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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 We construct a large class of projective threefolds with one node (aka non-degenerate quadratic singularity) such that their small resolutions are not projective. small resolution; Picard-Lefschetz theory; extremal ray Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.) On non-projective small 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 We study the depth properties of certain direct image sheaves on normal varieties. Let \(f: Y\rightarrow X\) be a proper morphism of relative dimension \(d\) from a smooth variety onto a normal variety such that the preimage \(E\) of the singular locus of \(X\) is a divisor. We show that for any integer \(m>0\), the higher direct image \(R^df_*\omega^{\otimes m}_Y(aE)\) modulo the torsion subsheaf is \(S_2\), provided that \(a\) is sufficiently large. In case \(f\) is birational, we give criteria on \(a\) for the direct image \(f_*\omega_Y(aE)\) to coincide with \(\omega_X\). We also introduce an index measuring the singularities of normal varieties. direct images of twisted pluricanonical sheaves; normal varieties; reflexive sheaves; index of singularities Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Sheaves in algebraic geometry On direct images of twisted pluricanonical sheaves on normal varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We present here a combinatorial formulation for the procedure of reduction
of singularities in terms of polyhedral systems. This combinatorial structure is free of restrictions of the characteristic and provides
a combinatorial support for the reduction of singularities of varieties, foliations,
vector fields and differential forms, among other possible objects.
Hironaka's characteristic polyhedra represent the combinatorial steps in almost any procedure of reduction of
singularities. This is implicit in
the formulation of the polyhedra game [\textit{H. Hironaka}, J. Math. Kyoto Univ. 7, 251--293 (1967; Zbl 0159.50502)], solved by \textit{M. Spivakovsky} [Prog. Math. 36, 419--432 (1983; Zbl 0531.14009)] and in many other papers about characteristic polyhedra.
The combinatorial features concerning the problem of reduction of singularities are reflected in polyhedra systems without losing the global aspects.'' polyhedra systems; maximal contact; resolution of singularities Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of holomorphic vector fields and foliations, Modifications; resolution of singularities (complex-analytic aspects) Combinatorial aspects of classical 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 For an n-dimensional algebroid hypersurface \(F=0,\) \(F\in k[[X_ 1,...,X_ n,Z]],\) it is known that, when \(char(k)=0\), the equimultiple locus is hyperplanar [see \textit{S. S. Abhyankar}, Adv. Math. (to appear)]. When \(n=2\) the same is true of the equimultiple curve for \(char(k)>0\)- see below for a characteristic free proof. The author has given elsewhere [Proc. Am. Math. Soc. 87, 403-408 (1983; Zbl 0521.13014)] an example of an equimultiple curve when \(n>2\) and \(char(k)=2\) which is not hyperplanar. In this note a similar example for the case \(n>2\) and \(char(k)>2\) is given. positive characteristic; complete intersection; not hyperplanar equimultiple curve; equimultiple locus Narasimhan R.: Monomial equimultiple curves in positive characteristic. Proc. Am. Math. Soc. 89, 402--413 (1983) Singularities of curves, local rings, Multiplicity theory and related topics, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Finite ground fields in algebraic geometry, Formal power series rings Monomial equimultiple curves in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors prove that the total homology Chern class of a toric variety \(X\) is given by \(c_ \bullet=\sum[\bar B]\) where one sums over the set of all orbits. They first prove it in the smooth case, then for the singular case they use the Chern classes defined by \textit{R. D. MacPherson} [Ann. Math., II. Ser. 100, 423-432 (1974; Zbl 0311.14001)]: these classes may be computed with the help of an equivariant resolution of the singularities.
At the end, in the case where \(X\) is projective of dimension 3, the authors prove that the Chern classes of \(X\) are in the image of the canonical homomorphism \(\omega: \mathbb{H}^{2n-\bullet}(X,\mathbb{Q})\to H_ \bullet(X,\mathbb{Q})\) of the intersection homology in the usual homology. total homology Chern class of a toric variety; equivariant resolution of the singularities; intersection homology G. Barthel, J.-P. Brasselet and K.-H. Fieseler, Classes de Chern des variétés toriques singulières, C. R. Math. Acad. Sci. Paris Sér. I 315 (1992), no. 2, 187-192. Toric varieties, Newton polyhedra, Okounkov bodies, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Characteristic classes and numbers in differential topology, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds Chern classes of singular toric varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0493.00004.]
