stack in the fppf topology or criteria (4) must be strengthened. This is similar to Artin's criteria where the functor is assumed to be an fppf-sheaf. This is crucial for his proof that formally universal deformations are formally étale, settling step (iv) for functors. This proof also depends on the existence of universal deformations and so does not extend to stacks with infinite or nonreduced stabilizers. Using a different approach, the authors extend this result to fppf stacks.
Artin only assumes that the groupoid is an étale stack. The authors express that they do not understand Artin's proof of step (iv) when \(S\) is not of finite type over \(\mathbb Z\) or a perfect field. This special case is solved by the techniques given in the present paper.
Step (iii), openness of formal versality: In the treatment of the present exposition, localization is only required when passing to nonclosed points of finite type. These points only exist when \(S\) is not Jacobson, that is if \(S=\operatorname{Spec} D\) of a DVR. The validation of the proof boils down to a matter of simple algebra, the criterion for the openness of the vanishing locus of half-exact functors that follows from the Ogus-Bergman Nakayama Lemma for half-exact functors. From step (ii) and (iv) it is proved that conditions (1)--(4) and (5a) at fields gives homogeneity for arbitrary integral morphisms, and so that \(\operatorname{Aut}_{X/S}(T,-),\;\operatorname{Def}_{X/S}(T,-)\) and \(\operatorname{Obs}_{X/S}(T,-)\) are additive functors.
The distinct advantage of the criterion in this paper is the dramatic weakening of the homogeneity. The ideas of the paper have lead to a criterion for a half-exact functor to be coherent, and this does not follow from any algebraicity criterion.
The authors recall the notion of homogeneity, limit preservation and extensions. They introduce homogeneity involving only artinian rings and show that residue field extensions are invariant for stacks in the fppf topology. They relate formal versality, formal smoothness and the vanishing of Exal, which is also defined. Then additive functions are considered, and their vanishing loci. This is applied to Exal, which assure that the locus of formal versality is open. Conditions are given on \(\operatorname{Def}\), \(\operatorname{Aut}\), \(\operatorname{Obs}\) that imply the corresponding conditions on Exal.
The general theory is applied to introduce \(n\)-step obstruction theories. These are formulated without using linear obstruction theories of Artin, and are applied to the study of effectivity.
This is a very far-reaching article, concatenating and giving a universal understanding of the correspondence and differences between algebraic stacks and obstruction theory. As there are very few authors that are experts in both fields, this article is a valuable contribution, with a lot of very important applications and comparisons. moduli scheme; moduli stack; quotient stack; representability; algebraic spaces; stacks; Exal; coherent functors; algebraicity; Picard scheme; Hilbert scheme; excellent scheme; locally of finite presentation; goupoid; fibred in groupoids; algebraic stack; preserves limits; weakly effective; fppf topology; deformations; \(\mathbf{Art}^{\text{triv}}\)-homogeneous; bounded aumorphisms and deformations; constructible automorphisms and deformations; Zariski local automorphisms and deformations; constructible obstructions; Zariski local obstructions; formal versality; openness of formal versality; étale stack; homogeneity; limit preservation; Exal; effectivity Formal methods and deformations in algebraic geometry, Stacks and moduli problems, Generalizations (algebraic spaces, stacks) Artin's criteria for algebraicity revisited | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We construct natural relative compactifications for the relative Jacobian over a family \(X/S\) of reduced curves. In contrast with all the available compactifications so far, ours admit a Poincaré sheaf after an étale base change. Our method consists of studying the étale sheaf \(F\) of simple, torsion-free, rank-1 sheaves on \(X/S\), and showing that certain open subsheaves of \(F\) have the completeness property. Strictly speaking, the functor \(F\) is only representable by an algebraic space, but we show that \(F\) is representable by a scheme after an étale base change. Finally, we use theta functions originating from vector bundles to compare our new compactifications with the available ones. compactifications for the relative Jacobian; Poincaré sheaf; étale base change; theta functions; family of reduced curves Busonero, S.: Compactified Picard schemes and Abel maps for singular curves, PhD thesis, Sapienza Università di Roma (2008) Algebraic moduli problems, moduli of vector bundles, Jacobians, Prym varieties, Vector bundles on curves and their moduli Compactifying the relative Jacobian over families of reduced curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This book is both a continuation and a complement to \textit{E. Freitag} and \textit{R. Kiehl}'s book on ``Étale cohomology and the Weil conjecture'' (1988; Zbl 0643.14012). Among other important things, the book gives a clear account of \textit{P. Deligne}'s generalized Weil conjecture [Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014)] mostly cited as ``Weil II''. This conjecture has had important consequences and applications, and many of them are presented in the book as well.
The book starts with the theory of \(\ell\)-adic sheaves and the proof of Deligne's generalized Weil conjecture. The conjecture is first proven for curves, following essentially \textit{G. Laumon}'s approach based on the Fourier transform for complexes of \(\ell\)-adic sheaves [Publ. Math., Inst. Hautes Étud. Sci. 65, 131-210 (1987; Zbl 0641.14009)]. Then the proof of the conjecture for a morphism (roughly speaking, that the higher direct images \(R^i f_!\) of a \(\tau\)-mixed sheaf are \(\tau\)-mixed) is given.
The second chapter is devoted to the formalism of derived categories, and it is a very nicely written account of the theory. The theory of abstract truncations for a triangulated category is given. This theory is used in the book to introduce truncation functors on the triangulated category \(D^b_c(x,\mathbb{Z}_\ell)\) of constructible complexes of étale \(\mathbb{Z}_\ell\)-sheaves, because this category, being not the derived category of an abelian category, does not have the natural simple truncation operators necessary to construct cohomology functors.
In the third chapter the theory of (middle) perverse sheaves is revisited; working in the category of perverse sheaves, the theory of global weights and the notion of purity of sheaves are then studied. Some important results, like Gabber's theorem on the semisimplicity of pure complexes are given. New variants of the Fourier transform, like the Deligne-Fourier transform, are introduced and used to simplify and enlighten the theory.
Among the applications given, we can list the Kazhdan-Lusztig polynomials, the hard Lefshetz theorem, the Radon-Brylinski transform, the theory of trigonometric sums and the Springer representations of Weyl groups of semisimple algebraic groups. Regarding trigonometric sums, they are exponential sums related with étale cohomology by means of the Deligne-Fourier transform. Katz and Laumon proved that the Deligne-Fourier transforms \(D^b_c(\mathbb{A}_{\mathbb{F}_p},\overline\mathbb{Q}_\ell)\to D^b_c(\mathbb{A}_{\mathbb{F}_p},\overline\mathbb{Q}_\ell)\) defined for every prime number \(p\), are uniform in a certain sense. The consequence is that those different Deligne-Fourier transforms behave as if there exists a global Deligne-Fourier transform \(D^b_c(\mathbb{A}_{\mathbb{Z}}, \overline\mathbb{Q}_{\ell})\to D^b_c(\mathbb{A}_{\mathbb{Z}}, \overline\mathbb{Q}_{\ell})\) inducing the previous ones for almost all prime numbers. This important result implies the existence of uniform estimates for the corresponding exponential sums. The book describes this remarkable theory in a very clear way.
Since the book uses freely many results included in the aforementioned book of Freitag and Kiehl, the reader is advised to read the latter first. Besides, the book under review assumes that the reader is familiar with some techniques like étale cohomology of schemes. The details of the proofs are quite technical and involve a great amount of abstract algebraic geometry, but the ideas behind them are very neatly explained and the book succeeds in giving both a global picture of the theory and a precise development with complete (and often difficult) proofs.
This book will be very useful for those interested in number theory from the viewpoint of algebraic geometry as well as for those who seek for an introduction to the topics covered by the book, like perverse sheaves, derived categories, \(t\)-structures or Deligne weights, among others. Weil conjectures; perverse sheaves; \(l\)-adic Fourier transform; Springer representations; derived categories; truncation functors; Deligne weights R. Kiehl and R. Weissauer, \textit{Weil conjectures, perverse sheaves and }l\textit{'adic Fourier transform}, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 42, Springer-Verlag, Berlin, 2001.Zbl 0988.14009 MR 1855066 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Local ground fields in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Varieties over finite and local fields, \(p\)-adic cohomology, crystalline cohomology, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Weil conjectures, perverse sheaves and \(l\)-adic Fourier transform | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The extensive framework of étale topology, sheaf theory, and cohomology was both initiated and largely developed by A. Grothendieck, M. Artin, J.-L. Verdier, and others in the early 1960s. Based on previous ideas due to J.-P. Serre, this algebro-geometric instrument was primarily created as a novel approach to tackle the famous Weil conjectures (from 1949) on varieties over finite fields and their zeta functions. In fact, the étale approach proved to be highly successful in arithmetic geometry ever since, especially when P. Deligne finally verified the Weil conjectures in this context around 1973. The fundamental concepts, methods, techniques, and results of étale topology and cohomology theory where first published in the volumes SGA 1, SGA 4, SGA \(4\frac12\) and SGA 5 of the series ``Séminaire de Géometrie Algébrique'' (SGA) by A. Grothendieck and his collaborators between 1960 and 1966, originally as mimeographed notes, but in the 1970s also as Lecture Notes in Mathematics by Springer Verlag. Among the classical standard textbooks on the subject are the popular primers [\textit{J. S. Milne}, Étale cohomology. Princeton, New Jersey: Princeton University Press (1980; Zbl 0433.14012)] and [\textit{E. Freitag} and \textit{R. Kiehl}, Étale cohomology and the Weil conjecture. With a historical introduction by J. A. Dieudonné. Berlin etc.: Springer-Verlag (1988; Zbl 0643.14012)], apart from the concise course notes [\textit{G. Tamme}, Introduction to étale cohomology. Translated by Manfred Kolster. Berlin: Springer-Verlag (1994; Zbl 0815.14012)].
The book under review is the second, revised edition of another, more recent introduction to the subject of étale cohomology theory. The original edition appeared in 2011 and has been very briefly reviewed back then [Zbl 1228.14001], but here we would like to be a little more precise concerning both its contents and its features. First of all, the book was written as a complete, basically self-contained preparation for a series of talks on Deligne's proof of the Weil conjectures, with the main focus on both the conceptual and technical toolkit of étale cohomology theory, and without any reference to the Weil conjectures themselves as for motivation or application of the machinery developed in the text. In this respect, the present book differs quite a bit form the above-mentioned classics by Milne and Freitag-Kiehl, respectively, but on the other hand it offers much more methodological completeness, detailedness of proofs, and technical rigor in presenting the foundational material as covered in the basic SGA volumes cited before. Accordingly, the current book may serve as a perfect companion to the venerable classics on étale cohomology, and as a profound introduction to the study of the original research works in the field, too. As for the precise contents, the book comprises ten chapters, each of which consists of several thematic sections. Each chapter begins with a list of references related to the respective contents, with strong emphasis on the relevant volumes of the SGA series as for further, more advanced and thorough reading.
Chapter 1 gives an introduction to descent theory, including the basics on flat modules and morphisms, descent properties of sheaves, morphisms and schemes, quasi-finite morphism, and passage-to-limit properties. Chapter 2 discusses sheaves of relative differentials, étale morphisms and smooth morphisms of schemes, infinitesimal liftings of morphisms, the concept of Henselization as well as direct and inverse limits in categories. Chapter 3 deals with the concept of étale fundamental group of a scheme and its functorial properties, whereas Chapter 4 briefly explains group cohomology, profinite groups and their cohomology, cohomological dimension, and Galois cohomology. Chapter 5 turns to the main subject of the book and provides the basic concepts of étale cohomology, thereby introducing Čech cohomology, étale sheaves, Grothendieck topologies, the notion of étale cohomology, calculation methods for étale cohomology, constructible sheaves, and the according passage-to-limit results. Chapter 6 deals with triangulated categories, derived categories and functors, together with their effective use in algebraic geometry, while Chapter 7 is devoted to various base change theorems in algebraic geometry, together with applications to particular cohomology theories and higher direct image sheaves. Chapter 8 treats duality theory, including extensions of Henselian discrete valuation rings, trace morphisms, duality for curves, the functor \(Rf^!\), Poincaré duality (à la SGA 4), and cohomology classes corresponding to algebraic cycles. Chapter 9 presents the finiteness theorems for constructible sheaves on Noetherian schemes and the so-called biduality theorem, complemented by sections on nearby cycles and vanishing cycles, sheaves with group actions, and generic local acyclicity, respectively, whereas Chapter 10 finally turns to the fundamentals of \(l\)-adic cohomology, especially regarding the adic formalism in general, the Grothendieck-Ogg-Shafarevich formula, Frobenius correspondences, the Lefschetz trace formula, and Grothendieck's formula for \(L\)-functions on compactifiable schemes over finite fields.
As for the prerequisites for reading this utmost comprehensive, however rather functional primer on étale cohomology theory, the reader is assumed to have a profound background knowledge of both basic commutative algebra and advanced modern algebraic geometry, the latter perhaps through R. Hartshorne's standard textbook or -- even better -- through the volumes EGA I--IV by A. Grothendieck and J. Dieudonné.
Summing up, this book is highly useful and valuable for any seasoned reader looking for a thorough introduction to the toolkit of étale cohomology with a view toward further study of its applications in both algebraic and arithmetic geometry. étale cohomology; \(l\)-adic cohomology; theory of descent; étale fundamental group; group cohomology; Galois cohomology; duality theory; Grothendieck topology; derived categories Fu, L., Etale cohomology theory, pp., (2015), World Scientific, Hackensack, NJ Research exposition (monographs, survey articles) pertaining to algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homotopy theory and fundamental groups in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Derived categories, triangulated categories Étale cohomology theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A Weil cohomology theory is a contravariant functor from the category of irreducible smooth projective varieties over an algebraically closed field to the category of finite-dimensional graded anticommutative algebras over a fixed coefficient field, satisfying certain sophisticated axioms. The notion was developed in the sixties by A.~Grothendieck and M.~Artin in attempts to prove the Weil conjectures (concerning the number of solutions of equations over finite fields and their relation to the topological properties of the variety defined by the corresponding equations over \(\mathbb C\)) and was later used by P.~Deligne in his proof of the conjectures. The main result of the paper is that, although the axioms for a Weil cohomology theory are formulated in terms of high level notions and so do not look like first order ones, they are, in a natural sense, first order. In particular, one can form ultraproducts of Weil cohomology theories and these are models of the axioms. An essential point is a study of first order aspects of intersection theory; in his analysis the author follows \textit{W. Fulton} [``Intersection theory'' (1984; Zbl 0541.14005)]. Then he gives a first order axiomatization of the notion of Weil cohomology theory essentially equivalent to \textit{S. Kleiman}'s axiomatization [in: Dix Exposés sur la cohomologie des Schémas, Adv. Stud. Pure Math. 3, 359--386 (1968; Zbl 0198.25902)]. He remarks that Grothendieck's standard conjectures, assumed for all Weil cohomology theories, imply strong uniformities in the theory of cycles, and, using an ultraproduct argument, demonstrates that for numerical equivalence of cycles. first order axiomatization; ultraproduct Macintyre A (2000) Weil cohomology and model theory. In: Macintyre A (ed) Connections between model theory and algebraic and analytic geometry. Quaderni di matematica, vol 6, pp 179--199. Arache, Rome Étale and other Grothendieck topologies and (co)homologies, Algebraic cycles, Model-theoretic algebra Weil cohomology and model theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In 1952 and 1954 \textit{M. Rosenlicht} published two papers on algebraic curves with singularities) and their Picard varieties (called generalized Jacobian varieties in the case of curves) [Ann. Math. (2) 56, 169--191 (1952; Zbl 0047.14503); 59, 505--530 (1954; Zbl 0058.37002)]. These papers were written with the help of the notions introduced in algebraic geometry by Weil. These two Bourbaki talks review the results of Rosenlicht in that language. The results of Rosenlicht remained very important in algebraic geometry. The best place to find the precise statements and their proof nowadays is the book of \textit{J.-P. Serre} [Groupes algébriques et corps de classes. Paris: Hermann (1959; Zbl 0097.35604)].
Joint review for this article and Zbl 0134.16505. algebraic geometry Algebraic geometry Rélations d'équivalence sur les courbes algébriques ayants des points multiples | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0717.00008.]
Let \(C'\) be a projective integral curve over a locally noetherian scheme \(S\), and \(C'\to C\) be a relatively birational morphism obtained by imposing a singularity \(Q\) on \(C'\). The difference between the compactified Jacobian variety \(J\) of \(C\) and \(J'\) of \(C'\) is measured by the concept of the presentation functor which was introduced by \textit{T. Oda} and \textit{C. S. Seshadri} [Trans. Am. Math. Soc. 253, 1-90 (1979; Zbl 0418.14019)] in a special case. \textit{H. Kleppe} [M.I.T. Thesis (1981)] generalized their results about the functor in the case where \(Q\) is a node, and he showed the representability of the functor and analyzed the structure. In the paper under review the authors show the corresponding theorem when \(Q\) is a cusp. This work is a continuation of some earlier papers by the authors [Bull. Am. Math. Soc. 82, 947-949 (1976; Zbl 0336.14008); Am. J. Math. 101, 10-41 (1979; Zbl 0427.14016); Adv. Math. 35, 50-112 (1980; Zbl 0427.14015)] and by \textit{H. Kleppe} and \textit{S. Kleiman} [Compos. Math. 43, 277-280 (1981; Zbl 0463.14008)]. compactified Jacobian; node; cusp A.B. Altman and S.L. Kleiman, \textit{The presentation functor and the compactified Jacobian}, in \textit{The Grothendieck Festschrift} , P. Cartier et al. eds., Birkhäuser Boston, Boston, U.S.A. (2007). Jacobians, Prym varieties, Singularities of curves, local rings The presentation functor and the compactified Jacobian | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In 1949 André Weil made his famous conjectures on solutions of equations over finite fields. These conjectures explained how a ``good'' cohomology theory (with values in a field of characteristic 0) would allow one to give amazingly precise information about the zeta function of a (smooth projective) variety over the finite field \(\mathbb{F}_ q\). These conjectures gave birth to a huge explosion of the most profound algebra at the hands of Serre, Grothendieck, M. Artin, and others, which led to the final proof of these conjectures by P. Deligne in 1973. But it is safe to say that these conjectures, and the algebra they generated, have been basic in much of the great successes of modern number theory and algebraic geometry.
