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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0747.00028.]
The purpose of this paper is to determine explicitly the local structure of the Hilbert scheme of curves in \(\mathbb{P}^ 3\) at certain points. Actually, the main body of this work is devoted to a quite detailed exposition of the various deformation theories of a closed subscheme \(Y\) of a projective scheme \(X = \text{Proj}(S)\), where \(K\) is a field and \(S\) is a graded Noetherian \(K\)-algebra, with \(S_ 0 = K\). The types of deformations are: the deformations of \(Y\) as a subscheme of \(X\), the deformations of the ideal sheaf \({\mathcal I}_ Y\) as a sheaf of \({\mathcal O}_ X\)-modules, the deformations of the ideal \(I(Y)\) as a homogeneous \(S\)-module, and ``conical deformations'' of \(Y\). This exposition builds on previous work by \textit{O. A. Laudal} [in Algebra, algebraic topology and their interactions, Proc. Conf., Stockholm 1983, Lect. Notes Math. 1183, 218-240 (1986; Zbl 0597.14010)] and \textit{J. O. Kleppe} [Math. Scand. 45, 205-231 (1979; Zbl 0436.14004)], but the material is presented here with a view towards explicit calculations. Various conditions are worked out under which these deformation theories are isomorphic. The theory is then applied to the calculation of the deformation theory of particular space curves, yielding explicit local equations for some singular points of the Hilbert scheme and explicit examples of obstructed curves (i.e. corresponding to singular points of the Hilbert scheme) of maximal rank. The same examples were found independently by \textit{Bolondi}, \textit{Kleppe} and \textit{Miró-Roig}.
The question remains whether curves with seminatural cohomology (i.e. curves \(C\) such that for each \(n\), the sheaf \({\mathcal J}_ C(n)\) has at most one nonzero cohomology group) are unobstructed. isomorphic deformation theories; deformation theory of space curves; Hilbert scheme of curves; deformation theories of a closed subscheme Charles H. Walter, Some examples of obstructed curves in \?³, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 324 -- 340. Plane and space curves, Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Deformations and infinitesimal methods in commutative ring theory Some examples of obstructed curves 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 In this survey we describe an interplay between Procesi bundles on symplectic resolutions of quotient singularities and Symplectic reflection algebras. Procesi bundles were constructed by \textit{M. Haiman} [J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)] and, in a greater generality, by \textit{R. V. Bezrukavnikov} and \textit{D. B. Kaledin} [Proc. Steklov Inst. Math. 246, 13-33 (2004; Zbl 1137.14301); translation from Tr. Mat. Inst. Steklova 246, 20-42 (2004)] Symplectic reflection algebras are deformations of skew-group algebras defined in complete generality by \textit{P. Etingof} and \textit{V. Ginzburg} [Invent. Math. 147, No. 2, 243--348 (2002; Zbl 1061.16032)]. We construct and classify Procesi bundles, prove an isomorphism between spherical symplectic reflection algebras, give a proof of wreath Macdonald positivity and of localization theorems for cyclotomic rational Cherednik algebras. McKay correspondence, Momentum maps; symplectic reduction, Deformation quantization, star products, Symmetric functions and generalizations, Representations of quivers and partially ordered sets, Representation theory of associative rings and algebras, Ordinary and skew polynomial rings and semigroup rings, Poisson algebras, Reflection and Coxeter groups (group-theoretic aspects) Procesi bundles and symplectic reflection 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 The paper deals with the following problem: Let \(X \subset \mathbb{P}^N_{\mathbb{C}}\) be an algebraic surface and let \(S_p \subset \mathbb{P}^2_{\mathbb{C}}\) be the branch locus of a generic projection \(p:X\to{\mathbb{P}^2_{\mathbb{C}}}\) and \(S \subset \mathbb{C}^2\) a generic affine part of \(S_p\). The question is to describe the fundamental groups \(\pi_1({\mathbb{C}}^2 \setminus S)\) and \(\pi_1({\mathbb{P}}_{\mathbb{C}}^2 \setminus S_p)\). In the 1930's, Zariski calculated \(\pi_1({\mathbb{P}}_{\mathbb{C}}^2 \setminus S_p)\) when \(X\) is a cubic hypersurface of \(\mathbb{P}^3_{\mathbb{C}}\). \textit{B. G. Moishezon} [in: Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math. 862, 107-192 (1981; Zbl 0476.14005)] generalized Zariski's result proving that, if \(X\) is a hypersurface of \(\mathbb{P}^3_{\mathbb{C}}\) of degree \(n \geq 3\), then \(\pi_1({\mathbb{C}}^2 \setminus S)\) is isomorphic to the braid group \(B_n\) of \(n\) strings and \(\pi_1({\mathbb{P}}_{\mathbb{C}}^2 \setminus S_p) \cong B_n / \text{Center}\).
In the paper under review, the author studies the fundamental groups \(\pi_1({\mathbb{C}}^2 \setminus S)\) and \(\pi_1({\mathbb{P}}_{\mathbb{C}}^2 \setminus S_p)\) when \(X\) is a complete intersection. More precisely, for \(n \in \mathbb{N}\), he introduces the group \(\widetilde {B_n}\), which is a quotient of \(B_n\) whose center is isomorphic to \(\mathbb{Z} \oplus \mathbb{Z}_2\). Let \(\eta\) and \(\mu\) be generators of infinite order and order 2, respectively. Then, the main result in the paper is the following:
If \(X\) is a smooth non-degenerate complete intersection of codimension greater than 1, then \(\pi_1({\mathbb{C}}^2 \setminus S) \cong {\widetilde {B_n}}\) and \(\pi_1({\mathbb{P}}_{\mathbb{C}}^2 \setminus S_p) \cong {\widetilde {B_n}} / \langle\mu^e \nu^m\rangle\) for some \(e \in \{0,1\}\) where \(m=\frac 1n \text{deg} S\) (resp. \(m=\frac 1{2n} \text{deg} S\)) if \(n\) is even (resp. odd). fundamental group; complete intersection; branch locus; braid group Robb, A.: On branch curves of algebraic surfaces. Studies in Advanced Mathematics, 5, 193--221 (1997) Homotopy theory and fundamental groups in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry On branch curves 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 This paper gives a characterization of a class of algebras called concealed-canonical algebras.
The original definition is the following: An algebra \(\Sigma\) is called concealed-canonical if it is obtained as the endomorphism ring of a tilted module, \(T\), over a canonical algebra, where all the indecomposable summands of \(T\) have strictly positive rank. This rank is defined via the Euler quadratic form.
There is a characterization of these algebras via the shape of their Auslander-Reiten quivers. One of the theorems says that they can be characterized via the fact that their A-R quivers have a sincere separating tubular family of stable tubes. Another equivalence is the existence of a small hereditary Noetherian \(k\)-category with an Artinian center \(k\) and without projectives and a torsion free tilting object \(T\), with endomorphism ring \(\Sigma\).
One of the main tools is the fact that the paper shows that in this case the indecomposables can be divided into three big classes, \(\text{mod}_+(\Sigma)\), \(\text{mod}_0(\Sigma)\) and \(\text{mod}_-(\Sigma)\), the modules in the first one have strictly positive rank, the ones in the middle one have zero rank, and the ones in the last one have negative rank. Also there are no nonzero maps between modules from the right to the left. Moreover, each one of them is a union of connected components of the Auslander-Reiten quiver. Also \(\text{mod}_0(\Sigma)\) is uniserial and decomposes into a coproduct of uniserial categories.
The paper uses various techniques from algebraic geometry. concealed-canonical algebras; separating exact subcategories; components of Auslander-Reiten quivers; quadratic forms; Artin algebras; tame hereditary algebras Lenzing, H.; de la Peña, J. A., Concealed-canonical algebras and separating tubular families, Proc. Lond. Math. Soc., 78, 513-540, (1999) Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Vector bundles on curves and their moduli, Representations of quivers and partially ordered sets Concealed-canonical algebras and separating tubular families. | 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 interesting paper, the authors compare the behavior of the test ideal to the behavior of the multiplier ideal. The test ideal \(\tau(f^t)\), which comes out of tight closure theory, is closely related to the multiplier ideal \(\mathcal{J}(f^t)\), which is defined by resolution of singularities in characteristic zero. See the paper by \textit{N. Hara} and \textit{K.-i. Yoshida} [Trans. Am. Math. Soc. 355, No. 8, 3143--3174 (2003; Zbl 1028.13003)] for background and definitions.
It turns out that many interesting and difficult-to-prove theorems about multiplier ideals have simple proofs for test ideals. However, as this paper shows, many basic facts about multiplier ideals fail spectacularly for test ideals. In particular, because multiplier ideals are defined by resolution of singularities, they are always integrally closed (furthermore, the local syzygies satisfy certain conditions, see the paper of \textit{R. Lazarsfeld} and \textit{K. Lee} [Invent. Math. 167, No. 2, 409--418 (2007; Zbl 1114.13013)]). However, the main result of this paper, stated below, shows that every ideal (including non-integrally closed ideals) is a test ideal.
Theorem 1.1. Suppose that \(R\) is an \(F\)-finite regular local ring of characteristic \(p > 0\). For every ideal \(I\) in \(R\), there is \(f \in R\) and \(c > 0\) such that \(I = \tau(f^c)\).
The authors also show several other common properties of multiplier ideals completely fail for test ideals (for example, \(\tau(f^c)\) need not be a radical ideal when \(c\) is the \(F\)-pure threshold). See the paper for details.
As the authors point out, most of the pathological behavior in characteristic \(p > 0\) occurs for test ideals \(\tau(f^c)\) when \(c\) is a rational number with \(p\) in the denominator. It may be that when the \(F\)-jumping numbers of \(\tau(f^c)\) do not have \(p\) in the denominator, the test ideal behaves more like the multiplier ideal. Test ideal; multiplier ideal; F-pure threshold; tight closure; log canonical threshold Mircea Mustaţă and Ken-Ichi Yoshida, Test ideals vs. multiplier ideals, Nagoya Math. J. 193 (2009), 111 -- 128. Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry Test ideals vs. multiplier ideals | 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 establish an explicit bound for the order of the torsion part of the divisor class group of an \(F\)-finite and strongly \(F\)-regular local Noetherian ring. The method is parallel to that of a previous paper by \textit{T. Polstra} [Int. Math. Res. Not. 2022, No. 3, 2086--2094 (2022; Zbl 1495.13013)] in which the finiteness of said torsion part is established. However, in this paper explicit calculation is done to achieve a quantitative result. \(F\)-signature; \(F\)-singularity; divisor; commutative algebra; algebraic geometry Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry The number of torsion divisors in a strongly \(F\)-regular ring is bounded by the reciprocal of \(F\)-signature | 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 prove that the `cohomology' of a Nakajima quiver varieties \(\mathfrak{M}\) is generated by the classes of tautological bundles naturally arising from their moduli of quiver representations presentation.
Here `cohomology' extends beyond just singualar cohomology over \(\mathbb{Z}\) and even includes the derived category of quasicoherent sheaves.
Classical results do not extend to this case since the ambient GIT space is singular.
To prove the result, the authors give an explicit modular compactification \(\overline{\mathfrak{M}}\) of \(\mathfrak{M}\).
They go on to describe the class of the graph of the inclusion \(\mathfrak{M} \hookrightarrow \overline{\mathfrak{M}}\) by a complex inspired by a construction due to Nakajima.
The fact that the complex is composed of terms build from external tensor product of tautological bundles allows the authors to deduce the results using purely topological arguments. GIT; cohomology; Nakajima quiver varieties Holomorphic symplectic varieties, hyper-Kähler varieties, Geometric invariant theory, Representations of quivers and partially ordered sets Kirwan surjectivity for quiver 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 This note points out a gap (the statements of Lemma 9.1 and Lemma 9.2 in the original article are incorrect) in the proof of the main theorem of the author's paper [ibid. 192, No. 3, 533--566 (2013; Zbl 1279.14019)], and provides a new proof of the theorem. Rationality questions in algebraic geometry, Multiplier ideals, Fano varieties, Rational and birational maps, Singularities in algebraic geometry, Adjunction problems Erratum to: ``Birationally rigid hypersurfaces'' | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{J.-F. Boutot} [Invent. Math. 88, 65-68 (1987; Zbl 0619.14029)] proved (in characteristic 0) that a pure subring of a ring with rational singularity has a rational singularity. ``The'' characteristic \(p\) analog of a ring with rational singularity is an \(F\)-rational ring. Thus it is natural to expect that the analog of Boutot's theorem holds, namely, that a pure subring of an \(F\)-rational ring is \(F\)-rational. This paper shows that a ring retract of an \(F\)-rational ring need not be \(F\)-rational. \(F\)-rational ring; characteristic \(p\); pure subring Wa3 K.-i.~Watanabe, \(F\)-rationality of certain Rees algebras and counterexamples to ``Boutot's theorem'' for \(F\)-rational rings, J. Pure Appl. Algebra \textbf 122 (1997), no. 3, 323--328. Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Local cohomology and commutative rings, Rational and birational maps \(F\)-rationality of certain Rees algebras and counterexamples to ``Boutot's theorem'' for \(F\)-rational rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0518.00005.]
In this paper the author intends to begin the study of the Chow ring for the moduli space \({\mathcal M}_ g\) of curves of genus g and for its compactification \(\bar {\mathcal M}_ g\), the moduli space of stable curves. The point is that, though \({\mathcal M}_ g\) itself is far more complicated than the Grassmann varieties (the former is not unirational for large g), its Chow ring seems to have a similar simple structure as that of the latter in lower degrees.
In part one, as preliminary, an intersection product is defined in the Chow ring A(\(\bar {\mathcal M}_ g)\) of \(\bar {\mathcal M}_ g\) by using two facts that it has only the quotient singularity and is globally the quotient of a Cohen-Macaulay variety, i.e. the moduli space of curves with level structure. Both the Grothendieck and Baum-Fulton-MacPherson forms of the Riemann-Roch theorem give the necessary tool. - In part II a sequence of what he calls ``tautological'' classes \(\kappa_ i\in A^ i(\bar {\mathcal M}_ g)\) is defined, and auxiliary classes \(\lambda_ i\) as follows: on \(\bar {\mathcal M}_ g\) there are a canonical family of curves \(\pi:\bar {\mathcal C}_ g\to \bar {\mathcal M}_ g\) whose fibre is C/Aut(C) for [C]\(\in \bar {\mathcal M}_ g\), and a sheaf \(\omega_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g}\) of relative 1-forms which is a Q-sheaf (i.e. quotient of an invertible sheaf by a finite action). With the results in part I one can define the intersection on Chow classes in \(A^.(\bar {\mathcal C}_ g)\) or in \(A^.(\bar {\mathcal M}_ g)\), hence we set \(K_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g}=c_ 1(\omega_{{\mathcal C}_{\bar gg}/\bar {\mathcal M}_ g})\in A^ 1(\bar {\mathcal C}_ g)\) and \(\kappa_ i=(\pi_*K^{i+1}_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g})\in A^ i(\bar {\mathcal M}_ g).\) Moreover for \(E=\pi_*(\omega_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g})\), set \(\lambda_ i=c_ i(E)\). - The author then studies relations between them in \(A^.({\mathcal M}_ g)\) and expresses certain subvarieties such as the hyperelliptic locus with them. One of the important results is that all classes \(\kappa_ i\), \(\lambda_ i\) are polynomials in \(\kappa_ 1,...,\kappa_{g-1}\) in \(A^.({\mathcal M}_ g)\). - In this direction \textit{S. Morita} has found a number of new interesting relations between them but in \(H^.({\mathcal M}_ g)\) with different, i.e. topological method [see Proc. Japan Acad., Ser. A 60, 373-376 (1984)]. Its geometric counterpart in \(A^.({\mathcal M}_ g)\) is unknown up to now. - In part III, as an example, the author works out \(A^.(\bar {\mathcal M}_ 2)\) completely. moduli space of curves; Chow ring; moduli space of stable curves; intersection product; Riemann-Roch theorem Mumford, D.: Towards an Enumerative Geometry of the Moduli Space of Curves, Arithmetic and Geometry, Vol. II, Progress in Mathematics, vol.~36, pp.~271-328. Birkhäuser Boston, Boston (1983) Families, moduli of curves (algebraic), Enumerative problems (combinatorial problems) in algebraic geometry, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Riemann-Roch theorems Towards an enumerative geometry of the moduli space of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C(D,r)\) be the Chow variety parametrizing nondegenerate irreducible curves of degree \(d\) in projective \(r\)-space. The authors show that the dimension of \(C(d,r)\) for \(r=3\) is given by a function \(\delta(d,3)\). Furthermore, they show that the component with the maximal dimension corresponding to those curves lies on a quadric with balanced bidegrees. The idea of the proof is to show that if the normal bundle of a curve has more than \(\delta(d,3)\) sections then the genus of the curve is larger than \((d^ 2/4)-2d+6\). Then using the classical Halphen bound on the genus of space curves, one sees that the curve must be contained in a quadric surface.
More generally the authors show that the dimension of \(C(d,r)\) is given by a function \(\delta(d,r)\) for \(d>4r^ 2-4r+3\). Again the components with the maximal dimension correspond to curves contained in a two dimensional rational normal scroll. dimension of Chow variety; Halphen bound Eisenbud, D.; Harris, J., The dimension of the Chow variety of curves, Compos. Math., 83, 3, 291-310, (1992) Parametrization (Chow and Hilbert schemes), Plane and space curves, Families, moduli of curves (algebraic), Projective techniques in algebraic geometry The dimension of the Chow variety of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective variety of dimension \(n\) over an algebraically closed field \(k\). Let \({\mathcal C}_d(X)\) be the \textit{Chow variety} parametrizing \(d\)-dimensional cycles on \(X\) (see, for example, the book of [\textit{J. Kollár}, Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 32. Berlin: Springer-Verlag (1995; Zbl 0877.14012)] for its construction). For \(a+b = n-1\), let \({\mathcal I}\subset {\mathcal C}_a(X)\times {\mathcal C}_b(X)\) be the incidence variety parametrizing the pairs \((A,B)\) with \(A\cap B \neq \emptyset\). When \(k = {\mathbb C}\), B. Mazur constructed, in 1993, using intersection theory operations on the universal cycles, a Weil divisor on \({\mathcal C}_a(X)\times {\mathcal C}_b(X)\) supported on \(\mathcal I\) and posed the problem whether \(\mathcal I\) is the support of a Cartier divisor, satisfying some additional properties. \textit{B. Wang} [Compos. Math. 115, No. 3, 303--327 (1999; Zbl 0982.14017)] showed that the Weil divisor \((n-1)!\, {\mathcal I}\) is Cartier.
In the paper under review, the author proposes a new approach to Mazur's question. Let \({\mathcal H}_d(X)\) be the \textit{Hilbert scheme} parametrizing \(d\)-dimensional subschemes of \(X\). Let \({\mathcal U}_a\), \({\mathcal U}_b\) be the closed subschemes of \(X\times {\mathcal H}_a(X)\times {\mathcal H}_b(X)\) obtained by pulling back the universal families over \({\mathcal H}_a(X)\) and \({\mathcal H}_b(X)\). Using the \textit{determinant functor} constructed by \textit{F. Knudsen} and \textit{D. Mumford} [Math. Scand. 39, No. 1, 19--55 (1976; Zbl 0343.14008)], one gets a line bundle \({\mathcal L} := \text{det}\, \text{R}\, pr_{23\ast}({\mathcal O}_{{\mathcal U}_a}\otimes^{\text{L}} {\mathcal O}_{{\mathcal U}_b})\) on \({\mathcal H}_a(X)\times {\mathcal H}_b(X)\). Let \(U\subset {\mathcal H}_a(X)\times {\mathcal H}_b(X)\) be the open subset over which the fibers of \({\mathcal U}_a\) and \({\mathcal U}_b\) are disjoint. One expects that, for \(a+b = n-1\), \(U\) is dense. Since \(\text{R}\, pr_{23\ast}({\mathcal O}_{{\mathcal U}_a}\otimes^{\text{L}} {\mathcal O}_{{\mathcal U}_b})\) is acyclic on \(U\), the ``Div'' construction of Knudsen and Mumford shows that \(\mathcal L\) is the invertible sheaf associated to a canonically defined Cartier divisor on \({\mathcal H}_a(X)\times {\mathcal H}_b(X)\).
One faces, now, the problem of showing that \(\mathcal L\) descends to a line bundle on \({\mathcal C}_a(X)\times {\mathcal C}_b(X)\) (via the product of the Hilbert-Chow morphisms). In order to solve this problem, the author studies the morphism of Picard groups induced by a seminormal proper hypercovering of a seminormal scheme, using some results of \textit{L. Barbieri-Viale} and \textit{V. Srinivas} [``Albanese and Picard 1-motives'', Mém. Soc. Math. Fr., Nouv. Sér. 87 (2001; Zbl 1085.14011)].