In this paper the authors start with the notion of a standard base of an ideal in a local ring (in definition 1.6, \(f^*_ 1,...,f^*_ r\) should be a minimal set of generators of \(J^*)\) and investigate various properties of a standard base, particularly in relation to the elements (resp. their initial forms) forming a regular sequence, and the Koszul complex corresponding to the initial forms. The paper concludes with some application and examples. tangent cone; minimal set of generators; standard base; Koszul complex ROBBIANO, L. and VALLA, G.: Free resolutions of special tangent cones.Commutative Algebra.Proceedings of the Trento Conference.Lect.Notes in Pure and Appl. Math.Series,84.Marcel Dekker(l983) Global theory and resolution of singularities (algebro-geometric aspects), Commutative rings and modules of finite generation or presentation; number of generators, Singularities in algebraic geometry Free resolutions for special tangent cones | 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 \(Y\) be a complex projective variety of positive dimension \(n\) with isolated singularities, \(\pi: X \rightarrow Y\) a resolution of singularities, and \(G = \pi^{-1}\text{Sing}(Y)\) the exceptional locus. It is well-known that in this case the natural morphism \(\gamma: H^{k-1}(G)\rightarrow H^k(Y,Y\setminus \text{Sing}(Y))\) vanishes for \(k>n\). However, the same is also true when either \(\dim G <n/2\), or when \(Y\) is normal variety, or when \(\pi\) is the blowing-up of \(Y\) along the singular locus with smooth and connected fibres, or when \(\pi\) admits a natural Gysin morphism, etc. The authors explain that the vanishing condition is, in fact, equivalent to the Decomposition Theorem valid in the bounded derived category of sheaves of \(\mathbb Q\)-vector spaces on \(Y\) in the sense of [\textit{G. Williamson}, in: Séminaire Bourbaki. Volume 2015/2016. Exposés 1104--1119. Avec table par noms d'auteurs de 1948/49 à 2015/16. Paris: Société Mathématique de France (SMF). 335--367, Exp. No. 1115 (2017; Zbl 1373.14010)]. As a result, they obtain a short and simple proof of this theorem in all cases when \(\gamma\) vanishes, discuss some useful relations with topological bivariant theory, and so on. projective variety; isolated singularities; resolution of singularities; derived category; intersection cohomology; decomposition theorem; bivariant theory; Gysin morphism; cohomology manifold Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Sheaves in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Topological properties in algebraic geometry, Global theory of complex singularities; cohomological properties, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Topological properties of mappings on manifolds On the topology of a resolution of isolated 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 To an arbitrary intersection of exceptional varieties of an embedded resolution we associate a finite number of congruences between naturally occurring multiplicities. This theory generalizes previous results concerning just one exceptional variety [see e.g. \textit{F. Loeser}, Am. J. Math. 110, 1-21 (1988; Zbl 0644.12007)]. Moreover we describe precise equalities which imply the congruences and we give some applications on the poles of Igusa's local zeta function. exceptional varieties of an embedded resolution; multiplicities; Igusa's local zeta function Veys, W.: More congruences for numerical data of an embedded resolution. Compositio Math 112, 313--331 (1998) Global theory and resolution of singularities (algebro-geometric aspects), Algebras and orders, and their zeta functions, Modifications; resolution of singularities (complex-analytic aspects), Local ground fields in algebraic geometry, Singularities in algebraic geometry More congruences for numerical data of an embedded resolution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A pair \((W,E)\) consists in a pure dimensional scheme \(W\) smooth over a field \(k\) with char \(k=0\) and a set of hypersurfaces \(E=\{H_1,\dots,H_r\}\) in \(W\) with normal crossings. Let \((W_0,E_0)=\{\emptyset\}\) be a pair and \(X_0\subset W_0\) a closed subscheme defined by \({\mathcal J}_0\subset {\mathcal O}_{W_0}\). Assume that Reg\((X_0)\) is dense in \(X_0\). Then there exists a finite sequence of so called transformation of pairs \((W_r,E_r)\rightarrow\ldots \rightarrow (W_0,E_0)\) inducing a proper birational morphism \(\pi_r:W_r\rightarrow W_0\) so that setting \(E_r=\{H_1,\ldots,H_s\}\) and letting \(X_r\subset W_r\) be the strict transform of \(X_0\) we have
(1) \(X_r\) is regular in \(W_r\) and \(W_r-\bigcup_{i=1}^rH_i\cong W_0-\)Sing\((X_0)\),
(2) \(X_r\) has normal crossings with \(E_r=\bigcup_{i=1}^rH_i\),
(3) the total transform of \({\mathcal J}(X_0)\subset {\mathcal O}_{W_0}\) factors as a product of ideals in \({\mathcal O}_{W_r}\):
\({\mathcal J}(X_0){\mathcal O}_{W_r}={\mathcal L}{\mathcal J}(X_r)\), where \({\mathcal J}(X_r)\subset {\mathcal O}_{W_r}\) is the sheaf defining \(X_r\), and \({\mathcal L}={\mathcal J}(H_1)^{a_1}\cdots (H_r)^{a_r}\) is an invertible sheaf of ideals supported by the exceptional locus of \(\pi_r\).