As is well known, cohomology comes in many different ``flavors''; one has Betti cohomology, de Rham cohomology, \(\ell\)-adic cohomology, \(p\)-adic cohomology, etc. All these cohomology theories tend to have certain things in common, such as Poincaré duality, cohomology classes of cycles and so on. One of Grothendieck's deepest -- and as yet not completely fulfilled -- programs was to use these properties to ``sculpt'' the category of varieties (over an arbitrary field) into a new category (called the category ``motives'') which, in some sense, is the ``universal'' cohomology theory. For instance, in a linear space, all projectors split and so one wants to impose the same property on motives and so on. In order to actually carry out these constructions geometrically one needs lots of cycles and so Grothendieck made his famous ``standard conjectures'' which are still unknown.
In the case where we are back to working over the finite field \(\mathbb{F}_ q\), one can give a construction of the category of motives via Deligne's proof of the Weil conjectures (one uses the Frobenius morphism and its various characteristic polynomials to construct the needed projectors). Upon assuming the Tate conjecture (relating cycles to the order of pole of the zeta function and to \(\ell\)-adic representations) one can give an ``almost entirely satisfactory'' description of the category of motives over finite fields. The paper being reviewed gives a very readable complete treatment of these motives and describes a reduction functor from the category of CM-motives (over the algebraic closure of \(\mathbb{Q}\)) to the category of motives over finite fields. Tannakian category; Grothendieck's standard conjectures; finite ground field; category of motives; category of CM-motives J. S. Milne, Motives over finite fields, in Motives (Seattle, WA, 1991), 401--459, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.Zbl 0811.14018 MR 1265538 (Co)homology theory in algebraic geometry, Generalizations (algebraic spaces, stacks), Finite ground fields in algebraic geometry, Monoidal categories (= multiplicative categories) [See also 19D23], Other (co)homology theories (cyclic, dihedral, etc.) [See also 19D55, 46L80, 58B30, 58G12], Drinfel'd modules; higher-dimensional motives, etc. Motives over finite fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth projective algebraic curve of genus \(g\) defined over a finite field \(F_{q}\) of \(q\) elements. Then Weil's bound, improved by Serre, states that number of its rational points \(N(C)\) is bounded by
\[
N(C)\leq q+1+g\lfloor{2\sqrt q}\rfloor .
\]
If we denote by \(N_q(g)\) (resp. \(M_q(g)\)) the maximum (resp. the minimum) of \(N(C)\) as \(C\) runs through all curves of genus \(g\) over \(F_q\), a natural question is: For which values of \(g\) is the difference between the Serre-Weil upper bound and \(N_q(g)\) bounded as \(q\) varies? This paper answers this question when \(g=3\) and for any \(q\).
For \(g=1\) and any \(q\), in [\textit{W. C. Waterhouse}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 2, 521-560 (1969; Zbl 0188.53001)], it is shown that the aforementioned difference is either \(0\) or \(1\). For \(g=2\) and for all \(q\), in [\textit{J.-P. Serre}, C. R. Acad. Sci., Paris, Sér. I 296, 397-402 (1983; Zbl 0538.14015)], it is proven that the difference in this case is less than or equal to \(3\). In this work, the authors show that for \(g=3\) and all \(q\), either the maximum or the minimum is within \(3\) of the Serre-Weil bound.
The proof of the main Theorem uses Serre's theory of Hermitian modules, which is detailed in the Appendix due to Serre. This theory gives an equivalence between the category of Abelian varieties over \(F_q\) which are isogenous over \(F_q\) to a product of copies of \(E\) (where \(E\) is an ordinary elliptic curve defined over \(F_q\)) and the category of torsion-free \(R_d\)-modules of finite type (where \(R_d=Z[\pi]\), \(\pi\) being the Frobenius of \(E\), and \(d\) is related with the number of rational point of \(E\)).
First, they make a list of the possible zeta functions of a curve whose number of rational points is close to Weil bound. The characteristic polynomial of the Frobenius acting on the Jacobian of the curve, which is the numerator of the zeta function, determines, via Tate's theorem, the isogeny type of an Abelian variety. For each zeta function the corresponding isogeny type is studied using the equivalence of categories mentioned before. Since the Jacobian of some curves are not isogenous to the product of copies of a unique elliptic curve, the authors also consider the glueing of polarizations on Abelian varieties of different isogeny types. In the cases when the glueing is not possible, the authors obtain improvements on the upper bounds.
The proof is divided into cases according to the existence or nonexistence of indecomposable Hermitian forms over certain rings of a given determinant. In the Appendix, polarizations of Abelian varieties are translated into positive definite Hermitian forms on \(R\)-modules, and it is stated that the polarization is principal if and only if the Hermitian form has discriminant one. For an absolutely irreducible curve, the canonical polarization corresponds to an indecomposable Hermitian module of discriminant one. This correspondence is used in both directions in the proof in order to determine if a curve of a certain type exists or not depending on the existence or not of an indecomposable Hermitian module of discriminant one. rational point; zeta function; Serre bound Lauter, K., \textit{the maximum or minimum number of rational points on genus three curves over finite fields, with an appendix by J.-P. Serre}, Compos. Math., 134, 87-111, (2002) Curves over finite and local fields, Rational points, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry The maximum or minimum number of rational points on genus three curves over finite fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Chevalley-Weil theorem is a powerful tool in Diophantine Geometry. It roughly asserts that given an unramified morphism of finite degree \(\pi \colon W \to V\) of projective algebraic varieties defined over a number field \(k\), one can lift the rational points \(p \in V(k)\) to points of \(W\) defined over a fixed number field.
One key point of this result is the assumption that the morphism is ``unramified'', since otherwise, if \(\operatorname{deg}(\pi) >1\) and if \(V\) has enough rational points, then one would expect that the field of definitions of the fibers \(\pi^{-1}(p)\) vary with \(p\), so as to generate a field of infinite degree.
A proof of this result was first given by \textit{C. Chevalley} and \textit{A. Weil} [C. R. Acad. Sci., Paris 195, 570--572 (1932; Zbl 0005.21611; JFM 58.0182.04)] in the case of curves. Since then, many proofs have appeared in the literature, also for arbitrary dimension (e.g. in the works of Bombieri and Gubler, Lang, Serre, etc.).
The main goal of the present note is to illustrate a self-contained proof of this theorem with a rather different presentation and assumptions of purely topological content (see Theorem 1.1). In order to this, the authors discuss and compare various concepts of ``ramification''. They finally adopt a purely topological notion of ramification via the notion of ``topological cover'', with which they are able to exploit the absence of it and deducing it for the number fields after specialization. In previous proofs, the notion ``unramified'' was always formulated in an algebraic way (e.g.~ ``étale'' or ``\(\Omega^1_{W/V} = 0\)'').
Finally, the authors apply Theorem 1.1 to the study of solutions of generalized Fermat equations.
Even though this note does not contribute to substantially new results, the notion of topological cover is new in this context, and it provides a more accessible statement of the Chevalley-Weil theorem which is easier to use for applications. Chevalley-Weil theorem; covers; ramification; Diophantine equations Rational points, Ramification and extension theory, Coverings in algebraic geometry Around the Chevalley-Weil theorem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Several decades ago, André Weil discovered that the classical explicit formulas of prime number theory can be generalized to the case of Artin-Hecke \(L\)-series over an arbitrary global field. Weil's generalized formulas connect arithmetic expressions, namely certain character sums, and analytic expressions, basically sums over the zeros of an Artin-Hecke \(L\)-series, in a fundamental way. Moreover, Weil's approach led to a statement on the positivity of a certain distribution, which is equivalent to the simultaneous validity of the Riemann hypothesis for Artin-Hecke \(L\)-series and the famous Artin conjecture [cf.: \textit{A. Weil}, Izv. Akad. Nauk SSSR Ser. Mat. 36, 3--18 (1972; Zbl 0245.12010)].
More recently, \textit{A. Connes} gave an interpretation of Weil's generalized explicit formulas as a geometric Lefschetz trace formula over the noncommutative space of adèle classes [Sel. Math., New Ser. 5, No. 1, 29--106 (1999; Zbl 0945.11015)], thereby making transparent a striking interplay between number theory and noncommutative geometry.
The problem of extending these results to \(L\)-functions of algebraic varieties seems to require an even more general theory that combines noncommutative spaces and algebraic motives. On the other hand, examples of an intriguing interaction between noncommutative geometry and the theory of motives have already been encountered in recent years, for instance in the context of perturbative renormalization in quantum field theory [cf.: \textit{A. Connes} and \textit{M. Marcolli}, in: Frontiers in Number Theory, Physics and Geometry, Vol. II, Springer-Verlag, 617--713 (2006; Zbl 1200.81113)].
In the treatise under review, the authors provide a first systematic account of such a common framework for noncommutative geometry and motives, mainly in order to obtain a cohomological interpretation of the special realization of zeros of \(L\)-functions.
After a comprehensive introduction to their work (Section 1), the authors first discuss the problem of properly defining morphisms for noncommutative spaces in Section 2. In this context, it is explained that there are, in fact, at least two well-developed constructions in noncommutative geometry that allow to define morphisms as correspondences reflecting the phenomenon of Morita equivalence, namely G. G. Kasparov's \(KK\)-theory, on the one hand, and the theory of modules over the cyclic category, including cyclic cohomology, on the other. Section 3 is to show how certain categories of motives can be embedded faithfully into categories of noncommutative spaces, with a special emphasis put on the category of Artin motives and its relations to a particular noncommutative space linked to adèle classes, the so-called Bost-Connes system. The authors' construction leads to a new category, the category of endomotives, whose objects are obtained from projective systems of Artin motives with actions of semigroups of endomorphisms.
In Section 4, the authors describe a very general cohomological procedure which associates to certain noncommutative spaces of type \(({\mathcal A},\varphi)\), where \({\mathcal A}\) is a unital involutive algebra over \(\mathbb C\) and \(\varphi\) is a so-called ``state'' on \({\mathcal A}\), a representation of the multiplicative group \(\mathbb R_+^*\). In the particular case of the Bost-Connes system, it turns out that the spectrum of this representation is precisely the set of nontrivial zeros of Hecke \(L\)-functions with ``Grössencharakter'' associated with this special set-up. Also, it is explained to what extent the action of the scaling group \(\mathbb R^*_+\) on the cyclic homology is analoguous the action of the Frobenius on the \(\ell\)-adic cohomology in characteristic zero, and how this construction is related to the physical aspects (thermodynamics) of noncommutative spaces.
More precisely, it is shown that the action of the scaling group is obtained through the thermodynamics of the quantum statistical system associated to an endomotive as defined in Section 2. In Section 5, the cohomological construction of the previous section is discussed in the context of general global fields and their Hecke \(L\)-functions. The authors mainly give a summary of their main results in this direction, the details of which will be provided in their forthcoming paper ``The Weil proof and the geometry of the adèle class space'' [\url{arXiv:math/0703392}]. Anyway, these results are to lay the foundations of a geometric framework in which one can begin to transpose Weil's particular approach to the Riemann hypothesis (via a special Riemann-Hilbert correspondence) in the case of positive characteristics to the case of general number fields, thereby obtaining the desired useful cohomological interpretation of the zeros of the Riemann zeta function.
Section 6 turns from the special zero-dimensional case of Artin motives to higher-dimensional cases, including motives of abelian varieties and Shimura varieties. In this context, the authors discuss the compatibility of morphisms of noncommutative spaces given by correspondences in the higher-dimensional setting, mainly by comparing correspondences defined by algebraic cycles with P. Baum's topological correspondences [cf.: \textit{A. Connes} and \textit{G. Skandalis}, Publ. Res. Inst. Math. Sci. 20, 1139--1183 (1984; Zbl 0575.58030)], and by reformulating the case of algebraic cycles as a particular case of the \(KK\) correspondence established by A. Connes and G. Skandalis in 1984.
The final Section 7 is devoted to \(L\)-functions of smooth projective varieties over a number field \(K\). The authors consider the archimedean local factors of the Hasse-Weil \(L\)-function \(L(H^m(X,\mathbb C),z)\) of such a variety \(X\) and establish a Lefschetz trace formula for those, where their approach is based on A. Connes's earlier geometric interpretation of A. Weil's generalization of the classical explicit formulas in prime number theory.
All together, this highly stimulating article provides a wealth of new ideas and insights concerning the interaction of number theory, noncommutative geometry, algebraic geometry, and mathematical physics. varieties over global fields; Riemann hypothesis; \(K\)-theory; quantum field theory; \(L\)-series A. Connes, C. Consani and M. Marcolli, ''Noncommutative geometry and motives: the thermodynamics of endomotives,'' Adv. Math. 214(2), 761--831 (2007). Noncommutative algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic cycles, Global ground fields in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Perturbative methods of renormalization applied to problems in quantum field theory Noncommutative geometry and motives: the thermodynamics of endomotives | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper generalizes a formula of Grothendieck, Ogg, and Shafarevich that expresses the Euler-Poincaré characteristic of a constructible sheaf of \(F_{\ell}\)-modules on a smooth, proper curve, over an algebraically closed field k of characteristic \(p>0,\) as a sum of local and global terms, where \(\ell \neq p\). The primary focus is on removing the restriction on \(\ell\). We begin with calculations for p-torsion sheaves trivialized by p-extensions, but using étale cohomology to give a unified proof for all primes \(\ell.\)
In the remainder of this work, only p-torsion sheaves are considered. We show the existence on \(X_{et}\), X a scheme of characteristic \(p,\) of a short exact sequence of sheaves, involving the tangent space at the identity of a finite, flat, height 1, commutative group scheme, and the subsheaf fixed by the p-th power endomorphism; the latter turns out to be an étale group scheme. A corollary gives complete results on the Euler- Poincaré characteristic of a constructible sheaf of \(F_ p\)-modules on a smooth, proper curve, over an algebraically closed field k of characteristic \(p>0,\) when the generic stalk has rank p.
Explicit computations are given for the Euler characteristics of such p- torsion sheaves on \(P^ 1\) and a result on elliptic surfaces is included. A study is made of the comparison of the p-ranks of abelian extensions of curves. Several examples of p-ranks for nonhyperelliptic curves are discussed. The paper concludes with a brief sketch of results on certain constructible sheaves of \(F_ q\)-modules, \(q=p^ r\), \(r\geq 1\). Euler-Poincaré characteristic of a constructible sheaf; p-torsion sheaves; étale group scheme; p-ranks of abelian extensions of curves Étale and other Grothendieck topologies and (co)homologies, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Group schemes, Coverings of curves, fundamental group The etale cohomology of p-torsion sheaves. I | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In 1976, P. Deligne introduced a Fourier transformation into algebraic geometry, i.e. an involution for \(\ell\)-adic sheaves on the affine line (more generally on a vector bundle over a scheme of finite type) over a field \(k\) of finite characteristic \(p\ne \ell\) (depending on a character \(\mathbb F_p\to\mathbb Q_{\ell})\). The properties of this Fourier transformation have been studied by \textit{N. M. Katz} and the author [Publ. Math., Inst. Hautes Étud. Sci. 62, 145--202 (1985; Zbl 0603.14015)].
In the present paper a local version of Fourier transformation is introduced, defined on \(\ell\)-adic representations of the Galois group of a local field of equal characteristic. At the end of the paper, these Fourier transformations are used to give a new proof of the main result of \textit{P. Deligne}'s paper [Publ. Math., Inst. Haut. Étud. Sci. 52, 137--252 (1980; Zbl 0456.14014)] which in turn implies the part of the Weil conjectures which is the analog of the Riemann hypothesis.
The main part of the paper is devoted to Grothendieck's \(L\)-function for a complex \(K\) of \(\ell\)-adic sheaves on a curve \(X\) over \(k\). This \(L\)-function satisfies a functional equation for \(t\to t^{-1}\) in which a constant \(\varepsilon(X,K)\) appears. The main result of the paper under review expresses this constant as a product of local constants in the places of the curve. The proof of this product formula reduces to the case \(X=\mathbb P^1\) and then uses the Fourier transformation to investigate a one parameter deformation of a certain complex.
The author also explains the relevance of his product formula for Langlands' conjectures on the correspondence between \(L\)-functions and automorphic representations. geometric Fourier transformation; involution for \(\ell \)-adic sheaves; Weil conjectures; Riemann hypothesis; L-function; Langlands' conjectures Laumon, G., Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil, Publ. Math. Inst. Hautes Études Sci., 65, 131-210, (1987) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Langlands-Weil conjectures, nonabelian class field theory, Arithmetic problems in algebraic geometry; Diophantine geometry Fourier transformat, constants of the functional equations and Weil conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Grothendieck duality theory on noetherian schemes plays a crucial role in various branches of algebraic and arithmetic geometry, ranging from the study of moduli spaces of algebraic curves up to the arithmetic theory of modular forms. About fourty years ago, \textit{A. Grothendieck} initiated this theory, the main goal of which was to produce a certain ``trace map'' in the cohomology theory of coherent sheaves generalizing the classical Serre duality for smooth schemes over a field. The foundations of what is now called Grothendieck duality theory are worked out in \textit{R. Hartshorne}'s celebrated monograph ``Residues and duality'', Lect. Notes Math. 20 (1966; Zbl 0212.26101) published more than thirty-five years ago.