Then, by analysing the Hilbert-Chow morphism for 0-cycles, using the results of [\textit{B. Iversen}, Linear determinants with applications to the Picard scheme of a family of algebraic curves.Lecture Notes in Mathematics. 174. Berlin-Heidelberg-New York: Springer-Verlag. (1970; Zbl 0205.50802)], and for codimension 1 cycles, using the results of Knudsen and Mumford, the author shows that, in the case \(a=0\), \(b = n-1\), \(\mathcal L\) descends from \({\mathcal H}_0(X)\times {\mathcal H}_{n-1}(X)\) to \({\mathcal C}_0(X)\times {\mathcal C}_{n-1}(X)\). Chow variety; incidence divisor; Hilbert-Chow morphism; determinant functor; seminormal variety; simplicial Picard functor Parametrization (Chow and Hilbert schemes), Picard groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The Hilbert-Chow morphism and the incidence divisor: zero-cycles and 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 In what follows k denotes the field of real or complex numbers. If \(D,0\subset k^ p\), 0 is the discriminant variety of the versal unfolding of an analytic function germ \(f: k^ n,0\to k,0\), it is of some interest to classify smooth functions h: D,0\(\to k,0.\)
In Commun. Pure Appl. Math. 29, 557-582 (1976; Zbl 0343.58003), \textit{V. I. Arnol'd} classified such germs in the case when they are generic and f is a simple singularity of type \(A_ k\), \(D_ k\), \(E_ k\). Full details were, however, only given in the case of \(A_ k\). In this paper, using the basic vector fields of \textit{K. Saito} [Invent. Math. 14, 123- 142 (1971; Zbl 0224.32011)] tangent to the discriminant D, we give another proof of Arnol'd's results; the method of proof here is fairly unsophisticated. We then generalize these results to cover a wide collection of weighted homogeneous functions, which include the 14 weighted homogeneous unimodal germs, as well as the simple singularities. Furthermore we use these basic fields to prove that the discriminants of the simple elliptic singularities \(\tilde E_ k\), for \(k=6,7,8\), are topologically trivial along the modulus parameter. We also discuss the existence of stable germs on these discriminants. In the appendix we give a self-contained proof that Saito's vector fields are tangent to the discriminant. discriminant variety; versal unfolding; analytic function germ; stable germs; Saito's vector fields DOI: 10.1112/jlms/s2-30.3.551 Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Singularities in algebraic geometry, Local complex singularities Functions on discriminants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a short survey on Cohen-Macaulay nilpotent schemes on a smooth variety. The author first presents geometric situations where multiple structures arise naturally. For double structures he considers Fossum's extension of a Cohen-Macaulay ring by a canonical module, which is a Gorenstein ring with a double multiplicity. This technique can be extended to obtain geometric structures, which are called Ferrand's doublings. For general multiple structures the author describes two different constructions. The first one uses ideal powers which was studied systematically by \textit{C. Banica} and \textit{O. Forster} [in: Algebraic geometry, Proc. Lefschetz Centen. Conf. Mexiko 1984, Contemp. Math. 58, 47--64 (1986; Zbl 0605.14026)]. The second one comes from linkage theory and was studied first by the author [Rev. Roum. Math. Pures Appl. 31, 563--575 (1986; Zbl 0607.14027)]. The last part of the survey gives examples on the relationship between multiple structures and vector bundles. multiple structure; doubling; linkage; stratification; vector bundle Manolache, N.: Cohen-Macaulay nilpotent schemes. In: Andrica, D., Blaga, P.A. (eds.) Recent Advances in Geometry and Topology, pp. 235-248. Cluj University Press, Cluj-Napoca (2004) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage, complete intersections and determinantal ideals, Parametrization (Chow and Hilbert schemes) Cohen-Macaulay nilpotent 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 Log canonical singularities is expected to be the largest class of singularities for the minimal model conjecture to hold. Typical examples of a log canonical singularity is the vertex of the projective cone over a smooth hypersurface of degree \(d\leq n+1\) in \(\mathbb{P}^n\). In this article, the author provides a generalization of the previous example for complete intersections to have log canonical singularities: Let \(X\subseteq\mathbb{P}^N\) be an irreducible complete intersection of multidegree \((d_1,\dots, d_r)\). If \(\dim\text{Sing}(X)\leq N-\sum_id_i\), then \(X\) is log canonical. Indeed, this condition is sharp, cf. Theorem 1.4 (ii).
The proof utilize the theory of Du Bois singularities, since a normal Gorenstein singularity is log canonical iff it is Du Bois. The advantage is that for complete varieties to be Du Bois, instead of going back to the original local definition via Du Bois complex, one can simply check it cohomologically (numerically Du Bois), cf. Theorem 2.3. The proof of theorem then follows easily by a cohomology computation. singularities; Du Bois; log canonical; complete intersections Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Minimal model program (Mori theory, extremal rays) Singularities of low degree complete intersections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a detailed study of the motivic Milnor fibre at infinity of a polynomial \(f\) of two variables in an algebraically closed field \(K\) of characteristic zero, in relation to its Newton polygon at infinity. The motivic Milnor fibre at infinity of \(f\), denoted by \(S_{f,\infty}\), has been defined by Matsui and Takeuchi, and independently by \textit{M. Raibaut} [Bull. Soc. Math. Fr. 140, No. 1, 51--100 (2012; Zbl 1266.14012)], by using the motivic integration introduced by Kontsevich and developed by Denef and Loeser. It follows from their work that \(S_{f,\infty}\) is a motivic incarnation of the topological Milnor fibre at infinity \(F_{\infty}\) of \(f\) endowed with its monodromy action \(T_{\infty}\).
The authors produce, in Theorems 3.8 and 3.23, some detailed formulas for the zeta-functions defining the motivic Milnor fibre at infinity \(S_{f,\infty}\), and the motivic nearby cycles at infinity \(S_{f,a}^{\infty}\), for some value \(a\in K\), in terms of certain motives associated to faces of the Newton polygon at infinity.
\textit{L. Fantini} and \textit{M. Raibaut} [in: Arc schemes and singularities. Hackensack, NJ: World Scientific. 197--220 (2020; Zbl 1440.14072)] proved over \(\mathbb C\) that the Euler characteristic of the motive \(S_{f,a}^{\infty}\) is equal to zero for all \(a\in \mathbb{C}\) except of finitely many values, for which it is equal, modulo a sign, to the Milnor-Lê jump invariant \(\lambda_{a}(f)\). This invariant has been defined and used by several authors before, see e.g. [\textit{M. Tibăr}, Polynomials and vanishing cycles. Cambridge: Cambridge University Press (2007; Zbl 1126.32026)] for more details and references. This leads to the definition of the motivic bifurcation locus \(B_f^{mot}\), in case \(f\) has isolated singularities, by taking into account both the affine singularities and these jumps at infinity.
Classically, the topological bifurcation locus \(B_f^{top}\) is defined as the finite set of values \(a\in \mathbb C\) for which the fibre \(f^{-1}(a)\) is singular, or it is not singular but \(\lambda_{a}(f) \not= 0\).
Here the authors define the Newton bifurcation locus \(B_f^{Newton}\) for any polynomial \(f\) depending effectively on two variables, whatever its Newton polygon might be.
The main results of the paper is the equality (Theorem 3.30) of the bifurcation loci in case of \(f: \mathbb C^{2}\to \mathbb C\) with isolated singularities: \(B_f^{mot} = B_f^{mot} = B_f^{Newton}\). The proof is developed in Section 3 of the paper over several technical lemmas, most of which holding for coefficients in \(K\). motivic Milnor fibre at infinity; Newton polygons; bifurcation values of polynomials; motivic nearby cycles at infinity Arcs and motivic integration, Local ground fields in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Singularities in algebraic geometry Newton transformations and motivic invariants at infinity of plane 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 We give some examples of four dimensional cusp singularities which are not of Hilbert modular type. We construct them, using quadratic cones and subgroups of reflection groups. cusp singularity; quadratic cone; reflection Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Singularities in algebraic geometry Examples of four dimensional cusp 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 0626.00011.]
This paper is an excellent survey, written mainly for non-experts, concerning the recent developments in the theory of Hilbert schemes of points. Four main themes are discussed, namely: (1) Punctual Hilbert scheme of a surface and of a curve lying on it, (2) The local punctual Hilbert scheme and the mapping germs, (3) The geometry of \(Hilb^ nY\) and foundations, (4) Extensions to modules, applications and vector bundles.
The author succeeded to give some feeling for this subject, illustrating the theory by many examples. The paper can also be considered as reflecting the actual state of affairs in the field. An extensive bibliography is included. Bibliography; zero cycles; symmetric products; Hilbert schemes [I3] Iarrobino, A.: Hilbert Scheme of Points: Overview of Last Ten Years. Proc. of Symp. in Pure Math. Vol.46 Part 2, Algebraic Geometry, Bowdoin 1987, 297--320 Parametrization (Chow and Hilbert schemes), Algebraic cycles, History of algebraic geometry, History of mathematics in the 20th century Hilbert scheme of points: Overview of last ten years | 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 \(R\) a reduced irreducible curve singularity over \(k\), i.e. \(R \supset k\) is a complete local domain of Krull dimension 1 having \(k\) as residue field. Any torsion free rank one module \(M\) can be seen as a submodule of the normalization \(R'=k[[t]]\) of \(R\) such that \(R \subseteq M \subseteq R'\). The aim of this paper is to construct coarse moduli spaces parametrizing the isomorphism classes of such modules corresponding to some fixed invariants as e.g. \(\delta(M)= \dim_ kR'/M\), \(\Gamma(M)=v(M)\), where \(v\) is the valuation of \(R'\). The idea is to show that the above isomorphism classes can be seen as orbits of the multiplicative group \(R'{}^*\) of invertible elements of \(R'\) (in fact of the Jordan group \(J \cong R'{}^*/k^*)\) acting on some Grassmannians. Then using the authors' results from Proc. Lond. Math. Soc., III. Ser. 67, No. 1, 75-105 (1993) it is enough to see that there exist geometric quotients for a certain stratification given by some invariants. The paper contains an algorithm for computation of these moduli spaces and many useful nice examples. irreducible curve singularity; coarse moduli spaces; Grassmannians Greuel, G.-M., Pfister, G.: Moduli spaces for torsion free modules on curve singularities I. J. of Algebraic Geometry2, 81--135 (1993) Families, moduli of curves (algebraic), Singularities of curves, local rings, Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry Moduli spaces for torsion free modules on curve singularities. 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 paper continues the authors' investigations [see \textit{K. Altmann} and \textit{J. A. Christophersen}, Manuscr. Math. 115, No. 3, 361--378 (2004; Zbl 1071.13008) and \textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] on the graph \({\mathcal G}\) of monomial ideals in the polynomial ring \(R=k[x_1,\dots,x_n]\), \(k\) a field. \({\mathcal G}\) is the infinite graph with the monomial ideals in \(R\) as vertex set. Two monomial ideals \(M_1\) and \(M_2\) are connected by an edge if there exists an ideal \(I\) in \(R\) such that the set of all initial monomial ideals of \(I\), with respect to all term orders, is precisely \(\{M_1,M_2\}\). \(I\) is called edge providing in this case. It is well known that \(I,M_1,M_2\) have many invariants in common. Each invariant yields a stratification of \({\mathcal G}\). A first proposition concerns the subgraph \({\mathcal G^r}\) obtained by restriction to artinian ideals of colength \(r\): Each such stratum is a connected component of \({\mathcal G}\). The main result (theorem 8) characterizes edge providing ideals as ``very homogeneous'':
There exists upto multiples a single \(c\in\mathbb Z^n\) such that \(I\) is \(A\)-graded for \(A=\mathbb Z^n/c\mathbb Z\).
This allows to define the Schubert scheme \(\Omega_c(M_1,M_2)=\Omega(M_1,M_2)\) of all \(A\)-homogeneous edge providing ideals connecting \(M_1\) and \(M_2\). A thorough analysis of the settings yields an algorithm that computes, for given \(M_1\) and \(M_2\), the direction \(c\) and \(\Omega(M_1,M_2)\) as affine scheme. More generally, the authors consider \(A\)-homogeneous ideals for arbitrary gradings \(\deg\:\mathbb Z^n\to A\) (including the standard one). Define \(h_I\:A\to \mathbb N\) as the Hilbert function of \(I\), i.e., \(h_I(a),\;a\in A\), is the \(k\)-dimension of the \(a\)-homogeneous part of \(I\). In the above situation, for \(I\in \Omega(M_1,M_2)\) the Hilbert functions of \(I\), \(M_1\) and \(M_2\) coincide. For positive gradings, i.e., \(\mathbb N^n\cap \text{ker}(\deg)=(0)\), this is also the general situation:
\(\Omega_c(M_1,M_2)\not=\emptyset\) implies \(\deg(c)=0\) and hence the Schubert schemes describe an essential part of the multigraded Hilbert scheme \(\text{Hilb}_h\) of all \(A\)-homogeneous ideals with given Hilbert function \(h\). This part is sufficient to detect connectedness: Over \(k=\mathbb R\) or \(k=\mathbb C\), \(\text{Hilb}_h\) is connected if and only if the induced subgraph \({\mathcal G}(\text{Hilb}_h)\) is connected.
Section 4 discusses properties of the Schubert schemes for square-free monomial ideals. The results are more technical and continue the investigations started by \textit{K. Altmann} and \textit{J. A. Christophersen} [loc. cit.]. In particular, it turns out that neighboring square-free ideals are connected by a generalization of the bistellar flip construction [see, e.g., \textit{O. Viro}, Proc. Workshop Differential Geometry Topology, Alghero 1992, World Scientific. 244--264 (1993; Zbl 0884.57015) or \textit{D. Maclagan} and \textit{R. R. Thomas}, Discrete Comput. Geom. 27, No. 2, 249--272 (2002; Zbl 1073.14503)]. The paper ends with a list of open problems about the graph \({\mathcal G}\) and the Schubert schemes. graph of monomial ideals; multigraded Hilbert scheme; Schubert scheme; Gröbner bases; Gröbner degenerations; Stanley-Reisner ideals DOI: 10.1016/j.jpaa.2004.12.030 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds The graph of monomial ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review studies the problem of classifying tuples of linear endomorphisms and linear functions on a finite dimensional vector space up to base change, which can be reinterpreted in the language of quivers, as classifying representations on finite dimensional vector spaces of the two-vertex quiver with as many loops on the first vertex as linear endomorphisms and as many arrows between the two vertices as the number of linear functions.
Such representations can be reinterpreted as framed representations in the sense of [\textit{M. Reineke}, J. Algebra 320, No. 1, 94--115 (2008; Zbl 1153.14033)], where it was proved, for quivers without oriented loops, that the quotient of the stable representations up to isomorphism (i.e. up to base change of the linear maps) is isomorphic to a Grassmannian of subrepresentations of a certain injective representation of the quiver. The general case does not provide a projective quotient, and the author studies the fibers of this quotient in [\textit{S. N. Fedotov}, Trans. Am. Math. Soc. 365, No. 8, 4153--4179 (2013; Zbl 1277.14010)]. It remains to describe a trivializing covering for the quotient map, which is the problem adressed in this article.
By generalizing Reineke's construction, the author associates a finite \textit{skeleton} to each stable representation, and each skeleton carries an open subset of the space parametrizing representations. As there are a finite number of possible skeletons, we find a finite open covering of the parametrizing space. The trivialization comes from the fact that each one of these open pieces defines a normal form of the representation, hence the open pieces are isomorphic to affine spaces. Besides, it is given a criterium to determine whether 2 representations are isomorphic, this is when they have a skeleton in common and the normal form associated to them coincides.
The paper finishes with a collection of examples where all the computation are explicitly described. framed moduli spaces; quiver moduli; tuples of operators; Grassmannians of representations; skeleton Local ground fields in algebraic geometry, Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds Framed moduli spaces and tuples of operators | 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 article brings together various definitions of a noncommutative algebraic geometry.
It starts off with the observation that comparing the representations of the \textit{Kronecker quiver}, the quiver with two nodes and two arrows \(x,y\) from the first node to the second, and the category of coherent sheaves over the projective line \(\mathbb P^1\), their Auslander-Reiten quivers look similar. As the Auslander-Reiten quiver describes the indecomposable objects of an exact category, this indicates that the similarity has an explanation: It is known that \(\mathcal G=\mathcal O_{\mathbb P^1}(-1)\oplus\mathcal O_{\mathbb P^1}\) is a \textit{tilting object} of the category \(\text{Coh}(\mathbb P^1)\). This means that \(\text{Ext}^1_{\mathbb P^1}(\mathcal G,\mathcal G)=0\) for all \(i>0\) and, for every nonzero morphism \(f:\mathcal F\rightarrow\mathcal F^\prime\) of coherent sheaves, \(\text{Hom}_{\mathcal P^1}(\mathcal G,f)\neq 0\), i.e. \(\mathcal G\) generates the derived category \(\mathcal D(\text{Coh}\;\mathbb P^1).\) This means in particular that \(\text{RHom}_{\mathbb P^1}(\mathcal G,-)\) gives an equivalence of the derived categories \(\mathcal D(\text{Coh}\;\mathbb P^1)\) and \(\mathcal D(A-\text{mod})\), \(A=\text{End}(\mathcal G)^{\text{op}}\), and then \(A\) is the path algebra of the Kronecker quiver. Because both categories \(A-\text{mod}\) and \(\text{Coh}\;\mathbb P^1\) are \textit{hereditary}, every indecomposable object of the derived category is just a shift of a module.
From Beilinson, it is known that \(\text{Coh}\;\mathbb P^n\) has a tilting sheaf \(\mathcal G=\bigoplus_{i=-n}^0\mathcal O_{\mathbb P^n}(i)\) so that it is derived equivalent to the category of representations of the finite dimensional algebra \(A=\text{End}(\mathcal G)^{\text{op}}\), which can be explicitly described. Analogues results was proved for other projective varieties construction tilting sheaves.
\textit{Y. A. Drozd} and \textit{G.-M. Greuel} [J. Algebra 246, No. 1, 1--54 (2001; Zbl 1065.14041)] noticed a resemblance between the categories of vector bundles over a class of singular curves and the categories of representations of some finite dimensional algebras. In particular for a nodal cubic \(C\) and the algebra \(A\) with quiver consisting of three nodes in sequence with arrows \(a,b\) from the first to the second and \(c,d\) from the second to the third, with relations \(da=cb=0\). This correspondence cannot be a result following from an equivalence of derived categories, because the algebra \(A\) is of global dimension \(2\) while \(\mathcal O_C\) is of infinite global dimension. This was explained by \textit{I. Burban} and \textit{Y. Drozd} [Math. Ann. 351, No. 3, 665--709 (2011; Zbl 1231.14011)] by considering a sheaf of noncommutative algebras \(\mathcal A\) (an \textit{Auslander sheaf}), of global dimension \(2\). They constructed a tilting sheaf over \(\mathcal A\) such that its endomorphism algebra was the algebra \(A\) of [Zbl 1065.14041]. The category of coherent sheaves over the initial curve is a \textit{Serre quotient} of the category of coherent sheaves over \(\mathcal A\) by a semi-simple subcategory, saying that their indecomposable objects are almost the same, i.e. there is a resemblance between the Auslander Reiten quivers.
The article gives a generalization of the results of [Zbl 1231.14011] to all singular curves. The authors construct for every curve \(X\) a sheaf of \(\mathcal O_X\)-algebras \(\mathcal R\) such that \(\mathcal R\) is of finite global dimension and there is a functor \(\mathsf{F}:\mathsf{Coh}\;\mathcal R\rightarrow\mathsf{Coh}\;X\) such that \(\mathsf{Coh}\;X\) is a \textit{bilocalization}, that is, both localization and colocalization, of \(\mathsf{Coh}\;\mathcal R\). The derived categories have the same property, and \(\mathcal R\) has special properties analogous to those of \textit{quasi-hereditary} algebras. The authors name \(\mathcal R\) the \textit{Königs resolution} of the curve \(X\). If \(X\) is rational, \(\mathcal R\) has a tilting complex \(\mathcal T\) establishing the equivalence between the derived category of \(\mathsf{Coh}\;\mathcal R\) and the derived category of a finite-dimensional quasi-hereditary algebra. This construction is considered as a \textit{categorical resolution} of the category \(\mathcal D(\mathsf{Coh}\;X)\). If \(X\) is Gorenstein, this categorical resolution is \textit{weakly crepant}. The construction above is also proved to be applicable to \textit{non-commutative} curves.
The main ingredient in the construction is the notion of \textit{minors} of noncommutative schemes. A minor \(\mathcal B\) of a a sheaf of algebras \(\mathcal A\) is the endomorphism sheaf of a locally projective sheaf of \(\mathcal A\)-modules. Then \(\mathsf{Qcoh}\;\mathcal B\) is a bilocalization of \(\mathsf{Qcoh}\;\mathcal A\), and the same for their derived categories. The main properties of these bilocalizations is studied, and specialized to the \textit{endomorphism construction}. Then this technique is applied to a special class of non-commutative schemes called \textit{quasi-hereditary}, generalizing the notion of quasi-hereditary algebras with many of the same features. In particular, a quasi-hereditary non-commutative scheme always has finite global dimension, and its derived category has good semi-orthogonal decompositions. The authors study general properties of non-commutative curves and their minors, in particular related to Cohen-Macaulay modules. They consider the Königs resolution and prove that it is quasi-hereditary. Also, they prove that in the commutative case, the functors of direct and inverse image coming from the normalization of the curve are compositions of the functors coming from the Königs resolution. The authors construct a tilting sheaf for the Königs resolution of a rational singular curve (might be non-commutative), and show that it gives a categorical resolution of \(\mathcal D(\mathsf{Qcoh}\;X)\) by a quasi-hereditary finite dimensional algebra. The case where all singularities of a curve are of ADE types is studied.
