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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 The paper deals with Hilbert scheme of \(n\) points on a \(K3\) surfaces, which is denoted \(K3^{[n]}\). The authors first recall that Hodge numbers of such Hilbert scheme can be obtained by Göttsche's formula and from it one can observe the asymptotic behaviour, when \(n\rightarrow \infty\), of the Euler number \(\chi(S^{[n]})\) (this holds for \(S\) a complex projective surface, not necessarily a \(K3\) surface).
The authors extend ideas and techniques used in previous papers [\textit{K. Bringmann} and \textit{J. Dousse},Trans. Am. Math. Soc. 368, No. 5, 3141--3155 (2016; Zbl 1331.05029)] and [\textit{K. Bringmann} and \textit{J. Manschot}, Commun. Number Theory Phys. 7, No. 3, 497--513 (2013; Zbl 1312.14106)] in order to determine the asymptotic behaviour of the \(\chi_y\)-genus of \(K3^{[n]}\). Given the Hodge polynomial \(\sum_{p,r=0}^{\dim_\mathbb C M}h^{p,r}(M)x^py^r\) of a smooth complex manifold \(M\), the \(\chi_y\)-genus is obtained by putting \(x=-1\):
\[
\chi_y(M):=\sum_{p,r=0}^{\dim_\mathbb C M}h^{p,r}(M)(-1)^py^r.
\]
The techniques used to prove the results in the paper originates from analytic number theory and the author can express the function generating the \(\chi_y\)-genus of \(K3^{[n]}\) using eta and theta functions. Hilbert scheme; \(K3\) surfaces; \(\chi_y\) genera; Hodge numbers Manschot, J.; Rolon, J. M.Z., The asymptotic profile of \(\chi_y\)-genera of Hilbert schemes of points on K3 surfaces, Commun. Number Theory Phys., 9, 2, 413-436, (2015) \(K3\) surfaces and Enriques surfaces The asymptotic profile of \(\chi_y\)-genera 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 We study the natural \(\mathrm{GL}_2\)-action on the Hilbert scheme of points in the plane, resp. \(\mathrm{SL}_2\)-action on the Calogero-Moser space. We describe the closure of the \(\mathrm{GL}_2\)-orbit, resp. \(\mathrm{SL}_2\)-orbit, of each point fixed by the corresponding diagonal torus. We also find the character of the representation of the group \(\mathrm{GL}_2\) in the fiber of the Procesi bundle and its Calogero-Moser analogue over the \(\mathrm{SL}_2\)-fixed point. Parametrization (Chow and Hilbert schemes), Deformations of associative rings, Modifications; resolution of singularities (complex-analytic aspects) \(\mathrm{SL}_{2}\)-action on Hilbert schemes and Calogero-Moser spaces | 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 \(\chi\) be a numerical polynomial. Here we prove the connectedness of the real locus of the Hilbert scheme parametrizing all the subschemes of \(\mathbb{P}^r\) with \(\chi\) as Hilbert polynomial. connectedness of the real locus; Hilbert scheme Topology of real algebraic varieties, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Connected and locally connected spaces (general aspects) The connectedness of the real locus of the Hilbert scheme of \(\mathbb{P}^ r\) | 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 \(K3\) surface, and consider a moduli space \(\mathcal M\) of semi-stable sheaves on \(X\). \textit{S. Mukai} [Invent. Math. 77, 101--116 (1984; Zbl 0565.14002)] proved that, in many cases, \(\mathcal M\) is a holomorphic symplectic variety. The motivating question for the article under review is: What happens if we iterate this process? In other words: Are there moduli spaces of sheaves on \(\mathcal M\) (or, more generally, on any higher-dimensional holomorphic symplectic variety) that carry a holomorphic symplectic structure?
As a first step in the study of this question, one needs examples of stable sheaves on \(\mathcal M\). The author provides such examples in the case that \(\mathcal M=X^{[2]}\) is the Hilbert scheme of two points on the \(K3\) surface \(X\).
To every sheaf \(\mathcal F\) on \(X\), one can associate the sheaf \(\mathcal F^{[2]}:= a_*b^* \mathcal F\), called tautological sheaf, on \(X^{[2]}\). Here,
\[
a: \Xi_2 \to X^{[2]}\quad \text{and}\quad b: \Xi_2\to X
\]
denote the projections from the universal family \(\Xi_2\subset X^{[2]}\times X\). Since \(a: \Xi_2\to X^{[2]}\) is flat and finite of degree 2, we have \(\text{rank} \mathcal F^{[2]}=2\cdot \text{rank}\mathcal F\). Furthermore, if \(\mathcal F\) is a vector bundle or torsion free, the same holds for \(\mathcal F^{[2]}\). In this article, it is shown that stable sheaves of low rank on \(X\) induce stable tautological sheaves on the moduli space \(X^{[2]}\).
More precisely, the results are as follows. Let \(X\) be a smooth projective surface with \(H^1(X,\mathcal O_X)=0\), and let \(H\) be a polarisation on \(X\). For \(N\in \mathbb N\) sufficiently large, we consider the polarisation \(H_N:=N\cdot H-\delta\) on \(X^{[2]}\) where we use the usual isomorphism \(\text{Pic}(X^{[2]})\cong \text{Pic}(X) \oplus \mathbb Z\cdot \delta\); see \textit{J. Fogarty} [Am.\ J. Math.\ 95, 660--687 (1973; Zbl 0299.14020)]. The author of the paper under review proves that, for \(\mathcal F\) a torsion free sheaf of rank 1 or a \(\mu_H\)-stable vector bundle of rank 2 and \(c_1(\mathcal F)\neq 0\), the tautological sheaf \(\mathcal F^{[2]}\) is \(\mu_{H_N}\)-stable for \(N\gg 0\).
The special feature of \(X^{[2]}\) that is used in the proof is the identification of the universal family \(\Xi_2\) with the blow-up of \(X\times X\) along the diagonal. In Section 1, this fact is recalled and several results concerning the geometry of the varieties involved are collected. In Section 2, tautological sheaves are introduced and formulae for their first Chern class and their slope are derived. It is shown in Section 3 that, for \(\mathcal F\neq \mathcal O_X\) a \(\mu_H\)-stable vector bundle, the tautological bundle \(\mathcal F^{[2]}\) does not contain any \(\mu_{H_N}\)-destabilising line bundles if \(N\gg 0\). This is used in the proofs of the main results which are carried out in Section 4. The final Section 5 shows that the assumption on the non-triviality of the first Chern class of \(\mathcal F\) is really necessary. More concretely, \(\mathcal O_{X^{[2]}}\) is a \(\mu_{H_N}\)-destabilising line bundle in \(\mathcal O_X^{[2]}\) for every \(N\gg0\).
The paper under review generalises a result of [\textit{U. Schlickewei}, Rend.\ Semin.\ Mat.\ Univ.\ Padova 124, 127--138 (2010; Zbl 1208.14036)]. The result concerning the stability of tautological bundles associated to line bundles was generalised to Hilbert schemes \(X^{[n]}\) of an arbitrary number of points \(n\) in the recent preprint [\textit{D. Stapleton}, ``Geometry and stability of tautological bundles on Hilbert schemes of points'', \url{arXiv:1409.8229}]. Hilbert squares of \(K3\) surfaces; tautological sheaves; stable sheaves on higher-dimensional holomorphic symplectic varieties Wandel, M, Stability of tautological bundles on the Hilbert scheme of two points on a surface, Nagoya Math. J., 214, 79-94, (2014) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Algebraic moduli problems, moduli of vector bundles, \(K3\) surfaces and Enriques surfaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Stability of tautological bundles on the Hilbert scheme of two points on a surface | 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 classical theory of adeles in algebraic number theory, which works also for curves in algebraic geometry, has been generalized over the past few years, first to smooth proper surfaces over a perfect field, and then to all noetherian schemes \(X\). A construction of adeles in the latter case was given by \textit{A. A. Bejlinson} [in Funct. Anal. Appl. 14, 34-35 (1980; translation from Funkts. Anal. Prilozh. 14, No. 1, 44-45 (1980; Zbl 0509.14018)]. This construction associates to a quasi-coherent sheaf \({\mathcal F}\) on \(X\) a complex (rather than a single ring as in the classical case) of adeles \(\mathbb{A}^*(X,{\mathcal F})\), functorially in \({\mathcal F}\), and it has several applications, principally because of the following theorem: The cohomology of the complex \(\mathbb{A}^*(X,{\mathcal F})\) is isomorphic to the cohomology of \({\mathcal F}\).
As Bejlinson's article is very short and concise, the present paper aims mainly at providing a more accessible exposition of this construction with proofs in the language of commutative algebra. The author also gives a new construction of rational (as opposed to Bejlinson's analytic) adeles and shows that the complex of rational adeles also computes the cohomology of \({\mathcal F}\). adele cohomology; Parshin-Beilinson adeles; complex of rational adeles H A.~Huber, \emph On the Parshin--Beilinson adeles for schemes, Abh. Math. Sem. Univ. Hamburg \textbf 61 (1991), 249--273. Étale and other Grothendieck topologies and (co)homologies, Adèle rings and groups On the Parshin-Beilinson adeles for 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 Let \(P\) be an isolated singularity of an algebraic variety \(V\). The target of the paper is a natural extension of the notion of ordinary singularities of curves to varieties of arbitrary dimension. The authors consider the case where the normalization of \(V\) at \(P\) is regular around the point, a situation for which they provide huge classes of examples. In this case, the authors say that \(P\) is an ordinary singularity of \(V\) when the projectivized tangent cone at \(P\) is reduced. For ordinary singularities, the authors prove that the projectivized tangent cone is the union of \(e\) linear spaces of the same dimension, \(e\) being the multiplicity of \(P\). When the linear spaces that compose the projectivized tangent cone are in general position, the authors prove that also the affine tangent cone at \(P\) turns out to be reduced. singularities Singularities of surfaces or higher-dimensional varieties Ordinary isolated singularities of algebraic 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 \(k\) be a field. Let \(m\in\mathbf{N}_{>0}\) be a positive integer. Let \(f\in k[x_1,\ldots,x_m]\) be a polynomial with degree \(d\geq 1\) and associated hypersurface \(H:=H(f):=\mathrm{Spec}(k[x_1, \ldots, x_m]/\langle f\rangle)\). In this article, we firstly provide a structure property of the weighted-homogeneity of \(f\) in terms of the jet schemes \(\mathscr{L}_H\) of \(H\). As a by-product, we deduce from this property a new and very effective method for the computation of the motivic Poincaré power series \(P_H (T):=\sum_{n\geq 0} [\mathscr{L}_n (H)]T^n \in K_0 (\mathrm{Var}_k)[[T]]\) associated with a homogeneous hypersurface \(H\) with a single isolated singularity at the origin \(\mathfrak{o}\) (and more generally with a specific class of isolated quasi-homogeneous hypersurface singularities). With this point of view we obtain various consequences. For the considered class of varieties, our method provides a characteristic-free proof of the rationality of \(P_H (T)\) which does not use motivic integration nor the existence of resolutions of singularities; we obtain a precise description of the numerator and the possible poles in the rational expression of \(P_H (T)\); when the field is assumed to be of characteristic zero, this allows us to prove the validity of the motivic monodromy conjecture. jet scheme; motivic monodromy conjecture; motivic zeta function; quasi-homogeneous hypersurface singularities Arcs and motivic integration, Singularities in algebraic geometry, Hypersurfaces and algebraic geometry, Commutative rings of differential operators and their modules Jet schemes of quasi-homogeneous hypersurfaces and motivic monodromy conjecture for isolated quasi-homogeneous hypersurface 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 \(\{f=0,0\}\) be a germ of a hypersurface singularity in \(({\mathbb{C}}^{n+1},0)\) with a 1-dimensional singular set \(\Sigma\). If x is a generic linear form, a little deformation \(f+\epsilon x^ N\) (\(\epsilon\) little, N big) has an isolated singularity. Let S be a local irreducibe component of \(\Sigma\) at 0. Along S-\(\{\) \(0\}\), f can be viewed as a \(\mu\)-constant deformation of the transversal section (which has an isolated singularity). These singularities have a monochromy (called horizontal) and the local system over S-\(\{\) \(0\}\) defines another monodromy (called vertical).
The main theorem of this paper relates the characteristic polynomials of the monodromies of f, \(f+\epsilon x^ N\), the vertical and the horizontal monodromies. The methods, polar curves and carroussel, are essentially topological.
Almost simultaneously, M. Saito has proved in this situation the Steenbrink conjecture [\textit{M. Saito}, Math. Ann. 289, 703-716 (1991)], which relates the spectra of f and of \(f+\epsilon x^ N\). The monodromy is characterized by the values mod \({\mathbb{Z}}\) of the spectrum. This better result is proved by the Saito theory of mixed Hodge modules. germ of a hypersurface singularity; characteristic polynomials; Steenbrink conjecture; monodromy D. Siersma, ''The Monodromy of a Series of Hypersurface Singularities,'' Comment. Math. Helv. 65, 181--197 (1990). Complex surface and hypersurface singularities, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Deformations of complex singularities; vanishing cycles, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Mixed Hodge theory of singular varieties (complex-analytic aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties The monodromy of a series of hypersurface 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 In this paper, the authors study the higher codimensional cycle structure of the Hilbert scheme of three points in the projective plane.
Let \((\mathbb P^{2})^{[n]}\) be the Hilbert scheme parametrizing length-\(n\) \(0\)-dimensional closed subschemes of the projective plane \(\mathbb P^{2}\). It is known that the homology and Chow ring of \((\mathbb P^{2})^{[n]}\) are isomorphic.
The main result of the paper determines the nef cones \(\mathrm{Nef}^2\big ((\mathbb P^{2})^{[3]} \big )\) and \(\mathrm{Nef}^3\big ((\mathbb P^{2})^{[3]} \big )\) of \((\mathbb P^{2})^{[3]}\) in codimensions \(2\) and \(3\) respectively. Equivalently, via the intersection pairing, the main result determines the effective cones \(\mathrm{Eff}_2\big ((\mathbb P^{2})^{[3]} \big )\) and \(\mathrm{Eff}_3\big ((\mathbb P^{2})^{[3]} \big )\) in dimensions \(2\) and \(3\) respectively. In addition, the Chern/Segre classes of all the tautological bundles on \((\mathbb P^{2})^{[3]}\) are computed in terms of the geometric bases defined by R.~Mallavibarrena and I.~Sols.
The main idea in the proofs utilizes the structure of the orbits of the group action on \((\mathbb P^{2})^{[3]}\) which is induced by the group action on \(\mathbb P^{2}\).
Section~2 includes the preliminaries such as Grassmannians, tautological bundles, bases of the Chow ring, cones of cycles, and exceptional bundles. Section~3 computes the Chern/Segre classes of the tautological bundles on \((\mathbb P^{2})^{[3]}\) corresponding to the line bundles on \(\mathbb P^{2}\).
Section~4 presents some criterions for nefness.
In Section~5, the authors work out the cycle structure of the orbits of the \(\mathrm{PGL}(3)\) action on \((\mathbb P^{2})^{[3]}\).
The main result is proved in Section~6.
Section~7 concerns \(2\)-very ampleness and the pliant cones. Hilbert scheme of points; nef cone; effective cone; tautological bundle Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Algebraic cycles Higher codimension cycles on the Hilbert scheme of three points on the projective 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 \(k\) be a complete discrete valuation field and \(k^0\) its ring of integers. In part I of this work [ibid. 115, No. 3, 539-571 (1994; Zbl 0791.14008)], the author constructed and studied the vanishing cycles functor for formal schemes of locally finite type over \(k^0\). In this part II the construction is extended to a broader class of formal schemes that includes, for example, formal completions of the above formal schemes along arbitrary subschemes of their closed fibres. The main result is a comparison theorem which states that if \({\mathcal X}\) is a scheme of finite type over a Henselian discrete valuation ring with the completion \(k^0\) and \({\mathcal Y}\) is a subscheme of the closed fibre \({\mathcal X}_s\), then the vanishing cycles sheaves of the formal completion \(\widehat {\mathcal X}_{/{\mathcal Y}}\) of \({\mathcal X}\) along \({\mathcal Y}\) are canonically isomorphic to the restrictions of the vanishing cycles sheaves of \({\mathcal X}\) to the subscheme \({\mathcal Y}\). In particular, the restrictions of the vanishing cycles sheaves of \({\mathcal X}\) to \({\mathcal Y}\) depend only on \(\widehat {\mathcal X}_{/{\mathcal Y}}\), and any morphism \(\varphi: \widehat {\mathcal X}'_{/{\mathcal Y}'}\to \widehat {\mathcal X}_{/{\mathcal Y}}\) induces a homomorphism from the pullback of the restrictions of the vanishing cycles sheaves of \({\mathcal X}\) to \({\mathcal Y}\) to those of \({\mathcal X}'\) to \({\mathcal Y}'\). -- One also proves that, given \(\widehat {\mathcal X}_{/{\mathcal Y}}\) and \(\widehat {\mathcal X}'_{/{\mathcal Y}'}\), one can find an ideal of definition of \(\widehat {\mathcal X}'_{/{\mathcal Y}'}\) such that if two morphisms \(\varphi, \psi: \widehat {\mathcal X}'_{/{\mathcal Y}'}\to \widehat {\mathcal X}_{/{\mathcal Y}}\) coincides modulo this ideal, then the homomorphisms between the vanishing cycles sheaves induced by \(\varphi\) and \(\psi\) coincide.
These facts generalize results of part I as well as results of \textit{G. Laumon} [``Charactéristique d'Euler-Poincaré et sommes exponentielles'' (Thèse, Université de Paris-Sud, Orsay 1983)], and the author [``Vanishing cycles for non-Archimedean analytic spaces'', J. Am. Math. Soc. 9, No. 4, 1187-1209 (1996)], where certain cases when \({\mathcal Y}\) is a closed point of \({\mathcal X}_s\) were considered. The main new ingredient in the proof of the comparison theorem is the recent stable reduction theorem of \textit{A. J. de Jong} [``Smoothness, semi-stability and alterations'' (preprint 1995)]. Furthermore, one proves a vanishing theorem which states that the \(q\)-dimensional étale cohomology groups of certain analytic spaces of dimension \(m\) are trivial for \(q> m\). This class of analytic spaces induces, for example, the finite étale coverings \(\Sigma^{d,n}\) of the Drinfeld half-plane \(\Omega^d\) [\textit{V. G. Drinfel'd}, Funct. Anal. Appl. 10, 107-115 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 29-40 (1976; Zbl 0346.14010)]. vanishing cycles functor; formal schemes; comparison theorem; vanishing theorem V.\ G. Berkovich, Vanishing cycles for formal schemes. II, Invent. Math. 125 (1996), no. 2, 367-390. Étale and other Grothendieck topologies and (co)homologies, Algebraic cycles, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Local ground fields in algebraic geometry, (Co)homology theory in algebraic geometry Vanishing cycles for formal 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 Bekanntlich hat Herr Cayley aus den Plücker'schen Gleichungen zwischen den Anzahlen der sechs einfachsten Singularitäten einer ebenen Curve sechs Gleichungen zwischen den Zahlen für die neun einfachsten Singularitäten einer Raumcurve abgeleitet, wobei zu der als Punktreihe aufzufassende abwickelbare Fläche hinzugerechnet werden muss, welche reciprok zu der Curve als die Einhüllende ihrer sämmtlichen Osculationsebenen erscheint. Darauf haben die Herrn Cayley, Salmon, Zeuthen vermittelst der neun Cayley'schen Zahlen andere Singularitätenzahlen für eine Curve und eine abwickelbare Fläche aufgestellt. In der vorliegenden Abhandlung beweist nun Herr Zeuthen diese von ihm frührer durch die Comptes rednus (siehe Fortsch. d. M. I. p. 231.) veröffentlichten Resultate, und fügt eine Reihe neuer hinzu. Die Singularitäten, deren Anzahlen hergeleitet werden, sind sämmtlich von der ersten Ordnung, das heisst, sie setzen nur Punkte, Ebenen oder Gerade mit der Curve in Beziehung.