The foundational framework developed in Hartshorne's book, based on residual complexes and the notion of a dualizing sheaf, makes this duality theory quite computable in terms of differential forms and residues, and this efficient computability has turned out to be extremely useful in many concrete situations in both algebraic and arithmetic geometry.
However, in Hartshorne's construction of Grothendieck duality theory there are some assumptions on compatibility conditions, together with some explications of abstract results, which are not rigorously proven and are, in fact, quite difficult to verify. The hardest compatibiliy condition in the theory, and also one of the most important, is the base change compatibility of the trace map in the case of proper Cohen-Macaulay morphisms with pure relative dimension. For example, this important special case occurs in the study of flat families of semi-stable curves and their moduli. Although there are simpler methods for obtaining duality theorems in the projective Cohen-Macaulay case, which allow to ignore the base change compatibility problem [\textit{A. Altman} and \textit{S. Kleiman}, ``Introduction to Grothendieck duality theory'', Lect. Notes Math. 146 (1970; Zbl 0215.37201)], there remains the fundamental question of whether the hard unproven compatibilities in the foundations elaborated by R. Hartshorne can really be verified.
The aim of the book under review is to give an affirmative answer to this long-standing problem, i.e., to provide rigorous proofs of those compatibility theorems, and to derive some important consequences and examples of this (finally established) abstract theory. In this vein, and as the author himself points out, the present book should therefore be viewed as a companion (and complement) to R. Hartshorne's classical monograph from 1966. Also, it is by no means a logically independent treatment of Grothendieck duality theory from the very beginning, as it actually (and often) appeals to results proven in Hartshorne's standard book.
As to the contents, the book under review consists of five chapters and two appendices.
Chapter 1, the introduction, provides a first overview, some motivation, and the definitions of most of the basic constructions in Hartshorne's approach to Grothendieck duality theory.
Chapter 2, entitled ``Basic compatibilities'', is concerned with verifying several important compatibility conditions underlying Hartshorne's approach. The basic functorial formalism needed to this end is developed and discussed in full detail.
Chapter 3 comes with the title ``Duality foundations'' and is devoted to a thorough discussion of Grothendieck's notion of a residual complex. The material covered here includes, among other topics, dualizing complexes, residual complexes, the general trace map, Grothendieck-Serre duality, dualizing sheaves, Cohen-Macaulay maps, and the general base change theory for dualizing sheaves.
Chapter 4 culminates in the proof of the general duality theorem for proper Cohen-Macaulay maps with pure relative dimension between noetherian schemes admitting a dualizing complex. This result, the proof of which is indeed rather involved and profound, completes Hartshorne's approach to Grothendieck duality theory and, relievingly, justifies it in a concluding manner.
Also, the author compares his result to the (classical) duality theorem of Verdier [in: \textit{J.-L. Verdier}, Algebr. Geom., Bombay Colloq. 1968, 393-408 (1969; Zbl 0202.19902)] in a very enlightening manner.
Chapter 5, simply entitled ``Examples'' makes the abstract derived category duality theorem (theorem 4.3.1. in the present book) somewhat concrete. This is done by recovering from the general theory (developed here) some of the most widely used consequences for duality of higher direct image sheaves and, in the second part, by deducing the classical results of M. Rosenlicht that describe the dualizing sheaf and the trace map on a reduced proper curve over an algebraically closed field in terms of the so-called regular differentials and residues.
Appendix A addresses the topic of the residue symbol utilized in R. Hartshorn's standard text on Grothendieck duality theory. After stating the main results on residues and cohomology with supports, the author provides full and detailed proofs for them.
Appendix B deals particularly with the theory of residues for, and the trace map on smooth curves. The discussion given here makes some of the corresponding results contained in Hartshorne's book more explicit and digestible, on the one hand, and throws the bridge to the related duality theory for Jacobian varieties, on the other hand.
Whenever appropriate in the course of the text, the author compares his approach to the different approach to duality by J. Lipman, which seems to be even more general and far-reaching, on the one hand, but which is even more abstract and ``categorically'' involved than the original approach by Grothendieck and Hartshorne [cf. \textit{L. Alonso}, \textit{A. Jeremias} and \textit{J. Lipman}, ``Studies in duality on noetherian formal schemes and non-noetherian ordinary schemes'', Contemp. Math. 244 (1999; Zbl 0927.00024)], on the other hand.
The book under review provides, altogether, an important and major contribution towards a better understanding of Grothendieck duality theory in its full generality. schemes; sheaves of differentials; dualizing sheaves; residues; duality theorems; base change theorems; Cohen-Macaulay maps; trace maps; algebraic curves; morphisms B. CONRAD, Grothendieck duality and base change. Lecture Notes in Math. 1750, Springer-Verlag (2000). Zbl0992.14001 MR1804902 Schemes and morphisms, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Grothendieck duality and base change | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In [Proc. Indian Acad. Sci., Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)] \textit{M. V. Nori} introduced the so-called fundamental group scheme of a proper connected reduced scheme \(X\) over a field \(k\) as the affine group scheme associated to the (neutral) Tannakian category of essentially finite vector bundles on \(X\) with a fixed fibre functor \(x\). He also conjectured that if \(X\) is a complete geometrically connected and reduced scheme over an algebraically closed field \(k\) and \(k\subseteq K\) is an arbitrary extension of algebraically closed fields then the natural map
\[
h_X: \pi_1(X\times_k \text{Spec}(K), x) \to \pi_1(X,x)\times_k \text{Spec}(K)
\]
is an isomorphism of affine group schemes over \(K\).''
However, this conjecture is false in general in any positive characteristic. Firstly, it has been disproven for curves with cuspidal singularities by \textit{V. B. Mehta} and \textit{S. Subramanian} [Invent. Math. 148, No. 1, 143--150 (2002; Zbl 1020.14006)]. More recently, \textit{C. Pauly} [Proc. Am. Math. Soc. 135, No. 9, 2707--2711 (2007; Zbl 1115.14026)] has given an example of a smooth curve in characteristic 2 where the conjectured base change property fails.
In the article under review the author is extending this result to smooth curves in any positive characteristic by providing explicit smooth examples where the base change property fails. Nori fundamental group; \(F\)-trivial vector bundles Coverings of curves, fundamental group, Vector bundles on curves and their moduli On base change of the fundamental group scheme | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collectin see Zbl 0717.00009.]
This paper gives an extension of the theory of generalized Jacobians (of Rosenlicht-Serre) to the relative case. If \(S\) is a scheme and if \(X\) is a smooth \(S\)-curve with a suitable compactification \(\hat X\), then a projective system of separated \(S\)-groups \(J_ n\) is constructed (relative generalized Jacobians). --- A main result is the existence of a natural isomorphism \(\displaystyle\underset {n}\varinjlim \text{Hom}_ s(J_ n,G)\cong G(X)\) for a smooth \(S\)-group scheme \(G\). generalized Jacobians Jacobians, Prym varieties Jacobiennes généralisées globale relatives. (Relative global generalized Jacobians) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{P. Deligne}'s proof of the Weil conjectures [Publ. Math., Inst. Hautes Étud. Sci. 52, 137--252 (1980; Zbl 0456.14014)] shows an example on how methods in analytic number theory can deeply influence algebraic geometry, and vice versa. This paper gives a systematic presentation of the methods that turn the algebraic-geometric theory of \(\ell\)-adic étale sheaves into applications in analytic number theory. The central object of study is trace functions associated to lisse \(\ell\)-adic sheaves on an open subscheme of the affine line over a finite prime field (Definition 3.5), that can be extended to the whole affine line in different manners (Remark 3.7); Such a data can also be encoded by a representation of the arithmetic Galois group (Definition 3.1). An interesting example of a trace function is the Kloosterman sum (\S2.2 and Theorem 4.4). The authors show how to apply classical results in étale cohomology, such as the Grothendieck-Lefschetz trace formula (Theorem 4.1), the Grothendieck-Ogg-Shafarevich formula (Theorem 4.2), Deligne's results on weights (Theorem 4.6), in order to obtain estimations of a number of invariants related to the trace functions. For example, in Section 15 the authors explained in much details how the Bombieri-Vinogradov theorem (Theorem 15.2), through the Goldston-Pintz-Yildirim estimation [\textit{D. Goldston} et al., Ann. Math. (2) 170, No. 2, 819--862 (2009; Zbl 1207.11096)]), lead to \textit{Y. Zhang}'s innovative breakthrough on bounded gaps between primes [Ann. Math. (2) 179, No. 3, 1121--1174 (2014; Zbl 1290.11128)]. étale sheaf; trace function; Weil conjectures; Kloosterman sum Modular and automorphic functions, Gauss and Kloosterman sums; generalizations, Étale and other Grothendieck topologies and (co)homologies Lectures on applied \(\ell\)-adic cohomology | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f:X \rightarrow Y\) be a Galois cover of compact Riemann surfaces of genus \(g_X\) and \(g_Y\) respectively. Let \(G=\mathrm{Aut}_X(Y)\) be the corresponding Galois group. The group \(G\) acts by pull-back on the space of holomorphic differential 1-forms of \(X\) giving a \(g_X\)-dimensional complex representation \(\rho_f: G^{\mathrm{op}} \rightarrow\mathrm{GL}(H^0(X,\Omega_X))\), called ``canonical representation'' of \(G\).
Let \(\rho\) be an irreducible representation of \(G\). \textit{C. Chevalley} and \textit{A. Weil} gave in [Abh. Math. Semin. Univ. Hamb. 10, 358--361 (1934; JFM 60.0098.01)] a formula for the multiplicity of \(\rho\) inside \(\rho_f\) in terms of \(g_Y\) and the ramification structure of \(f\). This formula is usually called ``Chevalley-Weil formula''.
This result was generalized in the more general setting of algebraic curves over an algebraically closed field of characteristic \(p \geq 0\) provided that \(p \nmid |G|\), and then also by \textit{S. Nakajima} [Invent. Math. 75, 1--8 (1984; Zbl 0612.14017)] to any coherent sheaf and any ramified cover of algebraic varieties over an any algebraically closed field.
In this paper, the Chevalley-Weil formula is generalized to ramified Galois covers of ``orbifold curves'' (also known as ``Deligne-Mumford curves'' or ``stacky curves'') under the assumption that the ramification locus is disjoint from the locus of orbifold points in \(Y\). The author analyzes the problem over the complex field \(\mathbb{C}\), but every argument applies over arbitrary algebraically closed field whose characteristic does not divide \(|G|\) (see Section 3).
Applications are also provided in the study of Fermat curves \(F_N\) (See Section 6). orbifold curves; automorphisms; modular curves; Fermat curves Coverings of curves, fundamental group, Automorphisms of curves, Special algebraic curves and curves of low genus The Chevalley-Weil formula for orbifold curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result of this paper states that the profinite fundamental group of a proper smooth curve of genus \(\geq 2\) over a number field determines the isomorphism class of the curve. This was conjectured by A. Grothendieck (for any hyperbolic curve) around 1980 and called the fundamental conjecture of anabelian algebraic geometry [see \textit{A. Grothendieck}, Lond. Math. Soc. Lect. Note Ser. 242, 49-58; English translation 285-293 (1997) and ibid., 5-48; English translation 243-283 (1997)]. The author deduces the above result using a previous work of \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)], where the cases of affine curves of all genera over number fields and finite fields are established.
The method is to control group theoretically the subgroups of \(\pi_1\) arising from tubular neighborhoods of irreducible components appearing in stable bad reductions (of covers) of a given proper curve (at infinitely many primes), whose main factors are related to the tame fundamental groups of affine curves over finite fields (treated by Tamagawa) obtained as the component curves minus nodes in those special fibres. In the process, the author introduces/proves two new versions of Grothendieck's conjecture: ``log-admissible version'' over finite fields and ``ordinary version'' over \(p\)-adic fields. The latter version is further generalized by [\textit{S. Mochizuki}, ``The local pro-\(p\) anabelian geometry of curves'', RIMS-1097, preprint 1996]. Grothendieck conjecture; anabelian geometry; profinite fundamental group; curve over a number field; isomorphism class Mochizuki S., The profinite Grothendieck conjecture for closed hyperbolic curves over number fields, J. Math. Sci. Tokyo 3 (1996), no. 3, 571-627. Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic ground fields for curves, Coverings of curves, fundamental group, Curves over finite and local fields The profinite Grothendieck conjecture for closed hyperbolic curves over number fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The questions of how to compactify the Jacobian of a singular curve and how to extend the universal Picard variety over various moduli spaces of curves have been studied extensively in the last several decades.
In this paper, we answer those questions for hyperelliptic curves. We give an explicit description of the moduli space of line bundles on hyperelliptic curves, including singular curves. We study the universal Jacobian \(J^{2,g,n}\) of degree \(n\) line bundles over the Hurwitz stack of double covers of \(\mathbb P^{1}\) by a curve of genus \(g.\) Our main results are: the construction of a smooth, irreducible, universally closed (but not separated) moduli compactification \(\overline{J}^{2,g,n}_{bd}\) of \(J^{2,g,n}\) whose points we describe simply and explicitly as sections of certain vector bundles on \(\mathbb P^{1};\) a description of the global geometry and moduli properties of these stacks; and a computation of the Picard groups of \(\overline{J}^{2,g,n}_{bd}\) and \(J^{2,g,n}\) in the cases when \(n - g\) is even. An important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms. This parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms, which in particular gives the group law on integral binary quadratic forms first discovered by Gauss. Jacobians; universal Jacobians; hyperelliptic curves; compactifications of moduli spaces; line bundles on hyperelliptic curves Generalizations (algebraic spaces, stacks), Picard groups, Algebraic moduli problems, moduli of vector bundles, Algebraic functions and function fields in algebraic geometry Gauss composition for \(\mathbb{P}^1\), and the universal Jacobian of the Hurwitz space of double covers | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is an extension of the results of the author with \textit{J. Sebag} in [Invent.\ Math.\ 168, No.\ 1, 133--173 (2007; Zbl 1136.14010)]. He introduces motivic integration, the motivic Serre invariant, and the Weil generating series for certain classes of formal schemes \(X\) (regular of pseudo-finite type over a complete d.v.r.). For this, he proves a trace formula, that states that the \(\ell\)-adic euler characteristic of the Serre invariant of the generic fiber of \(X\) is the trace of the action of a generator of the tame geometric monodromy group on the \(\ell\)-adic cohomology space of \(X\) base changed to the tame closure of the ground field. (Note: the tameness condition is really necessary).
The proof is ``by explicit computation'', or possibly by resolutions of singularities if the characteristic of the ground field is zero.
A typical application is the following comparison result: let \(f\) be a morphism from a smooth variety to the affine line. The motivic zeta function of \(f\) coincides with the Weil generating series of the formal completion of \(f\). rigid variety; motivic integration; Milnor fiber Abbes, A.: Réduction semi-stable de courbes d'après Artin, Deligne, Grothendieck, Mumford, Saito, Winters. In: Bost, J.-B., Loeser, F., Raynaud M. (eds.) Courbes semi-stables et groupe fondamental en géométrie algébrique. Progress in Mathematics, vol. 187, pp. 59-110. Birkhäuser, Basel (2000) Rigid analytic geometry A trace formula for rigid varieties, and motivic Weil generating series for formal schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{A. Weil} proved in [J. Math. Pures Appl. (9) 17, 47-87 (1938; Zbl 0018.06302)] that every irreducible vector bundle of degree 0 over a Riemann surface \(X\) comes from a representation of the fundamental group of \(X\). Furthermore, he also proved that a vector bundle \(E\) over \(X\) becomes isomorphic to a direct sum of the trivial bundle over an étale Galois cover of \(X\) if and only if there exist \(f,g\in\mathbb{N}[x]\) such that \(f(E)\simeq g(E)\). Vector bundles satisfying this property are called finite vector bundles.
More recently, \textit{M. V. Nori} proved [Compos. Math. 33, 29-41 (1976; Zbl 0337.14016)] that if \(X\) is a projective variety defined over a perfect field \(k\), then the finite bundles form an Abelian category that when endowed with the tensor and the ``restriction to the fiber over a rational point'' functor becomes a Tannaka category. Therefore one associates with it a profinite group scheme \(\pi(X/k.x_0)\) which generalizes the ``usual'' fundamental group.
In the paper under review the author starts by giving an alternative approach to the fundamental group scheme which works for reduced schemes over Dedekind schemes. Then he analyzes two situations. In the first one he considers the fundamental group scheme of a smooth projective curve defined over a \(p\)-adic field. In the second one the geometric object is an arithmetic surface.
In the first setup, if \(K\) denotes the \(p\)-adic field and \(\pi(X_K/K,x_0)^{[p]}\) the prime to \(p\) part of \(\pi(X_K/K,x_0)\), under the hypothesis that \(X_K\) has good reduction, he proves that the set of isomorphism classes of rational representations \(\rho:\pi(X_K/K,x_0)^{[p]}\to\text{GL}_{N,K}\) is finite.