The article is a very important contribution to non-commutative algebraic geometry as it connects the various definitions, and explain their resemblances. This article is written in very readable way, pinpointing a lot of important applications. bilocalization; categorical resolution; Königs resolution; derived categories; minors; non-commutative schemes; quasi-hereditary schemes; Auslander-Reiten quiver; endomorphism construction; tilting object; Auslander sheaf Burban, Igor; Drozd, Yuriy; Gavran, Volodymyr: Singular curves and quasi-hereditary algebras. (2015) Noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories Minors and categorical resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a companion paper of [the authors, ibid. 22, No. 5, 1071--1147 (2018; Zbl 1479.81043)]. We study Coulomb branches of unframed and framed quiver gauge theories of type \(ADE\). In the unframed case they are isomorphic to the moduli space of based rational maps from \(\mathbb{P}^1\) to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian. Yang-Mills and other gauge theories in quantum field theory, Representations of quivers and partially ordered sets, Fine and coarse moduli spaces, Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) Coulomb branches of \(3d\) \(\mathcal{N}=4\) quiver gauge theories and slices in the affine Grassmannian | 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 a necessary and sufficient condition for the Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in all dimensions by using Ashikaga's continued fractions. Moreover, we prove that any three dimensional Gorenstein abelian quotient singularity possesses a crepant Fujiki-Oka resolution as a corollary. This alternative proof of existence needs only simple computations compared with the results ever known. crepant resolutions; Fujiki-Oka resolutions; higher dimension; finite groups; abelian groups; Hirzebruch-Jung continued fractions; invariant theory; multidimensional continued fractions; quotient singularities; toric varieties Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, \(3\)-folds, \(4\)-folds, \(n\)-folds (\(n>4\)), Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Crepant property of Fujiki-Oka resolutions for Gorenstein abelian 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 We prove that the isolated invariant branches of a weak toric type generalized curve defined over a projective toric ambient surfaces extend to projective algebraic curves. To do it, we pass through the characterization of the weak toric type foliations in terms of ``Newton non-degeneracy'' conditions, in the classical sense of Kouchnirenko and Oka. Finally, under the strongest hypothesis of being a toric type foliation, we find that there is a dichotomy: Either it has rational first integral but does not have isolated invariant branches or it has finitely many global invariant curves and all of them are extending isolated invariant branches. singular foliations; invariant curves; Newton polygons; toric surfaces Singularities of holomorphic vector fields and foliations, Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Newton non-degenerate foliations on projective 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 In this paper we show that every object in the dg category of relative singularities \(\mathbf{Sing}(B,\underline{f})\) associated to a pair \((B,\underline{f})\), where \(B\) is a ring and \(\underline{f}\in B^n\), is equivalent to an homotopy retract of a \(K(B,\underline{f})\)-dg module concentrated in \(n+1\) degrees, where \(K(B,\underline{f})\) denotes the Koszul algebra associated to \((B,\underline{f})\). When \(n=1\), we show that Orlov's comparison theorem, which relates the dg category of relative singularities and that of matrix factorizations of an LG-model, holds true without any regularity assumption on the potential. dg categories of relative singularities; matrix factorizations; non commutative algebraic geometry Singularities in algebraic geometry, Derived categories, triangulated categories, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.) On the structure of dg-categories of relative 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 contains a complete list of all finite Auslander-Reiten quivers of local Gorenstein orders over a complete Dedekind domain of finite lattice type. For each translation quiver \(\Gamma\) in this list, a Gorenstein order \(\Lambda\) is explicitly indicated, with \(\Gamma\) as its Auslander-Reiten quiver. In each case, indecomposable \(\Lambda\)-lattices are described. finite Auslander-Reiten quivers; local Gorenstein orders; finite lattice type; indecomposable \(\Lambda \) -lattices Wiedemann, A.: Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve singularities. J. pure appl. Algebra 41, No. 2-3, 305-329 (1986) Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Representation theory of associative rings and algebras, Singularities in algebraic geometry Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number \(p\) for a certain class of varieties. This class contains projective homogeneous varieties which are either split or considered over a field admitting some form of resolution of singularities, for example any field of characteristic not \(p\). These reduced Steenrod operations are sufficient for some applications to the theory of quadratic forms. Steenrod operations; Riemann-Roch theorem; Chow groups O. Haution, Reduced Steenrod operations and resolution of singularities, J. K-Theory 9 (2012), no. 2, 269-290. Riemann-Roch theorems, Global theory and resolution of singularities (algebro-geometric aspects) Reduced Steenrod operations and resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a quadratic hypersurface over a field \(F\) of characteristic different from two. It is shown here that the torsion subgroup of the Chow group \(\text{CH}^ 3X\) is either trivial or \(\mathbb{Z}/2\mathbb{Z}\). On the other hand, it is known that the groups \(\text{CH}^ pX\), \(p=0,1\), have only trivial torsion, and the torsion part of \(\text{CH}^ 2X\) is either trivial or \(\mathbb{Z}/2\mathbb{Z}\) [the author, Leningr. Math. J. 2, No. 1, 119- 138 (1991); translation from Algebra Anal. 2, No. 1, 141-162 (1990)]. The author and \textit{A. S. Merkur'ev} [Leningr. Math. J. 2, No. 3, 655-671 (1991); translation from Algebra Anal. 2, No. 3, 218-235 (1990)] proved that with suitable choices for \(F\) and \(X\) the groups \(\text{CH}^ pX\), \(p\geq 4\), can have arbitrarily many elements of order two. Grothendieck groups; torsion subgroup of the Chow group Algebraic cycles, Parametrization (Chow and Hilbert schemes), Grothendieck groups (category-theoretic aspects) On cycles of codimension 3 on a projective quadric | 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 0614.00006.]
The purpose of the paper is to give information on a certain smooth compactification of the space of all morphisms of a given degree from \({\mathbb{P}}^ 1\) to a Grassmann variety. This scheme is the Grothendieck Quot scheme of quotients of a trivial vector bundle on \({\mathbb{P}}^ 1\). We compute the additve and the multiplicative structure of its Chow ring and identify the ample cone and the corresponding projective embeddings. families of rational curves; smooth compactification; Grassmann variety; Quot scheme; Chow ring Strømme, S A, On parametrized rational curves in Grassmann varieties., Lecture Notes in Math, 1266, 251-272, (1987) Grassmannians, Schubert varieties, flag manifolds, Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Algebraic moduli problems, moduli of vector bundles On parametrized rational curves in Grassmann 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 The main result of this article is the following theorem: Let \(V\) be a valuation ring of arbitrary characteristic. Then for any proper surjective morphism \(f\): \(X\to \mathrm{Spec}(V)\), the induced map \(V\to\mathbf{R}f_*O_X\) splits in the derived category of \(V\)-modules, in other words, \(V\) is a derived splinter in the sense of Bhatt. If \(V\) is a discrete valuation ring, this fact is well-known and thus the authors' main contribution is to generalize this fact to all non-Noetherian valuation rings. The authors provide three proofs of their theorem: two of which are essentially elementary in nature (either using the valuative criterion for properness or using that finitely presented \(V\)-modules have projective dimension at most \(1\)), the third proof depends on the derived direct summand theorem due to Bhatt. This article is very clerely and very nicely written, and it contains results on descdent properties of splinters and derived splinters (beyond the Noetherian set up), universally cohesive rings, ind-regular valuaiton rings, that are of independent interest. valuation rings; splinters; derived splinters; local uniformization Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Derived categories and commutative rings, Homological conjectures (intersection theorems) in commutative ring theory Valuation rings are derived splinters | 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 (\(X\), \(h\)) be a polarized holomorphic symplectic manifold of \(K3\) type, i.e., deformation equivalent to the Hilbert scheme \(S^{[n]}\) of length \(n\geq2\) subschemes of a \(K3\) surface \(S\). Building on the earlier results of \textit{A. Bayer} and \textit{E. Macrì} [Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011)], \textit{A. Bayer} et al. [Ann. Sci. Éc. Norm. Supér. (4) 48, No. 4, 941--950 (2015; Zbl 1375.14124)] gave an explicit description of the Mori cone of (\(X\), \(h\)) in \(H_2(X, \mathbb{R})_{\mathrm{alg}}\). In particular they obtained a classification of extremal birational contractions, up to the action of monodromy, for such manifolds.
In the paper under review the authors describe these results through concrete examples and prove the following two interesting results on holomorphic symplectic manifolds.
There exists a polarized \(K3\) surface \(S\) such that \(S^{[3]}\) admits an automorphism not arising from \(S\).
There exist polarized holomorphic symplectic manifolds of \(K3\) type, admitting an isomorphism between their Hodge structures, not preserving ample cones. extremal rays; automorphisms; symplectic manifolds Hassett, Brendan; Tschinkel, Yuri, Extremal rays and automorphisms of holomorphic symplectic varieties. K3 surfaces and their moduli, Progr. Math. 315, 73-95, (2016), Birkhäuser/Springer, [Cham] Automorphisms of surfaces and higher-dimensional varieties, Minimal model program (Mori theory, extremal rays), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Parametrization (Chow and Hilbert schemes) Extremal rays and automorphisms of holomorphic symplectic 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 Whenever we have an action of an algebraic group \(G\) on a smooth complex algebraic variety \(X\), we can consider \(G\)-equivariant \(\mathcal{D}\)-modules on such variety. These are coherent \(\mathcal{D}_X\)-modules that satisfy some additional properties related to the action of \(G\). They form the category \(\operatorname{mod}_G(\mathcal{D}_X)\), which is a full abelian subcategory of that of coherent \(\mathcal{D}_X\)-modules and is closed under taking subquotients. When the action consists of finitely many orbits, it is known thanks to \textit{R. Hotta} [``Equivariant \(D\)-modules'', Preprint, \url{arXiv:math/9805021}] that every \(G\)-equivariant \(\mathcal{D}_X\)-module is regular holonomic, in particular of finite length. Thus, understanding simple objects is key to understand the whole category \(\operatorname{mod}_G(\mathcal{D}_X)\).
The paper under review explores the case of the action of the general linear group \(G=\operatorname{GL}(V)\) on the space of alternating \(3\)-tensors \(X=\Lambda^3 V\) of a given \(6\)-dimensional vector space \(V\). This apparently odd setting is one of the few remaining examples of representations with finitely many orbits (the rest being already studied by the authors) in which the task of describing simple \(G\)-equivariant \(\mathcal{D}_X\)-modules was not accomplished.
More concretely, the authors compute the six existent simple \(G\)-equivariant \(\mathcal{D}_X\)-modules, each of them supported on the closure of one of the five orbits of the action of \(G\) on \(X\). Furthermore, they compute all their nonvanishing local cohomology modules with support in such closures and show \(\operatorname{mod}_G(\mathcal{D}_X)\) is equivalent to the category of finite-dimensional representations of certain quiver with relations. equivariant \(\mathcal{D}\)-modules; local cohomology; quiver representations Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Local cohomology and commutative rings, Actions of groups on commutative rings; invariant theory, Representations of quivers and partially ordered sets Equivariant \(\mathcal{D}\)-modules on alternating sentry 3-tensors | 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 [Transform. Groups 9, No. 2, 167--209 (2004; Zbl 1092.14042)], the author gave a unified construction for many moduli spaces over curves. He introduced appropriate stability concepts and used geometric invariant theory to construct these coarse moduli spaces. In the paper under review, these results are extended in two directions: the dimension of the base manifold is now arbitrary and the group \(\text{GL}(r)\) which enters the definition of the objects to be classified, is replaced by a product of such groups. This corresponds to considering moduli problems which involve more than one vector bundle. The methods used to prove these generalisations are similar to those used in his previous work. Therefore, the author confines himself to highlight the main steps needed to adapt the proofs to the more general situation. Thus, large part of the effort goes into a clear and consistent formulation of the stability concepts, the definition of the moduli functors and the formulation of the main results. Apart from these fundamental results and as a motivation for this work, two applications to completely different moduli problems are presented. The first one deals with integral Gorenstein covers of degree four over a fixed projective manifold. The second one comes from representation theory of finite dimensional algebras. A stability concept for representations of quivers is introduced and the existence of a coarse moduli space is shown. The moduli space of Higgs bundles can be seen as a special case of this construction. moduli space; GIT quotient; semi-stability; Hitchin map; quiver representation; decorated sheaf; path algebra; stable representation Schmitt A., Moduli for decorated tuples of sheaves and representation spaces for quivers, Proc. Indian Acad. Sci. Math. Sci., 2005, 115(1), 15--49 Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Vector bundles on curves and their moduli, Representations of quivers and partially ordered sets Moduli for decorated tuples of sheaves and representation spaces for quivers | 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 under review an answer, in characteristic zero, to a question posed by \textit{B. Teissier} [in: Valuation theory and its applications. Volume II. Proceedings of the international conference and workshop, University of Saskatchewan, Saskatoon, Canada, July 28--August 11, 1999. Providence, RI: American Mathematical Society (AMS). 361--459 (2003; Zbl 1061.14016)] is given. The problem is on the possibility to find a resolution of singularities of an embeddable algebraic variety (or of a suitable scheme) ``inside'' an equivariant morphism of toric varieties. More precisely, the main results of the paper are, essentially, as follows. For us, ``variety'' means an irreducible algebraic variety over a base field \(k\).
(1) Let \(X\) be a closed subvariety of a smooth variety \(S\) and consider an embedded resolution of \(X \subset S\), that is a projective birational morphism \(\pi :W \to S\) such that \(W\) and \(Y\) (the strict transform of \(Y\) in \(W\)) are smooth. We also assume that \(D\), the exceptional set of \(\pi\), is a simple normal crossings divisor of \(W\) and that \(Y\) intersects \(D\) transversally. Then, \(X\) can be embedded into \({\mathbb P}^N\) (for a suitable integer \(N\)), in such a way that there is an algebraic torus \({\mathbb G}\), a smooth toric variety \(Z\) (of torus \(\mathbb G\)) (which may be assumed complete) and an equivariant morphism \(p:Z \to {\mathbb P}^N\) so that \(Y\) can be identified to the strict transform of \(X \subset {\mathbb P}^N\) in \(Z\). Moreover, \(Y\) intersects the toric boundary of \(Z\) transversally. Here, \( \mathbb G = {\mathbb P}^N - (H_0 \cup \ldots \cup H_N\), where \(H_j:z_j=0\), for all \(j\), for suitable homogeneous coordinates \(z_0, \ldots, z_N\); \({\mathbb G} \approx {k^*}^N\).
(2) Now we assume \(k\) algebraically closed, of characteristic zero. Let \(X \subset S\) be as in (1). Then there is a closed embedding \(X \to {\mathbb P}^N\) (for a suitable \(N\)), homogeneous coordinates \(z_0, \ldots, z_N\) of \({\mathbb P}^N\) and a toric morphism of smooth toric varieties \({\tilde p}:{\tilde Z} \to {\mathbb P}^N\), such that: (a) if \(\mathbb G\) is as in (1), \(X \cup {\mathbb G}\) is dense in \(X\), (b) \({\tilde p}\) is the composition of monoidal transforms with equivariant smooth centers of codimension two, (c) the strict transform \(Y\) of \(X\) in \(Z\) is smooth and intersects the toric boundary of \(Z\) transversally.
In (2) we cannot assert that the induced resolution of singularities \(Y \to X\) is an isomorphism over the open set of regular points of \(Y\). Both in (1) and (2) the embedding \(X \to {\mathbb P}^N\) is induced by the embedding \(S \to {\mathbb P}^N\) defined by \(nL\), where \(L\) is any ample divisor of \(S\) and \(N\) is a suitable integer.
Statement (2) follows rather easily from (1), if one applies some known but hard results.
(2) partially answers a question of Teissier, which appears, e.g., in [``A viewpoint on local resolution of singularities'' [Oberwolfach Rep. 6, No. 3, 2405--2470 (2009; Zbl 1192.00066)]. The original formulation of the problem required that \(X=\mathrm{Spec}(A)\), where \(A\) is a noetherian equicharacteristic excellent local ring, with algebraically closed field, possibly of positive characteristic.
In the proof of (1) the author uses, among others, results he obtained in previous works, as well as some due to \textit{H. Hironaka} [Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)], \textit{M. Luxton} and \textit{Z. Qu} [Trans. Am. Math. Soc. 363, No. 9, 4853--4876 (2011; Zbl 1230.14014)], and \textit{C. De Concini} and \textit{C. Procesi} [Adv. Stud. Pure Math. 6, 481--513 (1985; Zbl 0596.14041)]. resolution of singularities; toric variety; toric morphism; transversality Tevelev, J.: On a question of B. Teissier. Collect. math. 65, No. 1, 61-66 (2014) Global theory and resolution of singularities (algebro-geometric aspects), Embeddings in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, Modifications; resolution of singularities (complex-analytic aspects) On a question of B. Teissier | 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 \(H_{d,g}\) denote the Hilbert scheme of locally Cohen-Macaulay curves in \(\mathbb{P}^3\). For any \(d>4\) and \(g\leq {d-3\choose 2}\), \(H_{d,g}\) has two well-understood irreducible families: There is a component \(E\subset H_{d,g}\) corresponding to extremal curves [see \textit{M. Martin-Deschamps} and \textit{D. Perrin}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 6, 757-785 (1996; Zbl 0892.14005), these are the curves with maximal Rao function] and \(S\), the family of subextremal curves [see \textit{S. Nollet}, Manuscr. Math. 94, No. 3, 303-317 (1997; Zbl 0918.14014), these have the next largest Rao function]. In this short note we show that \(S\cap E\neq \emptyset\) in \(H_{d,g}\) by constructing an explicit specialization (proposition 1). Our construction also works for ACM curves of genus \(g= {d-3\choose 2}+1\) (remark 2) and hence \(H_{d,g}\) is connected for \(g> {d-3\choose 2}\) (corollary 3). Hilbert scheme Nollet, S., A remark on connectedness in Hilbert scheme, Communications in Algebra, 28, 5745-5747, (2000) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) A remark on connectedness in 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 The author proves a theorem about symplectic varieties which says, that any symplectic variety admits a canonical stratification with a finite number of symplectic strata. Locally, near a stratum the variety in question admits a decomposition into the product of the stratum itself and a transversal slice. In the Poisson language this means that a symplectic variety considered as a Poisson space has a finite number of symplectic leaves. Natural group actions on a symplectic variety are studied and a strong restriction on the type of singularities a symplectic variety might have is proved, namely that locally a symplectic variety always admits a non-trivial action of the one-dimensional torus \({\mathbb G}_m.\) The author also considers the special case when a symplectic variety admits a crepant resolution of singularities and proves that the geometry of such a resolution is very restricted, it is always semismall and the Hodge structure on the cohomology of its fibers is pure and Hodge-Tate. symplectic manifold; Poisson bracket; singularities; resolution of singularities Kaledin, D., \textit{symplectic singularities from the Poisson point of view}, J. reine angew. Math., 600, 135-156, (2006) Poisson manifolds; Poisson groupoids and algebroids, Global theory and resolution of singularities (algebro-geometric aspects) Symplectic singularities from the Poisson point of view | 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 \((S, 0)\) be a germ of complex analytic normal surface singularity with \(\{E_i\}_{i \in I}\) the irreducible components of the exceptional divisor \(E\) in its minimal resolution \(\pi_m: ({\tilde{S}}_m, E) \rightarrow (S, 0)\). The Nash map associates to each irreducible component \(C_j\) of the pre-image of \(0\) in the space of arcs a unique divisor \(E_{i(j)}\), and it is an injective map [\textit{J. F. Nash, Jr.}, Duke Math. J. 81, No. 1, 31--38 (1996; Zbl 0880.14010)]. The Nash problem asks (in arbitrary dimension) for which classes of singularities it is bijective. There is a counter example in dimension 4 [\textit{S. Ishii} and \textit{J. Kollár}, Duke Math. J. 120, No. 3, 601--620 (2003; Zbl 1052.14011)], which generalizes easily in higher dimensions, but in dimensions 2 and 3 the problem still has an answer only in some special cases.
For surfaces these include \(A_n\) singularities [\textit{J. F. Nash, Jr.}, Duke Math. J. 81, No. 1, 31--38 (1996; Zbl 0880.14010)], \(D_n\) singularities [\textit{C. Plénat}, C. R. Math. Acad. Sci. Paris 340, No. 10, 747--750 (2005; Zbl 1072.14004)], minimal [\textit{A.-J. Reguera}, Manuscr. Math. 88, No. 3, 321--333 (1995; Zbl 0867.14012)] and sandwiched singularities [\textit{M. Lejeune-Jalabert} and \textit{A. J. Reguera-López}, Am. J. Math. 121, No. 6, 1191--1213 (1999; Zbl 0960.14015); \textit{A. J. Reguera}, C. R. Math. Acad. Sci. Paris 338, No. 5, 385--390 (2004; Zbl 1044.14032)]. In a recent preprint [\textit{M. Morales}, \url{arXiv:math.AG/0609629}], infinitely many classes of surface singularities for which the Nash problem has a positive answer are constructed, and some known results are improved. Until now there is no example giving a negative answer to the problem in dimensions 2 and 3.
The main result in this paper is the following one. If in the vector space with basis \(\{ E_i \}_{i \in I}\) each open half-space \(\{ \sum_i a_i E_i| ~a_i<a_j \}\) contains a divisor \(D \neq 0\), supported on the exceptional locus and such that \(D.E_i<0 ~\forall i \in I\), then the Nash problem has a positive answer for \((X, 0)\). This is a condition only on the intersection matrix of \(\pi_m\), but does not depend on the genera or smoothness of \(E_i\). The construction is based on two results. The first one is a sufficient criterion [\textit{C. Plénat}, Ann. Inst. Fourier 55, No. 3, 805--823 (2005; Zbl 1080.14021)] (in arbitrary dimension), proposed for rational surface singularities in [\textit{A.-J. Reguera}, Manuscr. Math. 88, No. 3, 321--333 (1995; Zbl 0867.14012)], which helps to determine if some \(E_i\) is in the image of the Nash map. This criterion plays a central role in almost all results obtained in dimension 2. The second one is a numerical criterion, which proof is based on a result of \textit{H. B. Laufer} [Am. J. Math. 94, 597--608 (1972; Zbl 0251.32002)], and gives a sufficient condition for an effective divisor with support on the exceptional locus of \(\pi_m\) to be the exceptional part of a regular function on \((S, 0)\). It is related to another result of \textit{H. B. Laufer} [in: Singularities. Proc. Sympos. Pure Math. 40, 1--29 (1983; Zbl 0568.14008)], but neither one follows from the other. As a corollary of the main result one has that if \(E.E_i <0 ~\forall i\), then the Nash map is bijective. Another corollary gives infinitely many families of pairwise topologically distinct non-rational surface singularities, for which the Nash problem has a positive answer.