Nach Einführung übersichtlicher symbolischer Bezeichnungen für die Anzahl der Gebilde, welche gewissen gegebenen Bedingungen genügen, fügt der Verfasser den gewöhnlichen neun Singularitäten noch drei andere hinzu, nämlich das Begabtsein der Curve mit Doppelpunkten, das der abwickelbaren Fläche mit Doppeltangentialebenen, und das Enthalten von Inflexionstangenten, d. h. Geraden, die durch drei aufeinanderfolgende Punkte der Curve gehen oder recoprok auf drei aufeinanderfolgenden Ebenen der abwickelbaren Fläche liegen. Die Zahlen für diese Singularitäten führt er in die Cayley'schen Formeln ein, indem er die Plücker'schen Formeln auf den Kegel, welcher die gegebene Curve von irgend einem Punkte aus projicirt, resp. auf die ebene Curve anwendet, in welcher die abwickelbare Fläche von irgend einer Ebene geschnitten wird. Die Herleitungen der Formeln für die neuen Singularitätenzahlen beruhen auf dem von Herrn Chasles in die Geometrie eingeführten algebraischen Princip der Correspondenz, welches aussagt, dass wenn in einer geraden Punktreihe einem beliebigen Punkte in gewisser Weise \(m\) Punkte entsprechen, von denen jeder zu \(n\) Punkten der entsprechenden ist, es \(m+n\) Stellen auf der Geraden geben muss, wo zwei einander so entsprechende Punkte zusammenfallen, und welches das Analoge von ebenen Strahlbüscheln behauptet. Die Schwierigkeiten, welche bei der Anwendung dieses Princips die Frage verursacht, wie vielfach jede Lösung zu rechnen ist, löst der Verfasser durch zwei Methoden. In der ersten sieht er die Entfernungen \(x\) und \(u\) zweier einander entsprechender Punkte einer Geraden von einem auf ihr liegenden festen Punkte als die Coordinaten einer Curve \(S\) an, deren Schnittpunkte mit der Geraden \(x=u\) dann mit den Punkten der Geraden correspondiren, in denen zwei einander entsprchende Punkte vereinigt sind. Eine Untersuchung, wieviel Punkte \(S\) und \((x=u)\) in jedem dieser Schnittpunkte gemeinsam haben, ergiebt dann, wievielmal entsprechende Punkte an der zugehörigen Stellen des Trägers der Punktreihe zusammenfassen. Die zweite vom Verfaser experimentell genannte Methode beruht darauf, die unbekannten Coefficienten durch Gleichsetzung von Ausdrücken zu bestimmen, die auf verschiedene Weise nach dem Princip der Correspondenz für dieselbe Zahl gefunden sind. Zur Auseinandersetzung dieser beiden Methoden wird zuerst die Zahl der Geraden bestimmt, welche zwei feste Gerade treffen, und ausserdem noch die gegebene Curve \(m^{\text{ter}}\) Ordnung in zwei verschiedenen Punkten schneiden, und daran sich anschliessend, die Zahl der Geraden, welche eine feste Gerade und dreimal die Curve treffen. Bei dieser letzten Aufgabe ergiebt sich, um ein Beispiel solcher Singularitätenformeln hier anzuführen,
\[
(m-2)\biggl[h- \frac{m(m- 1)}{6} \biggr],
\]
wo \(h\) die Zahl der Geraden ist, welche, von einem Punkte ausgehend, die gegebene Curve zweimal treffen. Hierauf folgen mehr als vierzig Bestimmungen der Anzahlen von Punkten, die Geraden und Ebenen, welche mit der gegebenen Curve in Beziehung stehen, und zwar sind diese Zahlen in ihrer Abhängigkeit von den früher erwähnten zwölf fundamentalen Singularitätenzahlen dargestellt. Zu den complicirteren der hier gelösten Aufgaben gehört zum Beispiel die Bestimmung der Anzahl der Dreiecke, deren Ecken auf der Curve liegen, und deren Seite dieselbe noch einmal treffen, und die Bestimmung der Anzahl der Dreiecke, deren drei Seiten je drei Schnitte mit der Curve haben, ohne dass eine Ecke auf sie fällt. singularities; Plücker relations; algebraic curves; algebraic surfaces Questions of classical algebraic geometry, Plane and space curves, Singularities of curves, local rings On the ordinary singularities of a space curve and a developpable surface | 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 On an equidimensional scheme X we consider the complex of étale sheaves in degrees 0 and 1 given by: \({\mathcal G}:\oplus _{y\in X^{(0)}}j_{y*}{\mathbb{G}}_{m/y}\to ^{\partial}\oplus _{x\in X^{(1)}}i_{x*}{\mathbb{Z}}\) and we set \({\mathcal G}(n)=R \underline{Hom}_ X({\mathbb{Z}}/n,{\mathcal G})\in D^ +(X,{\mathbb{Z}}/n)\). For regular X we have quasi-isomorphisms \({\mathbb{G}}_ m\to ^{\sim}{\mathcal G}\) and \(\mu _ n\to ^{\sim}{\mathcal G}(n)\). Results:
(a) Let X be a one-dimensional proper \({\mathbb{Z}}\)-scheme. Then for any constructible sheaf F on X the Yoneda-pairing \(H^ i(X,F)\times {\mathbb{E}}xt_ X^{3-i}(F,{\mathcal G})\to {\mathbb{H}}^ 3(X,{\mathcal G})\to ^{tr}{\mathbb{Q}}/{\mathbb{Z}}\) is a perfect pairing of finite groups, \(i\in {\mathbb{Z}}\). If X is irreducible the trace map is an isomorphism. (The 2- primary components may have to be excluded.) This extends the case where X is regular, i.e. Artin-Verdier duality for number and function fields. [cf. \textit{B. Mazur}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 6(1973), 521-552 (1974; Zbl 0282.14004) and the author, Math. Z. 188, 91-100 (1984; Zbl 0585.14014)].
(b) There is also a local version of (a) where X is the spectrum of a henselian one-dimensional noetherian local ring with finite residue field.
(c) Assume Y is separated, smooth of relative dimension \(d\) over an open subscheme of X as in (a). Then for constructible F on Y with \(nF=0\), n invertible on Y there is a perfect Yoneda duality on Y in the category of \({\mathbb{Z}}/n\)-sheaves with dualizing complex \({\mathcal G}(n)\otimes \mu _ n^{\otimes d}\).
(d) If \(\mu_ n\) is replaced by \({\mathcal G}(n)\) Poincaré duality remains valid for singular proper curves over algebraically closed fields.
Remarks: The fibres of a regular arithmetical surface can be singular. Therefore (a) should be useful for establishing arithmetical duality on such surfaces. Moreover (a), (b) suggest that higher dimensional arithmetical duality should not be confined to regular schemes. complex of étale sheaves; Yoneda-pairing; Artin-Verdier duality; Poincaré duality; arithmetical duality C. Deninger, ''Duality in the étale cohomology of one-dimensional proper schemes and generalizations,'' Math. Ann., vol. 277, iss. 3, pp. 529-541, 1987. Étale and other Grothendieck topologies and (co)homologies Duality in the étale cohomology of one-dimensional proper schemes and generalizations | 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 One of the major open problems regarding the Macdonald polynomials \(J_{\mu}(x;q,t)\), \(\mu\) being a partition of \(n\), is the conjecture that, expanded in terms of certain modified Schur symmetric functions, the coefficients of \(J_{\mu}(x;q,t)\) are polynomials in the two parameters \(q,t\) with nonnegative integer coefficients. \textit{A. M. Garsia} and \textit{M. Haiman} [Proc. Natl. Acad. Sci. USA 90, No. 8, 3607-3610 (1993; Zbl 0831.05062)] introduced some bigraded \(S_n\)-module \({\mathbf H}_{\mu}\) and conjectured that the dimension of \({\mathbf H}_{\mu}\) is \(n!\) and this is known as the \(n!\) conjecture. It is known that the \(n!\) conjecture implies the Macdonald positivity conjecture. Many computer experimensts and some partial results support the validity of the \(n!\) conjecture. With the intention of proving the \(n!\) conjecture by induction \textit{F. Bergeron} and \textit{A. M. Garsia} [CRM Proc. Lect. Notes. 22, 1-52 (1999; Zbl 0947.20009)] formulated several conjectures concerning intersections of modules \({\mathbf M}_{\nu}\) for partitions lying below a given partition \(\mu\) of \(n+1\). The paper under review gives an explanation of these conjectures of Bergeron and Garsia in an algebraic-geometric setting, interpreting them in the context of the Hilbert scheme of \(n\) points in the plane. The author constructs a coherent sheaf \({\mathcal P}\) on the Hilbert scheme and shows that the \(n!\) conjecture is true if and only if \({\mathcal P}\) is a locally free sheaf, i.e. a vector bundle. Then the author studies the restriction of \({\mathcal P}\) to subvarieties isomorphic to \(\mathbb{P}^{k-1}\) contained in the Hilbert scheme and reduces the series of conjectures of Bergeron and Garsia to one conjecture on the structure of this vector bundle restricted to a projective space \(\mathbb{P}^k\) embedded in the Hilbert scheme. Finally the author reinterprets geometric statements combinatorially. Macdonald polynomials; partitions; Hilbert scheme; projective varieties; vector bundles Combinatorial aspects of representation theory, Symmetric functions and generalizations, Parametrization (Chow and Hilbert schemes) The \(n\)! conjecture and a vector bundle on the Hilbert scheme of \(n\) points 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 The paper under review, which represents the Ph. D. thesis of the author, deals with the problem of reduction modulo \(p\) of the Hilbert-Blumenthal varieties with \(\Gamma_0 (p)\)-level structure, where \(p\) is a fixed prime natural number. One considers special kinds of Shimura varieties, namely Shimura varieties \(Sh_C (G,X)\) for which the \(p\)-primary factor \(C_p \subseteq G(\mathbb{Q}_p)\) of the subgroup \(C \subseteq G(\mathbb{A}_f)\) is of parabolic type. For these Shimura varieties the author proves results in connection with the following two problems:
(1) Describe the local structure of the model \(M_C/ \mathbb{Z}_{(p)}\) of these Shimura varieties, and
(2) Describe the reduction modulo \(p\) of this model (with special regard to the supersingular locus of \(M_C \otimes\mathbb{F}_p)\). reduction modulo \(p\); Hilbert-Blumenthal varieties; Shimura varieties Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties On the reduction of the Hilbert-Blumenthal moduli scheme with \(\Gamma_ 0(p)\)-level structure | 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 Fix a field \(k\) and consider the class of all varieties which are smooth over \(k\). If \(W @>\pi>> \text{Spec} (k)\) is within this class, we can attach to it a relative tangent bundle or, equivalently, a locally free sheaf \(\Omega^1_{W/k}\) of relative differentials. This tangent bundle, intrinsic to \(\pi\), plays a central role in algebraic geometry. For instance
(a) in studying the birational class of \(W\),
(b) in analyzing the singular locus of a closed embedded subscheme of \(W\) (e.g. jacobian ideals).
Essential for the development of some of these problems is the fact that the class is closed by blowing up regular centers, namely has the following property:
(P) Let \(W\) be smooth over \(k\), \(C\) a regular closed subscheme of \(W\), and \(W\leftarrow W_1\) the blow-up at \(C\). Then \(W_1\) is also in the class (is also smooth over the field \(k)\).
However, this property fails to hold if we consider now the class of smooth schemes over \(\mathbb{Z}\). In this work we replace \(\text{Spec} (k)\) by a Dedekind scheme of characteristic zero (for instance \(Y=\text{Spec} (\mathbb{Z}))\), and define a class of schemes over \(Y\) such that:
(1) the class includes the smooth schemes over \(Y\),
(2) to any \(W @>\pi>> Y\) in the class there is an intrinsically defined tangent bundle,
(3) the class is closed by blowing up convenient regular centers.
As an application, in \(\S 4\), we analyze the behaviour of jacobian ideals of embedded arithmetic schemes. sheaf of relative differentials; tangent bundle; birational class; singular locus; blowing up; smooth schemes; jacobian ideals of embedded arithmetic schemes Villamayor, O.: On smoothness and blowing ups of arithmetical schemes. Math. Z. 225, 317-332 (1997) Schemes and morphisms, Global theory and resolution of singularities (algebro-geometric aspects), Arithmetic problems in algebraic geometry; Diophantine geometry On smoothness and blowing ups of arithmetic 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 the entire collection see Zbl 0653.00009.]
Let X be the canonical resolution of one of the 3-dimensional simple singularities and E its exceptional locus. The author shows how to find the Picard group of the 3-dimensional formal scheme \(X_ E^{\wedge}\) in terms of the resolution diagram. More precisely, let \(E_ 1,...,E_ m\) be the irreducible components of E, \(\mu \in {\mathbb{Z}}^ m\) such that \(\mu\) \(E\hookrightarrow X\) is a negative embedding, then \(Pic(\mu E)=Pic(E)\). The paper contains also the computation of Pic(E) in the case of the singularity \(A_ n\). resolution; 3-dimensional simple singularities; Picard group Picard groups, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) The Picard group of the canonical resolution of a 3-dimensional simple 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 We give a combinatorial proof for a multivariable formula of the generating series of type D Young walls. Based on this we give a motivic refinement of a formula for the generating series of Euler characteristics of Hilbert schemes of points on the orbifold surface of type D. Hilbert scheme of points; Young walls; generating function Parametrization (Chow and Hilbert schemes), Combinatorial aspects of representation theory, Quantum groups (quantized enveloping algebras) and related deformations, Singularities in algebraic geometry Young walls and equivariant Hilbert schemes of points in type \(D\) | 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 study the Gromov-Witten theory of the Hilbert scheme \(X^{[2]}\) of two points on an elliptic surface \(X\). Assume that \(|K_X|\) contains an element supported on the smooth fibers of \(X\). By analyzing the degeneracy locus and localized virtual cycle arising from the cosection localization theory of Kiem and Li [Y. Kiem and J. Li, Gromov-Witten invariants of varieties with holomorphic 2-forms, preprint; Y. Kiem and J. Li, Localizing virtual cycles by cosections, \textit{J. Amer. Math. Soc.} \textbf{26} (2013) 1025-1050], we determine the \(1\)-point genus-\(0\) Gromov-Witten invariant \(\langle w\rangle_{0,d(\beta_f-2\beta_2)}^{X[2]}\) up to some rational number \(m(d,X)\) depending only on \(d\) and \(X\), where \(w\in H^4( X^{[2]},\mathbb{C})\), \(d\geq 1\), \(f\) is a smooth fiber of \(X,\beta_f= x_0+f\subset X^{[2]}\) with \(x_0\in X-f\) being a fixed point, and \(\beta_2=\{\xi\in X^{[2]}|\mathrm{Supp}(\xi)=\{x_0\}\}\). Moreover, we propose a conjecture regarding \(m(d,X)\), and prove that the conjecture is true for \(X=C\times E\) where \(E\) is an elliptic curve and \(C\) is a smooth curve. Gromov-Witten invariants; Hilbert schemes; cosection localization Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Gromov-Witten invariants of Hilbert schemes of two points on elliptic 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 We introduce symmetrizing operators of the polynomial ring \(A[x]\) in the variable \(x\) over a ring \(A\). When \(A\) is an algebra over a field \(k\) these operators are used to characterize the monic polynomials \(F(x)\) of degree \(n\) in \(A[x]\) such that \(A\otimes_k k[x]_{(x)}/(F(x))\) is a free \(A\)-module of rank \(n\). We use the characterization to determine the Hilbert scheme parameterizing subschemes of length \(n\) of \(k[x]_{(x)}\). local rings; symmetrizing operators; free quotient algebras; monic polynomials Laksov, D. andSkjelnes, R. M., The Hilbert scheme parameterizing finite length subschemes of the line with support at the origin,Compositio Math. 126 (2001), 323--334. Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Regular local rings The Hilbert scheme parameterizing finite length subschemes of the line with support at the origin. | 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 \(H\) denote a multigraded Hilbert scheme (i.e., \(H\) parametrizes those ideals of a graded polynomial ring \(K[x_1,\dots,x_n]\) that have a fixed multi graded Hilbert function). The torus \(T = (K^*)^n\) acts naturally on \(H\) induced by its action on \(\mathbb{A}^n\). Then the \(T\)-graph of \(H\) is the graph whose vertices are zero-dimensional orbits of \(T\) while the edges are the one-dimensional orbits of \(T\). The \(T\)-graph encodes a great deal of structural information about \(H\). For example \(H\) is connected if and only if its \(T\)-graph is connected.
In general, the vertices of the \(T\)-graph will correspond to monomial ideals in \(H\) and an edge will exist between two such ideals \(I\) and \(J\) if and only if there exists a one-dimensional torus whose orbit closure contains both \(I\) and \(J\). As monomial ideals are essentially purely combinatorial objects one expects to be able to resolve questions about the \(T\)-graph by purely combinatorial means. In practice however this is exceedingly difficult.
In this paper, the authors find a necessary combinatorial condition for when two vertices in the \(T\)-graph are connected by an edge. They also apply their results to the interesting case of the Hilbert scheme of points in the plane which allows them to, in this case, resolve a question posed by Altman and Sturmfels by showing the \(T\)-graph depends on the characteristic of the ground field.
The paper is thorough and the authors illustrate the technique with some very interesting and explicit examples. It is an excellent resource for anyone interested in the multigraded Hilbert scheme. Hilbert schemes; torus actions; Gröbner bases Parametrization (Chow and Hilbert schemes), Polynomial rings and ideals; rings of integer-valued polynomials, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) The T-graph of a multigraded Hilbert scheme | 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 of characteristic \(0\) and \(Z\) over \(k\) a normal surface. An order over \(Z\) is a coherent sheaf \(\mathcal{O}_X\) of \(\mathcal{O}_Z\)-algebras with generic fibre \(k(X):=\mathcal{O}_X\otimes k(Z)\), a central simple \(k(Z)\)-algebra. The order \(\mathcal{O}_X\) is considered as a non-commutative surface \(X\) with a finite map to the (commutative) surface \(Z\).
\textit{D. Chan} and \textit{C. Ingalls} generalized Mori's minimal model program to the case of orders over surfaces [Invent. Math. 161, No. 2, 427--452 (2005; Zbl 1078.14005)]. Canonical singularities of orders over surfaces are defined. They are non-commutative analogues of Kleinians singularities that arise naturally in the minimal model program. Canonical singularities of orders are classified using their minimal resolutions. They are explicitly described as invariant rings for the action of a finite group on a full matrix algebra over a regular local ring. It is proved that canonical singularities of orders are Gorenstein, their Auslander--Reiten quivers are described. A simple version of the McKay correspondence is given. canonical singularities; noncommutative surface; McKay correspondence; Gorenstein; Auslander-Reiten quiver Chan, D.; Hacking, P.; Ingalls, C., Canonical singularities of orders over surfaces, Proc. Lond. Math. Soc., 98, 83-111, (2009) Singularities in algebraic geometry, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), McKay correspondence Canonical singularities of orders over 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 It is proved by \textit{H. Esnault} [J. Reine Angew. Math. 362, 63--71 (1985; Zbl 0553.14016)] and \textit{M. Auslander} [Trans. Am. Math. Soc. 293, 511--531 (1986; Zbl 0594.20030)] that a normal surface singularity in characteristic zero is Cohen-Macaulay finite if and only if it is a quotient singularity. \textit{C. P. Kahn} shows [Math. Ann. 285, No. 1, 141--160 (1989; Zbl 0662.14022)] that a simple elliptic singularity is Cohen-Macaulay tame.
The purpose of the paper under review is to prove that a minimal elliptic singularity is Cohen-Macaulay tame if and only if it is either simple elliptic or cusp. Some explicit descriptions of indecomposable Cohen-Macaulay modules on these singularities are obtained. Cohen-Macaulay modules; minimally elliptic singularities; Cohen-Macaulay tame; wild rings; surface singularity Yu. A. Drozd, G.-M. Greuel, and I. Kashuba, ''On Cohen-Macaulay modules on surface singularities,'' Mosc. Math. J., 3:2 (2003), 397--418. Cohen-Macaulay modules, Structure, classification theorems for modules and ideals in commutative rings, Singularities of surfaces or higher-dimensional varieties On Cohen-Macaulay modules on 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 study the irreducibility and the smoothness of the generic plane curve of degree \(d\) passing through \(r\) generic points with prescribed multiplicities \(m_i\). We prove that, under the assumptions that the multiplicities are at most 3 and the degree is high enough, these curves are irreducible and smooth away from the prescribed singularities. plane curve; prescribed multiplicities; prescribed singularities Mignon, T., Systèmes linéaires de courbes planes à singularités ordinaires imposées, C. R. Acad. Sci. Paris Ser. I Math., 327, 7, 651-654, (1998) Singularities of curves, local rings, Plane and space curves Linear systems of plane curves with prescribed ordinary 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 Soient \(K\) un corps de nombres totalement réel de degré \(g\) et \(R\) l'anneau de ses entiers. Le but de cet article est de définir l'espace de modules \({\mathcal M}\) des variétés abéliennes de dimension \(g\), à multiplication par \(R\), munies de données auxiliaires. Cet espace \({\mathcal M}\) est un schéma sur \(\text{Spec} (\mathbb{Z} [\zeta_n] [1/n])\) et contient comme ouvert dense l'espace de modules \({\mathcal M}^R\) défini précédemment par \textit{M. Rapoport}. A la différence de \({\mathcal M}^R\), l'espace \({\mathcal M}\) admet une compactification toroïdale \(\overline {\mathcal M}\) qui est propre sur \(\text{Spec} (\mathbb{Z} [\zeta_n] [1/n])\), par contre \(\overline {\mathcal M}\) n'est pas lisse et les auteurs déterminent ses singularités, en particulier ils montrent que les fibres de \({\mathcal M} \to \text{Spec} (\mathbb{Z} [\zeta_n] [1/n])\) sont normales. Les auteurs en déduisent en toute caractéristique, le problème venant des caractéristiques \(p\) divisant le discriminant \(\Delta\) de \(K\) sur \(\mathbb{Q}\), l'irréductibilité des fibres géométriques de \(\overline {\mathcal M}\), donc de \({\mathcal M}^R\). -- Si \(L\) est un \(R\)-module inversible ``positif'', il est possible de définir ``un schéma abélien à multiplication réelle par \(R\), \(L\)-polarisé et muni d'une structure de niveau \(n\)'' et de considérer l'espace de modules associé \({\mathcal M}^L_n\). L'espace \({\mathcal M}\) cherché est une composante connexe de \({\mathcal M}^L_n\) pour \(L\) égal à la différente inverse \(D^{- 1}\) de \(R\) sur \(\mathbb{Z}\). Les auteurs obtiennent alors le résultat suivant:
Soit \(\Delta\) le discriminant de \(R\) sur \(\mathbb{Z}\). Le schéma \({\mathcal M}^L_n\) est lisse au dessus de \(\text{Spec} (\mathbb{Z} [1/n \Delta])\), plat d'intersection complète relative au-dessus de \(\text{Spec} (\mathbb{Z} [1/n])\) et, pour \(p\) premier à \(n\) divisant \(\Delta\), le lieu de non lissité de \({\mathcal M}^L_n\) en caractéristique \(p\) est de codimension 2 dans la fibre de caractéristique \(p\).