In the second half of the paper he considers representations of the fundamental group scheme of an arithmetic surface \(X\). They are related to a special kind of vector bundles which play a role analogue to that of the finite vector bundles in the geometric (function field) case. The set of the ``new'' rank \(N\) finite vector bundles over \(X\) is denoted \(\text{FVect}_N(X)\). Moreover, the author refines a construction of \textit{C. Gasbarri} [Forum Math. 12, 135-153 (2000; Zbl 0955.14018)], obtaining an intrinsic height on the moduli space of semistable vector bundles over a smooth projective curve defined over a number field \(K\). This relies on the connection between geometric invariant theory and Arakelov geometry (C. Gasbarri, loc. cit.). This construction is similar to that of intrinsic heights on projective varieties by \textit{J.-B. Bost} [Duke Math. J. 82, 21-70 (1996; Zbl 0867.14010)].
Let \(N\) and \(d\) be fixed positive integers (\(d\) sufficiently large), \(\mathcal{E}\) a semistable vector bundle over \(X_K\) of rank \(N\) and degree \(d\), \(\mathcal{W}_E\) the set of Hermitian \(\mathcal{O}_K\)-modules \(H\) such that \(H_K=H^0(X_K,\mathcal{E})\), \(x\in X_K(K)\) and \(H_x=\alpha_x(H)\), where \(\alpha_x:H^0(X_K,\mathcal{E})\to\mathcal{E}_{|x}\) is the surjective restriction map. The following result is proved.
There exist \(M\) points \(x_1,\cdots,x_M\in X_K(\overline{K})\) depending only on \(N\), \(d\), the genus of \(X\) and a constant \(A=A(d,n,X)\) such that for every semistable vector bundle \(\mathcal{E}\) of rank \(N\) and degree \(d\) over \(X\) the height of \(\mathcal{E}\) satisfies the inequality
\[
h(\mathcal{E}):=\inf_{\mathcal{W}_E}\left(\frac 1{[K:\mathbb{Q}]}\left(\sum_{i=1}^M\widehat{\deg}(H_{x_i})-NM\frac{\widehat{\deg}(H)} {\text{rk}(H)}\right)\right)\geq A.
\]
Moreover, if \(B\geq A\) is a real number the set of isomorphism classes of stable vector bundles \(\mathcal{E}\) of rank \(N\) and degree \(d\) defined over \(K\) and such that \(h(\mathcal{E})\leq B\) is finite.
So the function \(h(\mathcal{E})\) defines an intrinsic height on the moduli space. Fix an ample Hermitian line bundle \(\overline{L}\) on \(X\) (of sufficiently high degree) and a metric on the relative dualizing sheaf \(\omega_{X/\mathcal{O}_K}\). The author also proves that there exists a constant \(C=C(X,N,\overline{L},\omega_{X/\mathcal{O}_K})\) such that if \(E\in\text{FVect}_N(X)\) then \(h(E_K\otimes L_K)\leq C\). In particular, \(\text{FVect}_N(X)\) is finite. As a consequence finite bundles have bounded height (as torsion points on the Jacobian), and thus (by Theorem 4.1 in the paper) there are only finitely many \(K\)-isomorphism classes of representations of the fundamental group scheme of \(X\). height of vector bundle; fundamental group scheme; smooth projective curve over a \(p\)-adic field; arithmetic surface C. Gasbarri, Heights of Vector Bundles and the Fundamental Group Scheme of a Curve. Duke Math. J. 117 No.2 (2003), 287-311. Zbl1026.11057 MR1971295 Heights, Arithmetic varieties and schemes; Arakelov theory; heights, Group schemes, Varieties over global fields, Vector bundles on curves and their moduli, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Algebraic moduli problems, moduli of vector bundles Heights of vector bundles and the fundamental group scheme of a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f: X \rightarrow S\) be a smooth projective morphism of Noetherian schemes of relative dimension \(d\). Let \({\mathcal O}_X(1)\) be a relatively very ample line bundle and \(\omega_{X/S}\) be the relative dualising sheaf on \(X\). Let \(H\) be a coherent sheaf on \(X\). Let \(Q\) denote the relative Grothendieck Quot-scheme of flat quotients of \(H\) with a fixed Hilbert polynomial \(P\) with respect to \({\mathcal O}_X(1)\). Let \(p\) and \(q\) denote the projections from \(Q \times_S X\) to \(Q\) and \(X\), respectively. Let \(K\) and \(F\) be respectively the universal subsheaf and quotient sheaf on \(Q\times_S X\). The author has the following global description of the relative dualising sheaf \(\Omega _{Q/S}\).
Theorem 1. There is a natural isomorphism of \({\mathcal O}_Q\)-sheaves
\[
h : {\mathcal E}xt^d_p(F, K\otimes q^*\omega_{X/S}) \rightarrow \Omega_{Q/S}.
\]
A useful application of this result is a similar description of the cotangent sheaf of the moduli space \(M_{X/S}\) of stable sheaves on the fibres of \(f\). Assume further that \(f\) has geometrically integral fibres and \(S\) is of finite type over a field of characteristic 0. The moduli space is the quotient of an open subscheme \(R\) of a suitable Quot-scheme by \(PGL(N)\). Let \(F_R\) denote the universal quotient bundle on \(R\) and \(\pi: R \rightarrow M\) the canonical morphism.
Theorem 2. There is a natural \(GL(N)\)-equivariant isomorphism
\[
h': {\mathcal E}xt^{d-1}_p(F_R, F_R \otimes \omega) \rightarrow\pi^*\Omega_{M/S}.
\]
Since the centre acts trivially, the sheaves \({\mathcal E}xt^i_p(F_R, F_R\otimes \omega)\) descend to coherent sheaves \(E^i\) on \(M\). Theorem 2 implies that \(E^{d-1} \approx \Omega_{M/S}\). In particular, if there is a universal family \(F\) on \(M \times_S X\) then \({\mathcal E}xt^{d-1}_p (F, F\otimes \omega) \approx \Omega_{M/S}\). quot-scheme; cotangent sheaf; relative Ext-sheaf; moduli space Lehn, M., On the cotangent sheaf of quot-schemes, Int. J. math., 9, 4, 513-522, (1998) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Parametrization (Chow and Hilbert schemes) On the cotangent sheaf of quot-schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth curve over a finite field of characteristic \(p\) with function field \(K\). Let \(\mathcal{X}\) be a regular scheme and \(\mathcal{X} \to C\) a proper, flat map whose generic fiber \(X = \mathcal{X} \times_C K\) is smooth and generically connected.
If \(\mathcal{X}\) is a surface then it is a classical result of Artin and Grothendieck that the finiteness of the Brauer group of \(\mathcal{X}\) is equivalent to the finiteness of the Tate-Shafarevich group of the Jacobian \(A\) of \(X\) [Dix exposés sur la cohomologie des schémas. Exposés de J. Giraud, A. Grothendieck, S. L. Kleiman, M. Raynaud et J. Tate. North-Holland, Amsterdam (1968; Zbl 0192.57801)]. Moreover, if the groups are finite, then their orders can be related by a formula.
The main goal of the present article is to generalize these results to arbitrary relative dimension. The authors show that in this general setting the Brauer group of \(\mathcal{X}\) is finite if and only if the Tate-Shafarevich group of the Albanese variety of \(X\) is finite and Tate's conjecture for divisors on \(X\) holds. In the case of relative dimension one, Tate's conjecture for divisors is trivial hence one recovers the classical result of Artin-Grothendieck. Furthermore, if one assumes that the groups are finite, then the authors also relate their orders generalizing Artin-Grothendieck's formula. Brauer group; Tate-Shafarevich group; Tate's conjecture \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Positive characteristic ground fields in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Varieties over finite and local fields, Varieties over global fields Tate's conjecture and the Tate-Shafarevich group over global function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The geometric language of this vast work is largely that of \textit{A. Grothendieck} [Publ. Math., Inst. Hautes Étud. Sci. 4, 1--228 (1960; Zbl 0118.36206)] and while an acquaintance with his work is desirable for an understanding of the present paper almost all of his concepts which are used here are also defined. The proof of the resolution theorem follows the general lines of the proofs given by \textit{O. Zariski} [Ann. Math. (2) 45, 472--542 (1944; Zbl 0063.08361)] for the case of surfaces and threefolds, and so goes back to the early attempts of the italian geometers, especially \textit{B. Levi} [Torino Atti 33, 66--86 (1897; JFM 28.0557.01); Annali di Mat. (2) 26, 218--253 (1898; JFM 28.0557.02)]. However the present proof does not depend directly on results used in these earlier proofs. The local arguments require some knowledge of the theory of local rings, in particular the notions of the multiplicity of an ideal
in a local ring and of a regular system of parameters in a regular local ring.
Chapter 0 begins with a rapid summary of some of the basic concepts in the language of schemes, and a definition of the general notion of blowing-up in terms of a universal
mapping property. A proof of the existence theorem of blowing-up is outlined for certain categories. This leads to the definition and existence of monoidal transformations (and the centre of a monoidal transformation) in algebraic and analytic geometry. Let $B$ be a commutative ring with unity. A $\mathrm{Spec} (B)$-scheme $X$ is called an algebraic $B$-scheme if it is of finite type over $\mathrm{Spec} (B)$. A point $x$ of $X$ is said to be simple (muitiple) if the local ring $\mathcal O_{X,x}$, is (is not) regular. $X$ is said to be noningular if each point of $X$ is simple. The version of the resolution theorem (main theorem I) which is proved in the later chapters is formulated as follows. If $X (= X_0)$ is an algebraic $B$-scheme which is reduced and irreducible, where B is a field of characteristic zero, then there exists a finite sequence of monoidal transformations $f_i:X_{i+1}\to X_i$, $(i =0,1,\dots r-1)$ such that $X_r$ is non-singular and (a) the centre $D_i$ of $f_i$ is non-singular and (b) no point of $D_i$ is simple for $X_i$. Let $D$ be an algebraic subscheme of $X$ defined by the sheaf of ideals $\mathcal I$ on $X$, and let $\mathrm{gr}^p_D(X)$ be the quotient-sheaf $\mathcal I^p/\mathcal I^{p+1}$ restricted to $D$. $X$ is said to be normally flat along $D$ at a point $x$ of $D$ if the stalk of $\mathrm{gr}^p_D(X)$ at $x$ is a free $\mathcal O_{D,p}$-module for $p = 0,1,2,\dots X$ is said to be normally flat along $D$ if it is so at every point $x$ of $D$. It is natural to take the centre $D_i$ of the monoidal transformation $f_i$ to be equimultiple on $X_i$, the present proof of the resolution theorem imposes (c) $X_i$ is normally flat along $D$. It is shown later that (c) is in fact a stronger condition than equi-multiplicity. The other notion that plays an important role in the proof of the resolution theorem is that of normal crossings. Let $E$ be a reduced subscheme of a non-singular algebraic $B$-scheme $X$ which is everywhere of codimension one. $E$ is said to have only normal crossings at a point $x$ of $X$ if there exists a regular system of parameters $(z_1,\dots,z_n)$ of $\mathcal O_{X,x}$, such that the ideal in $O_{X,x}$ of each irreducible component of $E$ which contains $x$ is generated by one of the $z_i$, $E$ is said to have only normal crossings if it does so at every point of $X$. Running alongside the inductive proof of the resolution theorem is the inductive proof of the author's main theorem II which includes the following result. The complement of a Zariski open subset of a non-singular algebraic $B$-scheme (where $B$ is a field of characteristic zero) can be transformed by a finite sequence of monoidal transformations with non-singular centres into a subscheme which has only normal crossings.
Chapter 0 also contains a discussion of the analogues of main theorems I and II in the analytic case, and indicates how his results can be used to prove the resolution theorem for an arbitrary real analytic space. For a complex analytic space the passage from the local to the global resolution of singularities apparently introduces added difficulties, and in this case the author claims a proof of the corresponding theorem for a complex analytic space of dimension $\le 3$. The present methods make virtually no progress towards the resolution of singularities of algebraic $B$-schemes when $B$ is a field of positive characteristic.
Chapter I begins by restricting the ring $B$ to the class $\mathcal B$ of noetherian local rings $S$ with the properties (i) the residue field of $S$
has characteristic zero and (ii) if $A$ is an $S$-algebra of finite type and $\hat{S}$ denotes the completion of $S$ then, under the canonical morphism $\mathrm{Spec} (A\otimes_S \hat{S})\to\mathrm{Spec} (A)$, the singular locus of the former is the preimage of that of the latter spectrum. An algebraic $B$-scheme with $B$ in $\mathcal B$ is called an algebraic scheme. The two main theorems of Chapter 0 are reformulated in terms of two types of resolution data giving four fundamental theorems. The fundamental theorems are of two types; two of the theorems are separation theorems for the resolution data, while the other two are resolution theorems which imply the main theorems of Chapter 0. For technical reasons the fundamental theorems are concerned only with algebraic schemes which have a given irreducible non-singular ambient scheme; the resolution theorem without such an ambient
scheme is achieved by passing to the completion of a certain local ring, since every complete local ring in is a homomorphic image of a formal power series ring over a
field of characteristic zero.
Chapter II is a self-contained study of normal flatness, and is of an algebraic nature. The theorems proved when interpreted geometrically imply the following results, among others.
\begin{enumerate} \item[(1)] The set of points of a reduced subscheme $W$ of an algebraic scheme $V$ at which $V$ is normally flat along $W$ form an open dense subset of $W$.
\item[(2)] When $W$ is a non-singular irreducible subscheme of an algebraic scheme $V$, then for $V$ to be normally flat along $W$ it is necessary that $W$ should be equi-multiple on $V$.
\item[(3)] When $W,V$ are as in (2) and $V$ is embedded in a non-singular algebraic scheme $X$ such that the sheaf of ideals of $V$ on $X$ is locally everywhere generated by a single non-zero element, then for $V$ to be normally flat along $W$ it is necessary and sufficient that $W$ should be equi-muitiple on $V$.
\end{enumerate}
In Ch. III the local effect on singularities of permissible monoidal transformations is studied. Two local numerical characters are introduced as a measure of the severity of a singularity. In terms of these characters it is shown that a singularity is not made worse by any permissible monoidal transformation, and that any sequence of such transformations cannot make a singularity infinitely better. The aim of this local study is to prove the existence of a special coordinate system of the non-singular ambient scheme at a point and of a special base of the ideal defining the subscheme at the point, both of which have a certain stability with respect to the sequences of permissible monoidal transformations.