Reviewer's remark: This paper is an important contribution to the list of classes of varieties, for which the answer of the Nash problem is known. I think it is well organized and easy to understand. Similar results are obtained in the preprint of Morales cited above, but the differences are not clearly explained. space of arcs; Nash map; Nash problem Plénat, Camille; Popescu-Pampu, Patrick, A class of non-rational surface singularities with bijective Nash map, Bull. Soc. Math. France, 134, 3, 383-394, (2006) Singularities in algebraic geometry, Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects) A class of non-rational surface singularities with bijective Nash map | 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 the author uses linkage techniques to obtain many curves \(C\subset {\mathbb{P}}^ 3\) with \(H^ 1(N_ C(-2))=0\) (hence with normal bundle \(N_ C\) semistable). The results seem very useful. space curve; Hilbert scheme; deformation theory; liaison; semistable normal bundle; linkage D. PERRIN . - C. R. Acad. Sci. Paris, 300, Série I, N^\circ 2, 1985 , p. 39-42. MR 86h:14026b | Zbl 0586.14025 Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry Courbes gauches, fibré normal et liaison. (Space curves, normal bundle and linkage) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This survey starts with a leisurely exposition of the notion of integration with respect to the Euler characteristic; with the use of the right cohomology theory the Euler characteristic becomes an additive (bot not non-negative) function on the algebra of constructible sets, and can be used like a measure to integrate constructible functions. The integral is in fact a finite sum, but is has a Fubini theorem. Some examples of this point of view are presented, like the Riemann-Hurwitz formula and A'Campo's formula for the monodromy zeta-function.
Then generalisations are discussed: integration over the infinite-dimensional spaces of arcs and functions, with applications such as the definition and computation of Poincaré series of multi-index filtrations, and motivic integration. Euler charactersitic; zeta-function; Poincaré series; motivic integration Gusein-Zade, S. M., Integration with respect to the Euler characteristic and its applications, Uspekhi Matem. Nauk, 65, 5, (2010) Singularities in algebraic geometry, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Milnor fibration; relations with knot theory, Zeta functions and \(L\)-functions Integration with respect to the Euler characteristic and its applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f : (\mathbb{C}^{n+1},0) \to (\mathbb C,0)\) be a holomorphic function defining an isolated hypersurface singularity \((V(f),0) = (V,0) \subset (\mathbb C^{n+1},0)\) and define \(A^k(V) = \mathcal O_{n+1}/\langle f,\mathfrak m^k \mathrm{Jac}(f)\rangle\) for \(k\geq 0\); in case \(k=0\) this is the familiar Tjurina -- or moduli algebra encoding non-trivial infinitesimal deformations of \((V,0)\). The authors study two conjectural inequalities on the so-called Yau numbers \(\lambda^k(V)\) which are defined as the length of the Lie algebra of derivations
\[
L^k(V) = \mathrm{Der}(A^k(V),A^k(V)).
\]
These Lie algebras play an important role in the study of deformations of singularities, see e.g. [\textit{C. Seeley} and \textit{S. S. T. Yau}, Invent. Math. 99, No. 3, 545--565 (1990; Zbl 0666.14002); \textit{N. Hussain}, Methods Appl. Anal. 25, No. 4, 307--322 (2018; Zbl 1436.14008)].
The first inequality (Conjecture 1.1; originally from \textit{N. Hussain} et al [``\(k\)-th Yau number of isolated hyper-surface singularities and an inequality conjecture'', J. Aust. Math. Soc. (2019; \url{doi:10.1017/S1446788719000132})]) bounds these numbers from below by
\[
\lambda^{k+1}(V) > \lambda^k(V)
\]
for \(k\geq 0\), \(n >0\), and \(f\) of multiplicity at least three. In this paper, the conjecture is verified for ``trinomial singularities'' in the case \(k=1\), other cases being covered in earlier publications.
The second inequality (Conjecture 1.2; originally from [\textit{N. Hussain} et al., Math. Z. 294, No. 1--2, 331--358 (2020; Zbl 1456.14005)]) is on singularities which are quasi-homogeneous for some weights \(w=(w_0,\dots,w_n;d)\) and it bounds the number \(\lambda^k(V)\) from above by the \(k\)-th Yau number of the associated Brieskorn-Pham singularity, which is then a function of the weights \(w\) only. The authors verify this second conjecture for trinomial singularities in the case \(k=2\).
The proof of these inequalities proceeds by reduction to normal forms and subsequent case-by-case analysis. Not all of the calculations are printed here, but can only be found online: \url{http://archive.ymsc.tsinghua.edu.cn/pacm_download/89/11687-HYZ2020.pdf}. derivation; Lie algebra; isolated singularity; Yau algebra Singularities in algebraic geometry, Local complex singularities On two inequality conjectures for the \(k\)-th Yau numbers of isolated 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 \(Q\) be a quiver and let \(\text{GL}_n(\overline k)\) denote the direct product of general linear groups of the family of \(n\)-vector spaces of dimensions \(k_1,\dots,k_n\), respectively. The representation space of \(Q\) is the coordinate ring \(R(Q,\overline k)=\prod_{a\in A}\Hom_K(E_{h(a)},E_{t(A)})\) where \(a\) denotes an element in the set of edges \(A\) of the quiver \(Q\). The group \(\text{GL}_n(\overline k)\) acts on the space \(R(Q,\overline k)\) by \(g\cdot(y_a)_{a\in A}=(g_{t(a)}y_ag^{-1}_{h(a)})_{a\in A}\) for \(g=(g_1,\dots,g_n)\in\text{GL}_n(\overline k)\). This action is extended to the coordinate ring by the rule \(Y_{\overline k}(a)\to g^{-1}_{t(a)}Y_{\overline k}(a)g_{h(a)}\) for all \(a\in A\), where \(Y_{\overline k}(a)\) denotes the generic matrix \((y_{i,j}(a))_{1\leq j\leq k_{h(a)}, 1\leq i\leq k_{t(a)}}\). A precise description of generators for the finitely generated algebra of invariants \(K[R(Q,\overline k)]^{\text{GL}_n(\overline k)}\) was obtained by Donkin. The main result of the paper provides an extension of Donkin's result to the free algebra of invariants with countable number of generators. quivers; representations; invariants Zubkov, A.N.: The Razmyslov--Procesi theorem for quiver representations. Fundam. Prikl. Mat. 7(2), 387--421 (2001) (Russian) Representations of quivers and partially ordered sets, Trace rings and invariant theory (associative rings and algebras), Vector and tensor algebra, theory of invariants, Group actions on varieties or schemes (quotients) The Procesi-Razmyslov theorem for quiver representations | 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 text under review grew out of a series of lectures given by the author at the University of Bordeaux, France, during the 2000/2001 academic year. The author's aim is to provide the reader with a systematic introduction to the theory of constructible sheaves in modern algebraic topology and its various applications to singular complex varieties, and that in a comprehensible, concise and user-friendly way. Constructible sheaves, in their abstract categorical setting, appeared in algebraic topology at an early stage, first as local systems of coefficients, and were then developed, in the 1970s, as an important tool in the advanced study of the topology of (singular) complex spaces. Although the vast amount of spectacular applications of abstract sheaves and their cohomology has become sheerly overwhelming, during the last decades, and in spite of the existence of a number of excellent books devoted to this subject, many topologists and geometers seem to be still hesitant to adopt this utmost powerful theory. Admittedly, there is quite a huge formalism behind the modern abstract approach to sheaf theory, ranging from derived categories to perverse sheaves, but the recent developments in algebraic topology, algebraic geometry, complex analytic geometry, micro-local analysis, and mathematical physics can barely be followed up without a profound knowledge of it, on the other hand.
The book under review covers most of the basic notions and results in the theory of constructible sheaves on complex spaces, together with numerous concrete geometric examples and applications. In addition, and along this path, the text does the service of filling the yawning gap between the very complete and highly general monographs [such as the books by \textit{M. Kashiwara} and \textit{P. Shapira}, ``Sheaves on manifolds'' (Berlin 1990; Zbl 0709.18001), \textit{A. A. Beilinson}, \textit{J. Bernstein} and \textit{P. Deligne}, ``Faisceaux pervers'', Astérisque 100 (1982; Zbl 0536.14011) or \textit{J. Schürmann}: ``Topology of singular spaces and constructible sheaves'' (Basel 2003; Zbl 1041.55001)], a number of recent survey articles on the subject, and some related research papers.
As for the approach chosen in the present introductory treatise, the author tries to take the reader from the simpler, meanwhile classical parts of the theory up to some most recent, powerful and general results, which represent the forefront of research in this realm. In this vein, the reader will meet a number of theorems that appear, step-by-step, in increasing generality and applicability. This methodological strategy is very pleasant, in that it makes the text overall enlightening and instructive, without interrupting its steady flow, but it also forces the author to omit most of the proofs in the first five chapters. As the author points out, this choice of his is not only motivated by the goal to keep the text floating, digestible and concise, but also by the fact that there are enough excellent sources that the reader could (and should) consult for looking up those details. Among those references of nearly encyclopaedic character are the books by \textit{M. Kashiwara} and \textit{P. Shapira} and \textit{J. Schürmann} (loc. cit), which the present text frequently refers to.
However, some results come with full proofs, mainly those that seemed new to the author, and most examples and applications have been provided with full details and ample explanations.
The text consists of six chapters, each of which is divided into several sections.
Chapter 1 gives a brief survey of the theory of derived categories, according to J.-L. Verdier and his successors. Chapter 2 is entitled ``Derived categories in topology'' and treats generalities on sheaves, derived tensor products, direct and inverse images of sheaves, sheaf cohomology, the adjunction triangle in cohomology, and the basics on local systems (of coefficients) as needed for the construction of perverse sheaves later on.
Chapter 3 discusses the framework of Poincaré-Verdier duality and, in particular, Poincaré and Alexander duality in algebraic topology, together with an exemplification of Borel-Moore homology and a number of cohomological vanishing theorems (after Deligne and Esnault-Viehweg) for later use.
Throughout the text, the author clearly separates the categorical, topological and analytic aspects from each other, thereby helping the reader to grasp the power of the general approach to sheaves.
Chapter 4 turns to the theory of constructible sheaves. Apart from their definition and basic properties, the author explains the triangulated category of bounded constructible complexes on a complex algebraic variety, discusses the functors of nearby and vanishing cycles, relates them to the classical theory of singularities, and he ends this already more advanced chapter with the description of the characteristic variety and the characteristic cycle associated with a constructible sheaf on a smooth manifold. This includes many applications to the topology of manifolds and micro-local analysis.
The theory developed so far culminates in the study of perverse sheaves. This is done in chapter 5, where the reader gets acquainted with the formalism of \(t\)-structures, \(p\)-perverse sheaves, the category of perverse sheaves on a variety, a special case of the Artin vanishing theorem, the fundamentals of the theory of \(D\)-modules and the role of perverse sheaves in there, the Riemann-Hilbert correspondence in a special case, and the basics of intersection (co-)homology after Goresky-MacPherson. In this context, the author also touches upon the corresponding Lefschetz-type theorems and some comparison theorems for intersection cohomology and classical cohomology.
Chapter 6, the concluding part of the book, is the longest and most original chapter of the entire treatise. The author discusses various applications of the general theory of perverse sheaves to concrete geometric situations. This chapter offers old and new results likewise, mainly with a view to hypersurface singularities. Being one of the leading experts in singularity theory, the author (re-)considers the Milnor fibers and the monodromy of isolated singularities, the topology of deformations, the topology of polynomial functions, and the geometry of hyperplane and hypersurface arrangements. This chapter appears very elaborated and detailed, with full proofs and numerous important examples, and it provides many new results, improvements of old results, and comparisons to earlier approaches. Most of the main results here involve properties of constructible or perverse sheaves in an essential way, pointing out the unifying character, elegance, utility, and ubiquity of their theory.
There are also many exercises scattered throughout the text, which are to add useful details to the main text. These exercises mostly come with detailed hints and references for suitable further reading.
No doubt, this is a highly enlightening inviting and useful book. Written in a very lucid, detailed and rigorous style, with an abundance of concrete examples and applications, it serves its afore-mentioned purpose in a perfect way.
Also experts in the field will find quite a bit of novelties in there, especially in the last chapter, and even physicists might profit from studying this largely introductory text. sheaves; constructible sheaves; perverse sheaves; sheaf cohomology; derived categories; triangulated categories; intersection homology; singularities; \(D\)-modules; Poincaré-Verdier duality; vanishing theorems; intersection cohomology; hypersurface singularities A.~Dimca. {\em Sheaves in topology}. Universitext. Springer-Verlag, Berlin, 2004. zbl 1043.14003; MR2050072 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Vanishing theorems in algebraic geometry, Singularities in algebraic geometry, Topology of analytic spaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Generalizations of fiber spaces and bundles in algebraic topology Sheaves in topology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper the authors compute explicit closed formulae for Donaldson-Thomas type invariants of local elliptic surfaces with sections. Consider \(S\to B\) a non-trivial elliptic surface over a smooth projective curve, which admits sections and such that all singular fibers are irreducible rational nodal curves and denote by \(X=\mbox{Tot}(K_S)\) the canonical bundle of \(S\). For an effective curve class \(\beta\) in \(S\), let
\[
\mbox{Hilb}^{\beta,n}(X)=\{Z\subset X: [Z]=\beta, \chi(\mathcal{O}_Z)=n \}
\]
be the Hilbert scheme of proper subschemes \(Z\subset X\) with homology class \(\beta\) and holomorphic Euler characteristic \(n\). In this paper, two kind of DT-type invariants are considered. The first one is
\[
\mathrm{DT}_{\beta,n}(X)=e(\mathrm{Hilb}^{\beta,n}(X), \nu)=\sum_{k\in \mathbb{Z}}k e(\nu^{-1}(k))
\]
where \(e(\cdot)\) denotes topological Euler characteristic and \(\nu: \mathrm{Hilb}^{\beta,n}(X)\to \mathbb{Z}\) is Behrend constructible function [\textit{K. Behrend}, Ann. Math. (2) 170, No. 3, 1307--1338 (2009; Zbl 1191.14050)]. The second one is its unweighted version
\[
\widehat{\mathrm{DT}}_{\beta,n}(X)=e(\mathrm{Hilb}^{\beta,n}(X))
\]
After fixing a section \(B\subset S\) the authors investigate the case of classes \(\beta=B+dF\) and \(\beta=dF\), where \(B,F\) denote the class of the section and of the fiber, respectively. Define the partition functions
\[
\mathrm{DT}(X)=\sum_{d=0}^{\infty}\sum_{n\in \mathbb{Z}}\mathrm{DT}_{B+dF,n}(X)p^nq^n
\]
\[
\mathrm{DT}_{\mathrm{fib}}(X)=\sum_{d=0}^{\infty}\sum_{n\in \mathbb{Z}}\mathrm{DT}_{dF,n}(X)p^nq^n
\]
and similarly \( \widehat{\mathrm{DT}}(X), \widehat{\mathrm{DT}}_{\mathrm{fib}}(X)\). The main results of the paper are the closed formulae
\[
\widehat{\mathrm{DT}}(X)=\left(M(p)\prod_{d=1}^\infty\frac{ M(p,q^d)}{1-q^d} \right)^{e(S)}\left(\frac{1}{p^{1/2}-p^{-1/2}}\prod_{d=1}^\infty\frac{(1-q^d)}{(1-pq^d)(1-p^{-1}q^d)} \right)^{e(B)}
\]
\[
\widehat{\mathrm{DT}}_{\mathrm{fib}}(X)=\left(M(p)\prod_{d=1}^\infty M(p,q^d) \right)^{e(S)}\left(\prod_{d=1}^\infty\frac{1}{(1-q^d)} \right)^{e(B)}
\]
Moreover assuming a certain conjecture, the two invariants are related by \( \mathrm{DT}(X)=(-1)^{\chi(\mathcal{O}_S)}\widehat{\mathrm{DT}}(X)\) and \( \mathrm{DT}_{\mathrm{fib}}(X)=\widehat{\mathrm{DT}}_{\mathrm{fib}}(X)\) after a natural change of variable.
The formula for \(\widehat{\mathrm{DT}}_{\mathrm{fib}}(X) \) has already been proven with wall-crossing methods by Toda. Specializing to the case of elliptically fibered K3 surfaces, the authors recover Katz-Klemm-Vafa formula for primitive classes, independently of the pre-existing proof using Kawai-Yoshioka formula.
To achieve the result, motivic and toric methods are combined. By localization, it is enough to study its fixed locus, which is accurately described combinatorially by a stratification argument. Each of these strata is reduced then to Quot schemes of \(\mathbb{C}^3\), where localization theorem allows to write everything in term of the topological vertex. Donaldson-thomas invariants; Hilbert schemes Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Donaldson-Thomas invariants of local elliptic surfaces via the topological vertex | 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 some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces.
We introduce a motivic Chern class transformation \(mC_{y}: K_{0}\)(var/\(X\)) \(\rightarrow G_{0}(X) \otimes \mathbb Z[y]\), which generalizes the total \(\lambda \)-class \(\lambda _{y}(T^*X)\) of the cotangent bundle to singular spaces. Here \(K_{0}\)(var/\(X\)) is the relative Grothendieck group of complex algebraic varieties over \(X\) as introduced and studied by Looijenga and Bittner in relation to motivic integration, and \(G_{0}(X)\) is the Grothendieck group of coherent sheaves of \(\mathcal O_X\)-modules. A first construction of \(mC_{y}\) is based on resolution of singularities and a suitable ``blow-up'' relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mC \(_{y}\) is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito. We define a natural transformation \(T_{y^*} : K_{0}\)(var/\(X\)) \(\rightarrow H_{*}(X) \otimes \mathbb Q[y]\) commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. \(T_{y^*}\) is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for \(y = -1\)), the Todd class transformation in the singular Riemann-Roch theorem of Baum-Fulton-MacPherson (for \(y = 0\)) and the L-class transformation of Cappell-Shaneson (for \(y = 1\)). We also explain the relation among the ``stringy version'' of our characteristic classes, the elliptic class of Borisov-Libgober and the stringy Chern classes of Aluffi and De Fernex-Lupercio-Nevins-Uribe. All our results can be extended to varieties over a base field \(k\) of characteristic 0. characteristic classes; characteristic number; genus; singular space; motivic; additivity; Riemann-Roch; Grothendieck group; cobordism group; Hodge structure; mixed Hodge module Brasselet, J-P; Schürmann, J.; Yokura, S., Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal., 2, 1-55, (2010) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Riemann-Roch theorems, Global theory and resolution of singularities (algebro-geometric aspects), Global theory of complex singularities; cohomological properties, Mixed Hodge theory of singular varieties (complex-analytic aspects) Hirzebruch classes and motivic Chern classes for singular 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 Singularities of Gauss maps are studied for algebraic hypersurfaces of complex projective 4-space. Such a hypersurface \(M\subset {\mathbb{P}}^ 4\) is called \textit{Gauss-stable} if, for every point \(x\in M\), the germ at x of the Gauss map \(\gamma: M\to {\mathbb{P}}^{4*}\) is a stable Legendre map germ. It is proved that the set of Gauss-stable hypersurfaces of degree \(d\) is a nonempty Zariski open subset of the space of all smooth degree \(d\) hypersurfaces in \({\mathbb{P}}^ 4\). The singularities of the Gauss map of a Gauss-stable hypersurface are classified, and this classification is related to the differential geometry of the second fundamental form. The Thom-Boardman symbols of these singularities are \(\Sigma ^ 1,\Sigma ^ 2,\Sigma ^{1,1},\Sigma ^{1,1,1}\). They correspond, respectively, to singularities of tangent hyperplane sections of M of types \(A_ 2,D_ 4,A_ 3,A_ 4\). The degree of each singularity type is computed. For example, the number of points of a degree \(d\) Gauss-stable hypersurface at which the tangent hyperplane section has an \(A_ 4\) singularity is \(5d(d-2)(3d-7)(17d-36)\). The geometry of the curve of \(A_ 3\) points is studied in detail, as well as the Fano surface of the cubic threefold and its relation to the singularities of the Gauss map. singularities of the Gauss map; Gauss-stable hypersurface; Fano surface of the cubic threefold Clint McCrory, Theodore Shifrin, and Robert Varley, The Gauss map of a generic hypersurface in \?\(^{4}\), J. Differential Geom. 30 (1989), no. 3, 689 -- 759. \(3\)-folds, Global submanifolds, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties The Gauss map of a generic hypersurface in \({\mathbb{P}}^ 4\) | 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, \(A\) be a finitely generated associative \(k\)-algebra and \(\text{Rep}_A[n]\) be the functor Fields\(_k \to\) Sets, which sends a field \(K\) containing \(k\) to the set of isomorphism classes of representations of \(A_K\) of dimension at most \(n\). We study the asymptotic behavior of the essential dimension of this functor, i.e., the function \(r_A(n) := \text{ed}_k (\text{Rep}_A [n])\), as \(n \to \infty\). In particular, we show that the rate of growth of \(r_A(n)\) determines the representation type of \(A\). That is, \(r_A(n)\) is bounded from above if \(A\) is of finite representation type, grows linearly if \(A\) is of tame representation type, and grows quadratically if \(A\) is of wild representation type. Moreover, \(r_A(n)\) allows us to construct invariants of algebras which are finer than the representation type. representation type; essential dimension; quivers; algebraic stacks; gerbes Representation type (finite, tame, wild, etc.) of associative algebras, Representations of quivers and partially ordered sets, Stacks and moduli problems Essential dimension of representations of 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 [For the entire collection see Zbl 0723.00028.]