Pour avoir ce résultat, il faut étudier localement l'espace des modules \({\mathcal M} = {\mathcal M}^L_n\) au voisinage d'un point \(x\). Il existe un morphisme naturel de \({\mathcal M}\) dans un sous-schéma fermé \({\mathcal N}\) d'une grassmannienne convenable définie grâce au \({\mathcal O} \otimes R\)-module \(H_1^{\text{DR}} (A/{\mathcal M})\), où \(A\) est le schéma abélien universel sur \({\mathcal M}\). En utilisant la théorie cristalline des déformations on peut montrer que ce morphisme \({\mathcal M} \to {\mathcal N}\) est étale en \(x\). Il suffit alors de montrer le résultat pour le schéma \({\mathcal N}\). En utilisant la même méthode, il est possible d'étudier les singularités d'espaces formels de modules dans d'autres cas. moduli space of abelian varieties; crystalline cohomology; formal moduli space P. \textsc{Deligne} and G. \textsc{Pappas}, Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, \textit{Compos. Math.}, \textbf{90} (1994), 59-79. Algebraic moduli of abelian varieties, classification, \(p\)-adic cohomology, crystalline cohomology, Algebraic moduli problems, moduli of vector bundles Singularities of Hilbert moduli spaces in characteristics dividing the discriminant | 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 formulate a conjecture on actions of the multiplicative group in motivic homotopy theory. In short, if the multiplicative group \(G_m\) acts on a quasi-projective scheme \(U\) such that \(U\) is attracted as \(t\) approaches 0 in \(G_m\) to a closed subset \(Y\) in \(U\), then the inclusion from \(Y\) to \(U\) should be an \(A^1\)-homotopy equivalence. We prove several partial results. In particular, over the complex numbers, the inclusion is a homotopy equivalence on complex points. The proofs use an analog of Morse theory for singular varieties. Application: the Hilbert scheme of points on affine \(n\)-space is homotopy equivalent to the subspace consisting of schemes supported at the origin. torus action; Morse homology; Hilbert scheme of points; motivic homotopy theory Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes), Motivic cohomology; motivic homotopy theory, Discriminantal varieties and configuration spaces in algebraic topology Torus actions, Morse homology, and the Hilbert scheme of points on affine space | 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 Motivated by relations between Hilbert-Samuel multiplicity and \(F\)-thresholds, we conjecture an inequality that relates \(F\)-thresholds with Hilbert-Kunz multiplicity. In this article, we present several results that support the conjecture. In particular, we prove it for hypersurfaces and we give several consequences of this inequality. In addition, we extend previous results for the Hilbert-Samuel multiplicity. Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry Hilbert-Kunz multiplicities and \(F\)-thresholds | 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 article we introduce the notion of logarithmic differential forms with poles along a Cartier divisor given on a variety with singularities, discuss some properties of such forms, and describe highly efficient methods for computing the Poincaré series and generators of modules of logarithmic differential forms in various situations. We also examine several concrete examples by applying these methods to the study of divisors on varieties with singularities of many types, including quasi-homogeneous complete intersections, normal, determinantal, and rigid varieties, and so on. logarithmic differential forms; de Rham lemma; normal varieties; Poincaré series; complete intersections; determinantal singularities; fans; rigid singularities de Rham theory in global analysis, Analytic theory of abelian varieties; abelian integrals and differentials Logarithmic differential forms on varieties with 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 is a next step toward understanding the structure of proper birational morphisms of smooth algebraic varieties. The main result is: Let \(f:X \to Y\) be a proper birational morphism of regular schemes. If the exceptional divisor \(E_ f\) of \(f\) has two nonsingular components collapsing to a nonsingular irreducible subscheme of \(Y\) then \(f\) can be factored by two blowing ups with nonsingular centers. factorization of proper birational morphisms; regular schemes; exceptional divisor; blowing ups Z.\ H. Luo, Factorization of birational morphisms of regular schemes, Math. Z. 212 (1993), no. 4, 505-509. Rational and birational maps, Birational automorphisms, Cremona group and generalizations Factorization of birational morphisms of regular 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 Let \(k\) be a non-Archimedean field, and let \({\mathfrak X}\) be a formal scheme locally finitely presented over the ring of integers \(k^ 0\). In this work one constructs and studies the vanishing cycles functor from the category of étale sheaves on the generic fibres \({\mathfrak X}_ \eta\) of \({\mathfrak X}\) (which is a \(k\)-analytic space) to the category of étale sheaves on the closed fibre \({\mathfrak X}_{\overline s}\) of \({\mathfrak X}\) (which is a scheme over the residue field of the separable closure of \(k)\). One proves that if \({\mathfrak X}\) is the formal completion \(\hat {\mathcal X}\) of a scheme \({\mathcal X}\) finitely presented over \(k^ 0\) along the closed fibre, then the vanishing cycles sheaves of \(\hat {\mathcal X}\) are canonically isomorphic to those of \({\mathcal X}\) [as defined by \textit{P. Deligne} in Sémin. Géométrie algébrique, 1967-1969, SGA7 II, Lect. Notes Math. 340, Exposé XIII, 82-115 (1973; Zbl 0266.14008)]. In particular, the vanishing cycles sheaves of \({\mathcal X}\) depend only on \(\hat {\mathcal X}\), and any morphism \(\varphi:\hat {\mathcal Y} \to \hat {\mathcal X}\) induces a homomorphism from the pullback of the vanishing cycles sheaves of \({\mathcal X}\) under \(\varphi_{\overline s}:{\mathcal Y}_{\overline s} \to {\mathcal X}_{\overline s}\) to those of \({\mathcal Y}\). Furthermore, one proves that, for each \(\hat {\mathcal X}\), there exists a nontrivial ideal of \(k^ 0\) such that if two morphisms \(\varphi,\psi:\hat {\mathcal Y} \to \hat {\mathcal X}\) coincide modulo this ideal, then the homomorphisms between the vanishing cycles sheaves induced by \(\varphi\) and \(\psi\) coincide. These facts were conjectured by P. Deligne.
The second fact is deduced from a theorem on the continuity of the action of the set of morphisms between two analytic spaces on their étale cohomology groups. Its particular case states the following. Let \(X={\mathcal M} ({\mathcal A})\) be a \(k\)-affinoid space, and let \(f_ 1,\dots,f_ n\) be a \(k\)-affinoid generating system of elements of \({\mathcal A}\). Then for any discrete \(\text{Gal} (k^ s/k)\)-module \(\Lambda\) and any element of \(\alpha \in H^ q (X,\Lambda)\) there exist \(t_ 1, \dots,t_ n>0\) such that, for any pair of morphisms \(\varphi,\psi:Y \to X\) over \(k\) with \(\max_{y \in Y} | (\varphi^* f_ i-\psi^*f_ i)(y) | \leq t_ i\), \(1 \leq i \leq n\), one has \(\varphi^*(\alpha)=\psi^*(\alpha)\) in \(H^ q(Y,\Lambda)\). The essential ingredient of the proof is a generalization of the classical Krasner lemma. This result implies, in particular, the following fact. If a \(k\)-analytic group \(G\) acts on a \(k\)-analytic space \(X\), then the étale cohomology groups of \(X\) with compact support are discrete \(G(k)\)-modules. The present paper is based on the previous works of the author [``Spectral theory and analytic geometry over non-Archimedean fields'', Math. Surveys Monographs 33 (1990; Zbl 0715.14013) and ``Étale cohomology for non-Archimedean analytic spaces'', Publ. Math., Inst. Hautes Étud. Sci. 78, 5-171 (1993)]. analytic group; non-Archimedean field; formal scheme; vanishing cycles functor; étale sheaves V.\ G. Berkovich, Vanishing cycles for formal schemes, Invent. Math. 115 (1994), no. 3, 539-571. Étale and other Grothendieck topologies and (co)homologies, (Co)homology theory in algebraic geometry, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Local ground fields in algebraic geometry, Algebraic cycles Vanishing cycles for formal 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 Let \(X|k\) be an algebraic variety over a field \(k\) of characteristic \(0\). The question is studied whether the sheaf of Kähler differentials \(\Omega_X\) is reflexive or torsionfree if \(X\) has ``mild'' singularities.
As an example, if \(X\) is a normal local complete intersection, then \(\Omega_X\) is torsionfree. It is reflexive if and only if \(X\) is non-singular in codimensin \(2\). The following theorem is proved: Let \(Z\subset \mathbb{P}^n_k\) be a smooth projective variety and \(I\) the ideal sheaf defining \(Z\). Let \(X\subseteq \mathbb{A}^{n+1}\) be the affine cone over \(Z\). If \(H^1(\mathbb{P}^n, I^2(d))=0\) for all \(d\geq 0\) then \(\Omega_X\) is torsionfree. If in addition \(Z\) is projectively normal, then \(\Omega_X\) is torsionfree if and only if also the first infinitesimal neighbourhood of \(Z\) in \(\mathbb{P}^n_k\) is projectively normal. As an application it is proved that Gorenstein terminal singularities will have in general a sheaf of Kähler differential with torsion and cotorsion. sheaf of Kähler differentials; torsionfree; smooth projective variety; Gorenstein Greb, D.; Rollenske, S., Torsion and cotorsion in the sheaf of Kähler differentials on some mild singularities, Math. Res. Lett., 18, 1259-1269, (2011) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Commutative rings of differential operators and their modules, Singularities in algebraic geometry Torsion and cotorsion in the sheaf of Kähler differentials on some mild 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 classical Beauville-Bogomolov decomposition theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau- and holomorphic-symplectic manifolds. The decomposition of the simply-connected part corresponds to a decomposition of the tangent bundle into a direct sum whose summands are integrable and stable with respect to any polarisation.
Building on recent extension theorems for differential forms on singular spaces, we prove an analogous decomposition theorem for the tangent sheaf of projective varieties with canonical singularities and numerically trivial canonical class.
In view of recent progress in minimal model theory, this result can be seen as a first step towards a structure theory of manifolds with Kodaira dimension zero. Based on our main result, we argue that the natural building blocks for any structure theory are two classes of canonical varieties, which generalise the notions of irreducible Calabi-Yau- and irreducible holomorphic-symplectic manifolds, respectively. Beauville-Bogomolov decomposition theorem; varieties of Kodaira dimension zero; Calabi-Yau; holomorphic-symplectic varieties; minimal model theory Greb, D., Kebekus, S., Peternell, T.: Singular spaces with trivial canonical class. In: Minimal Models and Extremal Rays Proceedings of the Conference in Honor of Shigefumi Moris 60th Birthday, Advanced Studies in Pure Mathematics. Kinokuniya Publishing House, Tokyo (2011) (\textbf{to appear}) Calabi-Yau manifolds (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Compact Kähler manifolds: generalizations, classification Singular spaces with trivial canonical class | 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 an irreducible holomorphic symplectic manifold. Let \((\cdot,\cdot)\) be the Beauville-Bogomolov form on \(H^2(X,\mathbb{Z})\). Since \(H_2(X,\mathbb{Z})\) has finite index in \(H^2(X,\mathbb{Z})\) one can extends this form to a \(\mathbb{Q}\)-valued form on \(H_2(X,\mathbb{Z})\).
Suppose \(X\) contains a Lagrangian projective space \(\mathbb{P}^{\dim(X)/2}\). Let \(\ell\) be a line on this subspace. \textit{B. Hassett} and \textit{Y. Tschinkel} [Asian J. Math. 14, No. 3, 303--322 (2010; Zbl 1216.14012)] conjectured that if \(\dim X=2n\) then \((\ell,\ell)=-(n+3)/2\) holds.
In the paper under review the authors prove this conjecture for the case where \(n=3\) holds and such that \(X\) is deformation equivalent to the Hilbert scheme of length 3 subschemes of a \(K3\) surface. Furthermore, they show that in the cohomology ring of \(X\) one has the relation \([\mathbb{P}^3]=\frac{1}{48}(\rho^3+\rho^2c_2(X))\), with \(\rho=2\ell\in H^2(X,\mathbb{Z})\).
To determine the class of the projective space in the cohomology ring the authors use intersection properties of the projective space and the knowledge on the cohomology ring of \(X\). They use this to reduce the problem to a Diophantine problem, i.e., they show that the possible cohomology classes for the projective space correspond to non-trivial integral points on an explicit elliptic curve. They conclude by proving that this elliptic curve has exactly one such point. irreducible symplectic manifolds Harvey, D; Hassett, B; Tschinkel, Y, Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces, Commun. Pure Appl. Math., 65, 264-286, (2012) \(n\)-folds (\(n>4\)), \(K3\) surfaces and Enriques surfaces Characterizing projective spaces on deformations of Hilbert schemes of \(K3\) 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 [For the entire collection see Zbl 0721.00007.]
In J. Algebra 88, 89-133 (1984; Zbl 0531.13015), \textit{D. Eisenbud} and the author proved that regular local rings have finite Buchsbaum- representation type, i.e. they have just a finite number of nonisomorphic maximal indecomposable Buchsbaum modules. -- Here are determined the surface singularities of finite Buchsbaum-representation type. Let \(R\) be a Noetherian complete local ring with \(\dim(R)=2\) whose residue field is algebraically closed. Then \(R\) has finite Buchsbaum-representation type iff \(R\cong P/\kappa I\), where \((P,m)\) is a complete regular local ring with \(\dim(P)=3\), \(I\subset P\) is an ideal with \(\hbox{ht}(I)\geq 2\) and \(\kappa\) is an element from \(m\backslash m^ 2\). In particular \(R\) is regular iff \(R\) is unmixed of finite Buchsbaum-representation type. surface singularities of finite Buchsbaum-representation type Shiro Goto, Surface singularities of finite Buchsbaum-representation type, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 247 -- 263. Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities of surfaces or higher-dimensional varieties Surface singularities of finite Buchsbaum-representation type | 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 studies finitely generated graded modules over a polynomial ring \(S=k[x_1,\dots,x_n]\), where \(k\) is an infinite field. He determines in theorem 13 which chains of surjections of such modules may be deformed to one another. He also shows that when such deformations are possible, one can make them in such a way so as to do minimal damage to the depths of these modules in the process. The grandparent of the technique used in this paper is Reeves's proof in her thesis [R] that one may deform a subscheme of projective space to the ``lexicographic subscheme'' with the same Hilbert polynomial (which is uniquely determined once we have chosen and ordered the variables \(x_1,\dots,x_n)\) through a chain of no more than \(\text{deg} p(z)+2\) deformations over \(\mathbb{P}_k^1\). Thus, there is a chain of no more than \(2\deg p(z)+4\) such deformations linking any two subschemes of \(\mathbb{P}_k^{n-1}\) with Hilbert ???? deformation; graded modules over a polynomial ring Pardue, K.: Deformations of graded modules and connected loci on the Hilbert scheme. Queen's papers pure appl. Math. 105, 131-149 (1997) Deformations and infinitesimal methods in commutative ring theory, Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Graded rings Deformations of graded modules and connected loci on the Hilbert scheme | 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 either a nonsingular affine variety or a projective space \(\mathbb P^n\) over the field of complex numbers. Two closed subvarieties of \(A\) are said to be \textit{geometrically linked} if they have no common component and their union is a complete intersection in \(A\). If one of the varieties, say \(X\), is fixed and if the complete intersection is chosen in as general a way as possible containing \(X\), then the complement \(Y\) is called a \textit{generic link} of \(X\). The object of this paper is to study how singularities behave under generic linkage. The author gives a description of the Grauert-Riemenschneider canonical sheaf of \(Y\) in terms of the multiplier ideal sheaves associated to \(X\), and uses it to study the singularities of \(Y\). His work generalizes previous results of Chardin and Ulrich, among others. He gives several applications of his main theorem, for instance to the study of rational singularities and to the study of long canonical threshold under generic linkage. Finally, he applies it to generalize known results by \textit{T. De Fernex} and \textit{L. Ein} [Am. J. Math. 132, No. 5, 1205--1221 (2010; Zbl 1205.14020)], and by \textit{M. Chardin} and \textit{B. Ulrich} [Am. J. Math. 124, No. 6, 1103--1124 (2002; Zbl 1029.14016)], on the Castelnuovo-Mumford regularity bound for a projective variety. generic link; singularity; multiplier ideal; Castenlnuovo-Mumford regularity; log canonical threshold Niu, Wenbo, Singularities of generic linkage of algebraic varieties, Amer. J. Math., 136, 6, 1665-1691, (2014) Linkage, Singularities of surfaces or higher-dimensional varieties Singularities of generic linkage of algebraic 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 prove that the Hilbert scheme of 11 points on a smooth threefold is irreducible. In the course of the proof, we present several known and new techniques for producing curves on the Hilbert scheme. Douvropoulos, Theodosios; Jelisiejew, Joachim; Utstøl Nødland, Bernt Ivar; Teitler, Zach, The Hilbert scheme of 11 points in \(\mathbb{A}^3\) is irreducible, (Combinatorial Algebraic Geometry, Fields Inst. Commun., vol. 80, (2017), Fields Inst. Res. Math. Sci. Toronto, ON), 321-352 Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry The Hilbert scheme of 11 points in \(\mathbb{A}^3\) is irreducible | 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 result of [\textit{M. Auslander}, Lect. Notes Math. 1178, 194--242 (1986); identical with: Representations of algebras, Proc. 4th Int. Conf., Ottawa/Can. 1984, Vol. 1, Carleton-Ottawa Math. Lect. Note Ser. 1, W5, 49 p. (1985; Zbl 0633.13007)], proved in a more general situation by \textit{C. Huneke} and \textit{G. J. Leuschke} [Math. Ann. 324, No. 2, 391--404 (2002; Zbl 1007.13005)] says that a commutative local Cohen-Macaulay ring of finite Cohen-Macaulay type is an isolated singularity. In the paper under review, the author proves a non-commutative version of the result, by showing that if \(A\) is a graded Artin-Schelter Cohen-Macaulay algebra which is fully bounded noetherian and has finite Cohen-Macaulay type, then the non-commutative projective scheme determined by \(A\) is smooth. Artin-Schelter Cohen-Macaulay algebra; Artin-Schelter Gorenstein algebra; Cohen-Macaulay ring; fully bounded noetherian algebra; isolated singularity; maximal Cohen-Macaulay module; non-commutative projective scheme; punctured spectrum Jørgensen, P., Finite Cohen-Macaulay type and smooth non-commutative schemes, Canad. J. Math., 60, 379-390, (2008) Noncommutative algebraic geometry, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.), Graded rings and modules (associative rings and algebras) Finite Cohen-Macaulay type and smooth non-commutative 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 ``We give a canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite type over \(\mathbb C\) by taking ultraproducts of absolute integral closures in positive characteristic. This yields a new tight closure characterization of rational singularities in characteristic zero.''
Let \(R\) be a local domain of finite type over the field of complex numbers. It is known that certain characteristic \(p\) domains \(R_p\) can be associated to \(R\) [\textit{H. Schoutens}, Manuscr. Math. 111, No. 3, 379--412 (2003; Zbl 1082.13005)]. The quasi-hull \(\mathcal B(R)\) of \(R\) is defined to be the ultraproduct of the absolute integral closures \(R_p^+\) of \(R_p\), i.e. \(\mathcal B(R) = \text{ulim}_{p\to\infty}\,R_p^+\). It is shown that \(\mathcal B(R)\) is a balanced big Cohen-Macaulay \(R\)-algebra, and that this construction restores canonicity, is weakly functorial and preserves many of the good properties of the absolute integral closure. The \(\mathcal B\)-closure or plus-closure \(I^+\) of an ideal \(I\) of \(R\) is defined to be the ideal \(I\mathcal B(R) \cap R\). It is said that \(R\) is \(\mathcal B\)-rational if there is an ideal \(I\) generated by a system of parameters such that \(I = I^+\), and that \(R\) is \(\mathcal B\)-regular if \(I = I^+\) for every ideal \(I\). Some properties of them and relations between rational singularities are discussed, and the following theorem is given:
\(R\) has \(F\)-rational type \(\Leftrightarrow\) \(R\) is generically \(F\)-rational \(\Leftrightarrow\) \(R\) is \(\mathcal B\)-rational \(\Leftrightarrow\) \(R\) has rational singularities.
The following Briançon-Skoda type theorem is proven:
If \(I\) is an ideal of \(R\) generated by \(n\) elements, then the integral closure of \(I^{n+k}\) is contained in \((I^{k+1})^+\) for every \(k\).