Chapter IV is devoted to the inductive proofs of the four fundamental theorems. Here the arguments are of a more geometric nature and rely on suitable geometric interpretations of the algebraic results in the two preceding chapters. In conclusion it is worth noting that the inductive proofs could not be carried out should the schemes be restricted to algebraic schemes over fields of characteristic zero; i.e. an essential part of the present proof of the resolution theorem for algebraic singularities is that the corresponding theorem for algebroid singularities should be proved at the same time. algebraic geometry Hironaka, Heisuke, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) \textbf{79} (1964), 109--203; ibid. (2), 79, 205-326, (1964) Global theory and resolution of singularities (algebro-geometric aspects) Resolution of singularities of an algebraic variety over a field of characteristic zero. I | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be an integral projective curve over a field. For the most part, we work with flat families of such \(C\) over an arbitrary locally Noetherian base scheme. However, for the sake of simplicity in this introduction, we fix \(C\) and assume that the base field is algebraically closed of arbitrary characteristic. Given \(n\), let \(J^n_C\) denote the component of the Picard scheme parametrizing invertible sheaves of degree \(n\), and let \(\overline J^n_C\) denote its natural compactification by torsion-free rank-1 sheaves. These scheme, are (essentially) independent of \(n\), and they are often called the (generalized) Jacobian of \(C\) and its compactified Jacobian. To treat \(\overline J^n_C\), there are two main tools: Abel maps and presentation schemes. \textit{A. B. Altman} and \textit{S. L. Kleiman} studied Abel maps [Adv. Math. 35, 50--112 (1980; Zbl 0427.14015)], and presentation schemes [in: The Grothendieck Festschrift, Collect. Artic. Hon. 60th Birthday A. Grothendieck. Vol. I, Prog. Math. 86, 15--32 (1990; Zbl 0737.14007)]. In the present paper, we advance those studies, and tie them together. Notably, we develop the theory we need to be able to prove the following autoduality theorem [see J. Lond. Math. Soc., II. Ser. 65, No. 3, 591--610 (2002; Zbl 1060.14045)]: Given an invertible sheaf \({\mathcal I}\) of degree 1 on \(C\), the corresponding Abel map \(A_{{ \mathcal I},C}:C\to \overline J^0_C\) induces an isomorphism, \(A^*_{{\mathcal I},C}: \text{Pic}^0_{\overline J^0_C} @>\sim>> J^0_C\), and \(A^*_{{ \mathcal I},C}\) is independent of \({\mathcal I}\), at least when \(C\) has at worst double points. (We discuss the Abel map next, and we outline the proof of the autoduality theorem at the end of the introduction.) However, in the present paper, we develop the theory in its natural generality; moreover, this general theory is of independent interest. compactified Jacobian; Abel maps; presentation schemes; autoduality theorem E. Esteves, M. Gagné and S. Kleiman, Abel maps and presentation schemes, Available at http://xxx.lanl.gov/abs/math.AG/9911069, November, 1999. To appear in a special issue, dedicated to R. Hartshorne. Jacobians, Prym varieties, Families, moduli of curves (algebraic) Abel maps and presentation schemes. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors generalise results of Deligne and Laumon ([\textit{G. Laumon}, Astérisque 82--83, 173--219 (1981; Zbl 0504.14013)], 2.1.1) for relative smooth and separated curves to the case of smooth and separated morphisms of finite type. The main theorem is:
Theorem 4.3. Let \(k\) be an algebraically closed field of characteristic \(p > 0\), \(S\) be a \(k\)-scheme of finite type, \(f: X \to S\) a separated and smooth \(k\)-morphism of finite type, \(D\) an effective Cartier divisor on \(X\) relative to \(S\), \(U = X \setminus D\), \(j: U \hookrightarrow X\) the canonical open immersion and \(\mathcal{F}\) a locally constant and constructible sheaf of \(\mathbb{F}_\ell\)-modules on \(U\). Let \(\{S_\alpha\}_{\alpha \in J}\) be the set of irreducible components of \(S\), and \(f_\alpha: X_\alpha := X \times_S S_\alpha \to S_\alpha\), assume that for all \(\alpha \in J\) the base change \(D_\alpha:= D \times_S S_\alpha\) is the sum of effective divisors \(D_{\alpha i}\), \(i \in I_\alpha\), relative to \(S_\alpha\) such that each \(D_{\alpha i}\) is irreducible and \(f|_{D_{\alpha i}}: D_{\alpha i} \to S_\alpha\) is smooth. For each geometric point \(\bar{s} \to S\), denote by \(\rho_{\bar{s}}: X_{\bar{s}} \to X\) the base change of \(\bar{s} \to S\) by \(f\). For each \(\alpha \in J\), denote by \(\bar{\eta}_\alpha\) a geometric generic point of \(S_\alpha\). Define the total dimension divisor of \(\mathcal{F}\) by
\[
\mathrm{DT}(j_!\mathcal{F}|_{X_{\bar{\eta}_\alpha}}) = \sum_{i \in I}\mathrm{dimtot}_{E_i}(j_!\mathcal{F}|_{X_{\bar{\eta}_\alpha}}) \cdot E_i
\]
with \(\{E_i\}_{i \in I}\) the set of irreducible components of \(D_\alpha\). For a geometric point \(\xi_i\) of \(E_i\) and \(\bar{\xi}_i\) a geometric point above \(\xi_i\), \(\eta_i\) the generic point of the strict localisation \(X_{(\bar{\xi}_i)}\) and \(\bar{\eta}_i\) a geometric point above \(\eta_i\), \(\mathcal{F}|_{\eta_i}\) associated to a finite \(\mathbb{F}_\ell\)-module, denote by \(\mathrm{dimtot}_{E_i}(j_!\mathcal{F}|_{X_{\bar{\eta}_\alpha}})\) the total dimension
\[
\mathrm{dimtot}_K M = \sum_{r \geq 1}r \cdot \mathrm{dim}_{\mathbb{F}_\ell}M^{(r)}
\]
of \(\mathcal{F}|_{X_{\bar{\eta}_\alpha}}\) for \(K\) a complete discrete valued field, \(M\) a finitely generated \(\mathbb{F}_\ell\)-module on which the wild inertia subgroup of \(G_K\) acts through a finite quotient and \(M = \bigoplus_{r \in \mathbb{Q}}M^{(r)}\) the slope decomposition. For \(\alpha \in J\), define the generic total dimension divisor \(\mathrm{GDT}_{f_\alpha}(j_!\mathcal{F}|_{X_\alpha})\) of on \(X_\alpha\) as the unique Cartier divisor on \(X_\alpha\) supported on \(D_\alpha\) such that \(\rho^*_{\bar{\eta}_\alpha}(\mathrm{GDT}_{f_\alpha}(j_!\mathcal{F}|_{X_\alpha})) = \mathrm{DT}(j_!\mathcal{F}|_{X_{\bar{\eta}_\alpha}})\). For each geometric point \(\bar{s} \to S\), denote by \(J_{\bar{s}}\) the subset of \(J\) such that \(\bar{s}\) maps to \(S_\alpha\). Then there is a dense open subset \(V \subseteq S\) such that:
(i) For any geometric point \(\bar{s} \to V\) and any \(\alpha \in J_{\bar{s}}\), one has
\[
\rho^*_{\bar{s}}(\mathrm{GDT}_{f_\alpha}(j_!\mathcal{F}|_{X_\alpha})) = \mathrm{DT}_{X_{\bar{s}}}(j_!\mathcal{F}|_{X_{\bar{s}}});
\]
(ii) For any geometric point \(\bar{t} \to S \setminus V\) and any \(\alpha \in J_{\bar{t}}\), the difference
\[
\rho^*_{\bar{t}}(\mathrm{GDT}_{f_\alpha}(j_!\mathcal{F}|_{X_\alpha})) - \mathrm{DT}_{X_{\bar{t}}}(j_!\mathcal{F}|_{X_{\bar{t}}})
\]
is an effective Cartier divisor on \(X_{\bar{t}}\). étale and other Grothendieck topologies and cohomologies; Ramification and extension theory Hu, H., Yang, E.: Semi-continuity for total dimension divisors of étale sheaves. Int. J. Math. \textbf{28}(1), 1750001, 1-21 (2017) Étale and other Grothendieck topologies and (co)homologies, Ramification and extension theory Semi-continuity for total dimension divisors of étale sheaves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The origin of this paper lies in the question on étale cohomology for rigid analytic spaces posed in a paper by \textit{P. Schneider} and \textit{K. Stuhler} [Invent. Math. 105, No. 1, 47-122 (1991; Zbl 0751.14016)]. In that paper an étale site and a corresponding cohomology theory for analytic varieties are defined. We prove here that the axioms for an `abstract cohomology' (as stated in the cited paper) hold for this cohomology theory. In addition, we prove a (quasi-compact) base change theorem for rigid étale cohomology and a comparison theorem comparing rigid and algebraic étale cohomology of algebraic varieties. The main tools in this paper are analytic (resp. étale) points and rigid (resp. étale) overconvergent sheaves. In section 2 we (re)introduce some basic notations concerning analytic points and rigid overconvergent sheaves, which are needed later on. We (re)prove a number of folklore results most importantly: (1) Rigid cohomology agrees with Čech cohomology on quasi-compact spaces. (2) The cohomological dimension of a paracompact space is at most its dimension. (3) A base change theorem.
The rest of the paper deals with étale sites and étale cohomology. Étale points and étale overconvergent sheaves are introduced. A key point is the introduction of special étale morphisms of affinoids \(U\to X\), analogues to rational subdomains in the rigid case. Included in the paper is the proof by \textit{R. Huber} that any étale morphism of affinoids is special étale. This simplifies the original exposition somewhat. A structure theorem for étale morphisms (3.1.2) allows us to give a proof of the étale base change theorem following closely the proof in the rigid case. We calculate the cohomology groups of one dimensional spaces in section 4. This allows us to prove the basic results mentioned at the beginning of this introduction (sections 5, 6 and 7).
This paper may serve as an introduction to rigid and étale cohomology of rigid analytic spaces. étale cohomology for rigid analytic spaces; rigid overconvergent sheaves; Čech cohomology; cohomological dimension de Jong, Johan; van der Put, Marius, Étale cohomology of rigid analytic spaces, Doc. Math., 1, 01, 1-56, (1996) Local ground fields in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Non-Archimedean analysis Étale cohomology of rigid analytic spaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a fixed smooth projective variety over an algebraically closed field of characteristic zero. A Weil \(b\)-divisor on \(X\) is a collection of Néron-Severi classes, ranging over all smooth birational models of \(X\) satisfying natural compatibility conditions under pushforward morphisms. A Weil \(b\)-divisor is called Cartier, if it is determined on a single model.
Weil and Cartier \(b\)-divisors were first considered by Shokurov in the context of the minimal model program and since then have found many important applications in algebraic geometry, singularity theory, algebraic dynamical systems, differential and arithmetic geometry, and in the context of pluripotential theory of non-archimidean analytic spaces.
In the applications, a Weil \(b\)-divisor normally serves to encode refined information of the object one wants to study in an algebraic way. For example, it can encode the singularities of a singular psh metric on a line bundle [\textit{A. M. Botero} et al., ``Chern-Weil and Hilbert-Samuel formulae for singular Hermitian line bundles'', Preprint, \url{arXiv:2112.09007}].To transform this information into a numerical invariant, it is thus desirable to have an intersection theory of Weil \(b\)-divisor classes.
However, the space of Weil \(b\)-divisors on \(X\) forms an infinite-dimensional vector space, thus, defining an intersection pairing, which involves a limiting process, will in general not be well-defined. Hence, one must restrict to an appropriate class of ``positive'' \(b\)-divisors in order to have such a well-defined intersection pairing. It turns out that a suitable positivity notion to define an intersection pairing of \(b\)-divisors is ``nefness''.
Before, such an intersection theory of nef Weil \(b\)-divisors had been developed in special cases: for Cartier \(b\)-divisors, for relatively nef \(b\)-divisors over a closed point and over the spectrum of a valuation ring, and for so called ``toroidal'' \(b\)-divisors.In the present article, such an intersection theory is developed in general.
The key result in order to develop this intersection theory is Theorem A, which shows that any nef Weil \(b\)-divisor is a monotonic decreasing sequence of nef Cartier \(b\)-divisors. The proof relies on the asymptotic construction of multiplier ideals.
The authors then discuss higher codimensional \(b\)-cycle classes an study different positivity notions of such classes. As an application, several variants of the Hodge index theorem are shown, and also that any big and basepoint-free curve class is a power of a nef \(b\)-divisor. Finally, the authors consider different norms on the space of \(b\)-divisors and are able to relate the various Banach spaces induced form them. These Banach spaces were also considered by the authors in previous works. \(b\)-divisors; birational geometry; intersection theory Divisors, linear systems, invertible sheaves, Rational and birational maps Intersection theory of nef \(b\)-divisor classes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0672.00003.]
Among the several approaches to the Riemann-Schottky problem of characterizing Jacobian varieties of curves, there is one specific (geometric) attempt based on the Lie-Wirtinger-Poincaré-Tchebotarev theorem on non-developable double translation hypersurfaces in \(\mathbb{C}^ g\). This theorem, worked out in its full generality during the first half of our century, says that the double translation hypersurfaces with special parametrization characterize, in a certain sense, the theta divisors of non-hyperelliptic Jacobians of dimension \(g\). The author of the present paper has studied this approach to the Schottky problem in two previous articles [cf. Compos. Math. 49, 147-171 (1983; Zbl 0521.14017); J. Differ. Geom. 26, 253-272 (1987; Zbl 0599.14038)]. In the present article, he presents both a survey of this (so-called) approach via translation manifolds and a summary of some recent results that indicate possible relations to some other geometric characterizations of Jacobians. After a historical survey on the Lie-Wirtinger-Poincaré- Tchebotarev theorem, the author explains how these ideas can be used to characterize Jacobians by the existence of special parametrizations for their theta divisors. Then he gives a brief sketch of some of his very recent results. More precisely, he discusses the new observation that more general parametrizations for theta divisors also lead to the conclusion of the Lie-Wirtinger-Poincaré-Tchebotarev theorem. At the end of the paper, the author points out a striking connection between this approach and some other geometric approaches to the Schottky problem. He refers to the recent characterization of Jacobians by \textit{J. M. Muñoz Porras} [ Compos. Math. 61, 369-381 (1987; Zbl 0624.14020)] and explains the resulting indications for the fact that the double- translation-manifold property of the theta divisor could be related to the reducibility properties of \(\Theta\cap\Theta_ a\) and the Andreotti- Mayer condition \(\dim(\text{Sing}(\Theta))\geq g-4\) [cf. \textit{A. Beauville} and \textit{O. Debarre}, Invent. Math. 86, 195-207 (1986; Zbl 0659.14021)]. Riemann-Schottky problem; Jacobian varieties of curves; theta divisors of non-hyperelliptic Jacobians; translation manifolds; Lie-Wirtinger- Poincaré-Tchebotarev theorem Theta functions and curves; Schottky problem, Period matrices, variation of Hodge structure; degenerations, Jacobians, Prym varieties, Analytic theory of abelian varieties; abelian integrals and differentials Translation manifolds and the Schottky problem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians At the time when A. Grothendieck established his ingenious ideas about schemes as the foundation of algebraic geometry there grew out a deep interest in the study of section functors and its right derived functors. Motivated by questions about fundamental groups, Lefschetz theorems a.o., \textit{A.Grothendieck} [see ``Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux'', Sémin. géométrie algébrique 2 (SGA2) (1962; Zbl 0159.50402), see also the enlarged edition (Amsterdam 1968; Zbl 0197.47202)] developed the local cohomology theory in algebraic geometry. In his work as well as in R. Hartshorne's notes about A. Grothendieck's ideas [see \textit{R. Hartshorne}, Notes in the book by \textit{A. Grothendieck}: ``Local cohomology'', Lect. Notes Math. 41 (1967; Zbl 0185.49202)] it turned out that the power of these techniques is -- at least -- two-fold. Firstly it allows to transform the homological techniques invented by J.-P. Serre in algebraic geometry [see \textit{J.-P. Serre}, ``Faisceaux algébriques cohérents'', Ann. Math., II. Ser. 61, 197-278 (1955; Zbl 0067.16201)] to commutative ring theory. Secondly it opens a wide field for research activities on local cohomology in commutative algebra, as one might see by the literature over the last decades. In the mean time there are several approaches to local cohomology, among them those by \textit{W. Bruns} and \textit{J. Herzog} [``Cohen-Macaulay rings'' (revised edition 1998; see also the first edition 1993; Zbl 0788.13005)], \textit{D. Eisenbud} [``Commutative algebra. With a view toward algebraic geometry'' (1995; Zbl 0819.13001)], \textit{M. Herrmann, S. Ikeda} and \textit{U. Orbanz} [``Equimultiplicity and blowing up. An algebraic study'' (1988; Zbl 0649.13011)], \textit{M. Hochster} [``Notes on local cohomology'', Lect. Univ. Michigan, Ann Arbor], \textit{P. Schenzel} [``On the use of local cohomology in algebra and geometry'', Lect. Summer School Commutative Algebra and Algebraic Geometry, Bellaterra 1996 (Birkhäuser 1998)], and \textit{C. Weibel} [``An introduction to homological algebra'' (1994; Zbl 0797.18001)].
The basic motivations for the introduction under review are the following: 1. The authors feel a challenge for an algebraic introduction to Grothendieck's local cohomology theory originally invented by the aid of scheme theory and in that form not available yet. -- 2. The introduction is designed primarily to graduate students who have some experience of basic commutative and homological algebra. So the approach is homologically based on the fundamental `\(\delta\)-functor' technique. -- 3. A large part of the investigations follows algebraic properties of local cohomology, most of them for the first time available in a textbook, e.g., local duality, secondary representations of local cohomology modules, annihilator and finiteness results, graded local cohomology, Hilbert polynomials etc. -- 4. From an algebraic point of view the authors illustrate the geometric significance of various aspects of local cohomology, in particular by applications of local cohomology to connectivity, Castelnuovo-Mumford regularity, sheaf cohomology.
The authors expect that the interested reader should be familiar with the basic sections of the books by \textit{H. Matsumura} [``Commutative ring theory'' (1986; Zbl 0603.13001)] and \textit{J. J. Rotman} [``An introdution to homological algebra'' (1979; Zbl 0441.18018)]. Consequently they included expositions about Matlis duality, the indecomposable injective modules, Hilbert polynomials, foundations about \(\mathbb Z\)-graded rings and modules for the use in graded local cohomology theory. Besides of the authors' approach to the fundamental vanishing theorems on local cohomology, the Lichtenbaum-Hartshorne vanishing theorem a.o. there are very interesting chapters about the annihilation and finiteness theorems on local cohomology, Castelnuovo-Mumford regularity in geometry, connectivity in algebraic geometry, where research results are provided in a textbook form for the first time.
The text is carefully and clearly written. Very often it is completed by examples and easy exercises. They make it easier to a beginner to learn the subject. The connectivity results are -- at least for the reviewer -- the highlights of the book. The authors' use of local cohomology leads to proofs of major results involving connectivity, such as Grothendieck's connectedness theorem, the Bertini-Grothendieck connectivity theorem, the connectedness theorem for projective varieties due to W. Barth, to W. Fulton and W. Hansen, and to G. Faltings, as well as a ring theoretic version of Zariski's main theorem. By the authors' intention the characteristic \(p\) methods in local cohomology -- introduced by \textit{C. Peskine} and \textit{L. Szpiro} [Inst. Hautes Étud. Sci., Publ. Math. 42 (1972), 47-119 (1973; Zbl 0268.13008)] and further developed by M. Hochster and C. Huneke in the notion of tight closure and related research as well as results about big Cohen-Macaulay modules and the rôle of local cohomology in intersection theorems are beyond the scope of the book. For an introduction to this subject the interested reader, prepared by the basics of that book, might and should consult the third part of the book by \textit{W. Bruns} and \textit{J. Herzog}, ``Cohen-Macaulay rings'', cited above.