Elliptic complete intersection singularities are studied and classified by equations using the results of \textit{H. B. Laufer} [Am. J. Math. 99, 1257-1295 (1977; Zbl 0384.32003)] and \textit{Reid}: They are hypersurfaces of multiplicity 2 or 3 in \(\mathbb{C}^ 3\) or intersections of two hypersurfaces of multiplicity 2 in \(\mathbb{C}^ 4\). --- The modality is computed.
The quasihomogeneous case is specially created. complete intersection singularities; multiplicity; modality ---, Elliptic complete intersection singularities, In: Singularity theory and its ap- plications, Part I, Warwick 1989 (D. Mond and J. Montaldi, eds.), Lecture Notes in Math., 1462, Springer, Berlin etc., 1991, pp. 340-372. i i i i Singularities in algebraic geometry, Complete intersections Elliptic complete intersection 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 recent years, multiview varieties have been extensively studied from the viewpoint of algebraic geometry. A multiview variety \(X\) is the Zariski closure of all images from a sequence of \(n\) ``cameras'' which are typically projective maps onto low dimensional image spaces (projective planes in the classical scenary of photogrammetry or computer vision). The aim is to use knowledge about \(X\) for the reconstruction of the cameras, up to projective transformation, from measured data on~\(X\).
In this paper, the authors prove that the reconstruction is unique unless one uses \(n+1\) cameras onto images spaces of dimension one in which case two solutions exist. This is a known result by \textit{R. I. Hartley} and \textit{F. Schaffalitzky} [Lect. Notes Comput. Sci. 3021, 363--375 (2004; Zbl 1098.68775)] but with a new algebro-geometric proof. The authors also prove that the map that sends a camera configuration to the multiview variety is dominant onto an irreducible component of the Hilbert schemes of projective subschemes. computer vision; algebraic vision; multiview geometry; projective reconstruction; Hilbert scheme Geometric aspects of numerical algebraic geometry, Rational and birational maps, Machine vision and scene understanding, Numerical aspects of computer graphics, image analysis, and computational geometry, Parametrization (Chow and Hilbert schemes) Projective reconstruction in algebraic vision | 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 0575.00008.]
The classification of singular points which can occur on normal quartic surfaces has been known over seventy years [\textit{C. Jessop}, ''Quartic surfaces with singular points'' (Cambridge 1916)]. Still unknown are all possible combinations of the singularities. The author gives a partial answer to this problem by applying the theory of \textit{E. Looijenga} of rational surfaces with effective anti-canonical divisor [Ann. Math., II. Ser. 114, 267-322 (1981; Zbl 0509.14035)].
Let \(\pi: \bar X\to X\) be a minimal resolution of a normal quartic surface X. Then X is either a K3-surface, or a rational surface, or a ruled irrational surface. In the first case all singularities of X are double rational points. The author considers the second case in which X has one minimal elliptic singularity and double rational points. The fundamental cycle D of the elliptic singularity represents the anticanonical class \(-K_{\bar X}\). By technical reasons the author restricts himself to the case D is irreducible and \(D^ 2=-1\), \(-2\), or \(-3\). This corresponds to either simple elliptic singularities of type \(\tilde E_ 8, \tilde E_ 7, \tilde E_ 6\), or cusp singularities \(T_{2,3,7}, T_{2,4,5}, T_{3,3,4}\), or unimodal exceptional singularities \(E_{12}, Z_{11}, Q_{10}\). Let L be the orthogonal complement of \(K_{\bar X}\) in Pic\((\bar X)\) and let \(\lambda =\pi^*(H)\) where H is a plane section of X. Then \(\lambda\in L\), \(\lambda^ 2=4\) and \(\lambda \cdot f>2\) for every isotropic vector f in L. The exceptional curves of double rational singularities of X correspond to irreducible components of the finite root system in the sublattice \(L'=Ker(res:\;L\to Pic(D)).\) Together with Looijenga's theorem of Torelli type for rational surfaces this allows the author to reduce the problem to the classification of the vectors \(\lambda_ 0\) as above in the abstract lattice \(L_ 0\) isomorphic to L and symmetric root systems in the sublattice \(L_ 0/{\mathbb{Z}}\lambda_ 0\) of \(({\mathbb{Z}}\lambda_ 0)^{\perp}\otimes {\mathbb{Q}}\). The latter are derived by elementary transformations known in the Lie theory.
The following is an example of the results obtained in the paper: Assume that X has a singularity \(\tilde E_ 8\). Then its other singularities are double rational points corresponding to the Dynkin diagrams obtained from the Dynkin diagram of either type \(B_ 9\) or type \(E_ 8\) by elementary operations repeated twice such that the resulting set of vertices has no vertex associated to the short simple root.
The same approach is applied to the classification of singularities of plane sextics by considering the double cover of the plane branched over the sextic.
Note that the classification of singular quartic surfaces with triple points was given by more direct computational method in the works of \textit{T. Takahashi, K. Watanabe} and \textit{T. Higuchi} [Sci. Rep. Yokohama Natl. Univ., Sect. I 29, 47-70; 71-94 (1982; Zbl 0586.14030)]. Dynkin graphs; minimal resolution of a normal quartic surface; classification of singularities of plane sextics; classification of singular quartic surfaces Urabe, T.: Singularities in a certain class of quartic surfaces and sextic curves and Dynkin graphs. Proc. 1984 Vancouver Conf. Alg. Geom., CMS Conf. Proc.6, 477-497 (1986) Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Singularities in algebraic geometry Singularities in a certain class of quartic surfaces and sextic curves and Dynkin graphs | 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 Linear free divisors are free divisors, in the sense of K. Saito, with linear presentation matrix. Using techniques of deformation theory on representations of quivers, we exhibit families of such linear free divisors as discriminants in representation varieties for real Schur roots of a finite quiver. Along the way we review some basic results on representation varieties of quivers, their associated fundamental exact sequence and semi-invariants; explain in detail how to verify the occurring discriminant as a free divisor and how to determine its components and their equations. As an illustration, the linear free divisors that arise as the discriminant from the highest root of a Dynkin quiver are treated explicitly. Ragnar-Olaf Buchweitz and David Mond, Linear free divisors and quiver representations, Singularities and computer algebra, London Math. Soc. Lecture Note Ser., vol. 324, Cambridge Univ. Press, Cambridge, 2006, pp. 41 -- 77. Formal methods and deformations in algebraic geometry, Representations of quivers and partially ordered sets, Deformation of singularities, Divisors, linear systems, invertible sheaves, Actions of groups on commutative rings; invariant theory Linear free divisors and quiver representations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a lecture note on singularity theory. It is shown how to associate to a triple of positive integers \((p_1,p_2,p_3)\) a two-dimensional isolated graded singularity by an elementary procedure that works over any field \(k\) (with no assumptions on characteristic, algebraic closedness or cardinality). This assignment starts from the triangle singularity \(x_1^{p_1} + x_2^{p_2} + x_3^{p_3}\) and when applied to the Platonic (or Dynkin) triples, it produces the famous list of A-D-E-singularities. As another particular case, the procedure yields Arnold's famous strange duality list consisting of the 14 exceptional unimodular singularities (and an infinite sequence of further singularities not treated here in detail). It is shown that weighted projective lines and various triangulated categories attached to them play a key role in the study of the triangle and associated singularities. weighted projective line; (extended) canonical algebra; simple singularity; Arnold's strange duality; stable category of vector bundles Lenzing, H., Rings of singularities, Bull. Iranian Math. Soc., 37, 2, 235-271, (2011) Singularities of surfaces or higher-dimensional varieties, Representations of quivers and partially ordered sets, Cohen-Macaulay modules in associative algebras Rings of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under review studies Hilbert schemes of points of the total spaces \(X_{n}\) of line bundles \(\mathcal{O}_{\mathbb{P}^{1}}(-n)\) in terms of ADHM data and realizes them as irreducible connected components of moduli spaces of quiver representations. The surfaces \(X_{n}\) are minimal resolutions of toric singularities of type \(\frac{1}{n}(1,1)\) having Hizebruch surfaces \(\Sigma_{n}\) as projective compactifications, connecting with string theory in physics.
In the paper, the authors construct ADHM data for the Hilbert schemes \(\mathrm{Hilb}^{c}(X_{n})\), by going through the description of moduli spaces of framed sheaves of Hirzebruch surfaces \(\Sigma_{n}\), \(\mathcal{M}^{n}(1,0,n)\simeq \mathrm{Hilb}^{c}(X_{n})\). This ADHM data turns out to provide a principal bundle over the Hilbert scheme (c.f. Theorem 3.1).
In section \(4\), it is shown how these Hilbert schemes are irreducible connected components of GIT quotients of representation spaces of certain quivers for a suitable choice of the stability parameter (c.f. Theorem 4.5), therefore they can be seen as embedded components into quiver varieties associated, in a natural way, with ADHM data of the Hilbert schemes.
It is worth noting that the quiver varieties of this article fall outside the notion of a Nakajima quiver variety carrying naturally a simplectic structure, while \(\mathrm{Hilb}^{c}(X_{n})\) encodes a Poisson structure in general. The authors propose to study further the wall-crossing of the stability parameters to shed light on questions in geometric representation theory, and also to study the Poisson structure of these spaces. Hilbert schemes of points; quiver varieties; HIrzebruch surfaces; ADHM data; monads; Nakajima quivers; McKay quivers; quiver varieties, moduli spaces of quiver representations Bartocci, C.; Bruzzo, U.; Lanza, V.; Rava, C.L.S., Hilbert schemes of points of \(\mathcal{O}_{\mathbb{P}^1}(- n)\) as quiver varieties, J. pure appl. algebra, 221, 2132-2155, (2017) Algebraic moduli problems, moduli of vector bundles, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Representations of quivers and partially ordered sets Hilbert schemes of points of \(\mathcal{O}_{\mathbb{P}^1}(- n)\) as quiver 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 \(f\) be a polynomial in \(n\) variables \(x_1\),\dots, \(x_n\) over a field \(k\) with \(f(0)=0\) and let \(k[[t]]\) be the ring of formal power series in one variable over \(k\). The arc space of the germ of algebraic variety \((X,0)\) defined by \(f\) (i.e. \(X=\{(x_1,\dots,x_n)\in k^n\;/\;f(x_1,\dots,x_n)=0\}\)) is
\[
X_{\infty}:=\{(x_1(t),\dots,x_n(t))\in k[[t]]^n\;/ \;f(x_1(t),\dots,x_n(t))=0 \;\text{ and } x_i(0)=0, 1\leq i\leq n\}.
\]
By expanding \(f(x(t))\) as a power series in \(t\) gives
\[
f(x(t))=F_1t+F_2t^2+F_3t^3+\cdots
\]
where the \(F_i\) are polynomials in the coefficients of \(t\) in \(x(t)\). Thus \(x(t)\in X_{\infty}\) if and only if the set of its coefficients is a solution of the equations
\[
F_1=F_2=F_3=\cdots=0.
\]
Therefore \(X_{\infty}\) is a provariety, i.e., an algebraic variety defined by an infinite number of equations depending on an infinite number of variables. The coordinate algebra of \(X_{\infty}\) is the ring
\[
J_{\infty}(X):=k[x_j^{(i)}; 1\leq j\leq n, i\in\mathbb{N}_{>0}]/(F_1,F_2,\cdots).
\]
This algebra has a natural grading given by assigning the weight \(i\) to the variables \(x_1^{i}\),\dots, \(x_n^{i}\) that makes the polynomials \(F_l\) homogeneous of weight \(l\).
The authors of the paper under review introduce a new object, the Hilbert-Poincaré series of \((X,0)\), which is defined by
\[
HP_{J_{\infty}(X)}(t)=\sum_{j=0}^{\infty}\dim_k\left(J_{\infty}(X)\right)_j.t^j,
\]
where \(\left(J_{\infty}(X)\right)_j\) denotes the \(j\)-th homogeneous component of \(J_{\infty}(X)\).
This generating series is quite complicated to compute, even in basic cases. For instance, if \(X=\mathbb{A}^1\) is the affine line, then
\[
HP_{J_{\infty}(\mathbb{A}^1)}(t)=\prod_{i\geq 1}\frac{1}{1-t^i}.
\]
In general the Hilbert-Poincaré series of \((X,0)\) coincides with the Hilbert-Poincaré series of the algebra
\[
k[x_j^{(i)}; 1\leq j\leq n, i\in\mathbb{N}_{>0}]/L(I)
\]
where \(L(I)\) is the leading ideal of \(I=(F_1,F_2,\cdots)\) (with respect to a well chosen monomial ordering). The leading ideal is simpler than \(I\) since it is a monomial ideal. Thus for computing Hilbert-Poincaré series one need to compute \(L(I)\) and the Hilbert-Poincaré series of \(L(I)\). Computing \(L(I)\) is not easy in general since it is not a finitely generated ideal. Computing the Hilbert-Poincaré series of \(L(I)\) corresponds to counting partitions, leaving out those coming from weights of monomials in \(L(I)\). But even in basic cases, counting the corresponding partitions is not easy. For instance, in the case \(n=1\) and \(f=x_1^2\), the partitions appearing in this computation are the partitions without repeated or consecutive terms. One of the mains results obtained by the authors is the following (proved by using the Rogers-Ramanujan identity):
\[
HP_{J_{\infty}(x_1^2=0)}=\prod_{ i\geq 1,\;i\equiv 1,4 \text{ mod }5}\frac{1}{1-t^i}.
\]
The rest of the paper is devoted to compute the Hilbert-Poincaré series of some particular germs of algebraic varieties. arc spaces; Hilbert-Poincaré series; Rogers-Ramanujan identities; Gröbner bases Bruschek, C.; Mourtada, H.; Schepers, J., Arc spaces and the Rogers-Ramanujan identities, Ramanujan J., 30, 1, 9-38, (2013) Singularities in algebraic geometry, Partition identities; identities of Rogers-Ramanujan type, Combinatorial aspects of partitions of integers, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Arc spaces and the Rogers-Ramanujan identities | 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{K. Kato} proved a formula which compares the dimension of the space of vanishing cycles in a finite morphism between formal germs of curves over a complete discrete valuation ring [Duke Math. J. 55, 629--659 (1987; Zbl 0665.14005)]. Kato's formula is explicit only in the case where the morphism in question is generically separable on the level of special fibres.
In the paper under review, using formal patching techniques à la Harbater, the author proves an analogous explicit formula in the case of a Galois cover of degree \(p\) between formal germs of curves over a complete discrete valuation ring of unequal characteristic \((0,p)\). This formula includes the case when one has inseparability on the level of special fibres and has applications in the study of semi-stable reduction of Galois covers of curves. finite morphism; Galois cover; formal germs of curves; complete discrete valuation ring Saïdi, M.: Wild ramification and a vanishing cycles formula. J. algebra 273, No. 1, 108-128 (2004) Families, moduli, classification: algebraic theory, Families, moduli of curves (algebraic), Singularities in algebraic geometry Wild ramification and a vanishing cycles formula. | 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 confirm a conjecture of \textit{C. Li} [Math. Z. 289, No. 1--2, 491--513 (2018; Zbl 1423.14025)] which says that the minimizer of the normalized volume function for a klt singularity is unique up to rescaling. They also give applications such as a finite degree formula for normalized volumes and an effective bound on the order of the local fundamental group of the germ of a klt singularity. normalized volume; klt singularity; local fundamental group Singularities in algebraic geometry, Valuations and their generalizations for commutative rings, Minimal model program (Mori theory, extremal rays) Uniqueness of the minimizer of the normalized volume function | 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 considers the divisor of order 12 of inflection points of the tacnodal quartic curve showing that it is determined by a cubic curve only under strong conditions. inflection points of the tacnodal quartic curve Dalibor Klucký, Libuše Marková: A contribution to the theory of tacnodal quartics. Časopis pro p\?stování matematiky, roč. 110 (1985), str. 92-100. Singularities of curves, local rings, Singularities in algebraic geometry A contribution to the theory of tacnodal quartics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper fills in a gap in the proof of the Iskovskikh's theorem describing the group of birational automorphisms of the Fano variety \(V^ 3_ 6\) (i.e. a smooth complete intersection of a quadric and a cubic). To do this, one shows that, in certain conditions (satisfied for general \(V^ 3_ 6)\), there is no infinitely close maximal singularity on \(V^ 3_ 6\). group of birational automorphisms; Fano variety; maximal singularity A. V. Pukhlikov, ``Maximal singularities on the Fano variety \(V^3_6\)'', Moscow Univ. Math. Bull., 44:2 (1989), 70 -- 75 Fano varieties, Birational automorphisms, Cremona group and generalizations, Singularities in algebraic geometry Maximal singularities on the Fano variety \(V^ 3_ 6\) | 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 concept of a stability condition on a triangulated category \(\mathcal{T}\) was introduced by \textit{T. Bridgeland} [Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)]. This notion can be seen as a generalisation of classical \(\mu\)-stability for vector bundles on curves. Bridgeland proved that under some technical assumptions the space of stability conditions is a finite-dimensional complex manifold, denoted by \(\text{Stab}(\mathcal{T})\). In general we do not know much about the geometry of this manifold, for example whether it is connected. However, quite a few (partial) results exist in some specific cases, an important one being the case where \(\mathcal{T}\) is the bounded derived category of a smooth projective \(K3\) surface \(X\) and a distinguished connected component of \(\text{Stab}({ D}^{ b}(X))\) has been identified \textit{T. Bridgeland} [Duke Math. J. 141, No. 2, 241--291 (2008; Zbl 1138.14022)]. An open conjecture, which would provide information about the group of autoequivalences of \({\text D}^{\text b}(X)\), claims (in particular) that this connected component is in fact simply-connected.
In the paper under review, the authors investigate stability conditions on \(A_n\)-singularities. To be more precise, let \(f: X \rightarrow Y=\text{Spec}\, \mathbb{C}\left[x,y,z\right]/(xy+z^{n+1})\) be the minimal resolution of the \(A_n\)-singularity. We consider \(\mathcal{D}\), the bounded derived category of coherent sheaves on \(X\) supported at the exceptional set \(Z=C_1 \cup \ldots \cup C_n\), and \(\mathcal{C}\), its full triangulated subcategory consisting of objects \(E\) satisfying \(Rf_*E=0\). The main theorem of the paper states that \(\text{Stab}(\mathcal{C})\) is connected and that \(\text{Stab}(\mathcal{D})\) is connected and simply-connected. This can be seen as evidence for the above mentioned conjecture, since the categories \(\mathcal{C}\) and \(\mathcal{D}\) are local models for derived categories of \(K3\) surfaces.
The authors first prove the connectedness of \(\text{Stab}(\mathcal{D})\). The strategy is as follows. Denote by \(U\) the set consisting of stability conditions such that skyscraper sheaves of closed points \(x \in Z\) are stable. One proves that \(U\) is connected. Furthermore, the connected component containing \(U\) is preserved by the action of \(\text{Br}(\mathcal{D})\), the group of autoequivalences generated by the spherical twists associated to the sheaves \(\mathcal{O}_{C_i}(-1)\) and the dualizing sheaf of \(Z\). The authors then show that any other connected component has to contain a certain stability condition \(\sigma\) with the property that \(\Phi \sigma \in U\) for some \(\Phi \in \text{Br}(\mathcal{D})\), and this concludes the proof.
In the next step the authors prove that the affine braid group action on \(\mathcal{D}\) is faithful. To be slightly more precise, the authors prove homological mirror symmetry for \(A_n\)-singularities (in particular, the McKay correspondence is used), then establish the faithfulness in characteristic two and finally lift the latter result to any characteristic. By a theorem of \textit{T. Bridgeland} [Int. Math. Res. Not. 2009, No. 21, 4142--4157 (2009; Zbl 1228.14012)] the faithfulness implies the simply-connectedness of \(\text{Stab}(\mathcal{D})\).