And a new proof of \textit{J. Lipman} and \textit{B. Teissier}'s Briançon-Skoda type theorem [Mich. Math. J. 28, 97-116 (1981; Zbl 0464.13005)] is given. In the final section 7, the following theorem is shown:
If \(R\) has at most an isolated singularity and \(\text{Tor}^R_1(\mathcal B(R),k) = 0\) (\(k\) is the residue field of \(R\)), then \(R\) is regular.
As a corollary we have:
If \(\text{Tor}^R_1(\mathcal B(R),k) = 0\), then \(R\) has rational singularities. quasi-hull; ultraproduct; plus-closure; rational singularity; Briançon-Skoda theorem; balanced big Cohen-Macaulay algebra; tight closure; local domain of finite type; characteristic \(p\) domains H. Schoutens, Canonical big Cohen-Macaulay modules and rational singularities, Illinois Journal of Mathematics 41 (2004), 131--150. Cohen-Macaulay modules, Singularities in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Canonical big Cohen-Macaulay algebras and rational 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 \(S\) be a smooth complex projective surface.
It has long been known that the Hilbert scheme \(S^{[k]}\) of \(k\) points on \(S\) is smooth and projective of dimension \(2k\).
A line bundle \(H\) on \(S\) gives rise to an associated vector bundle \({\mathcal H}_{[k]}\) on \(S^{[k]}\) whose fiber at
\(Z \subset S\) is \(H^0(H|_Z)\).
Here \textit{A. S. Tikhomirov} [Aspects Math. E25, 183--203 (1994; Zbl 0819.14003)] proved that the Segre numbers
\[
s_k = \int_{S^{[k]}} s_{2k} (\mathcal H_{[k]})
\]
depend only on the intersection numbers \(\pi = H \cdot K_S, d=H^2, \kappa = K_S^2, e=c_2 (S)\), where
\(s_i (\mathcal H_{[k]})\) denotes the \(i\)th Segre class of \(\mathcal H_{[k]}\).
When \(S\) is a K3 surface and \(c_1 (H)^2 = 2g-2\),
\textit{A. Marian} et al. [Ann. Sci. Éc. Norm. Supér. (4) 50, No. 1, 239--267 (2017; Zbl 1453.14016)]
proved that \(s_k = 2^k \frac{g-2k+1}{k}\) and hence the vanishings \(s_k=0\) for \(g-2k+1 \geq 0\) and \(k>g-2k+1\).
The author gives an alternate proof of the vanishings in a restricted range using a simple geometric argument.
The main ingredient is an extension of work of \textit{R. Lazarsfeld} [J. Differ. Geom. 23, 299--307 (1986; Zbl 0608.14026)],
namely that if \(g > 2k-2\), then \(\mathcal H_{[k]}\) is generated by sections coming from \(H^0(S,H)\).
Similar arguments give analogous vanishings for the blow up of \(S\) at a point.
Using the restricted vanishings and considering the disjoint union \(S\) of another \(K3\) surface \(S^\prime\) and an abelian surface \(A\), the author uses induction on \(g\) to deduce the formula of Marian-Oprea-Pandharipande for the \(s_k\).
The author uses these results to reduce the \textit{M. Lehn} conjecture on generating functions for the Segre numbers \(s_k\) [Invent. Math. 136, 157--207 (1999; Zbl 0919.14001)] to a simpler statement. This statement was confirmed by
\textit{A. Marian} et al. [J. Math. Soc. Japan 71, No. 1, 299--308 (2019; Zbl 1422.14008)] to fully prove
Lehn's conjecture. Hilbert scheme of points;K3 surfaces;Segre classes;tautological bundles Parametrization (Chow and Hilbert schemes), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Sheaves in algebraic geometry, \(K3\) surfaces and Enriques surfaces, Enumerative problems (combinatorial problems) in algebraic geometry Segre classes of tautological bundles on Hilbert schemes of 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 The author studies the canonical map \(\varphi\colon X\rightarrow {\mathbb P}(H^0(X,\,\Omega^1_X)^\ast)\) of a pointed curve \(X\) over an algebraically closed field with nodal singularities with the help of its dual modular graph \(\Gamma=\Gamma(X).\) In particular, it is proved that \(\varphi\) has no base points if and only if \(\text{th}(\Gamma) \geq 2;\) the canonical map is an embedding if and only if \(\text{th}(\Gamma) \geq 3,\) the dimension of \( H^0(X,\,\Omega^1_X)\) is at least 2, and the curve \(X\) is not a generalized hyperelliptic curve. Here the thickness of \(\Gamma\) is denoted by \(\text{th}(\Gamma)\) [\textit{A. N. Tyurin}, ``Quantization, classical and quantum field theory and theta functions''. CRM Monograph Series 21 (2003; Zbl 1083.14037)]. In fact, the author completes the results by \textit{F. Knudsen} [Math. Scand. 52, 161--199 (1983; Zbl 0544.14020)] describing the multi-canonical map for Deligne-Mumford stable curves. He also remarks that analogous results for compact curves were earlier obtained by \textit{F. Catanese} et al. [Nagoya Math. J. 154, 185--220 (1999; Zbl 0933.14003)]. stable curves; pointed curves; dual modular graph; hyperelliptic curves; moduli variety Artamkin I.V.: Canonical maps of pointed nodal curves. Sb. Math. 195(5), 615--642 (2004) Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles Canonical maps of punctured curves with the simplest 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 a complete nonarchimedean valued field with a nontrivial valuation and let \(k^\circ\) be its ring of integers. The paper under review contains finiteness results for the étale cohomology of compact \(k\)-analytic spaces (in the sense of Berkovich) and for vanishing cycles on formal schemes over \(k^\circ\). We now give precise statements.
Let \(X\) be a compact \(k\)-analytic space and let \(F\) be an abelian constructible sheaf on \(X\) with torsion orders prime to the residue characteristic. Assume that \(k\) is algebraically closed. Then, the groups \(H^q(X,F)\) are finite for \(q\geq 0\).
Let \(\mathfrak{X}\) be a formal scheme locally topologically of finite presentation over \(k^\circ\) and let \(F\) be an abelian constructible sheaf on \(\mathfrak{X}_{\eta}\) with torsion orders prime to the residue characteristic. Then the vanishing cycles sheaves \(R^q \Psi_{\eta}(F)\) are constructible for \(q\geq 0\).
The last results concern a more general class of formal schemes, namely special formal schemes, i.e. locally of the form Spf\((A)\), where \(A\) is a quotient of an algebra of the form \(k^\circ\{T_{1},\dots,T_{n}\}[[S_{1},\dots,S_{m}]]\). In this setting, the above result still holds if \(k\) is assumed to be discretely valued and if the constructibility condition is replaced by the more restrictive notion of \(\mathfrak{X}\)-constructibility.
Let us mention that similar results were proven by \textit{R. Huber} [J. Algebr. Geom. 7, No. 2, 313--357 (1998; Zbl 1040.14008); J. Algebr. Geom. 7, No. 2, 359--403 (1998; Zbl 1013.14007)] under the assumption that \(k\) has characteristic 0.
Note that \textit{V. Berkovich} himself had already proven similar results under algebraicity assumptions in [Invent. Math. 115, No. 3, 539--571 (1994; Zbl 0791.14008); ibid. 125, No. 2, 367--390 (1996; Zbl 0852.14002)]. Actually the proofs in the present paper rely on those former results, the extra hypotheses being removed thanks to Gabber's weak uniformization (see [\textit{L. Illusie} (ed.) et al., Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l'École Polytechnique 2006--2008. Paris: Société Mathématique de France (SMF) (2014; Zbl 1297.14003)]) and Deligne's cohomological descent.
Finally, as regards weak uniformization, let us mention that the author proves results of this kind in the settings he considers: \(k\)-analytic spaces (covered by generic fibers \(\hat{\mathcal{Y}}_{\eta}\) of schemes \(\mathcal{Y}\) over \(k^\circ\) with \(\mathcal{Y}_{\eta}\) smooth over \(k\)) and special formal schemes over \(k^\circ\) (covered by completions of semi-stable schemes over rings of integers of finite extensions of \(k\)). For this purpose, applying Gabber's results directly is sometimes not enough and the author needs to go back to the proof and adapt the arguments. Berkovich spaces; formal schemes; étale cohomology; vanishing cycles; weak uniformization 10.1007/s11856-015-1249-6 Rigid analytic geometry, Étale and other Grothendieck topologies and (co)homologies Finiteness theorems for vanishing cycles of formal 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 The author constructs explicitly some resolutions of singularities in many relevant examples that appear often in moduli problems. The base is an algebraically closed field of characteristic \(p\geq 2\), which is then generalized to \(\text{Spec}({\mathbb{Z}}_p )\). The main new idea is to use compactification of symmetric spaces and line bundles on flag varieties. In the first part of the paper, which has an independent interest in itself, it is shown that the compactification of symmetric spaces \(X=G/H\) given by \textit{C. De Concini} and \textit{C. Procesi} [Invariant theory, Proc. 1st 1982 Sess. C. I. M. E., Montecatini/Italy, Lect. Notes Math. 996, 1-44 (1983; Zbl 0581.14041)] works in any characteristic. More precisely \(G\) is an adjoint semisimple Chevalley group and \(H\) is the invariant group of an involution of \(G\). This part relies on many other results in the literature, but it is almost self-contained and clearly written.
If \(\lambda\) is a dominant weight, the compactification \(\overline{X}_{\lambda}\) is defined as the closure of \(X\) in a representation space \({\mathbb{P}} (V({\lambda}^*))\). There is a smooth \(\overline{X}\) which dominates the other \(\overline{X}_{\lambda}\). The author shows that there is a Frobenius splitting of the above compactification (i.e. a section of the injection \({\mathcal O}\to \text{Frob}_* ({\mathcal O})\) ). The splitting induces compatible splittings on all strata. The cohomology of line bundles on \(\overline X\) can be computed in many cases by using this splitting. It follows by these cohomological computations and by a criterion of \textit{G. Kempf} [``Toroidal embeddings. I'', Lect. Notes Math. 339, 41-52 (1973; Zbl 0271.14017)] that \(\overline {X}_{\lambda}^{norm}\) has rational singularities, improving previous results of several authors that showed that these singularities are Cohen-Macaulay. Then the author checks the projective normality of \(\overline{X}_{\lambda}\) in many cases, e.g. for \(X\) equal to \(G\times G\) quotiented by the diagonal. This machinery is applied to several examples of singularities. Additional informations about the strata at infinity are also obtained. A basic type of singularity studied in the examples is that of two \(n\times n\) matrices \(B\) and \(C\) satisfying \(B\cdot C=C\cdot B=p\text{ Id}\). These singularities appear in the author's paper [\textit{G. Faltings}, Math. Ann. 304, 489-515 (1996; Zbl 0847.14018)] about moduli-stacks for bundles on semistable curves. Some analogous examples for the other classical groups are also considered. resolutions of singularities; characteristic \(p\); compactification of symmetric spaces; moduli space; rational singularities; moduli-stacks for bundles on semistable curves Faltings, G.: Explicit resolution of local singularities of moduli-spaces. J. reine angew. Math. 483, 183-196 (1997) Global theory and resolution of singularities (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Singularities of curves, local rings Explicit resolution of local singularities of moduli-spaces | 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 purely expository article is a summary of the author's lectures on topological, algebraic, and geometric properties of the zero schemes of sections of vector bundles. These lectures were delivered at the seminar Impanga at the Banach Center in Warsaw (2006), and at the METU in Ankara (December 11--16. 2006). A special emphasis is put on the connectedness of zero schemes of sections, and the ``point'' and ``diagonal'' properties in algebraic geometry and topology. An overview of recent results by \textit{V. Srinivas, V. Pati}, and the author [Diagonal subschemes and vector bundles, \url{math.AG/0609381}, to appear in the special volume Q. Pure Appl. Math. Q., dedicated to Jean-Pierre Serre on his 80th Birthday] these properties is given. Pragacz, P.: Miscellany on the zero schemes of sections of vector bundles. In: Algebraic Cycles, Sheaves, Shtukas, and Moduli. Trends in Mathematics, pp. 105--116. Birkhäuser, Basel (2007) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Miscellany on the zero schemes of sections of vector bundles | 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 complex quasiprojective variety. The Hilbert scheme \(\text{Hilb}^nX\) parametrizes zero-dimensional subschemes of length \(n\) of \(X\). If \(X\) is projective, \(\text{Hilb}^nX\) is a natural compactification of the parameter space of \(n\) distinct points of \(X\) and when \(X\) is a surface, it is, in fact, a smooth compactification. However, when the dimension of \(X\) is greater than two, the ``Hilbert scheme of points'' of length \(n\) on \(X\) is no longer smooth unless \(n\) is very small. Just as flag varieties are a natural generalization of grassmannians, it is natural to consider ``nested'' Hilbert schemes on \(X\) parametrizing nests \(Z_1 \subset Z_2 \subset \cdots \subset Z_m\) of zero-dimensional subschemes of \(X\) of given length.
By adapting the method of Ellingsrud and Strømme, we classify the nested Hilbert schemes on \(X\) which are smooth and obtain cellular decompositions for the smooth nested Hilbert schemes on affine and projective space as well as for the corresponding punctual nested Hilbert schemes. One deduces immediately that the Chow group and the (Borel-Moore) homology group of each of these spaces are isomorphic; odd homology vanishes and the \(2i\)-th homology group is the free abelian group generated by the classes of the closures of the \(i\)-cells. The number of cells of each dimension and hence the Betti numbers of the punctual nested Hilbert schemes are calculated here; the number of cells of each dimension of the smooth nested Hilbert schemes on affine and projective space can be read off formulae which are obtained in the sequels to this paper [see \textit{J. Cheah}, Math. Z. 227, No. 3, 479-504 (1998; Zbl 0890.14003) and J. Algebr. Geom. 5, No. 3, 479-511 (1996; Zbl 0889.14001)]. In fact, in the sequels, we use the cellular decompositions for the punctual nested Hilbert schemes obtained here to calculate the virtual Hodge polynomials of all the smooth nested Hilbert schemes on \(X\) (and of related varieties such as the universal families over some of these nested Hilbert schemes) in terms of the virtual Hodge polynomial of \(X\). Finally, following Göttsche, we study the Hilbert function strata of the punctual nested Hilbert schemes and cellular decompositions for these spaces. zero-dimensional subschemes; nested Hilbert schemes; virtual Holdge polynomials J. Cheah, ''Cellular Decompositions for Nested Hilbert Schemes of Points,'' Pac. J. Math. 183, 39--90 (1998). Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Cellular decompositions for nested Hilbert schemes of 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 We prove that the space of radical ideals of a ring \(R\), endowed with the hull-kernel topology, is a spectral space, and that it is canonically homeomorphic to the space of the non-empty Zariski closed subspaces of \((R)\), endowed with a Zariski-like topology. spectral space; spectral map; Zariski topology; inverse topology; hull-kernel topology; closure operation; radical ideal C.~A.~\textsc{Finocchiaro}, M.~\textsc{Fontana}, \textsc{and} D.~\textsc{Spirito}, A topological version of Hilbert's Nullstellensatz, J. Algebra \textbf{461} (2016), 25-41. Ideals and multiplicative ideal theory in commutative rings, Integral domains, Morphisms of commutative rings, Chain conditions, finiteness conditions in commutative ring theory, Relevant commutative algebra A topological version of Hilbert's Nullstellensatz | 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 \(G\) be a finite group acting on a quasi-projective smooth scheme \(X\) over a field \(k\). Suppose that \(g:\tilde{Y}\to X/G\) is a resolution of singularities of \(X/G\). Then by the McKay principal many aspects of the geometry of \(\tilde{Y}\) is reflected in the \(G\)-equivariant geometry of \(X\). Assume that \(E=g^{-1}(Z)_{\text{red}}\) where \(Z \subset X/G\) is the singular locus is a divisor with simple normal crossing. Denote by \(\Gamma(E)\) the dual complex associated to \(E\). For example, in the case of the Klein singularity \(\mathbb{C}^2/G\), \(\Gamma(E)\) is one of the ADE Dynkin diagrams.
The paper under review studies the homotopy type of \(\Gamma(E)\) which is known to be independent of the choice of the resolution. To express the result let \(\pi:X\to X/G\) be the projection, \(T=g^{-1}(Z)_{\text{red}}\), and \(f:\tilde{X}\to X\) be a proper birational \(G\)-equivariant morphism such that \(\tilde{X}\) is a smooth \(G\)-scheme and \(E_T=f^{-1}(T)_{\text{red}}\) is a \(G\)-strict simple normal crossing divisor and \(f\) is an isomorphism over \(X-T\). The main result of the paper under review proves that if \(k\) is perfect with characteristic zero, then there is a canonical map \(\phi: \Gamma(E_T)/G\to \Gamma(E)\) in the homotopy category of \(CW\)-complexes which induces isomorphisms on the homology and fundamental groups. This result has important implications such as
1) \(\Gamma(E_T)/G\) is contractible \(\Rightarrow\) \(\Gamma(E)\) is contractible.
2) \(X/G\) have isolated singularities \(\Rightarrow\) T is smooth \(\Rightarrow\) \(\Gamma(E)\) is contractible.
The question of the contractibility of \(\Gamma(E)\) is very important, for example if \(X/G\) has rational singularities then \(\Gamma(E)\) is contractible. The proof of the main result regarding the fundamental group uses a geometric interpretation of the fundamental group by means of the classifying group of \(cs\)-coverings of \(E\). And regarding the homology groups the proof is based on an equivariant weight homology theory introduced in the paper under review, and proving an analog of McKay principal for this weight homology. The proof of the latter relies on introducing another (arithmetic) homology theory in the paper under review called equivariant Kato homology. The other ingredients of the proof are cohomological Hasse principal, Deligne's theorem on Weil conjecture and Gabber's refinement of De Jong's alteration theorem. McKay principal; Equivariant geometry Global theory and resolution of singularities (algebro-geometric aspects), McKay correspondence, Geometric class field theory Cohomological Hasse principle and resolution of quotient 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 \((X,x)\) be a germ of normal singularity on an algebraic variety over \(\mathbb{C}\). We call a morphism \(\varphi: Y\to X\) a minimal (respectively canonical) model of \((X,x)\), if \(\varphi\) is proper, birational, (2) \(Y\) has at worst terminal (respectively canonical) singularities and (3) \(K_Y\) is \(\varphi\)-nef (respectively \(\varphi\)-ample). We call a morphism \(\varphi: Y\to X\) the log-canonical model, if (1) \(\varphi\) is proper, birational, (2) \((Y,E)\) has at worst log-canonical singularities, where \(E\) is the reduced exceptional divisor of \(\varphi\) and (3) \(K_Y+E\) is \(\varphi\)-ample.
By the definition, if a canonical model or a log-canonical model exists, then it is unique up to isomorphisms over \(X\). For a 2-dimensional singularity, a minimal model is also unique and conicides with the minimal resolution. For a 3-dimensional singularity, a minimal model, the canonical model and the log-canonical model exist. But the existence of higher dimensional models of the three kinds is not yet proved. In this paper, we show the existence of these three models for a hypersurface singularity defined by a non-degenerate polynomial. canonical model; log-canonical model; minimal model; hypersurface singularity S. Ishii, Minimal, canonical and log-canonical models of hypersurface singularities, Birational algebraic geometry (Baltimore, MD, 1996), 63--77, Contemp. Math., 207, Amer. Math. Soc., Providence, RI, 1997. Minimal model program (Mori theory, extremal rays), Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Hypersurfaces and algebraic geometry Minimal, canonical and log-canonical models of hypersurface 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 In this paper, we use singularity theory to study the entanglement nature of pure three-qutrit systems. We first consider the algebraic variety \(X\) of separable three-qutrit states within the projective Hilbert space \(\mathbb{P}(\mathcal{H})=\mathbb{P}^{26}\). Given a quantum pure state \(| \varphi \rangle \in \mathbb{P}(\mathcal{H})\) we define the \(X_{\varphi}\)-hypersuface by cutting \(X\) with a hyperplane \(H_{\varphi}\) defined by the linear form \(\langle \varphi |\) (the \(X_{\varphi}\)-hypersurface of \(X\) is \(X\cap {H}_{\varphi}\subset X\)). We prove that when \(| \varphi \rangle\) ranges over the stochastic local operation and classical communication entanglement classes, the 'worst' possible singular \(X_{\varphi}\)-hypersuface with isolated singularities, has a unique singular point of type \(D_{4}\). entanglement; three qutrit system; singularity theory F. Holweck and H. Jaffali, Three-qutrit entanglement and simple singularities, preprint (2016), arXiv:1606.05537. Quantum coherence, entanglement, quantum correlations, Projective techniques in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Three-qutrit entanglement and 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 Let \(O_{{\mathbb{F}}}\) be the ring of integers of a totally real field \({\mathbb{F}}\) of degree \(g\). We study the reduction of the moduli space of separably polarized abelian \(O_{{\mathbb{F}}}\)-varieties of dimension \(g\) modulo \(p\) for a fixed prime \(p\). The invariants and related conditions for the objects in the moduli space are discussed. We construct a scheme-theoretic stratification by \(a\)-types on the Rapoport locus and study the relation with the slope stratification. In particular, we recover the main results of \textit{E. Z. Goren} and \textit{F. Oort} [J. Algebr. Geom. 9, 111--154 (2000; Zbl 0973.14010)] on the stratifications when \(p\) is unramified in \(O_{{\mathbb{F}}}\). We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when \(p\) is totally ramified in \(O_{{\mathbb{F}}}\). Hilbert-Blumenthal varieties; Dieudonné modules; stratifications; deformations Yu, C.-F.: On reduction of Hilbert-Blumenthal varieties. Ann. Inst. Fourier (Grenoble) 53, 2105--2154 (2003) Modular and Shimura varieties, Formal groups, \(p\)-divisible groups On reduction of Hilbert-Blumenthal 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 smooth Fano variety of dimension \(n\). Let \(\bar{G}\subset \text{Aut}(X)\) be a compact subgroup. The \(\bar{G}\)-invariant \(\alpha\)-invariant of \(X\) can be defined by using a Kähler metric. A natural question is whether there exists a finite subgroup of the automorphism groups of \(\mathbb{P}^n\) whose \(\alpha\)-invariant is greater than \(1\). In the paper under review, the authors give an affirmative answer to the question for \(n\leq 4\) by studying exceptional quotient singularities. quotient singularities; log canonical threshold; alpha invariant; Fano varieties; Kähler-Einstein metric Cheltsov I., Shramov C., On exceptional quotient singularities, Geom. Topol., 2011, 15(4), 1843--1882 Singularities in algebraic geometry, Fano varieties, Kähler-Einstein manifolds, Birational geometry On exceptional quotient 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 [For the entire collection see Zbl 0551.00002.]