The value of the book under review consists in its consequent introductory nature, which may be welcome to the beginners of the subject. By the aid of illuminating examples and exercises the authors shead different colours on several subjects of commutative algebra and algebraic geometry. The interested reader will be guided to research problems connected to local cohomology as well as the power of the methods in algebraic geometry. ideal transform; local duality; Hilbert polynomials; reduction of ideals; connectivity; sheaf cohomology; vanishing theorems; annihilation; finiteness theorems on local cohomology; Castelnuovo-Mumford regularity M.P. Brodmann, R.Y. Sharp, \(Local Cohomology: An Algebraic Introduction with Geometric Applications\). Cambridge Studies in Advanced Mathematics, vol. 60 (Cambridge University Press, Cambridge, 1998). MR 1613627 (99h:13020) Local cohomology and commutative rings, Local cohomology and algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Homological methods in commutative ring theory Local cohomology. An algebraic introduction with geometric applications | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians These notes aim at providing a complete and systematic account of some foundational aspects of algebraic supergeometry, namely, the extension to the geometry of superschemes of many classical notions, techniques and results that make up the general backbone of algebraic geometry, most of them originating from Grothendieck's work. In particular, we extend to algebraic supergeometry such notions as projective and proper morphisms, finiteness of the cohomology, vector and projective bundles, cohomology base change, semicontinuity theorems, relative duality, Castelnuovo-Mumford regularity, flattening, Hilbert and Quot schemes, faithfully flat descent, quotient étale relations (notably, Picard schemes), among others. Some results may be found elsewhere, and, in particular, there is some overlap with [51]. However, many techniques and constructions are presented here for the first time, notably, a first development of Grothendieck relative duality for proper morphisms of superschemes, the construction of the Hilbert superscheme in a more general situation than the one already known (which in particular allows one to treat the case of sub-superschemes of supergrassmannians), and a rigorous construction of the Picard superscheme for a locally superprojective morphism of noetherian superschemes with geometrically integral fibres. Moreover, some of the proofs given here are new as well, even when restricted to ordinary schemes. In a final section we construct a period map from an open substack of the moduli of proper and smooth supercurves to the moduli stack of principally polarized abelian schemes. algebraic supergeometry; finiteness of cohomology in algebraic supergeometry; base change and semicontinuity for superschemes; relative Grothendieck duality; Hilbert and Picard superschemes; superperiod maps Supervarieties, Algebraic moduli of abelian varieties, classification, Fine and coarse moduli spaces, Families, moduli of curves (algebraic), String and superstring theories in gravitational theory Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians As is well-known, the concept of divisors lies in the center of algebraic geometry, and its theory is well established for normal integral schemes. In recent times, the needs are felt to have a more general theory of ``generalized divisors'' over nonnormal schemes, e.g., in the classification theory of space curves. The author developed previously such a theory for Gorenstein curves. In this paper, he describes a satisfying general theory of generalized divisors. The schemes \(X\) under considerations are not necessarily integral but are only assumed to satisfy the following condition: the local rings \({\mathfrak O}_{X,x}\) are Gorenstein whenever \(\text{depth} ({\mathfrak O}_{X,x}) \leq 1\), i.e., \({\mathfrak O}_{X,x}\) is ``quasi-normal'' in the sense of Vasconcelos. After necessary preliminaries concerning reflexive modules, Gorenstein rings and duality, a generalized divisor is defined to be a reflexive coherent \({\mathfrak O}_X\)-submodule \(I\) of \({\mathcal K}_X\), the sheaf of total quotient rings of \({\mathfrak O}_X\). A generalized divisor \(I\) is said to be Cartier (resp. almost Cartier) if it is invertible (resp. invertible out of a closed subset of codimension \(\geq 2)\). The sum and the inverse of generalized divisors are defined, but in general the set of generalized divisors does not form a group. The set \(\text{GPic} (X)\) and the groups \(\text{APic} (X)\), \(\text{Pic} (X)\) of linear equivalence classes of generalized divisors, almost Cartier divisors and Cartier divisors are defined, and the general properties of these set and groups are examined in detail. As an application (or a dessert?) of the general theory, the author brushes up the foundations of the theory of liaison (or linkage) of subschemes of projective space, which have recently attracted some attention both in algebraic geometry and commutative algebra. The general theory is illustrated in the case of curves, some singular surfaces in \(\mathbb{P}^3\), and ruled cubic surfaces in detail, and the divergences from the classical theory of divisors on normal schemes are carefully examined.
This clear exposition of the fundamental notions will be expected to provide useful tools in the study of algebraic geometry in future. generalized divisor; Cartier divisors; liaison; linkage Hartshorne, R.: Generalized divisors on Gorenstein schemes. In: Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), \textbf{8}, pp.~287-339 (1994) Divisors, linear systems, invertible sheaves, Linkage, Vector bundles on curves and their moduli, Linkage, complete intersections and determinantal ideals, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Generalized divisors on Gorenstein schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors construct, for any connected quasi-compact and quasi-separated scheme \(X\), a group scheme \(\pi_1(X)\) over \(X\) whose fibre over a geometric point \(\bar x\) identifies with Grothendieck's étale fundamental group \(\pi_1(X, \bar x)\). They also work out the existence of an algebraic universal cover which is a scheme and not just a pro-object as in SGA1 [\textit{A. Grothendieck} (ed.), Seminar on algebraic geometry at Bois Marie 1960-61. Documents Mathématiques (Paris) 3. Paris: Société Mathématique de France. (2003; Zbl 1039.14001)]. The possibility of such constructions was certainly known to Grothendieck, and an exposition can be found in \S 10 of [\textit{P. Deligne}, ``Le groupe fondamental de la droite projective moins trois points'', Galois groups over \(\mathbb{Q}\), Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, 79--297 (1989; Zbl 0742.14022)], under somewhat more restrictive assumptions. There is also an obvious connection to Nori's fundamental group scheme as constructed in [\textit{M. Nori}, ``The fundamental group-scheme'', Proc. Indian Acad. Sci., Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)] which the authors mention but do not work out. On the other hand, their exposition is very reader-friendly, emphasizes points that are not always stressed in other treatments and contains a number of worked-out examples which are helpful for the novice.
A somewhat puzzling feature of the text is the unusually large number of facts and examples that the authors claim to have heard from other colleagues (though many of them can be found in the literature). At points the paper resembles those justly popular websites that contain a wealth of information reflecting the joint effort of many excellent mathematicians but which often lack precise references. As such it is warmly recommended for beginners in the subject but some may wonder whether the place of such a text is in a refereed journal. fundamental group; group scheme; universal cover Vakil, R.; Wickelgren, K., Universal covering spaces and fundamental groups in algebraic geometry as schemes, J. Théor. Nombres Bordeaux, 23, 489-526, (2011) Homotopy theory and fundamental groups in algebraic geometry, Coverings of curves, fundamental group, Schemes and morphisms Universal covering spaces and fundamental groups in algebraic geometry as schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For schemes in positive characteristic \(p>0\) this books develops a new characteristic \(p\)-valued cohomological theory, the theory of \(A\)-crystals, and applies it to give a purely algebraic proof of a conjecture of Goss on the rationality of \(L\)-functions arising in the arithmetic of function fields.
In the late 1940s, Weil posed the challenge to create a cohomology theory for algebraic varieties \(X\) over a finite field \(k\). Such a theory should provide a tool for proving his conjectures on the Zeta-functions of such \(X\), namely its rationality (first part), the existence of a functional equation satisfied by it (second part), and finally certain estimates on its poles and zeroes (third part).
The first significant progress towards the Weil conjecture came, however, from another approach by Dwork, who resolved the first part and some cases of the second part by \(p\)-adic analytic methods. Only later Grothendieck, Deligne et. al. gave cohomological proofs of the full conjectures of Weil along the lines he had proposed. Key ingredients in these proofs are the cohomological theory of \(\ell\)-adic étale sheaves together with the Lefschetz trace formula.
Drinfeld initiated an arithmetic theory for objects over global fields of positive characteristic \(p\), called \(A\)-modules (with reference to a fixed Dedekind domain \(A\) that is finitely generated over the field \({\mathbb F}_p\) with \(p\) elements). Later on he introduced more general objects: elliptic sheaves, shtukas, and then Anderson created \(t\)-motives, which generalize to \(A\)-motives. The category of \(A\)-motives contains (via a contravariant embedding) that of Drinfeld \(A\)-modules, but has the advantage over it of having all the standard operations from linear algebra.
These \(A\)-motives bear many analogies to abelian varieties. For any place \(v\) of \(A\) one associates the \(v\)-adic Tate module to an \(A\)-motive. It carries a continuous action of the absolute Galois group of the finite base field \(k\). This Galois representation is completely described by the action of the Frobenius automorphism. Its main invariant is therefore the dual characteristic polynomial of Frobenius, an element in \(1+tA_v[t]\). If, more generally, we are given a family of \(A\)-motives over a base scheme \(X\) of finite type over \({\mathbb F}_p\) then following Weil we may form a suitable product, over the closed points of \(X\), of the above dual characteristic polynomials (suitably stretched) of Frobenius in the fibres, to obtain an \(L\)-function, a priori an element in \(1+tA[[t]]\). Inspired by Weil's conjectures, Goss conjectured that such an \(L\)-function should be rational as well.
This was proved in 1996 by Taguchi and Wan by a method inspired by Dwork's. With the paradigm of the development around the Weil conjectures, the main motivation for the authors of the present book was to develop a set of algebro geometric and cohomological tools to give a purely algebraic proof of Goss' rationality conjecture. Namely, they develop a cohomological theory of so called \(A\)-crystals which has functors \(f^*\), \(\otimes^{\mathbb L}\), \({\mathbb R}f_*\) for proper \(f\), and \({\mathbb R}f_!\) for compactifiable \(f\), and for which they can prove, as the central technical result, the trace formula.
The definition of an \(A\)-crystal is as follows. Let as before \(X\) be an algebraic variety over a finite field \(k\), but now let \(A\) denote an arbitrary localization of a finitely generated \(k\)-algebra. Write \(C=\text{Spec}(A)\). A coherent \(\tau\)-sheaf over \(A\) on \(X\) is a pair \(\underline{\mathcal F}=({\mathcal F},\tau_{{\mathcal F}})\) consisting of a coherent sheaf \({\mathcal F}\) on \(X\times C\) and an \({\mathcal O}_{X\times C}\)-linear homomorphism
\[
(\sigma\times\text{id})^*{\mathcal F}\overset{\tau_{\mathcal F}}{\longrightarrow}{\mathcal F}.
\]
With obvious morphisms one gets an abelian \(A\)-linear category \(\mathbf{Coh}_{\tau}(X,A)\). A coherent \(\tau\)-sheaf \(\underline{\mathcal F}=({\mathcal F},\tau_{{\mathcal F}})\) is called nilpotent if the iterated homomorphism \(\tau^n_{{\mathcal F}}:(\sigma^n\times\text{id})^*{\mathcal F}\to{\mathcal F}\) vanishes for some \(n>>0\). A homomorphism of coherent \(\tau\)-sheaves is called a nil-isomorphism if both its kernel and its cokernel are nilpotent. And now: The category \(\mathbf{Crys}_{\tau}(X,A)\) of \(A\)-crystals on \(X\) is the localization of \(\mathbf{Coh}_{\tau}(X,A)\) at the multiplicative system of nil-isomorphisms.
The point of inverting nil-isomorphisms is to allow the definition of an extension by zero functor \(j_!\). As the autors point out, the effect is that crystals behave more like constructible sheaves than like coherent sheaves. They construct a derived category of crystals and derived functors as indicated above. They associate what they call crystalline \(L\)-functions to complexes of crystals of finite Tor dimension, and for them they prove the relevant trace formula:
Theorem. Let \(f:Y\to X\) be a morphism of schemes of finite type over \(k\). Suppose that \(A\) is an integral domain that is finitely generated over \(k\). Then for any complex \(\underline{\mathcal F}^{\bullet}\) of finite Tor dimension of crystals on \(Y\), one has
\[
L^{\text{crys}}(Y,\underline{\mathcal F}^{\bullet},t)=L^{\text{crys}}(X,{\mathbb R}f_!\underline{\mathcal F}^{\bullet},t).
\]
Using this, a new proof of Goss' rationality conjecture is obtained. However, this book visibly contains very much more than just a new prove of this conjecture.
If \(A\) is finite, the \(\tau_{{\mathcal F}}\)-invariant sections of a coherent \(\tau\)-sheaf form a constructible étale sheaf of \(A\)-modules on \(X\), and this assignment passes to a functor
\[
\mathbf{Crys}_{\tau}(X,A)\longrightarrow \text{Ét}_c(X,A).
\]
It follows from an old theorem of Katz that this is an equivalence of categories.
For smooth schemes \(X\), a theory similar to the one presented in this book was developed independently by Emerton and Kisin, emphasizing rather the indicated relation with \(p\)-adic étale cohomology, culminating in a Riemann-Hilbert type correspondence.
A prerequisite to reading this book is a good working knowledge in algebraic geometry as well as familiarity with homological algebra and derived categories (although, for the convenience of the reader, many or all the required categorial concepts are assembled in the second section of the book). From that point on it is largely self contained.
The book can be warmly recommended to anyone interested or working in the modern arithmetic of function fields. It is written with great didactical care. All ten chapters and many of the sections and subsections begin with a helpful motivating text. Examples are included.
The reviewer did not spot a single misprint. The layout is very attractive, it is a pleasure to have this book in hands. A-motives; crystals; cohomology; trace formula; L-function; \(\tau\)-sheaves Böckle, G.; Pink, R., Cohomological theory of crystals over function fields, EMS Tracts in Mathematics, vol. 9, (2009), European Mathematical Society Research exposition (monographs, survey articles) pertaining to algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, Étale and other Grothendieck topologies and (co)homologies, Algebraic number theory: global fields Cohomological theory of crystals over function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author defines the fundamental group underlying the Weil-étale cohomology of number rings.
This interesting paper is the continuation of the work by the author [``The Weil-étale fundamental group of a number field. I'', Kyushu J. Math. 65, No. 1, 101--140 (2011; Zbl 1226.14029)].
In this paper, the author presented and extended results of his dissertation of the subject of Weil-étale topos [Sur le topos Weil-étale d'un corps de nombres, L'université Bordeaux I, École Doctorale de Mathématiques et Informatique (2008)].
In the paper under review, the Weil-étale topos is defined as a refinement of the Weil-étale sites introduced by \textit{S. Lichtenbaum} [Ann. Math. (2) 170, No. 2, 657--683 (2009; Zbl 1278.14029)]. The author shows that his definition of the (small) Weil-étale topos of a smooth projective curve is equivalent to the natural definition. Then the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring is defined. This fundamental group is a projective system of locally compact topological groups, which represents first degree cohomology with coefficients in locally compact abelian groups. The author applies this result to compute the Weil-étale cohomology in low degrees and to prove that the Weil-étale topos of a number ring satisfies the expected properties of the conjectural Lichtenbaum topos. Side by side with these the methods of the paper are inspired by the fundamental research of \textit{A. Grothendieck, M. Artin} and \textit{J. L. Verdier} [Théorie des Topos et cohomologie étale des schémas (SGA4). Lecture Notes in Mathematics. 269. Berlin-Heidelberg-New York: Springer-Verlag. (1972; Zbl 0234.00007), 270. (1972; Zbl 0237.00012), 305. (1973; Zbl 0245.00002)].
Contents: 1. Introduction. 2. Preliminaries. 3. The Weil-étale topos. 4. The Weil-étale fundamental group. 5. Weil-étale Cohomology with coefficients in \(\tilde{\mathbb{R}}\). 6. Consequences of the main result.
Let \(Y\) be an open subscheme of a smooth projective curve over a finite field \(k,\) and let \(S_{et} (W_{k}, {\overline Y})\) be the topos of \(W_k\)-equivariant étale sheaves on the geometric curve \({\overline Y}= Y{\otimes}_k {\overline k}\).
Theorem 1.1: There is an equivalence \(Y_W^{sm} \simeq S_{et} (W_k,\overline Y)\) where \(Y_W^{sm}\) is the (small) Weil-étale topos defined in this paper.
For a connected étale \({\overline X}\)-scheme \({\overline U}\) author defines its Weil-étale topos as the slice topos \({\overline U}_W := {\overline X}_W/{\gamma}^* {\overline U}\). Let \(K\) be the number field corresponding to the generic point of \({\overline U}\), and let \(q_{\overline U} : \mathrm{Spec}({\overline K}) \to {\overline U} \) be a geometric point. Similarly to the definition of the étale fundamental group as a (strict) projective system of finite quotients of the Galois group \(G_K\), author defines the analogous (strict) projective system \( {\underline W}({\overline U}, q_{\overline U})\) of locally compact quotients of the Weil group \(W_K\) .
Theorem 1.2: The Weil-étale topos \({\overline U}_W\) is connected and locally connected over the topos \(T\) of locally compact spaces. The geometric point \(q_{\overline U}\) defines a \(T-\)valued point \(p_{\overline U}\) of the topos \({\overline U}_W,\) and we have an isomorphism \(\pi_1 ({\overline U}_W, p_{\overline U } ) \simeq {\underline W}({\overline U}, q_{\overline U} )\) of topological pro-groups.