In the last section the connectedness of \(\text{Stab}(\mathcal{C})\) is established. In particular, homological mirror symmetry enters as the authors use topological arguments on the symplectic side to prove a certain technical statement on the algebraic side. In the appendix it is shown that any autoequivalence of \(\mathcal{D}\) is of Fourier--Mukai type. derived categories; stability conditions; \(A_n\)-singularities; homological mirror symmetry; McKay correspondence Ishii, Akira; Ueda, Kazushi; Uehara, Hokuto, Stability conditions on \(A_n\)-singularities, J. Differential Geom., 84, 1, 87-126, (2010) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Singularities in algebraic geometry, Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category Stability conditions on \(A_n\)-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 studies the singularities of jet schemes of homogeneous hypersurfaces of general type. We obtain the condition of the degree and the dimension for the singularities of the jet schemes to be of dense \(F\)-regular type. This provides us with examples of singular varieties whose m-jet schemes have rational singularities for every \(m\). Singularities in algebraic geometry, Arcs and motivic integration, Singularities of surfaces or higher-dimensional varieties Jet schemes of homogeneous hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author describes necessary conditions that must be satisfied by the generators of the group of birational automorphisms of a three- dimensional cubic. The conditions are described with respect to the ''maximal singularity'' of an automorphism of a three-dimensional cubic. The author shows that all the conditions described are effectively realised. generators of the group of birational automorphisms of a three- dimensional cubic; maximal singularity Rational and birational maps, Singularities in algebraic geometry, \(3\)-folds, Automorphisms of surfaces and higher-dimensional varieties On birational automorphisms of a three-dimensional cubic | 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 \((K,v)\) be a valued field, \(\bar K\) an algebraic closure of \(K\) and \(\bar v\) an extension of \(v\) to \(\bar K\). Let \((K^h,v^h)\) be the henselization of \((K,v)\) determined by the choice of \(\bar v\). The main result of this paper is the following theorem: Let \(\Lambda\) be a divisible ordered abelian group extending \(v(K^*)\). Consider the trees \(\mathcal{T}(K,\Lambda)\), \(\mathcal{T}(K^h,\Lambda)\) of all \(\Lambda-\)valued extensions of \(v\), \(v^h\) to \(K[x]\), \(K^h[x]\) respectively. Then the natural restriction mapping \(\mathcal{T}(K^h,\Lambda)\longrightarrow\mathcal{T}(K,\Lambda)\) is an isomorphism of posets. Similar results are proved for other situations. Henselization; key polynomial; valuation; valuative tree Valuations and their generalizations for commutative rings, General valuation theory for fields, Complete rings, completion, Global theory and resolution of singularities (algebro-geometric aspects) Rigidity of valuative trees under Henselization | 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 \({\mathcal M}_ g\) be the moduli space of stable curves of genus \(g\). The author first finds \(a={[(g^ 2-1)/4]}+3g-3\) generators for the Chow group \(A^ i_ \mathbb{Q}(\overline{{\mathcal M}}_ g-{\mathcal M}_ g)\) and uses this fact to show that \(a\) is an upper bound on the rank of the homology group \(H_{2(3g-3)-4}(\overline {\mathcal M}_ g-{\mathcal M}_ g,\mathbb{Q})\). The dual basis for \(H_ 4(\overline{\mathcal M}_ g)\) is seen to be algebraic. tautological class; intersection product; moduli space of stable curves; generators for the Chow group; rank of the homology group Edidin, D, The codimension-two homology of the moduli space of stable curves is algebraic, Duke Math. J., 67, 241-272, (1992) Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Étale and other Grothendieck topologies and (co)homologies The codimension-two homology of the moduli space of stable curves is algebraic | 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 \(G \subset \text{GL}(2,\mathbb{C})\) be a finite subgroup, and \(Y=G\text{-Hilb}(\mathbb{C}^2)\) be the Hilbert scheme of \(G\)-clusters \(\mathcal{Z}\subset \mathbb{C}^2 \). By definition \(\mathcal{Z}\) is a 0-dimensional subscheme of \(\mathbb{C}^2\) of length \(|G|\) such that \(H^0(\mathcal{O}_{\mathcal{Z}})\) is isomorphic to the regular representation of \(G\). It is known that \(Y\) is isomorphic to the minimal resolution of the singularity \( \mathbb{C}^2/G \).
The paper under review gives an explicit description of a natural affine open covering \(Y\) in the case that \(G\) is small binary dihedral group. The open covering is in bijection with the set of bases for \(H^0(\mathcal{O}_{\mathcal{Z}})\) which are called the \(G\)-graphs. All the possible \(G\)-graphs are explicitly calculated by interpreting the action of \(G\) as the cyclic action of its maximal normal abelian subgroup \(H\) of index 2 followed by a dihedral involution. As an application the classification of the \(G\)-graphs is used to list the special representations of any small binary dihedral group. \(G\)-graphs; special representations; binary dihedral groups; McKay correspondence Alvaro Nolla de Celis, \(G\)-graphs and special representations for binary dihedral groups in \(\mathrm{GL}(2,\mathbf{C})\). (to appear in Glasgow Mathematical Journal). McKay correspondence, Parametrization (Chow and Hilbert schemes) \(G\)-graphs and special representations for binary dihedral groups in \(\mathrm{GL}(2,\mathbb C)\) | 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 algebraic \((2n-1)\)-knot \(k\) is obtained as the link of an isolated singularity of a complex hypersurface in \(\mathbb{C}^{n+1}\), where the link is a topological \((2n-1)\)-sphere. In this paper the authors answer a question of Durfee: Are cobordant algebraic knots isotopic? For any \(n\geq 3\), the answer is no. The counter-examples are obtained by examining the Seifert forms of polynomials of the form \(g(z_ 0,z_ 1)+ z_ 2^ p+ z_ 3^ q+ z_ 4^ 2+\cdots+ z_ n^ 2\) for suitable \(g\), \(p\), \(q\), and using the Thom-Sebastiani theorem. Milnor fibre; isotopic knots; algebraic \((2n-1)\)-knot; link of an isolated singularity of a complex hypersurface; cobordant algebraic knots; Seifert forms Du Bois, Ph.; Michel, F.: Cobordism of algebraic knots via Seifert forms. Invent. math. 111, 151-169 (1993) Knots and links (in high dimensions) [For the low-dimensional case, see 57M25], Singularities in algebraic geometry, Hypersurfaces and algebraic geometry Cobordism of algebraic knots via Seifert forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth scheme of finite type over a field \(k\), equipped with a projective birational map \(\pi : X\rightarrow Y\) onto a normal irreducible affine scheme of finite type over \(k\). The main result of the paper under review asserts that if \(\text{char}\, k = 0\) and \(X\) is symplectic (i.e., it admits a non-degenerate closed 2-form \(\Omega \in H^0(X,{\Omega}^2_X)\)) then every point \(y\in Y\) has an étale neighborhood \(U_y \rightarrow Y\) such that there exists a vector bundle \(\mathcal E\) on the pullback \(X_y = X{\times}_YU_y\) which is a tilting generator of the derived category \(D^b_{\text{coh}}(X)\). The last assertion means that: (i) \({\text{Ext}}^i({\mathcal E},{\mathcal E}) = 0\), \(\forall i > 0\), and (ii) \(\forall {\mathcal F}^{\bullet} \in \text{Ob}\, D^-_{\text{coh}}(X_y)\), \({\text{RHom}}^{\bullet}({\mathcal E},{\mathcal F}^{\bullet}) \simeq 0\) implies \({\mathcal F}^{\bullet} \simeq 0\). In this case, \({\mathcal F}^ {\bullet} \mapsto {\text{RHom}}^{\bullet}({\mathcal E},{\mathcal F}^{\bullet})\) is an equivalence of categories \(D^b_{\text{coh}}(X_y) \rightarrow D^b_{\text{coh}}(R\text{-mod})\), where \(R = \text{End}({\mathcal E})\) and \(R\)-mod is the category of finitely generated left \(R\)-modules (which is abelian since it turns out that \(R\) is left noetherian).
The author also shows that if \({\pi}^{\prime} : X^{\prime} \rightarrow Y\) is another resolution of singularities and if the canonical bundles \(K_X\) and \(K_{X^{\prime}}\) are trivial, then every point \(y \in Y\) admits an étale neighborhood \(U_y \rightarrow Y\) such that \(D^b_{\text{coh}}(X_y)\) and \(D^b_{\text{coh}}(X^{\prime}_y)\) are equivalent.
The author proves these results by reduction to positive characteristic. He uses his results from \textit{D. Kaledin} [Math. Res. Lett. 13, No. 1, 99--107 (2006; Zbl 1090.53064)] about the existence of twistor deformations of line bundles on \(X\), and a quantization result in positive characteristic based on the techniques developed in \textit{R. Bezrukavnikov} and \textit{D. Kaledin} [J. Am. Math. Soc. 21, No. 2, 409--438 (2008; Zbl 1138.53067)]. Resolution of singularities; symplectic manifold; derived category; Poisson structure; quantized algebra; Frobenius endomorphism D. Kaledin, ''Derived equivalences by quantization,'' Geom. Funct. Anal., vol. 17, iss. 6, pp. 1968-2004, 2008. Global theory and resolution of singularities (algebro-geometric aspects), Poisson manifolds; Poisson groupoids and algebroids, Deformation quantization, star products, Derived categories, triangulated categories, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Derived equivalences by quantization | 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 Differential operators on smooth schemes have played a central role in the study of embedded desingularization. \textit{J. Giraud} [Math. Z. 137, 285--310 (1972; Zbl 0275.32003) and Ann. Sci. Éc. Norm. Supér. 8, 201--234 (1975; Zbl 0306.14004)] provided an alternative approach to the form of induction used by Hironaka in his desingularization theorem (over fields of characteristic zero). In doing so, Giraud introduced technics based on differential operators. This result was important for the development of algorithms of desingularization in the late 80's (i.e., for constructive proofs of Hironaka's theorem).
More recently, differential operators appear in the work of \textit{J. Wlodarczyk} [J. Am. Math. Soc. 18, No. 4, 779--822 (2005; Zbl 1084.14018)] and also on the notes of \textit{J. Kollár} [Lectures on resolution of singularities, Ann. Math. Stud. 166 (2007; Zbl 1113.14013)]. The form of induction used in Hironaka's desingularization theorem, which is a form of elimination of one variable, is called maximal contact. Unfortunately it can only be formulated over fields of characteristic zero. In this paper we report on an alternative approach to elimination of one variable, which makes use of higher differential operators. These results open the way to new invariants for singularities over fields of positive characteristic [\textit{O. Villamayor}, Adv. Math. 213, No. 2, 687--733 (2007; Zbl 1118.14016), preprint \url{arXiv:math.AG/0606796}]. Villamayor, O.: Differential operators on smooth schemes and embbeded singularities. Rev. Un. Mat. Argentina 46 (2005), no. 2, 1-18. Global theory and resolution of singularities (algebro-geometric aspects) Differential operators on smooth schemes and embedded 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 ultimately concerns one-point AG codes associated to so-called AMS curves. The authors show how to construct such codes using so-called \(\delta\)-sequences. These \(\delta\)-sequences introduced by the authors are finite sequences of integers satisfying certain arithmetic properties.
The paper does into some detail as to how they can arise naturally from curves, how they are related to weight or order functions, and develop their theory and classification. For their computation and practical use, they rely on the paper of \textit{D. Ruano} [J. Algebra 309, No. 2, 672--682 (2007; Zbl 1172.14308)].
This paper also explains how to construct (and compute the parameters of) the dual code of any evaluation code associated to a weight function defined by a \(\delta\)-sequences from the polynomial ring in two variables to a semigroup in \({\mathbb Z}^2\) or \({\mathbb R}\). A well-written and interesting paper. Galindo, C; Monserrat, F, \(\delta \)-sequences and evaluation codes defined by plane valuations at infinity, Proc. Lond. Math. Soc., 98, 714-740, (2009) Geometric methods (including applications of algebraic geometry) applied to coding theory, Singularities in algebraic geometry, Algebraic coding theory; cryptography (number-theoretic aspects) \({\delta}\)-sequences and evaluation codes defined by plane valuations at infinity | 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 ring \(R\) in characteristic \(p > 0\), a non-zero element \(f \subseteq R\) and a real number \(t \geq 0\), one can form the test ideal \(\tau(R, f^t)\) which is an ideal that measures both the singularities of \(R\) and \(R/f\). As \(t\) varies, these test ideals change, and the \(t\) at which the test ideal change are called \textit{\(F\)-jumping numbers} or \textit{\(F\)-thresholds}. Because these test ideals are closely related to multiplier ideals, it is hoped that these jumping numbers behave in a similarly to the jumping number of multiplier ideals (ie, are discrete and rational).
In [\textit{M. Blickle, M. Mustata, K. E. Smith}, Trans. Am. Math. Soc. 361, No. 12, 6549--6565 (2009; Zbl 1193.13003)], it was shown that these numbers are discrete and rational under the assumption that \(R\) is regular and \(F\)-finite (and thus excellent).
In the paper under review, the authors prove a generalization. They show that the \(F\)-jumping numbers are discrete and rational if \(R\) is a regular local (but possibly not \(F\)-finite) ring. The methods in this paper have some similarity to those of the aforementioned paper but on a key point they differ, this paper does not rely on the theory of \(D\)-modules. Also compare with [\textit{M. Blickle, M. Mustata, K. E. Smith}, Mich. Math. J. 57, 43--61 (2008; Zbl 1177.13013)]. test ideal; jumping number; F-threshold; multiplier ideal; Frobenius action; tight closure Katzman, M.; Lyubeznik, G.; Zhang, W., On the discreteness and rationality of \textit{F}-jumping coefficients, J. Algebra, 322, 9, 3238-3247, (2009) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry On the discreteness and rationality of \(F\)-jumping coefficients | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Recall that for a quasi-projective variety \(X\) over a field \(k\), an element of \(K_{0}(X)\) is defined to be algebraically equivalent to zero provided that it lies in the image of \(t_{0}^{*} - t_{1}^{*} : K_{0}(X \times T) \longrightarrow K_{0}(X)\) for some smooth, connected \(k\)-variety \(T\) with \(k\)-rational points \(t_{0}, t_{1}\). The authors denote by \(K_{0}^{\text{semi}}(X)\) the quotient of \(K_{0}(X)\) by the subgroup of elements algebraically equivalent to zero.
Now suppose that \(X\) is a weakly normal, quasi-projective complex variety. Using techniques from the theory of spaces which are ``algebras'' over \(E_{\infty}\)-operads (the \(E_{\infty}\)-operad in question here is a sort of complex analytic version of the classical linear isometries operad) the authors construct an infinite loopspace \({\mathcal K}^{\text{semi}}(X)\) having \(K_{0}^{\text{semi}}(X)\) as its space of components and sitting, as an infinite loopspace between the infinite loospace of Quillen algebraic K-theory \({\mathcal K}(X)\) and that of topological K-theory of the underlying analytic space \({\mathcal K}_{\text{top}}(X^{\text{an}})\). Just as \({\mathcal K}_{\text{top}}(X^{\text{an}})\) is related to Betti cohomology and \({\mathcal K}(X)\) is related to motivic cohomology via the Atiyah-Hirzebruch and Bloch-Lichtenbaum-Suslin spectral sequences, respectively, the authors conjecture that \({\mathcal K}^{\text{semi}}(X)\) is related to morphic cohomology. They make a number of conjectures about the relationships between these three types of K-theory and prove several of the standard K-theory properties (e.g. Mayer-Vietoris, projective bundle theorem, etc.) for \({\mathcal K}^{\text{semi}}(X)\).
This paper is one of a series by the authors on this topic and in a footnote they announce that they have been able, using work of Voevodsky-Suslin, to verify all their conjectures [\textit{E. M. Friedlander} and \textit{M. E. Walker}, Am. J. Math. 123, No.~5, 779-810 (2001; Zbl 1018.19001)]. K-theory; morphic cohomology; Segre classes; motivic cohomology Friedlander, Eric M.; Walker, Mark E., Semi-topological \(K\)-theory using function complexes, Topology, 41, 3, 591-644, (2002) Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Generalized (extraordinary) homology and cohomology theories in algebraic topology, Parametrization (Chow and Hilbert schemes), Semi-topological \(K\)-theory using function complexes | 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 moduli spaces of quiver representations from the point of view of Geometric Invariant Theory and Kählerian Reduction Theory. As the moduli spaces at hand can be obtained as GIT quotients of Hermitian vector spaces by a reductive group, results of \textit{R. Sjamaar} [Ann. Math. (2) 141, No. 1, 87--129 (1995; Zbl 0827.32030)] on the compatibility of GIT and Symplectic Reduction can be applied to conclude that the curvature form of the natural Hermitian metric on a descended line bundle is smooth along the orbit type stratification, where it coincides with the form obtained by Symplectic Reduction. representations of quivers; moduli spaces; moment maps; Hermitian line bundles Algebraic moduli problems, moduli of vector bundles, Representations of quivers and partially ordered sets, Momentum maps; symplectic reduction A natural Hermitian line bundle on the moduli space of semistable representations of a quiver | 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 present an analytic construction of multi-brane solutions with any integer brane number in cubic open string field theory (CSFT) on the basis of the \(KBc\) algebra. Our solution is given in the pure-gauge form \(\Psi=UQ_{\mathrm{B}}U^{-1}\) by a unitary string field \(U\), which we choose to satisfy two requirements. First, the energy density of the solution should reproduce that of the \((N+1)\)-branes. Second, the equations of motion (EOM) of the solution should hold against the solution itself. In spite of the pure-gauge form of \(\Psi\), these two conditions are non-trivial ones due to the singularity at \(K=0\). For the \((N+1)\)-brane solution, our \(U\) is specified by \([N/2]\) independent real parameters \(\alpha_k\). For the 2-brane (\(N=1\)), the solution is unique and reproduces the known one. We find that \(\alpha_k\) satisfying the two conditions indeed exist as far as we have tested for various integer values of \(N\) (\(=2, 3, 4, 5, \dots\)). Our multi-brane solutions consisting only of the elements of the \(KBc\) algebra have the problem that the EOM is not satisfied against the Fock states and therefore are not complete ones. However, our construction should be an important step toward understanding the topological nature of CSFT, which has similarities to the Chern-Simons theory in three dimensions. String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Open systems, reduced dynamics, master equations, decoherence, Operator algebra methods applied to problems in quantum theory, Yang-Mills and other gauge theories in quantum field theory, Singularities in algebraic geometry, Eta-invariants, Chern-Simons invariants Analytic construction of multi-brane solutions in cubic string field theory for any brane number | 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 article investigates a particular combinatorial structure known as an alternating strand diagram or Postnikov diagram. This is roughly a collection of strands defined on an oriented disc, each with a specified start and end point on the boundary, finitely many intersections with other strands, no self-intersections, and specific laws governing the orientation of intersections. Postnikov diagrams are useful for encoding geometric information, particularly with regard to the Grassmannian, and the main result of the paper concerns the associated dimer algebra of a connected Postnikov diagram.
In short, the main theorem of the article, Theorem 1, is a categorification result, whose underlying algebra is the cluster algebra \(\mathcal{A}_D\) of the associate ice quiver of a Postikov diagram \(D\). The theorem states that the category of Gorenstein-projective modules \(GP(B_D)\) over the boundary \(B_D\) of the dimer algebra of \(D\) can be realised as an additive categorification of this cluster algebra. Since this cluster algebra is closely related to the homogeneous coordinate ring of a positroid variety in the Grassmannian, this result is certainly useful from a geometric point of view. The key step in the argument is the second main result, Theorem 2, which states that the dimer algebra of \(D\) satisfies the appropriate Calabi-Yau property.
The paper begins with a rough outline of the main results and their motivation, focusing heavily on positroid varieties, the current state of their research in the community, and how the results of the article advance this research. After introducing some preliminary material in section 2, including of course the formal definition and basic properties of a Postnikov diagram, its associated ice quiver, the dimer algebra and the cluster algebra, the article goes on in section 3 to explore the Calabi-Yau property. This section focuses on the dimer algebra of a Postnikov diagram, and after describing some algebraic and combinatorial properties of this algebra, it concludes with a complete proof of Theorem 2 (here relabelled Theorem 3.7).
In section 4, the article explores some category theory, and the main theorem (Theorem 4.3) concerns a general algebra satisfying the appropriate Calabi-Yau property, and roughly speaking it states that the associated category of Gorenstein-projective modules is triangulated and satisfies many of the properties required of the categorification of a cluster algebra, and that it contains a specified cluster tilting object. Using Theorem 2, this result of course applies to the dimer algebra.
After proving in section 5 that the quiver associated to the cluster tilting objects in Theorem 4.3 contain no loops or cycles, the article goes on to use this fact to reduce to the situation previously explored by \textit{C. Fu} and \textit{B. Keller} [Trans. Am. Math. Soc. 362, No. 2, 859--895 (2010; Zbl 1201.18007)], and consequently prove Theorem 1 in section 6 (relabelled as Theorem 6.11). Finally, the paper concludes in section 7 with a discussion of the Grassmannian cluster category, and explores how it is closely related to the categorification explored throughout.
Overall, this paper should be of great interest to anyone working with quivers, cluster algebras and interested in their geometric and combinatorial properties. It follows on from previous work of many different authors, none of whom achieved results in as great a generality of this, so it should create a stir in the community. quivers; cluster-algebras; path-algebras; categorification; Grassmannians; triangulated-categories Representations of quivers and partially ordered sets, Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Abelian categories, Grothendieck categories, Derived categories, triangulated categories Calabi-Yau properties of Postnikov diagrams | 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 multi-graded Hilbert scheme parametrizes all ideals with a given Hilbert function. An important subclass is the class of all toric Hilbert schemes. This paper details the main component of a toric Hilbert scheme in case it contains a point corresponding to an affine toric variety. Here the component can be viewed as the set of all flat limits of this variety.
The main tool used in the paper is a computation of the fan of the toric variety. The treatment is very self-contained and thorough. Moreover, using a Chow morphism, a connection with certain GIT quotients of the toric variety is included. Some helpful examples are provided as well. toric Hilbert scheme; fiber polytope; toric Chow quotient Chuvashova O.V., The main component of the toric Hilbert scheme, Tôhoku Math. J., 2008, 60(3), 365--382 Parametrization (Chow and Hilbert schemes), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies The main component of the toric 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 This paper is a survey report on some recent results mainly concerning weakly normal algebraic varieties, and the linear systems contained in them. The results are fully stated, with complete references and some ideas of the proofs.
Recall that an algebraic variety X is said to be weakly normal (WN) if every birational morphism \(X'\to X\), which is also a universal homeomorphism is indeed an isomorphism. A natural problem is to understand whether the general hyperplane section of a WN variety embedded in a projective space is WN; the answer in this case is positive, and a number of different approaches are described. - The above problem has a natural generalization, obtained by adapting the classical theorem of Bertini (``the general member of a linear system S over an algebraic variety X (over the field of complex numbers) is non-singular but perhaps at the base points of S and at the singular points of X'').
There is an axiomatic approach to this problem, i.e. it is possible to show that Bertini's theorem holds for all local properties P which satisfy certain axioms. This implies Bertini's theorem for several properties, including \(P=WN\) in characteristic zero [the authors, J. Algebra 98, 171-182 (1986; Zbl 0613.14006)].