This note mostly summarizes known results on compressed algebras and on the punctual Hilbert scheme \({\mathcal H}=Hilb^ n({\mathbb{P}}^ r)\) (length n zero-dimensional subschemes of \({\mathbb{P}}^ r)\). If \(r<3\) or \(n<8\) we have \({\mathcal H}=\bar U\), where \({\mathcal U}={\mathcal U}(n,r)\) parametrizes non- singular subschemes of \({\mathcal H}\), i.e. sets of n distinct points in \({\mathbb{P}}^ r\). In general \({\mathcal H}\) may have several components. The author uses compressed algebras [cf. \textit{A. Iarrobino}, Trans. Am. Math. Soc. 285, 337-378 (1984; Zbl 0548.13009)] to show the existence of additional components of \({\mathcal H}\). A new result that is proved shows that compressed Gorenstein algebras have termwise maximal Hilbert function among Artin algebras of the same embedding dimension and the same socle degree. local Hilbert scheme; compressed algebras; punctual Hilbert scheme; maximal Hilbert function Iarrobino, A.: Compressed algebras and components of the punctual hubert scheme. Springer lecture notes #1124, 146-165 (1985) Parametrization (Chow and Hilbert schemes), Other special types of modules and ideals in commutative rings, Multiplicity theory and related topics Compressed algebras and components of the punctual Hilbert scheme | 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 show that principal bundles for a semisimple group on an arbitrary affine curve
over an algebraically closed field are trivial provided that the order of the \(\pi_1\) of the group is invertible in the ground field or the curve has semi-normal singularities. Several consequences and extensions of this result (and method) are given. As an application, we realize conformal blocks bundles on moduli stacks of stable curves as push-forwards of line bundles on (relative) moduli stacks of principal bundles on the universal curve. principal bundles; singular curves; \(B\)-reduction; Drinfeld-Simpson theorem; conformal blocks Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Singularities of curves, local rings Triviality properties of principal bundles on singular 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 Let k be an algebraically closed field of characteristic zero and \({\mathcal O}\) the ring of formal and convergent power series in one variable over k. Let \(A\subset {\mathcal O}\) be a k-subalgebra such that \(\dim _ k{\mathcal O}/A\) is finite. Then it was proved by Stafford and Smith that the rings of differential operators \({\mathcal D}({\mathcal O})\) and \({\mathcal D}(A)\)-modules are Morita equivalent i.e. the category of left \({\mathcal D}({\mathcal O})\)- modules is equivalent with the category of left \({\mathcal D}(A)\)-modules. - A \({\mathcal D}(A)\)-module N of finite type is called holonomic if it is of finite length over \({\mathcal D}(A)\). A holonomic \({\mathcal D}(A)\)-module N is said to have regular singularities if it possesses a good filtration such that Ann(gr(N)) is a radical ideal.
The main result of the paper states that under the Morita equivalence holonomic \({\mathcal D}\)-modules (respectively holonomic \({\mathcal D}\)-modules with regular singularities) correspond with holonomic \({\mathcal D}(A)\)- modules (respectively holonomic \({\mathcal D}(A)\)-modules with regular singularities). - As a byproduct of the whole analysis one obtains a criterion which decides if a holonomic \({\mathcal D}(A)\)-module has regular singularities; even in the case \(A={\mathcal O}\) this result seems to be new.
In a forthcoming paper the author generalizes the above results to \({\mathcal D}(A)\)-modules with irregular singularities. analytic algebra; rings of differential operators; Morita equivalence; holonomic D-modules; regular singularities Den Essen, A. Van: Modules with regular singularities on a curve. J. London math. Soc. (2) 40, 193-205 (1989) Power series rings, Modules of differentials, Analytic algebras and generalizations, preparation theorems, Formal power series rings, Valuations, completions, formal power series and related constructions (associative rings and algebras), Morphisms of commutative rings, Singularities of curves, local rings Modules with regular singularities on a 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 Let \(P_1,\dots,P_n \in \mathbb{C}^2\) be an ordered \(n\)-tuples of points in the plane. Assigning the points coordinates \(x_1,y_1,\dots,x_n,y_n\) the author identify the space \(E\) of all \(n\)-tuples \((P_1,\dots,P_n)\) with \(\mathbb{C}^{2n}\,.\) The locus \(V=\bigcup_{i<j} V_{ij}\) where two points coincide is a subspace arrangement in \(E\); its defining ideal \(I=I(V)=\bigcap _{i<j}(x_{i} - x_{j}, y_{i}-y_{j})\) is doubly homogeneous with respect to the double grading giving by degrees in the \({\mathbf x}=x_1,\ldots,x_n\) and \(\mathbf{y}=y_1,\dots,y_n\) variables separately. Moreover, the author shows that \(I^{m}=I^{(m)}\) and the Rees algebra \(R=\mathbb{C}[\mathbf{x},\mathbf{y}][tI]\) is Gorenstein. There are some open problems related to the study of rings of invariants and coinvariants for the action of the symmetric group \(S_n\) permuting the points among themselves. The author gives an exposition of some results by \textit{J.L. Martin} [Trans. Am. Math. Soc. 355, No.10, 4151--4169 (2003; Zbl 1029.05040)], related to the ideal of relations among the slopes of the lines that connect the \(n\) points pairwise. The appendix by E. Miller describes the Hilbert scheme \(H_n= \text{Hilb}^n(\mathbb{C}^2)\) of \(n\) points in the plane. Using results by Hartshorne, Haiman, Sturmfels, and Santos he shows that \(H_n\) is connected. The main result of this section says that \(H_n\) is a smooth and irreducible subvariety of the Grassmannian \(\text{Gr}^{n}(V_d)\) for \(d\geq n+1\). monomial ideal; Gorenstein ring; partition; complete graph Haiman, M.: Commutative algebra of n points in the plane. Trends Commut. Algebra, MSRI Publ. 51, 153--180 (2004) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage, complete intersections and determinantal ideals, Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Commutative algebra of \(n\) points in the plane. Appendix by Ezra Miller: Hilbert schemes of points 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 Suppose \(X\rightarrow Y\) is a double covering with \(Y\) smooth and \(X\) a normal complex surface. It is known that one can produce a birationally equivalent double covering \({\widetilde X} \rightarrow {\widetilde Y}\), with \({\widetilde X}\) smooth, by repeatedly blowing up singularities of the branch curve on \(Y\) and normalising the pull-back of \(X\). Here the authors prove that the same holds in the case of triple coverings of algebraic surfaces. The proof uses local computations which are based on results of \textit{R. Miranda} [Am. J. Math. 107, 1123--1158 (1985; Zbl 0611.14011)]. They also provide an example which shows that this procedure does not work in general for coverings of degree four. triple covering; normal surface; desingularisation; canonical resolution; blow-up Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties A canonical resolution of the singularities of a triple covering of algebraic 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 In this paper, the authors study degeneracy loci of maps of vector bundles on smooth ambient spaces, virtual fundamental cycles,
and their applications to nested Hilbert schemes.
Let \(X\) be a smooth complex quasi-projective variety and \(\sigma: E_0 \to E_1\) be a map of vector bundles of ranks \(e_0\) and \(e_1\) over \(X\).
For a positive integer \(r \le e_0\), the \(r\)-th degeneracy locus of \(\sigma\) is defined to be \(D_r(\sigma) = \{x \in X| \dim \ker (\sigma_x) \ge r\}\), i.e., the \((e_1 - e_0 + r - 1)\)-th Fitting scheme of the first cohomology sheaf \(h^1(E_*) =\mathrm{coker}(\sigma)\) of the complex \(E_*\).
The authors prove that the virtual resolution \(\widetilde D_r(\sigma)\) of \(D_r(\sigma)\) admits a perfect obstruction theory which depends only on the quasi-isomorphism class of the complex \(E_*\), and that the resulting virtual fundamental cycle \([\widetilde D_r(E_*)]^{\mathrm{vir}}\), when pushed forward to \(X\), is given by the Thom-Porteous formula.
When \(\widetilde D_r(\sigma)\) can be naturally embedded into the Grassmannian bundle \(\mathrm{Gr}(r,B)\) for some vector bundle
\(B\) over \(X\), the push-forward of \([\widetilde D_r(E_*)]^{\mathrm{vir}}\) to the Chow group \(A_*(\mathrm{Gr}(r,B))\) is also expressed
via the Thom-Porteous formula.
These results are applied to the nested Hilbert schemes \(S_\beta^{[n_1, n_2]} = \{I_1(-D) \subset I_2 \subset \mathcal O_S|[D] = \beta, \mathrm{length}(\mathcal O_S/I_i) = n_i \}\) of points and curves on a smooth projective surface \(S\) where \(\beta \in H^2(S, \mathbb Z)\) and \(n_1\) and \(n_2\) are non-negative integers.
The key observation is that these nested Hilbert schemes is identified with the virtual resolution \(\widetilde D_1(E_*)\) of the degeneracy locus \(D_1(E_*)\) for some \(2\)-term complex \(E_*\) over the product \(S^{[n_1]} \times S^{[n_2]}\) of two Hilbert schemes of points.
The earlier result describes the push-forward of \([S_\beta^{[n_1, n_2]}]^{\mathrm{vir}}\) to \(S^{[n_1]} \times S^{[n_2]} \times \mathbb P(H^0(X, \mathcal O_X(D)))\).
Similar result holds for the reduced class \([S_\beta^{[n_1, n_2]}]^{\mathrm{red}}\) as well. Moreover, a comparison theorem formulates \([S_\beta^{[n_1, n_2]}]^{\mathrm{vir}}\) and \([S_\beta^{[n_1, n_2]}]^{\mathrm{red}}\) in terms of the Carlsson-Okounkov K-theory class \(\mathrm{CO}_\beta^{[n_1, n_2]}\) on \(S^{[n_1]} \times S^{[n_2]} \times S_\beta\) where \(S_\beta\) denotes \(S_\beta^{[0, 0]}\). It follows that many integrals from the Vafa-Witten invariants of \(S\) can be evaluated in terms of the Seiberg-Witten invariants of \(S\) and
certain tautological virtual bundles over \(S^{[n_1]} \times S^{[n_2]} \times\mathrm{Pic}_\beta(S)\). Hilbert scheme; degeneracy locus; Thom-Porteous formula; local Donaldson-Thomas theory; Vafa-Witten invariants Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Degeneracy loci, virtual cycles and nested Hilbert 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 The Hilbert scheme of \(n\) points in \({\mathbb P}^2\) has a natural stratification by strata \(H(\phi)\) obtained from its Hilbert functions (or series) \(\phi\). The precise inclusion between the closures of the strata is in general unknown. The present paper gives necessary and sufficient conditions for the inclusion of \(H(\phi)\) in \(\overline {H(\psi)}\) where the Hilbert functions \(\phi\) and \( \psi\) are as close as possible and \(\phi \leq \psi\). This generalizes a result of \textit{F. Guerimand} [Thesis, University of Nice (2002)] which has the same result under an extra condition. In the case \(\phi\) is generic in some Brill-Noether locus, one may get the inclusion by using results of e.g. \textit{J. Brun} and \textit{A. Hirschowitz} [Ann. Sci. Éc. Norm. Supér. (4) 20, No. 2, 171--200 (1987; Zbl 0637.14002)].
Parts of the paper are a little technical, but the main ideas of the authors are clear and nice. They show that the necessary and sufficient conditions above are equivalent to a strict inequality of the dimension of certain \(\text{Ext}^1\)-groups where the smallest one is the dimension of the tangent space of deformations at some ideal of a non-commutative algebra. It turns, however, out that one may avoid ``non-commutative deformations'' and instead look to the Hilbert-flag scheme (of ``commutative'' deformations), or the incidence correspondence, of pairs of graded quotients of a polynomial ring and one of its projections. In the present case one needs to handle quotients for which the depth is zero, thus treat a slightly more general case than considered in several papers the reviewer [e.g., J. Algebra 311, No. 2, 665--701 (2007; Zbl 1129.14009)]. postulation Hilbert scheme; incidence; stratification; deformation Denaeghel, K.; Den Bergh, M. Van: On incidence between strata of the Hilbert scheme of points on P2, Math. Z. 255, 897-922 (2007) Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings, Deformations and infinitesimal methods in commutative ring theory, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On incidence between strata of the Hilbert scheme of points on \(\mathbb{P}^{2}\) | 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 \((R,m)\) be a Noetherian local ring, and let \(M\) denote a finitely generated \(R\)-module. A Buchsbaum \(R\)-module \(M\) is maximal if \(M\) has the same dimension as \(R\), and \(R\) has finite Buchsbaum representation type if there are only finitely many isomorphism classes of indecomposable maximal Buchsbaum \(R\)-modules.
In this paper after a short overview on the subject the author studies the case of curves; he proves under the assumptions that \(R\) is complete and \(R/m\) is infinite that the following conditions are equivalent:
(i) \(R\) has finite Buchsbaum representation type.
(ii) \(e(R) \leq 2\), \(\nu (R) \leq 2\) and the ring \(R/H^ 0_ m (R)\) is reduced \((\nu (R)\) is the embedding dimension).
The author obtains this theorem after the proof of the following: (same hypothesis) \(R\) is a Cohen-Macaulay ring of finite Buchsbaum representation type if and only if \(R\) is a reduced ring of \(e(R) \leq 2\). maximal Buchsbaum modules; Buchsbaum type; Noetherian local ring Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities of curves, local rings, Multiplicity theory and related topics, Commutative Noetherian rings and modules, Algebraic functions and function fields in algebraic geometry Curve singularities of finite Buchsbaum-representation type | 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 a smooth projective surface \(V\), let \(\mathrm{Hilb}^m_V\) denote the scheme parametrizing effective divisors \(D\) in \(V\) such that its first Chern class \(c_1({\mathcal O}_V(D))\) equals the fixed class \(m\in H^2(V,{\mathbb Z})\). In an earlier paper [Topology, 46, No. 3, 225--294 (2007; Zbl 1120.14034)] the authors constructed the virtual class \([[\mathrm{Hilb}^m_V]]\), which in the present paper is considered as a homology class by the cycle map.
Let \(C\subset V\) be an integral curve, and let \(c\) denote its first Chern class. By adding the curve to the divisors one gets a closed embedding \(\mathrm{Hilb}^{m}_V \to \mathrm{Hilb}^{m+c}_V\). The main result of the article relates the classes \([[\mathrm{Hilb}^{m-c}_V]]\) and \([[\mathrm{Hilb}^m_V]]\) when \(m\cdot c <0\), and the classes \([[ \mathrm{Hilb}^m_V]]\) and \([[\mathrm{Hilb}^{m+c}_V]]\) when \((k-m)\cdot c<0\), where \(k\) is the first Chern class of the tautological line bundle of the surface. Hilbert scheme; virtual fundamental class Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Relations for virtual fundamental classes of Hilbert schemes of curves on 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 The author surveys a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for the Macdonald symmetric functions, and the ``\(n!\)'' and ``\((n+1)^{n-1}\)'' conjectures which relate Macdonald polynomials to the characters of doubly graded \(S_n\)-modules.
In 1987 Macdonald unified the theory of Hall-Littlewood symmetric functions with that of spherical functions on symmetric spaces, introducing a class of symmetric functions, now known as Macdonald polynomials, with coefficients depending on two parameters \(q\) and \(t\). There are bivariate analogues of the Kostka-Foulkes polynomials, and Macdonald conjectured that these more general Kostka-Foulkes polynomials should have positive integer coefficients. In 1993 Garsia and the author introduced some bigraded \(S_n\)-module and conjectured that its dimension is \(n!\) and this is known as the \(n!\) conjecture. It is known that the \(n!\) conjecture implies the Macdonald positivity conjecture. The spaces figuring in the \(n!\) conjecture are quotients of the ring of coinvariants for the diagonal action of \(S_n\) on \({\mathbb C}^n\oplus{\mathbb C}^n\), and it was natural to investigate the characters of the full coinvariant ring. The space of coinvariants has dimension \((n+1)^{n-1}\). This involved \(q\)-analogues of this number and the Catalan numbers \(C_n\) in the data. A menagerie of things studied earlier by combinatorialists for their own sake turned up unexpectedly in this new context. Further, Procesi suggested that the diagonal coinvariants might be interpreted as sections of a vector bundle on the Hilbert scheme \(H_n\) of points in the plane. Understanding this geometric context has led the author to the proofs of the \(n!\) and \((n+1)^{n-1}\) conjectures. The full explanation depends on properties of the Hilbert scheme which were not known before, and had to be established from scratch in order to complete the picture. One might say, that the main results are not the \(n!\) and \((n+1)^{n-1}\) theorems, but new theorems in algebraic geometry.
In order to make the exposition self-contained, the author includes background from combinatorics, theory of symmetric functions, representation theory and geometry. At the end of the paper he discusses future directions, new conjectures, and related work of other mathematicians. Macdonald polynomials; symmetric polynomials; partitions; Hilbert scheme; \(n!\) conjecture Mark Haiman, Combinatorics, symmetric functions, and Hilbert schemes, Current developments in mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 39 -- 111. Symmetric functions and generalizations, Parametrization (Chow and Hilbert schemes) Combinatorics, symmetric functions, and Hilbert 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 This paper discusses the problem to decide whether or not a given basic sequence corresponds to a curve in \(P^ 3\) that satisfies certain conditions like smoothness, irreducibility, etc. For simple basic sequences (called neat) an existence theorem for smooth irreducible curves is proved. basic sequences; smooth irreducible curves Mutsumi Amasaki, Curves in \?³ whose ideals are simple in a certain numerical sense, Publ. Res. Inst. Math. Sci. 23 (1987), no. 6, 1017 -- 1052. Projective techniques in algebraic geometry, Curves in algebraic geometry Curves in \({\mathbb{P}}^ 3\) whose ideals are simple in a certain numerical sense | 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 \(G\) a split semi-simple group scheme and \(P\) a principal \(G\)-bundle on a relative curve \(X\rightarrow S\), we study a natural obstruction for the triviality of \(P\) on the complement of a relatively ample Cartier divisor \(D\subset X\). We show, by constructing explicit examples, that the obstruction is nontrivial if \(G\) is not simply connected, but it can be made to vanish by a faithfully flat base change, if \(S\) is the spectrum of a dvr (and some other hypotheses). The vanishing of this obstruction is shown to be a sufficient condition for étale local triviality if \(S\) is a smooth curve, and the singular locus of \(X-D\) is finite over \(S\).
For part I, see [\textit{P. Belkale} and \textit{N. Fakhruddin}, Algebr. Geom. 6, No. 2, 234--259 (2019; Zbl 1444.14036)]. principal bundle; singular curve; obstructions to triviality Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Singularities of curves, local rings Triviality properties of principal bundles on singular curves. 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 Let \(S\) be a smooth projective \(K3\) surface, and let \(\text{Hilb}^d(S)\) be the Hilbert scheme of \(d\) points on \(S\). For \(\beta \in H^2(S;\mathbb Z)\), and the point class \(p\) in \(S\) the classes \(C(\beta):= \mathfrak p_{-1}(\beta)\mathfrak p_{-1}(p)^{d-1} 1\) and \(A:=\mathfrak p_{-2}(p)\mathfrak p_{-1}(p)^{d-2}1\) (when \(d\geq 2\)), defined by means of Nakajima creation operators, span \(H_2(\text{Hilb}^d(S);\mathbb Z)\).
Now let \(\pi:S\to \mathbb P^1\) be an elliptically fibered \(K3\) surface and let \(\pi^{[d]}:\text{Hilb}^d(S)\to \text{Hilb}^d(\mathbb P^1)=\mathbb P^d\) be the induced Lagrangian fibration with generic fiber a smooth Lagrangian torus. Let \(L_z\) be the fiber of \(\pi^{[d]}\) over a point \(z\in \mathbb P^d\). Let \(\beta_h\) be a primitive effective curve class on \(S\) having intersection number 1 with the fiber class of \(\pi\) and self-intersection number \(2h-2\). For \(z_1, z_2\in \mathbb P^d\) and integer \(k\), let \(N_{d,h,k}:=\langle L_{z_1}, L_{z_2}\rangle_{C(\beta_h)+kA}\) be the 2-pointed Gromov-Witten invariant of \(\text{Hilb}^d(S)\) in class \(C(\beta_h)+kA\).