The author's concepts are applied to obtain a new prove of the reciprocity law of class field theory. He also gives other related propositions and consequences too complicated for quotation here. Weil-étale cohomology; topos; fundamental group; Dedekind zeta function B. Morin, The Weil-étale fundamental group of a number field II, Selecta Math. (N.S.) 17 (2011), 67-137. Étale and other Grothendieck topologies and (co)homologies, Homotopy theory and fundamental groups in algebraic geometry, Zeta functions and \(L\)-functions of number fields The Weil-étale fundamental group of a number field. II | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is a systematic introduction to Kummer's \(l\)-adic étale cohomology of log schemes on a standard logarithmic point and its logarithmic zeta function. Let \(Z\) be a variety defined on a local field \(K\) with a residue field \(\mathbb F_q\) of characteristic \(p\). In 1965, Serre defined the \(l\)-adic zeta function of \(Z\) in terms of the \(l\)-adic étale cohomology, and gave some fundamental conjectures about the existence of functional equations, zeros and poles. If \(Z\) has good reduction over the ring of integers of \(K\), then the fundamental conjectures were proved by Deligne, as a consequence of the proof of the Weil conjecture. However, in the case of bad reduction, the conjecture is still open and the structure of the \(l\)-adic étale cohomology \(H^*_c(Z_{\overline K},\mathbb Q_l)\) is mysterious. In particular, the group action of the Galois group \(G_K =\text{Gal}(\overline K/K)\) on \(H^*_c(Z_{\overline K},\mathbb Q_l)\) must be understood. For this purpose, some new technical tools are needed. Log geometry turns out to be the right framework; the \(l\)-adic étale cohomology can be interpreted as the Kummer's \(l\)-adic étale cohomology of log schemes. Vidal, I.: Monodromie locale et fonctions zêta des log schémas, 983-1038 (2004) \(p\)-adic cohomology, crystalline cohomology Local monodromy and zeta-functions of log schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given a projective smooth algebraic variety \(X_0\) over \(\mathbb F_q,\) with \(X\) its base change to an algebraic closure \(\mathbb F,\) Grothendieck has defined its (geometric) \(\ell\)-adic cohomology \(H^i(X,\mathbb Q_{\ell}),\) which is zero unless \(i\in[0,2\dim X_0],\) for any prime \(\ell\nmid q.\) These are finite dimensional \(\mathbb Q_{\ell}\)-vector spaces with continuous action by the absolute Galois group of \(\mathbb F_q,\) which is generated by the (geometric) Frobenius automorphism \(F:z\mapsto z^{1/q}.\)
The ``Riemann hypothesis'' part of the Weil conjectures, formulated certainly before the invention of the étale cohomology theory, can be reformulated in terms of étale cohomology groups: it is about the spectra of the operator \(F\) on various spaces \(H^i.\) Precisely, any eigenvalue of \(F\) on \(H^i(X,\mathbb Q_{\ell})\) should be an algebraic integer, all of whose conjugates under \(\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\) have complex absolute value \(q^{i/2}.\) This was proved by Deligne using a trick of Rankin and monodromy of Lefschetz pencils, and later by Laumon using \(\ell\)-adic Fourier transform.
The result of the article under review is that the Weil conjecture for a general projective smooth variety can be deduced from the special case of smooth hypersurfaces in projective spaces (which dates back to the 19th century), without using the technical tools mentioned above. The main tools used by the author are the spectral sequence of Rapoport-Zink, de Jong's alteration, and Deligne's monodromy theorem for pure sheaves on curves. Here is a sketch of the idea of proof. Any algebraic variety is birational to a hypersurface, namely they have an open dense part in common, so their cohomologies are almost the same, up to contributions from lower dimensional varieties, which causes no trouble by induction hypothesis on dimension. But the problem is that the hypersurface is singular in general. So one deforms it to a family of generically smooth hypersurfaces, and takes an alteration that is a semi-stable family, the cohomology of the generic fiber of which satisfies the weight-monodromy conjecture by Deligne's local monodromy theorem. Then one applies Rapoport-Zink's spectral sequence associated to this semi-stable family. The argument is birational in nature, so in fact one can deduce the Weil conjecture for proper smooth algebraic spaces, or even proper smooth Deligne-Mumford stacks over \(\mathbb F_q,\) from the case of smooth hypersurfaces in projective spaces.
Even though the author does not prove anything new, this short article gives clear exposition on many relevant concepts. Weil conjectures; alteration; Rapoport-Zink spectral sequence; local monodromy theorem; weight Deligne, P.: Les constantes des équations fonctionnelles des fonctions \(L\). In: Modular Functions of One Variable, II (Proceedings International Summer School, University of Antwerp, Antwerp, 1972. Lecture Notes in Mathematics), vol. 349, pp. 501-597. Springer, Berlin (1973) Finite ground fields in algebraic geometry, Varieties over finite and local fields, Étale and other Grothendieck topologies and (co)homologies Hypersurfaces and the Weil conjectures | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{V. Voevodsky} [Ann. Math. Stud. 143, 188--238 (2000, Zbl 1019.14009)] has defined a triangulated category of geometrical effective motives over a perfect field. The main purpose of the paper under review is to provide a detailed exposition of this construction over a general Noetherian and separated basis.
In the first section a thorough review of relative cycles is given. If \(X\) is an \(S\)-scheme, then the group of relative algebraic cycles is the free abelian group generated by the closed integral subschemes \(Z\) of \(X\) which are of finite type over \(S\) and dominate an irreducible component of \(S\). This group is graded by relative dimension and one often works with rational coefficients or with certain subgroups. The main objective of the first section is to study the base change morphism for relative cycles. Namely, given a map \(T\rightarrow S\), one, roughly, wants to define a relative cycle on \(X\times_S T\) over \(T\) which coincides with flat base change whenever the latter is defined. In fact, one can only do this for the subgroup of so-called universally rational cycles and only if one works over \(\mathbb{Q}\).
Section 2 deals with the explicit computation of the multiplicities that appear through base change. If the base scheme is regular, the intersection multiplicities defined by Serre can be used to introduce a suitable notion of base change of relative cycles, and the explicit computations allow the author to compare the more general definition with Serre's.
In Section 3 the story continues with the study of operations on relative cycles. One of the main constructions is the \(\text{Cor}\) operation which plays an important role in Section 4, where finite correspondences are studied. In particular, one can consider the category \(\text{SmCor}_S\), whose objects are smooth \(S\)-schemes of finite type and the morphisms are finite correspondences. The triangulated category of effective geometrical motives \(DM^{\text{eff}}_{\text{gm}}(S)\) over \(S\) is then defined in 2 steps. First, we take the Verdier quotient of the bounded homotopy category \(K^b(\text{SmCor}_S)\) by a thick triangulated subcategory \(E_{\text{gm}}\) which is generated by certain classes of complexes. In the second step one takes the idempotent completion of the quotient. In particular, any smooth \(S\)-scheme of finite type \(X\) has a geometrical motive, namely its image under the canonical functor \(\text{SmCor}_S\rightarrow DM^{\text{eff}}_{\text{gm}}(S)\).
The next section deals with the embedding theorem. Namely, the category \(DM^{\text{eff}}_{\text{gm}}(S)\) can be embedded into a larger category \(DM^{\text{eff}}_-(S)\), which is better suited for computations and is built from Nisnevich sheaves with transfers.
In the two appendices some results on equidimensional morphisms and cd-structures and topologies are presented. relative cycles; triangulated category; geometrical effective motives; finite correspondences Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Motivic cohomology; motivic homotopy theory Triangulated motives over Noetherian separated schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In an important paper [K-Theory 8, No. 3, 287--339 (1994; Zbl 0826.14005)], \textit{R. Hartshorne} developed the notion of \textit{generalized divisors}, extending the classical theory of divisors to schemes \(X\) satisfying property \(G_1\) (Gorenstein in codimension 1) and satisfying the condition \(S_2\) of Serre. He showed that most of the usual properties of divisors hold also in this more general setting, and in particular the applications to liaison theory extend. The set of all generalized divisors on \(X\), denoted GDiv(\(X\)), corresponds to the set of reflexive coherent \(O_X\)-modules which are locally free of rank one at every generic point of \(X\). A subclass of this set of reflexive \(O_X\)-modules is the class of \textit{totally reflexive} sheaves, which are defined in a technical way coming from module theory. This paper addresses the question of which generalized divisors correspond to totally reflexive sheaves. The authors give the name \textit{Gorenstein divisors} to this subset of the set of generalized divisors, and show that this class fits as follows in the hierarchy of divisors: \(\{\text{principal divisors}\} \subseteq \{ \text{Cartier divisors}\} \subseteq \{ \text{Gorenstein divisors} \} \subseteq \{ \text{generalized divisors} \}\). This paper gives a careful study of this new class of divisors, including both general results (mostly of an algebraic flavor) and examples. divisor; generalized divisor; Gorenstein divisor; Gorenstein scheme; perfect module; totally reflexive sheaf Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], (Co)homology theory in algebraic geometry, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Divisors, linear systems, invertible sheaves Generalized divisors and total reflexivity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In \textit{A. Grothendieck's} [Séminaire de géométrie algèbrique Du Bois-Marie 1967--1969 (SGA 7 I, Lect. Notes Math. 288, Berlin: Springer-Verlag) (1972; Zbl 0237.00013)], precisely in Exposé XV, there has been established a certain Picard-Lefschetz formula describing the local variation of an ordinary quadratic singularity. Its version with respect to étale cohomology was proved, back then, by means of transcendental arguments. Now, in the paper under review, the author presents a purely algebraic proof of this particular formula of Picard-Lefschetz type. Inspired by related works of \textit{M. Rapoport} and \textit{T. Zink} [Invent. Math. 68, 21--101 (1982; Zbl 0498.14010)] and \textit{J. H. M. Steenbrink} [in: Real and compl. Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 525--563 (1977; Zbl 0373.14007)], the author's new (algebraic) proof is based on an explicit description of the variation via the logarithm of the monodromy for semi-stable families and, above all, a reduction to the case of just two branches of the singularity.
Moreover, this approach is applied, more generally, to further classes of homogeneous singularities and their variation. singularities; families of singularities; Picard-Lefschetz theorems; étale cohomology; monodromy; semi-stable reduction Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies About the Picard-Lefschetz formula | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result of this paper is the nonabelian generalization of proper base change theorem in étale cohomology, which works for étale sheaves valued in spaces that satisfy certain finiteness condition named truncated coherent étale sheaves. There are various application of this theorem, including a proof that profinite étale homotopy type commutes with finite product and symmetric power of proper algebraic spaces.
The contents in more detail:
In Section 1 the author states the main theorem and applications as well as the strategy of the proof.
In Section 2 the author gives a review of shapes and profinite completions in the infinity categorical context. Given an \(\infty\)-topos \(\mathcal{X}\), there is an essentially unique geometric morphism \(\pi_*:\mathcal{X}\longrightarrow\mathcal{S}\), the composition \(\pi_*\pi^*\) is a pro-space \(\mathrm{Sh}(\mathcal{X})\) named the shape of \(\mathcal{X}\). This is a generalization of the étale homotopy type to general topoi.
In Section 3 the author studies limits of \(\infty\)-topoi and proves that being a truncated pullback guarantees that the global section in the limit topos is equivalent to the colimit of global sections in each topos.
In Section 4 the author proves the proper base change theorem. The proof combines the standard technology in the abelian setting with the continuity properties of truncated objects.
In Section 5 and 6 the author proves that profinite shapes functor commute with finite products and symmetric powers for proper schemes by the main theorem. The basic idea is that the main theorem allows us to directly identify \(\mathrm{Sh}(X)\circ\mathrm{Sh}(Y)\) with \(\mathrm{Sh}(X\times Y)\) upon profinite completion, while \(\mathrm{Sh}(X)\circ\mathrm{Sh}(Y)\) is equivalent \( \mathrm{Sh}(X)\times \mathrm{Sh}(Y)\) on finite spaces by straightforward computation. proper base change; étale homotopy; infinity-categories; étale topologies; shape theory Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Homotopy theory and fundamental groups in algebraic geometry, \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories, Étale and other Grothendieck topologies and (co)homologies, Shape theory Proper base change for étale sheaves of spaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the late 1940's, André Weil started looking for a cohomology theory for general algebraic varieties (over arbitrary fields), which could be defined in a purely algebraic way, and which would admit those ``good properties'' that were known from the classical De Rham cohomology for differentiable manifolds. Such a ``Weil cohomology theory'', being intended as a powerful tool for attacking the famous Weil conjectures, was invented by A. Grothendieck in the late 1950's and, in the sequel, developed by himself, M. Artin, and J.-L. Verdier into the so-called \(\ell\)-adic étale cohomology theory. Grothendieck also outlined another cohomology theory, the so-called cristalline cohomology, which complemented the \(\ell\)-adic étale theory in the case of \(\ell= \text{char} (k)\), \(k\) being the groundfield. The common properties and interrelations between these (and other) Weil cohomology theories led A. Grothendieck, in the mid sixties, to the abstract categorical concept of algebraic motives. The so-called pure Grothendieck motives are objects of a certain semi-simple abelian category, which functorially incorporate all possible Weil cohomology groups in such a way that concrete Weil cohomologies are ``realizations'' of this abstract ``motivic'' cohomology.
In the past thirty years, the theory of motives has undergone a tremendous development, mainly in connection with the just as spectacular progress made in the theory of algebraic cycles; higher algebraic \(K\)-theory, and algebraic Hodge theory. However, in spite of this fact, the theory of motives is still a subject familiar only to the very specialists in the field, and even Groethendieck's fundamental ideas and constructions are yet far from being a common knowledge of all algebraic geometers.
The present paper grew out of a survey lecture delivered at the University of Rome III in 1993, during the decennial meeting of the project ``Geometria algebraica''. The author provides a brief but lucid introduction to Grothendieck's construction of the category of pure motives, together with some of its variants obtained by using various equivalence relations for cycles on algebraic varieties, the related standard conjectures on algebraic cycle classes, some concrete examples (motives of curves), and the discussion of a few very recent results on motivic categories. Most of the recent developments discussed here are published in the encyclopedic conference proceedings ``Motives'', edited by \textit{U. Jannsen}, \textit{S. Kleiman} and \textit{J.-P. Serre} [Proc. Summer Res. Conf., Washington 1991; Proc. Symp. Pure Math. 55, Part I and II (1994; Zbl 0788.00053 and Zbl 0788.00054)] and, as far as the author's own contributions are concerned, in the original papers by \textit{C. Deninger} and \textit{J. Murre} [J. Reine Angew. Math. 422, 201-219 (1991; Zbl 0745.14003)], and \textit{J. P. Murre} himself [J. Reine Angew. Math. 409, 190-204 (1990; Zbl 0698.14032) and Indag. Math., New Ser. 4, 177-201 (1993; Zbl 0805.14001)].
Of course, this brief overview of some topics in a vast field of contemporary algebraic geometry contains no proofs of results, but the author, one of the most active researchers in the field over the past decades, provides a concise, enlightening and informative account on the basic ideas, concepts, methods, and consequences of the universal theory of algebraic motives. algebraic \(K\)-theory; pure Grothendieck motives; Weil cohomologies; algebraic cycles; algebraic Hodge theory Murre J.P.: Introduction to the theory of motives. Bolletino U.M.I. 10-A, 477--489 (1996) Generalizations (algebraic spaces, stacks), Applications of methods of algebraic \(K\)-theory in algebraic geometry, (Equivariant) Chow groups and rings; motives, Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects) Introduction to the theory of motives | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Suppose \(f:Y \to X\) is a finite morphism of smooth absolutely irreducible projective algebraic curves over a finite field \(k\). The Weil conjectures for curves [proven in \textit{A. Weil}, Sur les courbes algébriques et les variétés qui s'en déduisent, Hermann \& Cie, Paris (1948; Zbl 0061.06407)] show that then we have \(|\# Y(k)- \# X(k) |\leq 2 (g_Y - g_X) \sqrt \# k\), where \(g_X\) and \(g_Y\) are the geometric genera of the curves \(X\) and \(Y\), respectively.
The authors of the present note prove a generalized version of this inequality that holds for certain maps between possibly-singular curves. Namely, they show that if \(f:Y\to X\) is a finite flat morphism of reduced absolutely irreducible projective algebraic curves over \(k\), then
\[
\bigl|\# Y(k)- \# X(k) \bigr|\leq 2 (\pi_Y- \pi_X) \sqrt {\#k},
\]
where \(\pi_X\) and \(\pi_Y\) denote the arithmetic genera of the curves \(X\) and \(Y\), respectively. They present some examples to show that their statement would be false without the hypothesis that \(f\) be flat, and they close with a quick application of their result to the estimation of character sums over finite fields. coverings of singular curves; arithmetic genus; projective algebraic curves over a finite field; Weil conjectures; estimation of character sums over finite fields Aubry Y., Perret M.: A Weil theorem for singular curves, Proceedings of Arithmetic, Geometry and Coding Theory IV, ed. Pellikaan, Perret, Vląduţ. De Gruyter, (1995) Curves over finite and local fields, Coverings of curves, fundamental group, Finite ground fields in algebraic geometry, Singularities of curves, local rings, Other character sums and Gauss sums Coverings of singular curves over finite fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Grothendieck conjecture is a conjecture that the arithmetic fundamental group of a hyperbolic algebraic curve completely determines the algebraic structure of the curve. Let \(X\) be an algebraic variety defined over a field \(K\), and let \(\pi_1(X)\) denote the ``arithmetic'' fundamental group of \(X\). Let \(\text{Gal}(K):=\text{Gal}(\overline K/K)\) denote the absolute Galois group of \(K\) where \(\overline K\) is the separable closure of \(K\). There is an exact sequence of groups
\[
1 \longrightarrow \pi_1(X_{\overline K})\longrightarrow \pi_1(X) @>\text{pr }x>> \text{Gal}(K)\longrightarrow 1
\]
which gives rise to a homomorphism called an outer Galois representation: \(\rho_X: \text{Gal}(K)\to \text{Out}(\pi_1(X_{\overline K}))\). \textit{A. Grothendieck} [``La longue marche à travers de la théorie de Galois'' (1981), in preparation by J. Malgoire and the articles in ``Geometric Galois actions. 1'', Lond. Math. Soc. Lect. Note Ser. 242, 5--48 and 49--58 (1997; Zbl 0901.14001 and Zbl 0901.14002)] formulated a collection of conjectures, the so-called anabelian conjectures, for algebraic varieties \(X\) over base fields \(K\), where \(X\) are anabelian and \(K\) are finitely generated over the prime field.
(GC1) The Fundamental Conjecture: An anabelian algebraic variety \(X\) over a field \(K\) which is finitely generated over the prime field may be reconstituted from the structure of \(\pi_1(X)\) as a topological group equipped with its associated surjection \(\text{pr}_X: \pi_1(X)\to \text{Gal}(K)\).
For algebraic curves in characteristic \(0\), there is a more precise formulation of the conjecture.