In positive characteristic it can be shown that there are WN projective varieties, whose general hyperplane sections are not WN [the authors, Proc. Am. Math. Soc. 106, No.1, 37-42 (1989; Zbl 0699.14063)]. However by using the axiomatic approach one can prove Bertini's theorem for the property \(P=WN1+S_ 2\) (where WN1 means, roughly, WN \(+\) ``well behaved in codimension 1'') and, in particular, \(P=WN+Gorenstein\) [the authors, J. Algebra 128, No.2, 488-496 (1990)]. weakly normal algebraic varieties; linear systems; birational morphism; Bertini's theorem Rational and birational maps, Divisors, linear systems, invertible sheaves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities in algebraic geometry Weakly normal algebraic varieties and Bertini theorems | 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 \(G\) denote either a special orthogonal group or a symplectic group defined over the complex numbers. We prove the following saturation result for \(G\): given dominant weights \(\lambda^1,\dots,\lambda^r\) such that the tensor product \(V_{N\lambda^1}\otimes\cdots\otimes V_{N\lambda^r}\) contains nonzero \(G\)-invariants for some \(N\geqslant 1\), we show that the tensor product \(V_{2\lambda^1}\otimes\cdots\otimes V_{2\lambda^r}\) also contains nonzero \(G\)-invariants. This extends results of Kapovich-Millson and Belkale-Kumar and complements similar results for the general linear group due to Knutson-Tao and Derksen-Weyman. Our techniques involve the invariant theory of quivers equipped with an involution and the generic representation theory of certain quivers with relations. reductive groups; irreducible representations; semi-invariants of quivers; tensor product multiplicities; classical groups; invariant theory Sam, S, Symmetric quivers, invariant theory, and saturation theorems for the classical groups, Adv. Math., 229, 1104-1135, (2012) Representation theory for linear algebraic groups, Representations of quivers and partially ordered sets, Geometric invariant theory, Vector and tensor algebra, theory of invariants, Actions of groups on commutative rings; invariant theory Symmetric quivers, invariant theory, and saturation theorems for the classical groups. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a compact connected Riemann surface of genus \(g\), with \(g\geq 2\), and let \(\mathcal{O}_{X}\) denote the sheaf of holomorphic functions on \(X\). Fix positive integers \(r\) and \(d\) and let \(\mathcal{Q}(r,d)\) be the Quot scheme parametrizing all torsion coherent quotients of \(\mathcal{O}^{\oplus r}_{X}\) of degree \(d\). We prove that \(\mathcal{Q}(r,d)\) does not admit a Kähler metric whose holomorphic bisectional curvatures are all nonnegative.
For Part I, cf. [the authors, ibid. 57, No. 4, 1019--1024 (2013; Zbl 1304.14012)]. Biswas, I.; Seshadri, H., On the Kähler structures over quot schemes II, Ill. J. math., 58, 689-695, (2014) Divisors, linear systems, invertible sheaves, Computational aspects of algebraic curves, Positive curvature complex manifolds, Vector bundles on curves and their moduli, Parametrization (Chow and Hilbert schemes) On the Kähler structures over Quot schemes. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper gives irreducibility results on Hilbert schemes of space curves. Let \(H_{d,g}\) denote the Hilbert scheme parametrizing smooth connected curves \(C \subset \mathbb P^3\) of degree \(d\) and genus \(g\). Confirming part of a claim of Severi, \textit{L. Ein} showed that \(H_{d,g}\) is irreducible for \(d \geq g+3\) [Ann. Scient. Ec. Norm. Sup. 19, 469--478 (1986; Zbl 0606.14003)]. The authors proved irreducibility of \(H_{d,g}\) for \(d=g+2, g \geq 5\) and \(d=g+1, g \geq 11\) [J. Algebra 145, 240--248 (1992; Zbl 0783.14002)] and \textit{H. Iliev} proved that \(H_{g,g}\) is irreducible for \(g \geq 13\) [Proc. Amer. Math. Soc. 134, 2823--2832 (2006; Zbl 1097.14022)].
Here the authors complete Iliev's result, showing that every non-empty \(H_{g,g}\) is irreducible. Noting that \(H_{g,g}\) is empty for \(1 \leq g \leq 7\) and irreducible for \(g=8\) and \(9\) by \textit{K. Dasaratha} [``The reducibility and dimension of Hilbert schemes of complex projective curves'', undergraduate thesis, Harvard University, Department of Mathematics, available at \url{http://www.math.harvard.edu/theses/senior/dasaratha/dasaratha}], the authors need only consider \(g \geq 10\). Here they build on the proof of Iliev [loc. cit.], making a careful study of an irreducible component \(\mathcal G\) of the space \(\mathcal G_g^3\) of pairs \((C,D)\) with \(C\) a smooth connected curve and \(D\) a linear series of degree \(g\) and dimension \(3\) whose general element is very ample. The key result here is that \(D\) is in fact a complete linear system of dimension \(4g-15\) and that the general member of the component of the residual series is base point free, complete, and birationally very ample. Hilbert schemes; space curves \textsc{C. Keem and Y.-H. Kim}, Irreducibility of the Hilbert scheme of smooth curves in \({\mathbb{P}}^3\), Arch. Math. \textbf{108} (2017), 593-600. Plane and space curves, Parametrization (Chow and Hilbert schemes) Irreducibility of the Hilbert scheme of smooth curves in \(\mathbb {P}^3\) of degree \(g\) 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 Let $R$ be a noetherian ring of positive characteristic $p$ which is an F-pure standard graded algebra over a field $K$ such that $[K:K^p]<\infty$. Let $a_i(R)$ be the $i$-th $a$-invariant of $R$, that is the degree of the highest nonzero part of the $i$-th local cohomology with support in $\mathfrak{m}$. For $d=\dim(R)$, the invariant $a_d(R)$ is the usual well-known $a$-invariant of $R.$ On the other hand, let $\mathrm{fpt}(R)$ be the F-pure threshold with respect to $\mathfrak{m}$ and $c^{\mathfrak{m}}(R)$ be the diagonal threshold of $R$. The authors are proving several statements that were conjectured by \textit{D. Hirose} et al. [Commun. Algebra 42, No. 6, 2704--2720 (2014; Zbl 1314.13011)]. Thus they show that if $R$ is a standard graded $K$-algebra which is F-pure and F-finite, then: \par (i) $\mathrm{fpt}(R)\leq -a_i(R),\ \forall i\in\mathbb{N}$; \par (ii) If $a_i(R)\neq-\infty,$ then $-a_i(R)\leq c^{\mathfrak{m}}(R)$; \par (iii) If $R$ is Gorenstein, then $\mathrm{fpt}(R)=-a_d(R),$ where $d=\dim(R)$. \par Also, analogous statements in characteristic zero are proved. a-invariant; F-pure threshold; diagonal F-threshold; F-purity. Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities in algebraic geometry \(F\)-thresholds of graded rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We classify spherical modules and full exceptional sequences of modules over the Auslander algebra of \(k[x]/(x^t)\). We categorify the left and right symmetric group actions on these exceptional sequences to two braid group actions: of spherical twists along simple modules, and of right mutations. In particular, every such exceptional sequence is obtained by spherical twists from a standard sequence, and likewise for right mutations. Auslander algebra; full exceptional sequence; exceptional module; spherical module Module categories in associative algebras, Representations of quivers and partially ordered sets, Rings arising from noncommutative algebraic geometry, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Derived categories, triangulated categories Exceptional sequences and spherical modules for the Auslander algebra of \(k[x]/(x^t)\) | 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 prove new interesting relations among various invariants associated to complex analytic function germs (or to pairs of such germs) defined on a singular analytic set: the Milnor number, (first) Teissier number, local Euler obstruction, Brasselet number. These invariants are also related to the number of Morse critical points occurring in a partial Morsefication of the given germ, generalizing in this way the formulas due to Lê and Greuel. Euler obstruction; constructible function Dutertre, N., Grulha Jr., N.G.: Lê-Greuel type formula for the Euler obstruction and applications. Preprint (2011). arXiv:1109.5802 Singularities in algebraic geometry, Topology of analytic spaces, Singularities of vector fields, topological aspects Lê-Greuel type formula for the Euler obstruction and applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Soit X une variété de Fano de dimension 3 plongée dans \({\mathbb{P}}^{g+1}\) par \(| -K_ X|\), où \(2(g-1)=c^ 3_ 1(X)\). Soit \(\Sigma\) la variété des coniques de \({\mathbb{P}}^{g+1}\) contenues dans X, supposée être une surface lisse. Soit \(I:=\{(C,C')\in \Sigma \times \Sigma | \#C\cap C'\geq 2\}\). Alors: \(I=\emptyset\) si, par exemple: \(b_ 2=1\), \(r=2\), \(g\geq 24\) ou si: \(b_ 2=1\), \(r=1\), \(g\geq 7\) si r est l'indice de X. Soit \(\Phi\) : Alb(\(\Sigma\))\(\to JX\) l'application d'Abel-Jacobi où JX est la Jacobienne intermédiaire de X, de polarisation \(\Theta\). Si \(I=\emptyset\) et si \(\chi({\mathcal O}_{\Sigma})= \binom{k-1}{2}\) avec \(k:=h^{1,2}(X)\), il est établi en particulier (sous des hypothèses générales) que \(\Phi\) est un isomorphisme, et que: \([\Phi(\Sigma)]= \Theta^{k-2}/(k-2)!\) Lorsque X est une cubique \((k=5)\), on retrouve donc certaines résultats de Clemens-Griffiths.
La méthode utilisée est très différente de la leur, et repose sur l'étude géométrique de \(D:=^ t\Phi \circ \Phi: Alb(\Sigma)\to Pic^ 0(\Sigma)\), aussi induite par \(\tilde D: \Sigma\to Pic(\Sigma)\) telle que: \(\tilde D(\Sigma)= \{C'\in \Sigma | C'\cap C\neq \emptyset \}\). - Des résultats analogues sont aussi obtenus lorsque I est involutives (dans un sens précise). conics on Fano threefolds; Abel-Jacobi mapping; intermediate Jacobian Fano varieties, Parametrization (Chow and Hilbert schemes), Picard schemes, higher Jacobians, \(3\)-folds Relations dans l'anneau de Chow de la surface des coniques des variétés de Fano, et applications. (Relations in the Chow ring of the surface of conics of Fano varieties and applications) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let H be the Hilbert scheme of projective space \({\mathbb{P}}^ n\). It is well known that H is a disjoint union of subschemes \(H_ Q\) of finite type, each parametrizing coherent sheaves of ideals with a given Hilbert polynomial Q. In fact, this is the way H is constructed by \textit{A. Grothendieck} [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV'', Sém. Bourbaki 13 (1960/61), Exposé 221 (1969; Zbl 0236.14003). \textit{R. Hartshorne} proved that the \(H_ Q\) are connected [Publ. Math., Inst. Hautes Étud. Sci. 29, 5-48 (1966; Zbl 0171.415)]. The paper under review generalizes this connectedness result in the following way:
For a given ideal \(I\subseteq {\mathcal O}_{P^ n}\), let h(I) be the Hilbert function of I. Introduce a partial order on the set of all functions \({\mathbb{N}}\to {\mathbb{N}}\) by putting \(f\leq g\) if and only if f(n)\(\leq g(n)\) for all \(n\in {\mathbb{N}}\). Then by standard semicontinuity results, h(I) is a semicontinuous function of I, i.e., if \(f:\quad {\mathbb{N}}\to {\mathbb{N}}\) is given, then the set \(H_{\geq f}=\{I\in H_ Q| \quad h(I)_{\geq f}\}\) is a closed subset of \(H_ Q\). The main theorem of the present paper is that the subschemes \(H_{\geq f}\) are connected in characteristic zero. - The proof makes use of the action of the upper triangular subgroup of \(GL(n+1)\) (and its subgroups) on \({\mathbb{P}}^ n\) and the induced action on \(H_ Q\). Several kinds of specializations are considered (arising from the group actions) and a careful study of the behaviour of the Hilbert function under such specialization is undertaken. Hilbert scheme; Hilbert function Gotzmann, G.: Durch Hilbertfunktionen definierte Unterschemata des Hilbertschemas. Comment. Math. Helv.63, 114--149 (1988) Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients) Durch Hilbertfunktionen definierte Unterschemata des Hilbertschemas. (Subschemes of the Hilbert scheme defined by Hilbert functions) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Using \textit{F. S. Macaulay}'s correspondence [Proc. Lond. Math. Soc. (2) 26, 531--555 (1927; JFM 53.0104.01)] we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for the dimension of cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur. cactus rank; Artinian Gorenstein local algebra Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Computational aspects of higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On polynomials with given Hilbert function and applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the famous article [Compos. Math. 110, No. 1, 65--126 (1998; Zbl 0894.18005)] \textit{E. Getzler} and \textit{M. Kapranov} studied operadic type structures related to the moduli space of algebraic curve. Intuitively, algebraic curves with marked points can be glued along the marked points generating operations on the moduli spaces. However, when considering arbitrary genuses of curves, the classical operadic picture, in which operations are labeled by trees, is replaced by operation labeled instead by graphs. They call this operadic structure modular. Moreover, moduli of curves with marked points do have typically (for instance when considering genus \(0\) curves) an extra cyclic symmetry obtained by permuting the punctures. In the same paper, they show how to construct a Feynman transform on the category of dg-modular operads and how to compute its Euler characteristic in terms of the Wick's theorem, hence highlighting the relation of this operad with mathematical physics.
In this paper the authors present a generalization of these results for curves with marked points \(k-\log\) canonically embedded, meaning admitting a projective embedding by a complete linear system. The study of log canonical models for curves has been central in the study of moduli spaces of curves and for its relationships to the Minimal Model program. There are three results presented: first, they show that for \(k\geq 5\) the moduli spaces of \(k-\log\) canonically embedded curves assemble together in a modular operad in Deligne-Mumford stacks. Second, they show that for \(k\geq 1\) the moduli spaces of \(k-\log\) canonically embedded curves of genus \(0\) assemble together in a cyclic operad in schemes. Third, they show that for \(k\geq 2\) the moduli spaces of \(k-\log\) canonically embedded curves assemble together in a stable cyclic operad in Deligne-Mumford stacks. In order to prove these results, they construct morphisms on these moduli spaces corresponding to the gluing of two embedded curves and to the gluing of two points together on the same embedded curve.
The proofs of these statements appear correct. Would be interesting, as a follow up work, to understand weather the construction of Getzler and Kapranov of the Feynman transform could be generalized to this setting. modular operad; log-canonical Hilbert scheme Families, moduli of curves (algebraic), , Parametrization (Chow and Hilbert schemes) Modular operads of embedded 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 If \(G\) is a complex simple Lie group and \(g\) is its Lie algebra, a nilpotent orbit \(O_x \subset g\) is the orbit of a nilpotent element \(x \in g\) under the adjoint action of \(G\) on \(g\). The Kostant-Kirillov symplectic form on \(O_x\) always extends to a holomorphic symplectic form on certain resolution of the closure \(\bar{O}_x\) of \(O_x\); and such resolution is called symplectic if the extended symplectic form on it remains everywhere non-degenerate. For any parabolic subgroup \(P\) of \(G\) there exists a unique nilpotent orbit \(O\) (the Richardson orbit for \(P\)), intersecting the nilradical \(n(p)\) of the Lie algebra \(p\) of \(P\) in an open dense subset of \(n(p)\); ant then \(P\) is called a polarization of \(O\). For the Richardson orbit \(O\) of the parabolic subgroup \(P\) there exists a natural finite proper map \(\mu: T^*(G/P) \rightarrow \bar{O}\), where \(T^*(G/P)\) is the cotangent bundle of \(G/P\); and in the particular case when \(\deg(\mu) = 1\), the map \(\mu\) becomes a symplectic resolution of \(\bar{O}\), called the Springer resolution of \(\bar{O}\). By a recent result of \textit{B. Fu} [Invent. Math. 151, No. 1, 167--186 (2003; Zbl 1072.14058)], any symplectic resolution of a nilpotent orbit \(O\), if such resolution exists, coincides with the Springer resolution of some polarization of \(O\).
Not all nilpotent orbit have a projective symplectic resolution, and if such resolution exists it may be not unique. In the case when \(g\) is a classical simple Lie algebra, earlier results of \textit{W. Hesselink} [Math. Z. 160, 217--234 (1978; Zbl 0364.20048)] and \textit{N. Spaltenstein} [``Classes unipotentes et sous-groupes de Borel'', Lect. Notes Math. 946 (1982; Zbl 0486.20025)] describe necessary and sufficient conditions a nilpotent orbit \(O\) to have a symplectic resolution and also give the number of parabolics \(P\), up to conjugation, that give Springer resolutions of \(O\).
The present paper deals with arbitrary simple Lie algebras, thus incorporating in the study of symplectic resolutions also the exceptional Lie algebras. After describing an equivalence relation in the set of parabolic subgroups of \(G\) in terms of marked Dynkin diagrams, the author proves the main result of the paper (Theorem 6.1) which states that if the closure of a nilpotent orbit \(O\) in a complex simple Lie algebra has a polarization \(P_o\), and hence a Springer resolution \(T^*(G/P_o)\), then any equivalent parabolic \(P\) to \(P_o\) also gives a Springer resolution \(T^*(G/P)\) of \(\bar{O}\). Moreover all projective symplectic resolutions of \(\bar{O}\) are of this form, and if two polarizations of \(O\) are equivalent then they are connected by Mukai flops of type \(A\), \(D\) and \(E_6\). The proof of Theorem 6.1 is written in an abstract way so that it is valid also for exceptional Lie algebras. One can also find a more explicit treatment in Example 6.5 when the Lie algebra \(g\) is classical. Another result of the paper is the affirmative answer to a conjecture of the author and Fu, stating that any two symplectic resolutions of a normal symplectic variety \(W\) are deformation equivalent, in all cases when \(W\) is the normalization of a nilpotent orbit closure in a simple Lie algebra \(g\). In an earlier paper of the author with \textit{B. Fu}
[Ann. Inst. Fourier 54, No. 1, 1--19 (2004; Zbl 1063.14018)] this conjecture has been proved in the particular case when \(g = sl(n)\). Y. Namikawa, \textit{Birational geometry of symplectic resolutions of nilpotent orbits}, in \textit{Moduli spaces and arithmetic geometry}, vol. 45 of \textit{Adv. Stud. Pure Math.}, Math. Soc. Japan, Tokyo, Japan (2006), pp. 75 [math/0404072]. Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Simple, semisimple, reductive (super)algebras Birational geometry of symplectic resolutions of nilpotent orbits | 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 \(f=(f_1, \ldots, f_k):(\mathbb C^n, 0)\to (\mathbb C^k, 0)\) be a germ of a complex analytic map defining an isolated complete intersection singularity \((V,0)=f^{-1}(0)\) and let \(w=\sum A_i dx_i\) be the germ of a holomorphic \(1\)-form on \((\mathbb C^n, 0)\) whose restriction to \(V\) has no singular points in a punctured neighbourhood of \(0\). Let \(I_{V, w}\) be the ideal generated by \(f_1, \ldots, f_k\) and the \((k+1)\)--minors of the matrix
\[
\left( \begin{matrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n}\\ \vdots & & \\ \frac{\partial f_k}{\partial x_1} & \cdots & \frac{\partial f_k}{\partial x_n}\\ A_1 &\cdots & A_n \end{matrix} \right)
\]
and let \(\mathcal A_{V, w}=\mathcal O_{\mathbb C^n, 0}/I_{V,w}\). Suppose \(f\) and \(w\) are real, i.e., \(f(\mathbb R^n)\subseteq \mathbb R^k\) and \(w\) has real values on the tangent space \(T_x\mathbb R^n\). Let \(\text{ind}_{V,0,w}\) be the index of \(w\) on the real part \((V_\mathbb R,0)\) of \((V,0)\). The authors construct a family \(Q_\varepsilon\) of quadratic forms such that \(Q_\varepsilon\) is nondegenerate for \(\varepsilon \notin \Sigma\), the bifurcation diagram of the map. For real \(\varepsilon\) the form \(Q_\varepsilon\) is real and its index is \(\text{ind}_{V,0,w}+(\chi(V_\varepsilon)-1)\) where \(V_\varepsilon=f^{-1}(\varepsilon)\cap\mathbb R^n\cap B_\delta\) for \(0<\varepsilon<\!<\delta\). It is proved that this construction gives a nontrivial quadratic form on the algebra \(\mathcal A_{V, w}\). isolated complete intersection singularity; quadratic form; 1-form W. Ebeling and S. M. Gusein-Zade, ''Quadratic Forms for a 1-Form on an Isolated Complete Intersection Singularity,'' Math. Z. 252, 755--766 (2006). Invariants of analytic local rings, Singularities in algebraic geometry, Differential forms in global analysis Quadratic forms for a 1-form on an isolated complete intersection 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 Let \(X \subset \mathbb{C}^3\) be a rational double point singularity, and let \(\Gamma\) be its Newton dual fan in \(\mathbb{R}^3\). Because \(X\) is Newton non-degenerate, any regular subdivision of \(\Gamma\) induces a toric embedded resolution of \(X\). Inspired by the Nash problem, the authors define weights in \(\mathbb{R}^3\) for certain components of the jet schemes of \(X\), and they construct a regular subdivision of \(\Gamma\) whose rays are generated by these weights. In other words, they construct a map from a certain set of jet scheme components to the set of boundary components of a certain toric embedded resolution of \(X \subset \mathbb{C}^3\). In a case by case analysis for each singularity type, the authors show that this map is a bijection, and they show that in all cases except the \(E_8\) singularity, the constructed toric embedded resolution is minimal in the sense that each boundary divisor's valuation is centered at some boundary divisor in any other toric embedded resolution of \(X \subset \mathbb{C}^3\). embedded Nash problem; resolution of singularities; toric geometry Global theory and resolution of singularities (algebro-geometric aspects), Arcs and motivic integration, Toric varieties, Newton polyhedra, Okounkov bodies Jet schemes and minimal toric embedded resolutions of rational double point 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 primary concern of this paper is to investigate the possible deformations of so-called ``thick'' points in \({\mathbb{C}}^ 3\), in this case, zero dimensional analytic spaces corresponding to algebras of the form \(X_{\alpha}={\mathbb{C}}[X,Y,Z]/I_{\alpha}\), where \(I_{\alpha}\) is an ideal generated by three linearly independent homogeneous quadratic forms. Here the investigation is restricted to the essentially generic case where \(I=(x^ 2+2xy,y^ 2+xz,z^ 2+2\alpha xy)\) with \(\alpha\) not equal to 0, 1 or -1/8. The nets of quadrics in \(P^ 2\) corresponding to these forms have no base points.