One of the main results of the paper under review proves that
\[
\sum_{h\geq 0}\sum_{k\in \mathbb Z}N_{d,h,k}y^k q^{h-1}=\frac{F(z,\tau)^{2d-2}}{\Delta(\tau)}
\]
where \(F(z,\tau)\) is the Jacobi theta function considered as a formal power series in the variables \(y=-e^{2\pi i z}\) and \(q=e^{2\pi i \tau}\) with \(|q|<1\). This formula specializes the famous Yau-Zaslow formula when \(d=1\). The paper under review also computes the genus 0 Gromov-Witten invariants of \(\text{Hilb}^d(S)\) for a few other natural incidence conditions in terms of Jacobi forms.
In the case that \(\pi\) admits a section \(B\) and \(\beta_h=B+hF\) for the fiber class \(F\) the paper under review proves an explicit formula for the generating series of genus 0 arbitrary 2-pointed GW invariants of \(\text{Hilb}^2(S)\) in terms of a quasi-Jacobi form of index 1 and weight \(\leq\) 6. It is conjectured in this paper that the same formula holds for all \(d\) and this time the quasi-Jacobi form is of index \(d- 1\) and weight \(\leq 2 + 2d\).
A couple of interesting applications are presented at the end. First, is a proof of a conjecture fir \(d=2\) relating genus 1 invariants of \(\text{Hilb}^d(S)\) to the Igusa cusp form. Second, a conjectural formula is given for the number of hyperelliptic curves on a \(K3\) surface passing through 2 general points. \(K3\) surfaces; Gromov-Witten invariants Oberdieck, G.: Gromov-Witten invariants of the Hilbert scheme of points of a \(K3\) surface. Geom. Topol. (2017, to appear). arXiv:1406.1139 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), \(K3\) surfaces and Enriques surfaces, Jacobi forms Gromov-Witten invariants of the Hilbert schemes of points of a \(K3\) surface | 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 0742.00065.]
This paper is devoted to prove that general facts about the theory of deformations of hypersurface singularities can be used to obtain information on the structure of the subschemes \({\mathcal N}_ d=\{(A,\Theta)\in{\mathcal A}_ g:\dim\Theta_{sing}\geq d\}\) of the moduli scheme, \({\mathcal A}_ g\), of principally polarized abelian varieties (p.p.a.v.) introduced \textit{A. Andreotti} and \textit{A. L. Mayer} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 21, 189-238 (1967; Zbl 0222.14024)].
As a corollary of the study of the tangent cones to discriminant loci for families of hypersurfaces, the authors deduce the result of Andreotti- Mayer on the Schottky problem, the Green theorem on rank four quadrics and the Beauville result on the Schottky problem for genus \(g=4\). Following the same techniques the authors also prove that given \(g',g''\geq 1\) and \(g=g'+g''\), the locus \({\mathcal A}(g',g'')\) of p.p.a.v.'s of dimension \(g\) which are product of p.p.a.v.'s of dimensions \(g'\) and \(g''\) is an irreducible component of \({\mathcal N}_{g-2}\). singularities of \(\Theta\)-divisors; deformations of hypersurface singularities; Schottky problem Smith, Roy; Varley, Robert, Singularity theory applied to \(\Theta \)-divisors.Algebraic geometry, Chicago, IL, 1989, Lecture Notes in Math. 1479, 238-257, (1991), Springer, Berlin Theta functions and abelian varieties, Singularities in algebraic geometry, Theta functions and curves; Schottky problem, Singularities of surfaces or higher-dimensional varieties Singularity theory applied to \(\Theta\)-divisors | 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 moduli space of triples of the form \((E,\theta,s)\) is considered, where \((E,\theta)\) is a Higgs bundle on a fixed Riemann surface \(X\), and \(s\) is a nonzero holomorphic section of \(E\). Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting \(s\). If \((Y,L)\) is the spectral data for the Higgs bundle \((E,\theta)\), then \(s\) defines a section of the line bundle \(L\) over \(Y\). The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle \(K_X\), since \(Y\) is a curve on \(K_X\). The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple \((E,\theta,s)\) to the divisor of the corresponding section of the line bundle on the spectral curve. Vector bundles on curves and their moduli, Parametrization (Chow and Hilbert schemes), Symplectic structures of moduli spaces Symplectic structures of moduli space of Higgs bundles over a curve and Hilbert scheme of points on the canonical bundle. | 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 Given a filtered ring, we give bounds of its multiplicity in terms of the data of the tangent cone using the technique of the filtered blowing-up. Applying it to each simple \(K3\) singularity of multiplicity two, we find a good coordinate where the Newton boundary of the defining equation contains the point \((1,1,1,1)\in\mathbb{R}^4\). In the course of the proof, we classify simple \(K3\) singularities of multiplicity two into 48 weight types. Furthermore, we prove that the weight type of the singularity stays the same under arbitrary one-parameter (FG)-deformations. Newton boundary; (FG)-deformation Tomari, M., Multiplicity of filtered rings and simple K3 singularities of multiplicity two, Publ. Res. Inst. Math. Sci., 38, 693-724, (2002) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, \(K3\) surfaces and Enriques surfaces Multiplicity of filtered rings and simple \(K3\) singularities of multiplicity 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 Many authors have investigated the claim of \textit{F. Severi} [Vorlesungen über algebraische Geometrie. (Übersetzung von E. Löffler.). Leipzig u. Berlin: B. G. Teubner (1921; JFM 48.0687.01)] that for \(d,g\) and \(r \geq 3\), the Hilbert scheme \({\mathcal H}_{d,g,r}\) of smooth connected curves of degree \(d\) and genus \(g\) in \(\mathbb P^r\) is irreducible for \(d \geq g+r\). \textit{L. Ein} proved the claim for \(r=3\) and \(r=4\) [Ann. Sci. Éc. Norm. Supér., IV. Sér., 19, 469--478 (1986; Zbl 0606.14003); Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], but for \(r \geq 6\) there are counterexamples. It has been suggested by \textit{C. Ciliberto} and \textit{E. Sernesi} [in: Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co.. 428--499 (1989; Zbl 0800.14002)] and Lopez in his Mathematical Review of an earlier paper of \textit{C. Keem} [Proc. Am. Math. Soc. 122, No. 2, 349--354 (1994; Zbl 0860.14003)] that Severi only intended his claim for linearly normal curves in the Brill-Noether range.
The authors of the current paper consider the Hilbert scheme \({\mathcal H}_{d,g,r}^{\mathcal L} \subset {\mathcal H}_{d,g,r}\) consisting of irreducible components whose general member is linearly normal. The main result states that \({\mathcal H}_{g+1,g,4}^{\mathcal L}\) is irreducible whenever it is non-empty, except when \(g=9\). Most of the paper deals with the low genus cases \(g \leq 12\) before a uniform approach covers the rest of the possibilities. The arguments consist mainly of dimension counts of components of moduli spaces of linear series on curves [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume I. Berlin: Springer (1985; Zbl 0559.14017)] and \textit{E. Arbarello} et al. [Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris. Berlin: Springer (2011; Zbl 1235.14002)] and use irreducibility of the Severi variety of plane curves [\textit{J. Harris}, Invent. Math. 84, 445--461 (1986; Zbl 0596.14017)] and work of \textit{H. Iliev} [Proc. Am. Math. Soc. 134, No. 10, 2823--2832 (2006; Zbl 1097.14022)]. Hilbert scheme; algebraic curves; linear series Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) On the Hilbert scheme of linearly normal curves in \(\mathbb{P}^4\) of degree \(d = g+1\) and genus \(g\) | 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 Conditions characterizing the membership of the ideal of a subvariety \({\mathfrak S}\) arising from (effective) divisors in a product complex space \(Y \times X\) are given. For the algebra \({\mathcal O}_Y [V]\) of relative regular functions on an algebraic variety \(V\), the strict stability is proved, in the case where \(Y\) is a normal space, and the Noether stability is established under a weakened condition. As a consequence (for both general and complete intersections) a global Nullstellensatz is derived for divisors in \(Y\times {\mathbb C}^N\) and \(Y\times {\mathbb P}^N ({\mathbb C})\), respectively. Also obtained are a principal ideal theorem for relative divisors, a generalization of the Gauss decomposition rule, and characterizations of solid pseudospherical harmonics on a semi-Riemann domain. A result towards a more general case is as follows: Let \({\mathfrak D}_j\), \(1 \leq j\leq p\), be principal divisors in \(X\) associated to the components of a \(q\)-weakly normal map \(g = (g_1,\ldots,g_p) : X \to {\mathbb C}^p\), and \(S := \bigcap {\mathfrak S}_{|{\mathfrak D}_j|}\). Then for any proper slicing \((\varphi,g,D)\) of \(\{{\mathfrak D}_j\}_{1\leq j\leq p}\) (where \(D\subset X\) is a relatively compact open subset), there exists an explicitly determined Hilbert exponent \({\mathfrak h}_{_{{\mathfrak D}_1 \cdots {\mathfrak D}_p,D}}\) for the ideal of the subvariety \({\mathfrak S} = Y\times (S\cap D)\). Noether stability; Hilbert number; Hilbert exponent; mapping multiplicity; intersection number Tung, C, On Noether and strict stability, Hilbert exponent, and relative nullstellensatz, Ann. Pol. Math., 107, 1-28, (2013) Complex spaces, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Analytic subsets and submanifolds On Noether and strict stability, Hilbert exponent, and relative Nullstellensatz | 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 begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion-free. In addition, when the codimension of the singular locus is at least two, it is shown that being reduced is equivalent to the reflexivity of the tangent sheaf. Our main results state on one hand, that the tangent sheaf of a codimension one foliation in \(\mathbb P^{3}\) is locally free if and only if the singular scheme is a curve, and that it splits if and only if it is arithmetically Cohen-Macaulay. On the other hand, we discuss when a split foliation in \(\mathbb P^{3}\) is determined by its singular scheme. holomorphic foliations; reflexive sheaves; split vector bundles Giraldo, L; Pan-Collantes, AJ, On the singular scheme of codimension one holomorphic foliations in \({\mathbb{P}}^3\), Intern. J. Math., 7, 843-858, (2010) Singularities of holomorphic vector fields and foliations, Dynamical aspects of holomorphic foliations and vector fields, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On the singular scheme of codimension one holomorphic foliations in \(\mathbb P^{3}\) | 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 provide a technique to detect the singularities of rational planar curves and to compute the correct order of each singularity including the infinitely near singularities without resorting to blow ups. Our approach employs the given parametrization of the curve and uses a \(\mu \)-basis for the parametrization to construct two planar algebraic curves whose intersection points correspond to the parameters of the singularities including infinitely near singularities with proper multiplicity. This approach extends Abhyankar's method of \(t\)-resultants from planar polynomial curves to rational planar curves. We also derive the classical result that for a rational planar curve of degree \(n\) the sum of all the singularities with proper multiplicity is \((n - 1)(n - 2)/2\). Examples are provided to flesh out our results. rational planar curve; singularities; infinitely near points; blow up; intersection multiplicity; \(\mu \)-basis; \(\delta \)-invariant Jia, X.; Goldman, R., \textit{\({\mu}\)}-bases and singularities of rational planar curves, Comput. Aided Geom. Des., 26, 9, 970-988, (2009) Singularities of curves, local rings, Plane and space curves \(\mu \)-bases and singularities of rational planar 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 There is a natural class of topological rings which contains both the affinoid rings (from rigid analytic geometry) and the noetherian adic rings. Namely the class of topological rings which have an open adic subring with a finitely generated ideal of definition. We call such a ring \(f\)-adic. For every \(f\)-adic ring \(A\) the topological space \(\text{Cont }A\) of all continuous valuations of \(A\) is a spectral space. If \(A\) has an open noetherian adic subring or if for every \(n \in N\) the ring \(A\langle X_ 1,\dots, X_ n\rangle\) of restricted power series in \(n\) variables over \(A\) is noetherian then there is a natural sheaf \({\mathcal O}_ A\) of topological rings on \(\text{Cont }A\). All stalks of \({\mathcal O}_ A\) are local rings. We call a topologically and locally ringed space which is locally isomorphic to some \(\text{Spa }A := (\text{Cont }A,{\mathcal O}_ A)\) adic. There are natural fully faithful functors from the category of rigid analytic varieties and the category of locally noetherian formal schemes to the category of adic spaces. In the first case one assigns to an affinoid rigid analytic variety \(\text{Sp }A\) the adic space \(\text{Spa }A\) and in the latter case one assigns to an affine noetherian formal scheme \(\text{Spf }A\) the adic space \(\text{Spa }A\). affinoid rings; adic rings; \(f\)-adic ring; continuous valuations; formal schemes; rigid analytic variety R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), no. 4, 513-551. Local ground fields in algebraic geometry, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Formal groups, \(p\)-divisible groups, Global topological rings A generalization of formal schemes and rigid analytic 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 $C$ be a smooth irreducible projective curve of genus $g$$\geq3$, $U_C^s(2,d)$ the moduli space of stable, degree $d$, rank $2$ vector bundles on $C$ and $B_d^k$ the Brill-Noether locus of vector bundles $\mathcal{F}$ of degree $d$ with $h^0(\mathcal{F})$$\geq{k}$. In the present paper the authors strenghten the result they proved in [\textit{Y. Choi} et al., Proc. Am. Math. Soc. 146, No. 8, 3233--3248 (2018; Zbl 1392.14003)]. where they showed that for $3$ $\leq$ $\nu$ $\leq$ $(g + 8 )/4$ and $3g - 1$ $\leq$ $d$ $\leq$ $4g - 6 -2$$\nu$ , $B^k_d$ $\cap$ $U^s_C(2,d)$ has exactly two reduced components $B_r$ and $B$$_s$ where $C$ is a general $\nu$-gonal curve. In this paper they improve the conditions on $\nu$ and $d$ and give more information on the components $B_r$ and $B_s$. Furthermore, they apply their main theorem to the study of Hilbert schemes of smooth surface scrolls in projective space. [\textit{C. Ciliberto} and \textit{F. Flamini}, Rev. Roum. Math. Pures Appl. 60, No. 3, 201--255 (2015; Zbl 1389.14013)], [\textit{M. Teixidor i Bigas}, Tohoku Math. J. (2) 43, No. 1, 123--126 (1991; Zbl 0702.14009)]. stable rank 2 bundles; Brill-Noether loci; general \(\nu \)-gonal curves; Hilbert schemes Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Rational and ruled surfaces, Parametrization (Chow and Hilbert schemes), Special divisors on curves (gonality, Brill-Noether theory), Special algebraic curves and curves of low genus Moduli spaces of bundles and Hilbert schemes of scrolls over \(\nu \)-gonal 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 Let \(X \to Y\) be a \(Y\)-scheme; a proper \(Y\)-scheme \(Z@>p>>Y\) is called a compactification of the \(Y\)-scheme \(X\) if there exist an open dense immersion \(X@>i>>Z\) over \(Y\). -- \textit{M. Nagata} proved earlier (1962) that if \(Y\) is a noetherian scheme and \(X\to Y\) is separated and of finite type, then it admits a \(Y\)-compactification. In this article, this result is proved by alternative methods. In the first part of the paper are given some results closed to blowing-ups and compactification problem on extensions of morphisms. The result of Nagata is a consequence of these preliminary results. The paper also contains many results useful in birational geometry. open immersion; blowing-ups; compactification; extensions of morphisms; birational geometry Lütkebohmert, W., On compactification of schemes, Manuscripta Math., 80, 95-111, (1993) Rational and birational maps, Global theory and resolution of singularities (algebro-geometric aspects), Schemes and morphisms On compactification of 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 Let \(X\) be a smooth projective toric surface, and \(\mathbb H^d(X)\) the Hilbert scheme parametrizing the zero-dimensional subschemes of length \(d\) in \(X\). In this paper, the rational Chow ring \(A^*(\mathbb H^d(X))_{\mathbb Q}\) is computed. The main tool used for this is the technique of equivariant Chow rings. The author first computes in terms of generators and relations the \(T\)-equivariant Chow ring \(A^*_T(\mathbb H^d(X))_{\mathbb Q}\), where \(T\subset X\) is the two-dimensional torus embedded in \(X\). For this, results due to \textit{M. Brion} [Transform. Groups 2, 225--267 (1997; Zbl 0916.14003)] on equivariant Chow rings of toric varieties are applied. The usual Chow ring is then obtained as the quotient of \(A^*_T(\mathbb H^d(X))_{\mathbb Q}\) by a certain explicitly described ideal. In the last section of the paper, this method is illustrated by the computation of the equivariant and the usual Chow rings of the Hilbert scheme \(\mathbb H^3(\mathbb P^2)\). Chow rings; Hilbert schemes; equivariant Chow rings; toric varieties [9] L. Evain, &The Chow ring of punctual Hilbert schemes on toric surfaces&#xTransform. Groups12 (2007) no. 2, p. 227Article | &MR 23 | &Zbl 1128. Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies The Chow ring of punctual Hilbert schemes on toric 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 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 |
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 goal of this work is to develop tools to better understand singularities in mixed characteristic. As multiplier and test ideals have been critical to the understanding of singularities in equicharacteristic, the authors extend the big Cohen-Macaulay test ideals developed by \textit{F. Perez} and \textit{R. R.G.} [Trans. Amer. Math. Soc., Ser. B 8, 754--787 (2021; Zbl 1487.13024)] to the mixed characteristic setting. They prove that these BCM-test ideals are independent of the big Cohen-Macaulay $R^+$-algebra $B$ chosen as long as $B$ is sufficiently large. They show that the characteristic $p>0$ notions are in fact equivalent to the mixed characteristic notions and they establish relationships with characteristic $0$, which we illustrate in the following table:
\begin{align*}
\text{Characteristic } p>0 \qquad&\text{Mixed characteristic } (0,p) & \text{Characteristic } 0\\
F\text{-regular}\qquad& \mathrm{BCM}_B\text{-regular}& \text{KLT}\\
F\text{-rational} \qquad & \mathrm{BCM}_B\text{-rational}& \text{pseudorational} \\
\tau(R, \Delta) \qquad & \tau_B(R, \Delta)& \mathcal{J}(R, \Delta)\\
\tau(\omega_R)\qquad & \tau_B(\omega_R)& \mathcal{J}(\omega_R)
\end{align*}
As with (characteristic $p$) test ideals and multiplier ideals, they show that $\mathrm{BCM}_B$-test ideals are stable under small perturbations. They also show that $\mathrm{BCM}_B$-test ideals behave similar to characteristic $p$ test ideals under finite maps. As \textit{M. Hochster} and \textit{C. Huneke} [Mém. Soc. Math. Fr., Nouv. Sér. 38, 119--133 (1989; Zbl 0699.13003)] used localization to produce elements in the characteristic $p$ test ideal, localization can also help us to determine elements in the $\mathrm{BCM}_B$-test ideal $\tau_B(R, \Delta).$ They also prove that these test ideals restrict well when $R$ is a complete normal ring; in particular, if $h \in R$ such that $R/hR$ is normal; $\Delta$ is a $\mathbb{Q}$-divisor so that $K_R+ \Delta$ has index not divisible by $p$; and $\Delta$ and $V(h)$ have no common components, then for $1 > \epsilon >0$ and for every $R^+$-algebra $B$ there is an $(R/hR)^+$-algebra $C$ so that $\tau_C(R/hR, \Delta|_{R/hR}) \subseteq \tau_B(R, \Delta+(1-\epsilon) \mathrm{div}_R(h))R/hR$. They also address how $F$-rational and $F$-regular singularities vary as one varies the characteristic, including some helpful examples and pictures. big Cohen-Macaulay; F-rational; F-regular; log terminal; multiplier ideal; perfectoid; rational; singularities; test ideal Homological conjectures (intersection theorems) in commutative ring theory, Singularities in algebraic geometry, Arithmetic ground fields (finite, local, global) and families or fibrations, Arithmetic algebraic geometry (Diophantine geometry), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Multiplier ideals Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras | 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 \textit{O. Zariski} [Am. J. Math. 60, 151-204 (1938; Zbl 0018.20101)] showed that in a two-dimensional regular local ring every complete ideal has a unique factorization into product-irreducible complete ideals. Later, \textit{J. Lipman} [in: Algebraic geometry and commutative algebra, in honor of M. Nagata, 203-231 (1988; Zbl 0693.13011)] generalized Zariski's result in higher dimensions to finitely supported complete ideals. He found a kind of unique factorization for complete ideals into special complete ideals that are associated with finite sequences of infinitely near points to a smooth point, but allows negative exponents.