(GC2) The Hom Conjecture: For hyperbolic algebraic curves \(X, Y\) over a field \(K\) which is finitely generated over \({\mathbb Q}\), the natural map
\[
\Hom_K(X,Y)\to \Hom_{\text{Gal}(K)}(\pi_1(X),\pi_1(Y))/\sim
\]
defines a bijective correspondence between dominant \(K\)-morphisms and equivalence classes of \(\text{Gal}(K)\)-compatible open homomorphisms.
(GC2) claims that open homomorphisms of the arithmetic fundamental groups always comes from algebro-geometric morphisms. (GC2) is similar to the Tate conjecture [proved by \textit{G. Faltings}, Invent. Math. 73, 349--366 (1983; Zbl 0588.14026)] for abelian varieties and one-dimensional étale homology groups. (However, it was pointed out in this article that the Grothendieck conjectures (GC1), (GC2) are different in essential ways from the Tate conjecture.)
(GC3) The Section Conjecture: Let \(X\) be a hyperbolic algebraic curve over a field \(K\) which is finitely generated over \({\mathbb Q}\). Then every section homomorphism \(\alpha: \text{Gal}(K)\to \pi_1(X)\) of the projection \(\text{pr}_X:\pi_1(X)\to \text{Gal}(K)\) arises either from a \(K\)-rational point of \(X\), or from the \(K\)-rational points at infinity of \(X\).
The three authors of the paper under review were able to prove the conjectures (GC1) and (GC2). The conjecture (GC3) is still unsolved. This article reports on how the authors have succeeded in proving the Grothendieck conjecture.
\textit{Y. Ihara} [Ann Math. (2) 123, 43--106 (1986; Zbl 0595.12003)] and \textit{G. W. Anderson} and \textit{Y. Ihara} [Ann. Math. (2) 128, No. 2, 271--293 (1988; Zbl 0692.14018) and Int. J. Math. 1, No. 2, 119--148 (1990; Zbl 0715.14021)] investigated the pro-\(\ell\) outer Galois representation on \(\pi_1(X)\), for each prime \(\ell\), associated to a genus zero curve \(X:={\mathbb{P}}^1-\Lambda\) where \(\Lambda\subset{\mathbb{P}}^1(K)\) is a finite set containing \(0,1,\infty\). They gave a description of the subfield \(K_X^{(\ell)}\) of \(\overline K\) which arises naturally from the pro-\(\ell\) outer Galois representation \(\rho_X^{(\ell)}\) by means of a system of ``numbers'' obtained from the set \(\Lambda\) of ramification points.
However, further works were needed for an efficient book-keeping of these ``numbers''. This was accommodated by a series of papers by Nakamura translating the pro-\(\ell\) outer Galois representations on \(\pi_1(X)\) into a group-theoretic language involving Galois ``permutations of the pro-cusp points'' over \(\Lambda\) distributed in the ``rim'' of the pro-\(\ell\) universal covering of \({\mathbb{P}}^1-\Lambda\). This reduced the task of understanding the outer Galois representation to that of more accessible Galois permutations of the cusps. This idea was powerful enough to control inertia groups and decomposition groups. The first breakthrough by \textit{H. Nakamura} [J. Reine Angew. Math. 405, 117--130 (1990; Zbl 0687.14028)] was to prove the finiteness theorem that there are only finitely many subsets \(\Lambda\subset{\mathbb{P}}^1(K)\) that give rise to the same arithmetic fundamental group and that fundamental group already determines a curve of the form \({\mathbb{P}}^1-\{4\) points\}. Further, Nakamura reduced the problem of reconstructing (from their arithmetic fundamental groups) curves \({\mathbb{P}}^1-\{n\) points\} of genus \(0\) to the case where \(n=4\). Consequently \textit{H. Nakamura} [J. Reine Angew. Math. 411, 205--210 (1990; Zbl 0702.14024)] was able to prove that the hyperbolic algebraic curves of genus \(0\) over a field which is finitely generated over \({\mathbb Q}\) may be reconstituted from their arithmetic fundamental groups, thereby establishing (GC1) in this case. Nakamura's method also gave a group-theoretic characterization of those section homomorphisms in (GC3), which are also discussed in later work of Tamagawa and Mochizuki.
Now we describe Tamagawa's contribution. Let \(k\) be a finite field, and let \(X\) be a nonsingular affine curve over \(k\). \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135-194 (1997; Zbl 0899.14007)] addressed the analogue of the Grothendieck conjecture in positive characteristic. One of his main results was that the scheme \(X\) may be recovered from \(\pi_1(X)\). Tamagawa's proof was modeled on the work of \textit{K. Uchida} [Ann. Math. (2) 106, 589--598 (1977; Zbl 0372.12017)] and the key steps were
(i) the group-theoretic characterization of the decomposition groups of each closed point of \(X^*\) (the nonsingular compactification of \(X\)),
(ii) the reconstruction of the multiplicative group \(k(X)^{\times}\) and
(iii) the reconstruction of the additive structure on \(k(X)=k(X)^{\times}\cup \{0\}\).
In addition if \(X\) is hyperbolic, then the results are still valid for \(\pi_1(X)^{\text{tame}}\) (which is a quotient of \(\pi_1(X)\)). Further, the isomorphism version of (GC2) for affine hyperbolic curves over fields which are finitely generated over \({\mathbb Q}\) may be derived from the results about the tame fundamental group of affine hyperbolic curves over finite field. Let \(K\) be a number field, and \(X\) be an affine hyperbolic curve over \(K\). The main idea of Tamagawa is to show how to recover group-theoretically the tame fundamental group of the reduction of \(X\) at each finite prime of \(K\) from the arithmetic fundamental group of \(X\) itself. This established the fact that the isomorphism version of (GC2) for affine hyperbolic curves over number fields may be derived from the results about tame fundamental groups of affine hyperbolic curves over finite fields. If \(X_1\) and \(X_2\) are two affine hyperbolic curves over a number field \(K\) such that \(\pi_1(X_1)\simeq \pi_1(X_2)\) over Gal\((K)\), then there is an induced isomorphism \(\pi_1^{\text{tame}}((X_1)_{k_v})\simeq \pi_1^{\text{tame}}((X_2)_{k_v})\) at almost all of the primes \(v\) of \(K\). Thus there is an isomorphism \((X_1)_{k_v}\simeq (X_2)_{k_v}\). From the hyperbolicity, one has \(\text{Isom}(X_1,X_2)\simeq \text{Isom}((X_1)_{k_v}, (X_2)_{k_v})\), and finally this implies that \(X_1\simeq X_2\).
Mochizuki was the one to supply the final piece of the work to settle the conjectures (GC1) and (GC2) in the affirmative, which we now describe briefly. \textit{S. Mochizuki} in his series of papers [J. Math. Sci., Tokyo 3, No. 3, 571--627 (1996; Zbl 0889.11020); ``The local pro-\(p\) Grothendieck conjecture for hyperbolic curves'', RIMS Preprint 1045 (Kyoto Univ. 1995); Invent. Math. 138, No. 2, 319--423 (1999; Zbl 0935.14019); ``A Grothendieck conjecture-type result for certain hyperbolic surfaces'' (to appear)] introduced a totally new way of looking at the Grothendieck conjecture. A priori, the conjecture was formulated for objects defined over global fields. Mochizuki's striking idea was to look at this conjecture as a \(p\)-adic analytic phenomenon whose natural base is a local, not a global field. Mochizuki's insight (from global to \(p\)-adic fields) was indeed very powerful culminating in the proof of a more general result and (GC2) over fields which are finitely generated over \({\mathbb Q}\) can be deduced as a special case. To begin with, Mochizuki formulated and proved the \(p\)-adic analogue of the Grothendieck conjecture.
Theorem: For any smooth algebraic variety \(S\) and any hyperbolic curve \(X\) (both) over a sub-\(p\)-adic field \(K\), the natural maps
\[
\Hom^{\text{dom}}_K(S,X)\to \Hom^{\text{open}}_{\text{Gal}(K)}(\pi_1(S),\pi_1(X)) \to \Hom^{\text{open}}_{\text{Gal}(K)}(\pi_1^{(p)}(S),\pi_1^{(p)}(X))
\]
are bijections. Here \(\Hom_K^{\text{dom}}\) denotes the ``set of all dominant \(K\)-morphisms''; \(\Hom^{\text{open}}_{\text{Gal}(K)}\) denotes the ``set of all equivalent classes of open homomorphisms which are compatible with the projections to \(\text{Gal}(K)\); and \(\pi_1^{(p)}(V)\) is the natural pro-\(p\) analogue of \(\pi_1(V)\).
This theorem may be regarded as an analogue of the uniformization theorem of hyperbolic Riemann surfaces. This theorem resolves (GC2) in a fairly strong form.
As a corollary, the birational version of the Grothendieck conjecture was proved.
Corollary: For (regular) function fields \(L\) and \(M\) of arbitrary dimension over a field of constants \(K\) which is sub-\(p\)-adic, the natural map
\[
\Hom_K(M,L)\to \Hom^{\text{open}}_{\text{Gal}(K)}(\text{Gal}(L),\text{Gal}(M))
\]
is bijective. Here \(\Hom_K\) denotes the ``set of ring homomorphisms over \(K\)''; and \(\Hom^{\text{open}}_{\text{Gal}(K)}\) denotes the ``set of equivalence classes of open homomorphisms which are compatible with the projections to \(\text{Gal}(K)\).
A sketch of proof of the above theorem is presented. Let \(K\) be a finite extension of \({\mathbb Q}_p\), and let \(X\) and \(S\) be proper, non-hyperelliptic hyperbolic curves. The most essential problem was to reconstruct \(X\) from \(\pi_1^{(p)}(X)\to \text{Gal}(K)\) group-theoretically, and Mochizuki did this using \(p\)-adic analysis. Grothendieck's conjecture; hyperbolic algebraic curves; étale fundamental groups; arithmetic fundamental groups; anabelian conjectures; \(p\)-adic Grothendieck conjecture; birational Grothendieck conjecture Nakamura, H.; Tamagawa, A.; Mochizuki, S.: The Grothendieck conjecture on the fundamental groups of algebraic curves. Sugaku expositions 14, 31-53 (2001) Coverings of curves, fundamental group, Arithmetic ground fields for curves, Homotopy theory and fundamental groups in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Curves over finite and local fields, Curves of arbitrary genus or genus \(\ne 1\) over global fields The Grothendieck conjecture on the fundamental groups of algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is difficult to sum up all the results appearing in this nice, wide-ranging, and rich paper. We will use in this review the introduction given by the author, adding some commentaries and complements when necessary. Submersive morphisms of schemes are morphisms of schemes inducing the quotient topology on the target. They appear naturally in many situations, such as when studying quotients, homology, descent and the fundamental group of schemes. Questions related to submersive morphisms of schemes can often be resolved by using topological methods using the description of schemes as locally ringed spaces. Corresponding questions for algebraic spaces are much harder, because an algebraic space is not fully described as a locally ringed space. The first proper treatment of submersive morphisms seems to be due to \textit{A. Grothendieck} [Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1). Revêtements étales et groupe fondamental. Lecture Notes in Mathematics. 224. Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0234.14002)], with applications to the fundamental group of a scheme. He showed that submersive morphisms are morphisms of descent for the fibered category of étale morphisms, with effectiveness for the fibered category of quasi-compact and separated étale morphisms, in some special cases of universally submersive morphisms. The main result of the paper consists in several very general effectiveness results, extending significantly Grothendieck's work. For example, any universal submersion of Noetherian schemes is a morphism of effective descent for quasi-compact étale morphisms. As an application, these effectiveness results imply that strongly geometric quotients are categorical in the category of algebraic spaces.
Later on, \textit{G. Picavet} singled out a subclass of submersive morphisms [``Submersion et descente'', J. Algebra 103, 527--591 (1986; Zbl 0626.14014)]. This paper is written in the affine scheme context and defines subtrusive morphisms as morphisms such that specializations of points have a lifting from the target space to the domain space. For arbitrary morphisms of schemes, Rydh adds the condition that these morphisms are submersive in the constructible topology. The class of subtrusive morphisms is natural in many respects. For example, over a locally Noetherian scheme, every submersive morphism is subtrusive. Picavet gave an example showing that a finitely presented universally submersive morphism needs not to be subtrusive.
A key result of Rydh's paper, missing in Picavet's paper, is that every finitely presented universally subtrusive morphism is a limit of finitely presented submersive morphisms of Noetherian schemes, allowing the author to eliminate Noetherian hypotheses. These facts show that the class of subtrusive morphisms is an important and very natural extension of submersive morphisms between Noetherian schemes.
A crucial tool in this article is the structure theorem for universally subtrusive morphisms: Let \(f: X \to Y\) be a universally subtrusive morphism of finite presentation. Then there are a morphism \(g: X' \to X\) and a factorization \(f_2 \circ f_1\) of \(f\circ g\), where \(f_1\) is faithfully flat of finite presentation and \(f_2\) is proper, surjective, and of finite presentation. The author derives from this result many other useful results.
As a first application, it is shown in Section 4 that universally subtrusive morphisms of finite presentation are morphisms of effective descent for locally closed subsets.
Section 5 establishes that quasi-compact universally subtrusive morphisms are morphisms of effective descent for the fibered category of quasi-compact and separated étale schemes. This result hold also for algebraic spaces, not necessarily separated.
In Section 6, the author shows that the class of subtrusive morphisms is stable under inverse limits and that subtrusive (universally open) morphisms descend under inverse limits.
The rest of the paper is mainly devoted to descent results, in particular to weakly normal descent for universally submersive morphisms. The results of Appendix A about étale morphisms and Henselian pairs are the core of the proof that proper morphisms are morphisms of effective descent for étale morphisms. Appendix B describes properties of absolute weak normalizations.
To end we observe that some results are generalizations to the non-Noetherian case of \textit{V. Voevodsky}'s results in [``Homology of schemes'', Sel. Math., New Ser. 2, No. 1, 111--153 (1996; Zbl 0871.14016)]. algebraic space; blow-up; constructible topology; patch topology; effective descent; étale morphism; finitely presented morphism; flatification; geometric quotient; \(h\)-topology; integral morphism; schematically dominant morphism; scheme; proper morphism; \(S\)-topology; submersion; subtrusion; universally closed; totally integrally closed scheme; (absolute) weak normalization Rydh, D., Submersions and effective descent of étale morphisms, Bull. Soc. Math. France, 138, 2, 181-230, (2010) Schemes and morphisms, Integral dependence in commutative rings; going up, going down, Integral closure of commutative rings and ideals, Étale and flat extensions; Henselization; Artin approximation, Étale and other Grothendieck topologies and (co)homologies, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Submersions and effective descent of étale morphisms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be a simple algebraic group over a field \(k\), defined and split over the finite field \(\mathbb F_p\) for a prime \(p\). The Lie algebra \(\mathfrak g\) of \(G\) admits the structure of a \(p\)-restricted Lie algebra with restricted enveloping algebra \(u(\mathfrak g)\). Let \(G(\mathbb F_q)\) denote the finite Chevalley group of \(\mathbb F_q\)-rational points of \(G\) (for \(q=p^d\)). Given a rational \(G\)-module \(M\), by restriction, \(M\) can be considered as either a \(kG(\mathbb F_q)\)-module or a \(u(\mathfrak g)\)-module.
The first main result, which extends work of \textit{J. F. Carlson, Z. Lin} and \textit{D. K. Nakano} [Trans. Am. Math. Soc. 360, No. 4, 1879-1906 (2008; Zbl 1182.20039)] for \(q=p\) to an arbitrary \(q=p^d\), is a relationship between the (projectivized) cohomological support variety of \(G(\mathbb F_q)\) and that of \(u(\mathfrak g_{\mathbb F_q}^{\oplus d})\) as long as \(p\) is at least the Coxeter number of \(G\). A crucial tool in proving this relationship is use of the Weil restriction functor. For a Galois extension of fields \(E/F\), the Weil restriction functor is a functor from affine \(E\)-schemes to affine \(F\)-schemes that is right adjoint to base change from \(F\) to \(E\). The Weil restriction functor is applied for example to \(G_{\mathbb F_q}\) relative to the extension \(\mathbb F_q/\mathbb F_p\).
A second main result is that the complexity of a rational \(G\)-module \(M\) when considered as a \(kG(\mathbb F_q)\)-module is bounded by one-half its complexity when considered as a \(u(\mathfrak g_{\mathbb F_q}\otimes_{\mathbb F_p}k)\)-module. From this it follows that if \(M\) is projective upon restriction to \(u(\mathfrak g_{\mathbb F_q}\otimes_{\mathbb F_p}k)\), then it is necessarily projective upon restriction to \(kG(\mathbb F_q)\). These results extend to arbitrary \(q\) results for \(q=p\) of \textit{Z. Lin} and \textit{D. K. Nakano} [Invent. Math. 138, No. 1, 85-101 (1999; Zbl 0937.17006)].
Lastly, the author obtains a comparison theorem on non-maximal support varieties for a rational \(G\)-module with ``small'' high weights; comparing the support over \(kG(\mathbb F_p)\) with that over \(u(\mathfrak g)\). It follows that if such a module has constant Jordan type as a \(u(\mathfrak g)\)-module, then it has constant Jordan type as a \(kG(\mathbb F_p)\)-module. group schemes; cohomological support varieties; restricted Lie algebras; complexity; \(\pi\)-points; Weil restriction; projectivity; finite Chevalley groups; non-maximal support varieties; constant Jordan type Eric M. Friedlander, Weil restriction and support varieties, J. Reine Angew. Math. 648 (2010), 183 -- 200. Representations of finite groups of Lie type, Modular Lie (super)algebras, Cohomology of Lie (super)algebras, Modular representations and characters, Cohomology of groups, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Group schemes Weil restriction and support varieties. | 0 |
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