It is shown that to each \(\alpha\) there is a one parameter family of singularities into which the given one can be deformed, and that each such family contains infinitely many non-isomorphic spaces. Further the family is independent of \(\alpha\). In fact a complete and explicit list of the singularities that arise is presented.
The approach is to work with an explicit 10 dimensional versal deformation of \(X_{\alpha}\), and, using general results on deformations, to reduce the problem to the analysis of the basepoints (with their algebraic structure) of those nets of quadrics in \(P^ 3\) which intersect the given net in \(X_{\alpha}\) at infinity. Ultimately this involves many detailed computations. thick points; homogeneous quadratic forms; nets of quadrics; one parameter family of singularities; versal deformation Deformations of singularities, Pencils, nets, webs in algebraic geometry, Formal methods and deformations in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities in algebraic geometry Deformation dicker Punkte und Netze von Quadriken. (Deformation of thick points and nets of quadrics) | 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 how superisolated surface singularities can be used to find a counterexample to the following conjecture by \textit{A. H. Durfee} [Invent. Math. 28, 231--241 (1975; Zbl 0278.14010)]: for a complex polynomial \(f(x,y,z)\) in three variables vanishing at 0 with an isolated singularity there, ``the local complex algebraic monodromy is of finite order if and only if a resolution of the
germ \((\{f=0\},0)\) has no cycles''. A Zariski pair is given whose corresponding superisolated surface singularities, one has complex algebraic monodromy of finite order and the other not (answering a question by J. Stevens). Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings On a conjecture by A. Durfee | 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 \(F(t,z)\) be a polynomial in \(z=(z_0,z_1,\dots,z_n)\) with coefficients which are smooth complex valued functions of \(t\in I= [0,1]\) such that \(F(t,0)=0\) and that for each \(t\in I\), the polynomials \((\partial F/\partial z_i)(t,z)\) in \(z\) have an isolated zero at \(0\). Assume more over that the integer \(\mu_t=\dim_{\mathbb{C}}\mathbb{C}/(\partial f/\partial z_0)(t,z),\dots,(\partial f/\partial z_n)(t,z))\) is independent of \(t\). The authors prove that the monodromy fibrations of the singularities of \(F(0,z) =0\) and \(F(1, z) =0\) at \(0\) are of the same fiber homotopy and, if further \(n\neq 2\), these fibrations are even differentially isomorphic and the topological types of the singularities are the same. The hypothesis \(n\neq 2\) comes from using h-cobordism theorem. This gives a proof of the Hironaka's conjecture for \(n=1\) in the more general case of a \(C^{\infty}\) family of \(n\)-dimensional hypersurfaces of dimension \(n\neq 2\). Making use of the results of \textit{K. Brauner} [Abh. math. Semin. Hamburg Univ. 6, 1--55 (1928; JFM 54.0373.01)], \textit{W. Burau} [Abh. Math. Semin. Hamb. Univ. 9, 125--133 (1932; Zbl 0006.03402; JFM 58.0615.01)] and \textit{O. Zariski} [Am. J. Math. 54, 453--465 (1932; Zbl 0004.36902; JFM 58.0614.02)], the authors prove that Puiseux pairs of an analytically irreducible plane curve singularity depends only on the topology of the singularity. Dũng Tràng Lê & Chakravarthi P. Ramanujam, ``The invariance of Milnor number implies the invariance of the topological type'', Am. J. Math.98 (1976), p. 67-78 Complex singularities, Singularities in algebraic geometry The invariance of Milnor's number implies the invariance of the topological 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 In his book ``Vorlesungen über algebraische Geometrie'' (Leipzig 1921; F 48.068701), \textit{F. Severi} has asserted with an incomplete proof that the subscheme \({\mathcal I}_{d, g,r}'\) which is the union of the irreducible components of the Hilbert scheme \({\mathcal H}_{d, g,r}\) whose general points correspond to smooth, irreducible, and nondegenerate curves of degree \(d\) and genus \(g\) in \(\mathbb{P}^r\) is irreducible if \(d\geq g+r\). Also \textit{J. Harris} [``Curves in projective space'', Sem. Math. Sup. 85 (Montreal 1982; Zbl 0511.14014)] has conjectured that \({\mathcal I}_{d,g,r}\) is irreducible if the Brill-Noether number \(\rho (d,g, r): =g-(r+1) (g-d+r)\) is positive.
In this paper we demonstrate various reducible examples of the subscheme \({\mathcal I}_{d,g,r}'\) with positive Brill-Noether number. Indeed an example of a reducible \({\mathcal I}_{d,g,r}'\) with positive \(\rho (d,g,r)\), namely the example \({\mathcal I}_{2g-8,g,g-8}'\) (or other variations of it), has been known to some people (including the author), but it seems to have first appeared in the literature in a paper by \textit{D. Eisenbud} and \textit{J. Harris} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 33-53 (1989; Zbl 0691.14006)]. The purpose of the paper under review is to add a wider class of examples to the list of such reducible examples by using general \(k\)-gonal curves. We also show that \({\mathcal I}_{d,g,r}'\) is irreducible for the range of \(d\geq 2g-7\) and \(g-d+r \leq 0\). Throughout we will be working over the field of complex numbers. reducible Hilbert scheme; F 48.068701; irreducible components of the Hilbert scheme; positive Brill-Noether number Keem, C, Reducible Hilbert scheme of smooth curves with positive brill-Noether number, Proc. Amer. Math. Soc., 122, 349-354, (1994) Parametrization (Chow and Hilbert schemes), Curves in algebraic geometry Reducible Hilbert scheme of smooth curves with positive Brill-Noether number | 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 addresses the problem of classifying point systems (or \(N\)- uples of points) in the projective plane. More precisely, if \(Z\subset\mathbb{P}^ 2\) is a finite subscheme of degree \(N\) and \(d\) is a positive integer, consider the linear system \(I_ Z(d)\) of plane curves of degree \(d\) containing \(Z\). The subscheme \(Z\) is called superabundant (for the linear system of plane curves of degree \(d)\) if the dimension of \(I_ Z(d)\) is strictly greater than the expected dimension \({d+2\choose 2}-1-N\). The set of superabundant subschemes \(Z\) is a subscheme \(W_ N[d]\) of the Hilbert scheme \(\text{Hilb}^ N\mathbb{P}^ 2\). The author's main result is to give the dimension of \(W_ N[d]\) and determine when it is irreducible, in terms of \(d\) and \(N\). classifying point systems; superabundant subschemes; Hilbert scheme Coppo, M. -A.: Familles maximales de systèmes de points surabondants dans P2. Math. ann. 291, 725-735 (1991) Projective techniques in algebraic geometry, Parametrization (Chow and Hilbert schemes), Divisors, linear systems, invertible sheaves Maximal families of superabundant point systems in \(\mathbb{P}^ 2\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) an algebraically closed field. The idea of solving singularities by toroidal transformations was considered by several authors: Khovansky, Varchenko, \textit{M. Merle} and \textit{B. Teissier} [in: Sémin. sur les singularités des surfaces, Cent. Math. Éc. Polytechn., Palaiseau 1976-77, Lect. Notes Math. 777, 229-245 (1980; Zbl 0461.14009)] and the reviewer [Bull. Soc. Math. Fr. 112, 325-341 (1984; Zbl 0564.32006)] in the case of nondegenerate complete intersections. \textit{M. Oka} [in: Algebraic geometry and singularities, Proc. 3rd internat. Conf. algebraic geometry, La Rábida 1991, Proc. Math. 134, 95-121 (1996; Zbl 0857.14014)] introduced the idea that this toroidal process can be used in general. He proves [with \textit{Lê Dũng Trang}, Kodai Math. J. 18, No. 1, 1-36 (1995; Zbl 0844.14010)] that this is the case for plane curve singularities and they define the complexity of a curve and show it is a topological invariant of the singularity. In the paper under review, the author continues this program in the case of space curve singularities. He computes the complexity of the curve using a special parametrisation of the curve, the Hamburger-Noether matrix. toroidal blow up; space curve singularities; complexity; Hamburger-Noether matrix Global theory and resolution of singularities (algebro-geometric aspects), Plane and space curves, Singularities of curves, local rings, Toric varieties, Newton polyhedra, Okounkov bodies Resolution of space curves complexity | 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 Cremona group \(\mathrm{Cr}_3({\mathbb{C}})\) is the group of birational automorphisms of the projective space \({\mathbb{P}}^3\). The authors investigate the following two questions and are able to answer some cases. The first question was raised by J.-P. Serre in 2010: What are normalizers in \(\mathrm{Cr}_3({\mathbb{C}})\) of finite simple non-abelian subgroups in \(\mathrm{Cr}_3({\mathbb{C}})\)? The second question is: How to decide whether or not two finite isomorphic subgroups in \(\mathrm{Cr}_3({\mathbb{C}})\) are conjugate?
The main result is Theorem 1.5.
{ Theorem 1.5.} Up to conjugation, there are at least 5 subgroups in \(\mathrm{Cr}_3({\mathbb{C}})\) that are isomorphic to \(A_6\). For three of these non-conjugated subgroups, the normalizer in \(\mathrm{Cr}_3({\mathbb{C}})\) is \(S_6\), and for one it is the free product of \(S_6\) and \(S_6\) with an amalgamated subgroup \(A_6\).
The strategy to prove the main theorem is to translate the two questions into the geometric language. Let \(\bar{G}\) be a finite subgroup, and let \(\tau: \bar{G}\rightarrow \mathrm{Cr}_3({\mathbb{C}})\) be a monomorphism. Then there is a birational map \(\xi: V \dashrightarrow {\mathbb{P}}^3\) such that
1. the threefold \(V\) is normal and has terminal singularities;
2. there exists a monomorphism \(v: \bar{G} \rightarrow \mathrm{Aut}(V)\);
3. \(\forall g\in \bar{G}\), we have \(\tau (g)=\xi\circ v(g)\circ \xi^{-1}\);
4. there is a \(v( \bar{G})-\)Mori fibration \(\pi: V\rightarrow S\), i.e., a non-birational \(v(\bar{G})-\) equivariant surjective morphism with connected fibers such that the divisor \(-K_V\) is \(\pi\)-ample, and for every \(v(\bar{G})-\)invariant Weil divisor \(D\) on \(V\), there is \(\delta\in {\mathbb{Q}}\) such that
\[
\delta K_V+D\sim_{\mathbb{Q}} \pi^*(H)
\]
for some \({\mathbb{Q}}\)-Cartier divisor \(H\) on the variety \(S\).
The quadruple \((V, \xi, v, \pi)\) is called a Mori regularization of the pair \((\bar{G}, \tau)\).
If \(S\) is a point, then \(V\) is called \(v(\bar{G})\)-birationally rigid if for every Mori regularization \((V', \xi', v', \pi')\) of the the pair \((\bar{G}, \tau)\), we have \(V'\cong V\), \(\pi'(V')\) is a point, and the subgroups \(v(\bar{G})\) and \(v'(\bar{G})\) are conjugate in \(\mathrm{Aut}(V)\cong \mathrm{Aut}(V')\).
If \(S\) is a point, then \(V\) is called \(v(\bar{G})\)-birationally superrigid if for every Mori regularization \((V', \xi', v', \pi')\) of the pair \((\bar{G}, \tau)\), the map \(\xi^{-1}\circ \xi'\) is biregular.
Theorem 1.5 is an application of Theorem 1.24.
{ Theorem 1.24.} Suppose that \(\bar{G}=A_6\). If \(V\cong {\mathbb{P}}^3\), then the threefold \(V\) is \(\bar{G}\)-birationally rigid (but not \(\bar{G}\)-birationally superrigid) and \(\mathrm{Bir}^{\bar{G}} (V)\) is a free product of two copies of \(S_6\) with amalgamated subgroup \(A_6\). If \(V\) is either the Segre cubic or a smooth quadric threefold, then \(V\) is \(\bar{G}\)-birationally superrigid and \(\mathrm{Bir}^{\bar{G}} (V)\cong S_6\).
The authors introduce a new technique to prove Theorem 1.24. The main idea of the proof of Theorem 1.24 is to use the machinery of multiplier ideal (Kawamata subadjunction theorem, Nadel-Shokurov vanishing theorem, the Riemann-Roch theorem, the Clifford theorem and Castelnuovo bound). Prokhorov, Yu.G.: Fields of invariants of finite linear groups. In: Bogomolov, F., Tschinkel, Yu. (eds.) Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol. 282, pp. 245-273. Birkhäuser, Boston (2010) \(3\)-folds, Hypersurfaces and algebraic geometry, Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial), Singularities in algebraic geometry Five embeddings of one simple group | 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 \(S = \mathbb A^2\) be the affine plane and denote by \(S^{[n]}\) the Hilbert scheme parametrizing zero dimensional subschemes of length \(n\). The Chow ring \(A^*(S^{[n]}, \mathbb Q)\) has been studied from different vantage points over the last twenty years. For example, [\textit{H. Nakajima}, Lectures on Hilbert schemes of points on surfaces. University Lecture Series. 18. Providence, RI: American Mathematical Society (1999; Zbl 0949.14001)] and [\textit{M. Lehn}, Invent. Math. 136, No. 1, 157--207 (1999; Zbl 0919.14001)] gave a basis and described the ring structure by considering linear operators on the direct sum \(\bigoplus_{n \in \mathbb N} A^*(S^{[n]}, \mathbb Q)\) and their commutativity relations. \textit{G. Ellingsrud} and \textit{S.-A. Strømme} gave a basis [Invent. Math. 87, 343--352 (1987; Zbl 0625.14002)] by using the action of the 2-torus \(T = (k^*)^2\) and the theorem of \textit{A. Białnicki-Birula} [Some properties of the decompositions of algebraic varieties determined by the actions of a torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24, 667--674 (1976; Zbl 0355.14015)].
Here the authors restrict to fixed points for the action of the torus \(T\) on \(S = \mathbb A^2\), using operators (creation/destruction operators \(q_i\), the boundary operator \(\partial\) and an auxiliary operator \(\rho\)) on the equivariant Chow ring \(\bigoplus_{n \in \mathbb N} A_T^* (S^{[n]}) \otimes_{A_T^* (\mathrm{pt})} K\), where \(K\) is the fraction field of \(A_T^* (\mathrm{pt})\). By manipulating linear combinations of Young diagrams they recover equivariant analogs of some commutativity relations obtained by \textit{Lehn} and \textit{Nakajima} [loc. cit.]. They also show that the Chow ring bases of \textit{Nakajima} and \textit{Ellingsrud-Strømme} [loc. cit.] are equal up to sign and a normalizing constant. As part of their process, they compute the tangent space to the Hilbert schemes \(S^{[n,n+1]}\) of flags \(z_n \subset z_{n+1}\) of zero dimensional subschemes of length \(n\) and \(n+1\). While the spaces \(S^{[n,n+1]}\) are known to be irreducible by work of \textit{J. Cheah} [Pac. J. Math. 183, No. 1, 39--90 (1998; Zbl 0904.14001)], the authors show that the space \(S_0^{[n,n+1]}\) consisting of subschemes supported at the origin is irreducible. Examples show that the spaces \(S_0^{[p,q]}\) are not irreducible in general. equivariant cohomology; Hilbert schemes; Chow ring [5] Pierre-Emmanuel Chaput &aLaurent Evain, &On the equivariant cohomology of Hilbert schemes of points in the plane&#xhttp://arxiv.org/abs/1205.5470 (Equivariant) Chow groups and rings; motives, Parametrization (Chow and Hilbert schemes) On the equivariant cohomology of 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 Let \({\mathbb P}^{2[n]}\) be the Hilbert scheme parametrizing zero dimensional subschemes of \({\mathbb P}^2_{\mathbb C}\) of length \(n\). \({\mathbb P}^{2[n]}\) is a smooth irreducible projective variety of dimension \(2n\). In this paper, the authors study the birational geometry of \({\mathbb P}^{2[n]}\). They show that \({\mathbb P}^{2[n]}\) is a Mori-Dream space (in particular \(R(D):=\bigoplus _{m\geq 0}H^0(\mathcal O (mD))\) is finitely generated for any integral divisor \(D\) on \({\mathbb P}^{2[n]}\)). They characterize the effective cone (for many values of \(n\)), and investigate its stable base locus decomposition (into finitely many rational polyhedral cones) and the birational models (corresponding to \({\text{Proj}}(R(D))\) for \(D\) in the big cone). For \(n\leq 9\) they determine the Mori cone decomposition of the cone of big divisors corresponding to different birational models \({\text{Proj}}(R(D))\) and the birational maps (flips and divisorial contractions) between models of adjacent chambers (wall crossings). They also give a modular interpretation in terms of the moduli spaces of Bridgeland semi-stable objects and a description as a moduli space of quiver representations using G.I.T. Hilbert scheme; minimal model program; quiver representations; Bridgeland stability conditions Arcara, D.; Bertram, A.; Coskun, I.; Huizenga, J., The minimal model program for the Hilbert scheme of points on \(\mathbb{P}^2\) and Bridgeland stability, Adv. Math., 235, 580-626, (2013) Minimal model program (Mori theory, extremal rays), Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Stacks and moduli problems The minimal model program for the Hilbert scheme of points on \(\mathbb P^2\) and Bridgeland stability | 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 main proposition, Theorem 1.2, is the existence for excellent Deligne-Mumford champ of characteristic zero of a resolution functor independent of the resolution process itself. Received wisdom was that this was impossible, but the counterexamples overlooked the possibility of using weighted blow ups. The fundamental local calculations take place in complete local rings, and are elementary in nature, while being self contained and wholly independent of Hironaka's methods and all derivatives thereof, i.e. existing technology. Nevertheless \textit{D. Abramovich} et al. [``Functorial embedded resolution via weighted blowing ups'', Preprint, \url{arXiv:1906.07106}], have varied existing technology to obtain even shorter proofs of all the main theorems in the pure dimensional geometric case. Excellent patching is more technical than varieties over a field, and so easier geometric arguments are pointed out when they exist. Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry Very functorial, very fast, and very easy resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The simplest, and for many the most familiar singularities, are those presented by plane curves. Their local topology was first investigated in the late 1920's, and they have provided an excellent proving ground for theories and theorems ever since. This paper addresses three central problems concerning plane curve singularities. Suppose we are given a natural number \(d\) and a finite list of types of plane curve singularities. We can then ask:
(1) Is there an (irreducible) curve of degree \(d\) in the complex projective plane \(\mathbb{C}\mathbb{P}^2\) whose singularities are precisely those in the given list?
(2) Is the family of such curves smooth, and of the expected dimension?
(3) Is this set connected?
These questions have a long history, appearing in the works of Plücker, Severi, Segre and Zariski. They are part of a more general problem: that of moving from local information to global results. The authors have made spectacular recent progress, which is surveyed in this paper. They extend their discussion to curves on more general surfaces. Combined with recent ideas of Viro their results also have application in real algebraic geometry. equisingular families of plane curves; curves on surfaces; real algebraic geometry; plane curve singularities Gert-Martin Greuel and Eugenii Shustin, Geometry of equisingular families of curves, Singularity theory (Liverpool, 1996) London Math. Soc. Lecture Note Ser., vol. 263, Cambridge Univ. Press, Cambridge, 1999, pp. xvi, 79 -- 108. Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Families, moduli of curves (algebraic), Plane and space curves, Topology of real algebraic varieties Geometry of equisingular families of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study the improved evaluation codes, introduced by \textit{H. E. Andersen} [Finite Fields Appl. 14, No. 1, 92--123 (2008; Zbl 1136.94010)], for the case of finitely generated order structure given by plane valuations [see, \textit{O. Zariski} and \textit{P. Samuel}, Commutative algebra. Vol. II. (The University Series in Higher Mathematics.) Princeton, N.J.-Toronto- London-New York: D. Van Nostrand Company, Inc. (1960; Zbl 0121.27801); \textit{M. Spivakovsky}, Am. J. Math. 112, No. 1, 107--156 (1990; Zbl 0716.13003)].
In Section 2, some generalities about plane valuation are exposed. Section 3 is the main of the paper and is devoted to study the parameters (length, dimension, and minimum distance) of the improved evaluation codes. Section 4 provides with a comparing example of improved evaluation codes and Section 5 describes, in a detailed manner, the minimal generating sets of the value semigroups of those valuations that are used in the paper. Finite fields; evaluation codes; valuations; weight functions; order structures Geometric methods (including applications of algebraic geometry) applied to coding theory, Singularities in algebraic geometry, Algebraic coding theory; cryptography (number-theoretic aspects) Improved evaluation codes defined by plane valuations | 0 |
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