The paper under review deals with the factorization of complete ideals of an excellent Noetherian regular local ring of dimension \(\geq 3\). Sufficient conditions are given in terms of the Lipman multiplicities and linear proximity equalities in order to have Lipman's unique factorization using positive exponents. factorization of complete ideals; excellent Noetherian regular local ring; Lipman multiplicities; linear proximity Ideals and multiplicative ideal theory in commutative rings, Excellent rings, Plane and space curves, Singularities of curves, local rings Complete ideals and singularities of space 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 The authors first review work of themselves and others [for example, \textit{Yu. A. Drozd} and \textit{G.-M. Greuel}, J. Algebra 246, 1--54 (2001; Zbl 1065.14041)] on the classification of indecomposable vector bundles on chains and cycles of projective lines (where adjacent curves in the configuration intersect transversally). Then they introduce (complicated!) combinatorial objects called strings and bands, and prove (Theorem 3.2) that there is a one-to-one correspondence between isomorphism classes of indecomposable objects in the derived category \(D^{-}(Coh_X)\) and equivalence classes of strings and bands, where \(X\) is a cycle of projective lines and \(Coh_X\) is the category of coherent sheaves on \(X\). Later this result is used to describe the various types of coherent sheaves on these varieties (including the case of a cycle of length one, which is the rational curve with one node). vector bundles; coherent sheaves; projective line Polishchuk, A.: Moduli of curves with nonspecial divisors and relative moduli of \(A_\infty \)-structures. Preprint (2015), arXiv:1511.03797 Derived categories, triangulated categories, Vector bundles on curves and their moduli, Representations of quivers and partially ordered sets, Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations Coherent sheaves on rational curves with simple double points and transversal 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 It seems natural to relate the study of (moduli of) vector bundles over smooth complex, projective varieties with the study of the corresponding (families of) embedded projective bundles, when the vector bundle is very ample. Inspired by the case of rank-two vector bundles over curves (see the Introduction of the paper under review and references therein) this paper takes part of a series of papers which deals with the case of rank-two vector bundles over Hirzebruch surfaces \(\mathbb{F}_e\). As a natural continuation of previous works, the authors consider the case in which \(\mathcal{E}\) is a vector bundle over \(\mathbb{F}_e\) with first Chern class \(c_1(\mathcal{E})=4C_0+\lambda f\) (\(C_0\) stands for the minimal section and \(f\) for the fiber of \(\mathbb{F}_e\)) which is known to be very ample.
The authors prove, see Theorem 4.2, that when \(e \leq 2\) the corresponding ruled 3-folds are smooth points of the proper component of the Hilbert scheme, which in fact has the expected dimension. Moreover, see Theorem 4.6, these scrolls fill up either their whole component (\(e=0,1\)) or a codimension one subvariety of their component (\(e=2\)). Some interesting questions -- general elements of the components of the Hilbert schemes, degenerations in terms of vector bundles -- beyond the generalizations to \(e\geq 3\), are also provided (see Section 5). vector bundles; rational ruled surfaces; ruled threefolds; Hilbert schemes; moduli spaces \(3\)-folds, Rational and ruled surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Varieties of low degree, Adjunction problems On families of rank-\(2\) uniform bundles on Hirzebruch surfaces and Hilbert schemes of their scrolls | 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 An elementary proof of the real Nullstellensatz is given which does not use the place extension theorem or the Tarski-Seidenberg theorem. real Hilbert ring; real Jacobson semi-simple rings; real Nullstellensatz Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Real algebraic and real-analytic geometry, Relevant commutative algebra Real Hilbert rings and the real Nullstellensatz | 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, the author continues the study of Gorenstein three dimensional singularities coming from finite group quotients. The article of \textit{S. S.-T. Yau} and \textit{Y. Yu} [``Gorenstein quotient singularities in dimension three'', Mem. Am. Math. Soc. 505 (1993; Zbl 0799.14001)] describes the action of \(G\subset SL(3, \mathbb{C})\), a finite group, on \(\mathbb{C}^3\) and classifies the Gorenstein quotients which arise out of this procedure, by classifying the finite groups of the necessary type. The author's work is towards settling the following conjecture:
Let \(G\) be a finite subgroup of \(SL(3, \mathbb{C})\) acting on \(\mathbb{C}^3\). Then there exists a resolution of singularities \(\sigma: \widetilde X\to \mathbb{C}^3/G\) with \(\omega_{\widetilde X} = {\mathcal O}_{\widetilde X}\) and \(\chi (\widetilde X)= \#\)\{conjugacy classes of \(G\}\).
This is apparently of some interest to the physicists. Various cases of this conjecture had been proved earlier and the author gives a summary of what was known. The author had settled the case of trihedral groups [\textit{Y. Ito}, Proc. Japan Acad., Ser. A 70, No. 5, 131-136 (1994; Zbl 0831.14006) and Int. J. Math. 6, No. 1, 33-43 (1995; Zbl 0831.14005)]. In the present article more cases are settled from types (B) and (D), as opposed to the trihedral groups which are of type (C), in the classification table of Yau and Yu. The author also mentions a later preprint of \textit{S.-S. Roan} [Inst. Math., Acad. Sinica, preprint R940606-1 (1994)] where the rest of the cases which were not covered earlier are treated, thereby proving the conjecture in full. action of group on complex 3-space; Gorenstein three dimensional singularities; finite group quotients; resolution of singularities Ito, Y.: Gorenstein quotient singularities of monomial type in dimension three. J. math. Sci. univ. Tokyo 2, 419-440 (1995) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Modifications; resolution of singularities (complex-analytic aspects), Low codimension problems in algebraic geometry, Homogeneous spaces and generalizations, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects) Gorenstein quotient singularities of monomial type in dimension three | 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 Threedimensional terminal singularities have been classified by Morrison- Stevens and Mori, using Reid's result saying, that they are cyclic quotients of smooth or cDV points. The authors use a criterium, due to S. Mori, and complicated combinatorical arguments to give a complete classification and explicit description of all threedimensional canonical singularities which arise as cyclic quotients of hypersurface singularities. canonical models; Threedimensional terminal singularities; cyclic quotients of hypersurface singularities Singularities in algebraic geometry, \(3\)-folds, Minimal model program (Mori theory, extremal rays) On canonical singularities of dimension three | 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 is a short account of enumerative results on threefolds with ordinary singularities in \({\mathbb{P}}^ 4\). No proofs are given. enumerative results on threefolds; ordinary singularities \(3\)-folds, Enumerative problems (combinatorial problems) in algebraic geometry Introduction to the algebraic geometry of complex projective threefolds with ordinary 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 A real spherical space is, by definition, the quotient space of a real reductive algebraic group by a closed algebraic subgroup, such that the action of any minimal parabolic subgroup defined over the real field admits an open orbit. The authors study the large scale geometry of a real spherical space by proving that such a space admits a polar decomposition. The main tool used to prove this result is called simple compactification, which means a compactification with a unique closed orbit.
After recalling the basic geometry of real spherical spaces in Section 2, they construct in Section 3 an explicit representation space which gives rise to an embedding of a given real spherical space into a projective space. They show that the closure of the image is a simple compactification. Using this embedding (and with a simple reduction on the closed subgroup), they prove the main result in Section 4, providing a preliminary result on a polar decomposition of real spherical spaces. In Section 5, they refine this decomposition result using the idea of `compression cone'.
In the last section, they use the main result to prove a fact about a special kind of real spherical spaces called `wavefront', and suggest that this kind of spaces is related to the lattice counting problem. spherical spaces; polar decomposition; compactification Knop, F.; Krötz, B.; Sayag, E.; etal., Simple compactifications and polar decomposition for real spherical spaces, Selecta Math. (N.S.), 21, 1071-1097, (2015) Compactifications; symmetric and spherical varieties, Homogeneous spaces, General properties and structure of real Lie groups, Representations of Lie and linear algebraic groups over real fields: analytic methods Simple compactifications and polar decomposition of homogeneous real spherical spaces | 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 provide a construction of Saito primitive forms for Gepner singularity by studying the relation between Saito primitive forms for Gepner singularities and primitive forms for singularities of the form \(F_{k, n} = \sum_{i = 1}^n x_i^k\) invariant under the natural \(S_n\)-action. Frobenius manifolds; Saito primitive form; Saito structures; singularity theory Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Deformation quantization, star products, Topological field theories in quantum mechanics, Grassmannians, Schubert varieties, flag manifolds Primitive forms for Gepner 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 On a smooth complex projective threefold \(X\) there are two curve counting theories, which are conjecturally equivalent: Donaldson-Thomas (DT) invariants, studied by \textit{D. Maulik}, \textit{N. Nekrasov}, \textit{A. Okounkov} and \textit{R. Pandharipande} in [Compos. Math. 142, No. 5, 1263--1285 (2006; Zbl 1108.14046)], and Pandharipande-Thomas (PT) invariants, studied by \textit{R. Pandharipande} and \textit{R. P. Thomas} in [Invent. Math. 178, No. 2, 407--447 (2009; Zbl 1204.14026)].
If \(\beta\in H_{2}(X,\mathbb{Z})\) and \(n\in\mathbb{Z}\), let \(I_{n}(X,\beta)\) be the Hilbert scheme of subschemes \(Z\) of \(X\) in the class \([Z]=\beta\) with holomorphic Euler characteristic \(\chi(\mathcal{O}_{Z})=n\), and \(I_{n,\beta}:=e(I_{n}(X,\beta))\) be its Euler characteristic. Moreover, let \(P_{n}(X,\beta)\) be the moduli space of stable pairs \((F,s)\), where \(F\) is a pure sheaf on \(X\) with Chern character \((0,0,\beta,-n+\beta\cdot c_{1}(X)/2)\) and \(s\) is a section of \(F\) with \(0\)-dimensional cokernel, and let \(P_{n,\beta}:=e(P_{n}(X,\beta))\). Letting \(Z^{I}_{\beta}(X)\) and \(Z^{P}_{\beta}(X)\) be the generating series of the \(I_{n,\beta}\) and of the \(P_{n,\beta}\) respectively, the author prove that \(Z^{P}_{\beta}(X)=Z^{I}_{\beta}(X)/Z^{I}_{0}(X)\), which is a topological version of the DT/PT-correspondence. For Calabi-Yau threefolds, this was obtained by \textit{Y. Toda} in [J. Am. Math. Soc. 23, No. 4, 1119--1157 (2010; Zbl 1207.14020)] and by \textit{T. Bridgeland} in [J. Am. Math. Soc. 24, No. 4, 969--998 (2011; Zbl 1234.14039)] with different methods. The strategy of the proof is the following: let \(C\) be any Cohen-Macaulay curve in \(X\) and \(I_{n,C}\) (resp. \(P_{n,C}\)) the Euler characteristic of the subset of \(I_{n}(X,\beta)\) (resp. \(P_{n}(X,\beta)\)) of subschemes whose underlying Cohen-Macaulay curve is \(C\) (resp. of pairs supported at \(C\)), and \(Z^{I}_{C}(X)\) (resp. \(Z^{P}_{C}(X)\)) their generating series. The authors show that \(Z^{I}_{C}(X)=Z^{P}_{C}(X)\cdot Z^{I}_{0}(X)\): integrating over all \(C\), one gets the previous statement involving \(Z^{I}_{\beta}(X)\) and \(Z^{P}_{\beta}(X)\). In order to relate \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\), the authors provide a GIT wall-crossing between \(I_{n}(X,\beta)\) and \(P_{n}(X,\beta)\), and they study the relation between the Euler characteristic of the fibers of this wall-crossing using the Ringel-Hall algebra machinery of Joyce.
In section 2, the authors provide a GIT construction of \(P_{n}(X,\beta)\): Le Potier's construction of the moduli space of stable coherent systems presents \(P_{n}(X,\beta)\) as a quotient of a Quot scheme \(Q:=\mathrm{Quot}(\mathcal{H},P_{\beta})\), where \(\mathcal{H}=H^{0}(F(m))\otimes\mathcal{O}_{X}(-m)\). Here \(m\) is chosen so that for every pair \((F,s)\) the sheaf \(F(m)\) is globally generated, and we let \(V:=H^{0}(F(m))\).
Let \(R_{m}:=H^{0}(\mathcal{O}_{X}(m))\): for any pair \((F,s)\) there is an inclusion \(H^{0}(F)\subseteq V\otimes R_{m}^{*}\). As \(s\) spans a one-dimensional subspace of \(H^{0}(F)\), the pair \((F,s)\) corresponds to a point of the projective space \(\mathbb{P}(V\otimes R_{m}^{*})\). The authors construct a suitable closed subscheme \(\mathcal{N}\) of \(\mathbb{P}(V\otimes R^{*}_{m})\times Q\) parameterizing stable pairs, together with a natural action of \(\mathrm{SL}(V)\) and two ample \(\mathbb{Q}\)-linearizations \(\mathcal{L}_{0}\) and \(\mathcal{L}_{1}\) for the action of \(\mathrm{SL}(V)\). Using the Hilbert-Mumford criterion, the authors show that \(P_{n}(X,\beta)=\mathcal{N}^{s}//_{\mathcal{L}_{1}}\mathrm{SL}(V)\) and \(I_{n}(X,\beta)=\mathcal{N}^{s}//_{\mathcal{L}_{0}}\mathrm{SL}(V)\). Letting \(\mathcal{L}_{t}:=(1-t)\mathcal{L}_{0}+t\mathcal{L}_{1}\), the authors show that the quotient \(SS_{n}(X,\beta)=\mathcal{N}^{ss}//_{\mathcal{L}_{t*}}\mathrm{SL}(V)\) (for some \(0<t^{*}<1\)) has a stratification \(SS_{n}(X,\beta)=\coprod_{k=0}^{n}I^{\mathrm{pur}}_{n-k}(X,\beta)\times S^{k}(X)\), where \(I^{\mathrm{pur}}_{n-k}(X,\beta)\) is the locus of the Cohen-Macaulay closed subschemes of \(I_{n-k}(X,\beta)\), and \(S^{k}(X)\) is the \(k\)-th symmetric product of \(X\). Moreover, there are two morphisms \(\varphi_{P}:P_{n}(X,\beta)\longrightarrow SS_{n}(X,\beta)\) and \(\varphi_{I}:I_{n}(X,\beta)\longrightarrow SS_{n}(X,\beta)\), which are isomorphism on the subschemes of pairs with surjective sections and pure support respectively. This is the GIT wall-crossing between \(I_{n}(X,\beta)\) and \(P_{n}(X,\beta)\).
Now, letting \(I_{n}(X,C)=\varphi_{I}^{-1}(C,S^{n}(X))\), \(P_{n}(X,C)=\varphi^{-1}_{P}(C,S^{n}(X))\) for any Cohen-Macaulay curve \(C\) in \(X\), and \(I_{n,C}=e(I_{n}(X,C))\), \(P_{n,C}=e(P_{n}(X,C))\), the authors show that \(I_{n,C}=P_{n,C}+e(X)P_{n-1,C}+e(\mathrm{Hilb}^{2}(X))P_{n-2,C}+\dots+e(\mathrm{Hilb}^{n}(X))P_{0,C}\), which implies the relation between the generating series \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\). This is obtained by using the Ringel-Hall algebra machinery of Joyce: if \(\mathcal{T}\) is the stack of \(0\)-dimensional sheaves, Joyce provides a Ringel-Hall algebra \(H(\mathcal{T})\) together with an integration map \(P_{q}:H(\mathcal{T})\longrightarrow\mathbb{Q}(q^{1/2})[t]\). The generating series \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\) are the limit (for \(q\rightarrow 1\)) of the integration map \(P_{q}\) computed over some stacks mapping to \(\mathcal{T}\) (namely: for \(Z^{I}_{C}(X)\) the stack \(\Hom(\mathcal{I}_{C},-)\), whose fiber over \(T\) is \(\Hom(\mathcal{I}_{C},T)\), and for \(Z^{P}_{C}(X)\) the stack \(Ext^{1}(-,\mathcal{O}_{C})\), whose fiber over \(T\) is \(Ext^{1}(T,\mathcal{O}_{C})\)). Using convolutions and relations between these stacks, one concludes with the relation between \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\). Turaev, V.G.: The Conway and Kauffman modules of a solid torus. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) \textbf{167}(Issled. Topol. 6), 7989 (1988) (Russian) [English translation: J. Soviet Math. \textbf{52}, 27992805 (1990)] Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory Hilbert schemes and stable pairs: GIT and derived category wall crossings | 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 codimension \(r+1\) scheme \(X \subset {\mathbb P}^n\) is called a standard determinantal scheme if \(I_X\) is the ideal generated by the maximal minors of a homogeneous \(t\times (t+r)\)-matrix \(A\). The determinantal scheme \(X\) will be called good if after performing some row operations on \(A\), the resulting matrix contains a \((t-1)\times (t+r)\)-submatrix whose ideal of maximal minors defines a scheme of codimension \(r+2\). This paper gives several characterizations of standard and good determinantal schemes.
The main results can be summarized as follows. Let \(X\) be a subscheme of \({\mathbb P}^n\) with codimension \(\geq 2\). Then the following conditions are equivalent:
(a) \(X\) is a good determinantal scheme of codimension \(r+1\),
(b) \(X\) is the zero-locus of a regular section of the dual of a first Buchsbaum-Rim sheaf of rank \(r+1\),
(c) \(X\) is standard determinantal and locally a complete intersection outside a subscheme \(Y \subset X\) of codimension \(r+2\).
Furthermore, for any good determinantal subscheme \(X\) of codimension \(r+1\) there is a good determinantal subscheme \(S\) of codimension \(r\) such that \(X\) sits in \(S\) in a nice way.
As an application, the authors show that for a zero-scheme \(X \subset {\mathbb P}^3\), being good determinantal is equivalent to the existence of an arithmetically Cohen-Macaulay curve \(S\) which is a local completion such that \(X\) is a subcanonical Cartier divisor on \(S\). This generalizes an earlier result of \textit{M. Kreuzer} on 0-dimensional complete intersection [Math. Ann. 292, No. 1, 43-58 (1992; Zbl 0741.14030)]. The paper closes with a number of examples. standard determinantal scheme; Buchsbaum-Rim sheaf; Cartier divisor; good determinantal schemes; arithmetically Cohen-Macaulay curve; complete intersection Kreuzer, M; Migliore, J; Nagel, U; Peterson, C, Determinantal schemes and Buchsbaum-rim sheaves, J. Pure Appl. Algebra, 150, 155-174, (2000) Determinantal varieties, Complete intersections, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Syzygies, resolutions, complexes and commutative rings, Divisors, linear systems, invertible sheaves, Linkage, complete intersections and determinantal ideals Determinantal schemes and Buchsbaum-Rim sheaves | 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 his previous work [\textit{W. Wang}, Duke Math. J. 103, 1--23 (2000; Zbl 0947.19004)], the author has indicated the deep analogy and connections between (A) the theory of the Hilbert scheme \(X^{[n]}\) of \(n\) points on a (quasi-) projective surface \(X\) [cf. e.g. \textit{H. Nakajima}, Lectures on Hilbert schemes of points on surfaces (University Lecture Series 18, Providence, RI: AMS)(1999; Zbl 0949.14001)], and (B) the theory of wreath products \(\Gamma_{[n]} =\Gamma^n\rtimes S_n\) between a power \(\Gamma^n\) of a finite group \(\Gamma\) and the symmetric group \(S_n\) [cf. e.g. \textit{A. Zelevinsky}, Representations of finite classical groups. A Hopf algebra approach (Lecture Notes in Mathematics 869, Berlin: Springer) (1981; Zbl 0465.20009)]. The reviewed article is a survey that is related to many questions and works of this topic: first of all, the construction of the Heisenberg and Virasoro algebras in the framework of (A), with description of the cohomology ring of \(X^{[x]}\) and in the framework of (B), with description of vertex representations of affine and toroidal Lie algebras, whose Dynkin diagrams correspond to finite subgroups \(\Gamma\) of \(\text{SL}_2(\mathbb{C})\), a.o.
If \(Y\) is a quasi-projective surface acted by a finite group \(\Gamma\), and a resolution of singularities \(X\to Y/\Gamma\) (1) is given, then, according to the author [loc.cit.], one obtains the resolution of singularities \(X^{[n]} \to Y^n/\Gamma_{[n]}\) (2) that can preserve many properties of the resolution (1), in particular, some of them that are concerned with the Euler and Hodge numbers of the corresponding orbifolds.
Under some additional conditions, equivalence between the bounded derived categories \(D_{\Gamma_{[n]}}(Y^n)\) and \(D(X^{[n]})\) of \(\Gamma_{[n]}\)-equivariant coherent sheaves on \(Y^n\) and coherent sheaves on \(X^{[n]}\) respectively is established. The case \(Y= \mathbb{C}^2\) is considered in more detail, in particular, the question of replacement of \(X^{[n]}\) (in case of a certain known minimal \(X\) in (1)) by a special subvariety \(Y_{\Gamma,n}\) of \((\mathbb{C}^2)^{[nN]}\) (where \(N\) is the order of \(\Gamma\)) that admits a quiver description.
Finally, a short dictionary for comparison of analogous notions in the theories (A) and (B) is presented. algebraic surfaces; action of finite groups; resolution of singularities; Heisenberg algebra; Virasoro algebra; representation of Lie algebras; vertex algebras; equivariant \(K\)-theory of schemes W. Wang, Algebraic structures behind Hilbert schemes and wreath products, in: S. Berman et al. (Eds.), Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, Charlottesville, VA, 2000; Contemp. Math. 297 (2002) 271-295. Parametrization (Chow and Hilbert schemes), Virasoro and related algebras, Vertex operators; vertex operator algebras and related structures, Extensions, wreath products, and other compositions of groups Algebraic structures behind Hilbert schemes and wreath products. | 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 While connectedness of the full Hilbert scheme was proved long ago in Hartshorne's thesis [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)], connectedness of the Hilbert scheme \(H_{d,g}\) of locally Cohen-Macaulay curves of degree \(d\) and arithmetic genus \(g\) in \(\mathbb P^3\) remains an open problem. The extremal curves (those whose higher ideal sheaf cohomology has maximum dimensions) form an irreducible component \(E \subset H_{d,g}\) [\textit{M. Martin-Deschamps} and \textit{D. Perrin}, Ann. Sci. Éc. Norm. Supér. (4) 29, No. 6, 757--785 (1996; Zbl 0892.14005)], so a natural strategy consists of trying to flatly deform given curves to an extremal curve. This strategy works for \(g \geq {{d-1}\choose{2}}-1\) [\textit{S. Nollet}, Commun. Algebra 28, No. 12, 5745--5747 (2000; Zbl 0991.14002) and \textit{I. Sabadini}, Matematiche 55, No. 2, 517--531 (2000, Zbl 1165.14311)] and \(d=3\) [\textit{S. Nollet}, Ann. Sci. Éc. Norm. Supér. (4) 30, No. 3, 367--384 (1997; Zbl 0892.14004)], but fails for \(d=4\), though \(H_{4,g}\) turns out to be connected by other means [\textit{S. Nollet} and \textit{E. Schlesinger}, Compos. Math. 139, No. 2, 169--196 (2003; Zbl 1053.14035)]. Moreover, Koszul curves deform to extremal curves [\textit{D. Perrin}, Collect. Math. 52, No. 3, 295-319 (2001; Zbl 1074.14500)], as do smooth rational curves, ACM curves, smooth curves with \(d \geq g+3\) [\textit{R. Hartshorne}, Commun. Algebra 28, No. 12, 6059--6077 (2000; Zbl 0994.14002)]. In this last reference Hartshorne set out a challenge to prove or disprove it for families of \(d \geq 4\) skew lines on a smooth quadric surface.
The paper under review rises to this challenge, exhibiting such a deformation. Their technique is interesting, consisting of two observations. Let \(C\) be a curve consisting of \(d\) disjoint general rulings on a smooth quadric surface \(Q\). Deforming to the initial ideal of \(C\) with respect to the standard weight vector results in a curve with embedded points, but the authors show that the initial ideal of \(C\) with respect to the non-standard weight vector \(\omega = (d,2,1,1)\) takes the form
\[
(x^2,xy,y^{d-1},xG-y^{d-1}F)
\]
where \(x,y\) are linear and \(F,G\) are forms of respective degrees \({{d-2}\choose{2}}-g, {{d-1}\choose{2}}-g\). If the lines of \(C\) are sufficiently general, this defines a curve which remarkably has no embedded points; in fact it is an extremal curve of the same degree and genus as \(C\), thereby connecting this family to the extremal component. Hilbert scheme; locally Cohen-Macaulay curve; initial ideal; weight vector; Gröbner basis Lella, P., Schlesinger, E.: The Hilbert schemes of locally Cohen-Macaulay curves in \(\mathbb{P}^3\) may after all be connected. Collect. Math. \textbf{64}(3), 363-372 (2013) Parametrization (Chow and Hilbert schemes), Plane and space curves, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Formal methods and deformations in algebraic geometry, Families, moduli of curves (algebraic), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The Hilbert schemes of locally Cohen-Macaulay curves in \(\mathbb{P}^3\) may after all be connected | 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. A \(k\)--algebra \(R\) is called strongly Noetherian if \(R\otimes_k A\) is Noetherian for any commutative Noetherian \(k\)--algebra \(A\). If \(X\) is a projective scheme, \(\sigma\in \text{Aut}_k(X)\) and \(\mathcal{L}\) is an invertible sheaf on \(X\), the twisted homogeneous coordinate ring is \(\displaystyle B(X, \mathcal{L}, \sigma)=\bigoplus_{n\geq 0} H^0(X,\mathcal{L}\otimes \mathcal{L}^\sigma\otimes \cdots\otimes \mathcal{L}^ {\sigma^{n-1}}),\,\, \mathcal{L}^\sigma=\sigma\mathcal{L}\). If \(\mathcal{L}\) is \(\sigma\)--ample then \(B(X, \mathcal{L}, \sigma)\) is strongly Noetherian.
\textit{D. Rogalski} and \textit{J. J. Zhang} [Math. Z. 259, No. 2, 433--455 (2008; Zbl 1170.16021)] proved the following result. Let \(R=k\oplus R_1 \oplus R_2\oplus\cdots\) be a strongly Noetherian graded algebra generated in degree \(1\). Then there is a \(k\)--algebra homomorphism \(g: R\to B(X, \mathcal{L}, \sigma)\) for some projective scheme \(X\), \(\sigma\in \text{Aut}_k(X)\) and \(\mathcal {L}\) a \(\sigma\)--invertible sheaf on \(X\). This result is extended to a large class of Noetherian algebras. point space; noncommutative Hilbert scheme; naive blowup algebra Nevins, T.A., Sierra, S.J.: Naive blowups and canonical birationally commutative factors. arXiv:1206.0760 (2012) Noncommutative algebraic geometry, Rational and birational maps, Parametrization (Chow and Hilbert schemes), Noetherian rings and modules (associative rings and algebras), Rings arising from noncommutative algebraic geometry, Graded rings and modules (associative rings and algebras), Fine and coarse moduli spaces, Stacks and moduli problems Naïve blowups and canonical birationally commutative factors | 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 authors compute two point extremal Gromov-Witten invariants of Hilbert schemes of points on surfaces, and determine the extremal quantum boundary operator. The main ideas are to use the localized virtual fundamental cycle and to apply the results of Okounkov-Pandharipande on the quantum cohomology of Hilbert schemes of points on the affine plane \(\mathbb C^2\).
In Section 2, the authors review standard results about the Hilbert schemes \(X^{[n]}\) of points on smooth algebraic surfaces \(X\), and mentioned Ruan's Cohomological Crepant Resolution Conjecture. In Section 3, based on methods of Kiem-J. Li, some reduction lemmas about the localized virtual fundamental cycle of stable maps to \(X^{[n]}\) were proved. These stable maps are extremal in the sense that their images are contracted to points by the Hilbert-Chow morphisms. In Section 4, a universal formula about two point extremal Gromov-Witten invariants of \(X^{[n]}\) is established. In Section 5 and Section 6, the authors studies the extremal quantum boundary operator, and determines it when \(X\) is the projective plane \(\mathbb P^2\). In Section 7, two point extremal Gromov-Witten invariants and the extremal quantum boundary operator for an arbitrary surface \(X\) are computed. These results are then applied to partially confirm Ruan's conjecture. Hilbert schemes; Gromov-Witten invariants; localization technique Li, J; Li, W-P, Two point extremal Gromov-Witten invariants of Hilbert schemes of points on surfaces, Math. Ann., 349, 839-869, (2011) Parametrization (Chow and Hilbert schemes), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vertex operators; vertex operator algebras and related structures Two point extremal Gromov-Witten invariants of Hilbert schemes of points on 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 The authors consider the following problem: characterize all pairs \((C,p)\) which have constant moduli, where \(C\) is a plane curve and \(p \in {\mathbb P}^2_K\). This means that \((C,p)\) are such that if \(m= \text{mult}_p C\) and \(\deg C = d+m\), \(d\geq 3\), then for every two lines \(L_1,L_2\) containing \(p\), with \((C-\{p\})\cap L_i= \{p_{i,1},\dots,p_{i,d}\}\), \(i=1,2\), we can number the \(p_{i,j}\)'s in such a way that all the lines \(p_{1,j}p_{2,j}\) are concurrent. The main result is that if char\(K =0\), then \((C,p)\) has constant moduli if and only if there exists a birational map \(\varphi\) on \({\mathbb P}^2_K\) (which is an automorphism if \(m=0\)) and a form \(H(Y,Z)\) such that \(\varphi(p)=(1:0:0)\), almost every line \(l\) through \(p\) is such that \(\varphi (l)\) is a line through \((1:0:0)\) and \(\overline{\varphi (C)}\) has equation:
\[
\Pi_{t=1}^{d/k}(X^kZ^{n-k}-\alpha_tH(Y,Z) \quad \quad \text{or } \quad X\Pi_{t=1}^{(d/1)/k}(X^kZ^{n-k}-\alpha_tH(Y,Z),
\]
where \(k\in {\mathbb N}\) divides either \(d\) or \(d-1\) and \(\alpha_t\in \overline{K}\), \(\forall t\). This results allow to give a complete geometric description of \((C,p)\) when \(\deg C =3,4\) and to describe some subgroups of Aut(\(C\)). Notice that to every pair \((C,p)\) we can associate a surface fibred in hyperelliptic curves which are the double cover of lines \(L\) through \(p\), ramified on \(L\cap C\) or on \(L\cap (C-\{p\})\). The pair \((C,p)\) has constant moduli if and only if the family of hyperelliptic curves is locally trivial. plane curves; hyperelliptic curves Families, moduli of curves (algebraic), Plane and space curves Locally trivial families of hyperelliptic curves: the geometry of the Weierstrass scheme | 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 \textit{Rees algebra} over an algebraic variety \(W\) (over a field \(k\), of any characteristic) is defined by a sequence of coherent sheaves of ideals \(I_j \subset {\mathcal O}_{W}\) such that there is an affine cover \( \{ U_i \} \) of \(W\) such that, for all \(i\), \(\bigoplus I_n(U_i) \, W^i\) ( a subring of the polynomial ring \({\mathcal O}_{Z}(U_i) [W]\)) is a finitely generated \({\mathcal O}_Z(U_i)\)-algebra. We shall assume \(I_{j+1}\subset I_j\) for all \(j\), which is harmless if we work modulo integral closure. If \(W\) is smooth over \(k\) and the Rees algebra \(\mathcal G\) as above also satisfies the condition: ``for all \(i\), if \(h \in {\mathcal O}_{Z}(U_i)\) and \(D\) is any differential operator over \(k\), defined on \(U_i\), of order \(\leq r \leq n\) then \(D(h) \in I_{n-r}\)'', we say that \(\mathcal G\) is a \textit{differential algebra}, relative to \(k\). (The reference to \(k\) will be omitted in the sequel, if it is clear). It is hoped that differential algebras will play a relevant role in the study of resolution of singularities in arbitrary characteristic, namely that they will provide a useful substitute in certain constructions carried out in characteristic zero with the aid of derivatives.
The present paper discusses important foundational results on the theory of Rees and differential algebras. Some of the topics that are covered include: (1) the basic, purely algebraic theory, (2) the basic concepts in the context of Algebraic Geometry, (3) the notion of the differential algebra \(G(\mathcal G)\) generated by a Rees algebra \(\mathcal G\) over a smooth \(k\)-variety, (4) the restriction of a differential algebra \(\mathcal G\) to a smooth subvariety, (5), the singular set of a Rees (or differential) algebra, and related results, (6) integral closures of these algebras. Another concept (probably very important in inductive steps in resolution of singularities) is that of coefficients. If \(Z\) is a smooth subvariety of the \(k\)-smooth variety \(V\), \(\mathcal G\) is a Rees algebra over \(V\), \(x\) is a closed point of \(Z\), in the presence of local retraction \(\pi:V \to Z\) (defined near \(x\)) one may define a Rees algebra over \(Z\), the algebra of coefficients Coeff(\(\mathcal G\)). At the level of completions of local rings this algebra can be described in terms of coefficients in certain power series expansions. The assumption on the existence of the local retraction \(\pi\) is not restrictive if one works with the etale topology. A number of properties of this concept are shown, specially the fact that, if \(\mathcal G\) is a differential algebra, then \(G({\text{Coeff}}(\mathcal G))\) is independent of the local retraction used.
The author has used these results in other works, e.g., in \textit{O. Villamayor} [Adv. in Math. 213, 687--733 (2007; Zbl 1118.14016)]. Rees algebra; differential algebra; integral closure; differential operator; coefficients Villamayor U. O.E.: Rees algebras on smooth schemes: integral closure and higher differential operators. Rev. Mat. Iberoamericana 24(1), 213--242 (2008) Global theory and resolution of singularities (algebro-geometric aspects), Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Modifications; resolution of singularities (complex-analytic aspects), Commutative rings of differential operators and their modules Rees algebras on smooth schemes: Integral closure and higher differential operator | 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 Fix an integral plane curve \(C\). Here we study the postulation and the
homogeneous ideal of general unions of u double and v simple points whose support
is contained in \(C\) (at least when \(\deg(C) \geq (3u + v)/2)\). Vector bundles on surfaces and higher-dimensional varieties, and their moduli On the homogeneous ideal of general unions of simple and double points with support on a fixed plane 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 \textit{G. Ellingsrud} and \textit{S. A. Strømme} [Invent. Math. 91, No.2, 365-370 (1988; Zbl 0654.14003)] have found a cell decomposition for the space \(Hilb^ d(P^ 2)\), the Hilbert scheme of length \(d\) subschemes of \({\mathbb{P}}^ 2\). This implies that the Chow group \(A_*(Hilb^ d(P^ 2))\) is a free abelian group. However, the general element in a cell of this decomposition will correspond to a nonreduced scheme in \({\mathbb{P}}^ 2.\)
The authors give a new basis for the Chow group which avoids the difficulty above. In the case \(d=3\), this is essentially the basis given by \textit{G. Elencwajg} and \textit{P. Le Barz} in an earlier paper [see e.g. C. R. Acad. Sci., Paris, Sér. I 301, 635-638 (1985; Zbl 0597.14005)]. Hilbert scheme; basis for the Chow group R. Mallavibarrena et I. Sols , A base of the homology groups of the Hilbert scheme of points in the plane Comp. Math, à paraître. Parametrization (Chow and Hilbert schemes), Topological properties in algebraic geometry Bases for the homology groups of the Hilbert scheme of points 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 The Hilbert scheme \(S^{[n]}\) of zero-dimensional subschemes of length \(n\) on a smooth projective surface \(S\) is a smooth projective variety of dimension \(2n\). The direct sum over all \(n \geq 0\) of the rational cohomology space for each \(S^{[n]}\) is naturally a Fock space \(H\) over a Heisenberg Lie algebra modelled on the rational cohomology space of the surface \(S\) [see e.g.\ \textit{H.~Nakajima}, Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)].
The main result of this article is a description of the total Chern class and the Chern character of the tangent bundles of the Hilbert schemes \(S^{[n]}\) in terms of a closed formula in the creation operators of the Fock space \(H\) applied on the vacuum vector in \(H\) provided that \(S\) is the affine plane \({\mathbb C}^2\). The result is proven by firstly deriving an general formula for the total Chern class and the Chern character with unknown coefficients that holds true for any surface \(S\). By means of equivariant cohomology, the coefficients are then calculated for \(S = {\mathbb C}^2\). Hilbert scheme. Fock space; universal formulas; Chern classes; tangent bundle Boissière, J. Algebraic Geom. 14 pp 761-- (2005) Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Characteristic classes and numbers in differential topology Chern classes of the tangent bundle on the Hilbert scheme of points on the affine 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 Denote by \(H^S_{d,g}\) the Hilbert scheme of smooth reduced and irreducible algebraic space curves of degree \(d\) and genus \(g\). \textit{L. Gruson} and \textit{C. Peskine} [Ann. Sci. Éc. Norm. Supér. (4) 15, 401--418 (1982; Zbl 0517.14007)] have determined all pairs \((d,g)\) for which \(H^S_{d,g}\) is not empty, and among other things they also presented several examples in which all components of \(H^S_{d,g}\) were described, and in particular those of \(H^S_{16,30}\). In the article under review, the author conducts a detailed study of \(H^S_{d,3d-18}\) for \(6 \leq d \leq 16\), as he identifies all of its irreducible components and the general element of each of them in terms of generators of the Picard group of the normalization of a cubic surface, liaison classes, or resolution of ideal sheaf of the curve. The author gives a complete proof in the case \(d = 16\), as it is the most difficult one. In particular, he proves that: \(H^S_{16,30}\) has \(5\) distinct irreducible components \(H^S_{16,30} = V^{(64)}_{1} \cup V^{(64)}_{2} \cup V^{(64)}_{3} \cup V^{(68)}_{4} \cup V^{(69)}_{5}\), where the first three of them are of dimension \(64\) and the remaining two are of dimension \(68\) and \(69\), correspondingly; \(H^S_{16,30}\) is generically smooth along \(V^{(64)}_{1}\) and \(V^{(64)}_{2}\), and nonreduced along the other three; a general element \(X\) of \(V^{(64)}_{1}\) is found on a quintic surface and its ideal sheaf \({\mathcal I}_X\) has the resolution \(0 \to {\mathcal O}(-7) \oplus {\mathcal O}(-6)^3 \to {\mathcal O}(-5)^5 \to {\mathcal I}_X \to 0\); a general element \(X\) of \(V^{(64)}_{2}\) (\(V^{(64)}_{3}\)) lies on a smooth cubic surface and \(X \sim 12 \ell - 5 e_1 - 3e_2 - 3e_3 - 3e_4 - 3e_5 - 3e_6\) (\(X \sim 12 \ell - 4e_1 - 4e_2 - 4e_3 - 3e_4 - 3e_5 - 2e_6\)), where \(\ell\) and \(e_1, \dots, e_6\) are the standard generators of the Picard group of a smooth cubic surface in \({\mathbb{P}}^3\); a general element \(X\) of \(V^{(68)}_{4}\) (\(V^{(69)}_{5}\)) lies on a non-normal cubic surface and not a cone, say \(S\), whose normalization is \({\mathbb{P}} ({\mathcal O}_{\mathbb{P}^1} \oplus {\mathcal O}_{\mathbb{P}^1} (1))\), and its strict transform is in the class of \(6h - 2f\) (\(5h + f\)), where \(h\) is the pull back of \({\mathcal O}_S (1)\) and \(f\) is the fiber.
For the proof the author studies consequently the cases when a smooth curve \(X\) of genus \(30\) and degree \(16\) is contained in a quintic, quartic, or cubic surface. In the first case the author classifies the curves according to normal generation and finds their linkage classes. The normally generated ones have ideal sheaves with the above mentioned resolution and their closure is the component \(V^{(64)}_{1}\), while the closure of the set of non-normally generated curves has dimension \(62\), which refines the result of Gruson and Peskine. Studying further the cases when \(X\) is contained in a quartic, or a cubic surface and calculating \(h^0 (N_{X,{\mathbb{P}}^3})\) the author concludes the remaining statements in the theorem.
In addition, the author also proves that for \(d \geq 91\), \(H^S_{d,3d-18}\) has a unique component whose general element is contained in a smooth cubic surface, and identifies its equivalence in terms of generators of the Picard group of the surface. The proof is similar to the construction used by Gruson and Peskine. Hilbert scheme; space curves Parametrization (Chow and Hilbert schemes), Special algebraic curves and curves of low genus, Plane and space curves The Hilbert scheme of space curves of degree \(d\) and genus \(3d-18\) | 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 Borel-fixed ideals play a key role in the study of Hilbert schemes. Indeed each component and each intersection of components of a Hilbert scheme contains at least one Borel-fixed point, i.e. a point corresponding to a subscheme defined by a Borel-fixed ideal. Moreover Borel-fixed ideals have good combinatorial properties, which make them very interesting in an algorithmic perspective. In this paper, we propose an implementation of the algorithm computing all the saturated Borel-fixed ideals with number of variables and Hilbert polynomial assigned, introduced from a theoretical point of view in [\textit{F. Cioffi} et al., Discrete Math. 311, No. 20, 2238--2252 (2011; Zbl 1243.14007)]. Hilbert polynomial; Hilbert scheme; Borel-fixed ideals Lella, P., An efficient implementation of the algorithm computing the Borel-fixed points of a Hilbert scheme, (Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, (2012), ACM), 242-248 Symbolic computation and algebraic computation, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Parametrization (Chow and Hilbert schemes) An efficient implementation of the algorithm computing the Borel-fixed points of a Hilbert scheme | 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 give an existence theorem for integral curves \(C\) contained in a smooth projective surface \(S\) and with as only singularities prescribed ordinary multiple points at general points of \(S\). The proof heavily use the proof of the case \(S = \mathbb{P}^2\) given by \textit{T. Mignon} [J. Algebr. Geom. 10, No. 2, 281--297 (2001; Zbl 0987.14019)]. Plane and space curves, Families, moduli of curves (algebraic), Surfaces and higher-dimensional varieties Curves in a projective surface with prescribed ordinary 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 This paper follows from results by \textit{O. Riemenschneider} [Arch. Math. 37, 406-417 (1981; Zbl 0456.14006)] and is a generalization of the author's previous paper [\textit{T. De Jong}, Compos. Math. 113, No. 1, 67-90 (1998; Zbl 0923.14024)]. Having the dual resolution graph \(\Gamma\) of a quasideterminantal rational surface singularity, the author writes down equations, in quasideterminantal form, which define a rational surface singularity with this prescribed graph \(\Gamma\). He proves that all rational surface singularities with dual resolution graph \(\Gamma\) can be given by the equations constructed in the paper involving the quasiminors of the special quasimatrix. quasideterminantal rational surface singularity; dual resolution graph Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Determinantal varieties, Complex surface and hypersurface singularities Quasi-determinantal rational surface singularities | 0 |
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