The properties (1), (2) follow also by Hironaka's Theorem but (3) not. principalization of ideals; weak and strict transform; strong factorizing desingularization Bravo A., Villamayor O.: A strengthening of resolution of singularities in characteristic zero. Proc. Lond. Math. Soc. 86(2), 327--357 (2003) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Rational and birational maps, Global theory of complex singularities; cohomological properties A strengthening of resolution of singularities in characteristic zero. | 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 \(C_r\) denote the affine cone over a rational normal curve (or Veronese curve) of degree \(r\). It has an isolated singularity at the origin, and the main result of the paper shows that this type of singularity occurs quiet frequently in orbit closures in module varieties. To be more precise, recall that for a finitely generated \(k\)-algebra (where \(k\) is an algebraically closed field), the set of \(k\)-algebra homomorphisms from \(A\) into the algebra of \(d\times d\) matrices has the structure of an affine GL\((d)\)-variety, such that the GL\((d)\)-orbits are in a one-to-one correspondence with the isomorphism classes of \(d\)-dimensional \(A\)-modules. Denote by \(O_M\) the GL\((d)\)-orbit corresponding to the module \(M\). Assume that \(O_N\) belongs to the closure \({\overline{O_M}}\) of \(O_M\), and the codimension of \(O_N\) in \({\overline{O_M}}\) is at least two (note that orbit closures are known to be non-singular in codimension one, thanks to an earlier paper of the author). Under some additional conditions on \(N\), the author proves that the pointed variety \(({\overline{O_M}},N)\) is smoothly equivalent to \((C_r,0)\) for some \(r\geq 1\). The conditions mentioned above hold for example when both \(M\) and \(N\) are preprojective modules. module variety; orbit closure; isolated singularity; smooth equivalence Zwara, G.: Singularities of orbit closures in module varieties and cones over rational normal curves. J. Lond. Math. Soc. (2) 74(3), 623--638 (2006) Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Representations of associative Artinian rings, Representations of quivers and partially ordered sets Singularities of orbit closures in module varieties and cones over rational normal curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper deals with local and global properties of the resolution arising in the canonical way by blowing up closed points [for details see the authors' forthcoming paper in Prepr., Neue Folge, Humboldt-Univ. Berl., Sekt. Math. (to appear)]. simple singularity; resolution Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Untersuchungen zur Struktur der kanonischen Auflösungen der 4- dimensionalen einfachen Singularitäten. (Investigations on the structure of the canonical resolutions of four dimensional simple singularities) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an algebraic variety over the algebraically closed field \(k\) and \(x\in X\) a closed point. \(RZ(X,x)\) denotes the set of valuation rings of its function field \(K\) which are dominating \(\mathcal{O}_{X,x}\).
As introduced by Zariski, the space \(\mathcal X\) of all valuations of \(K\) extending the trivial valuation of \(k\) has a topology with a basis of open sets \(\mathrm{E} (A) = \{ \nu \in \mathcal{X}, A\subseteq R_{\nu} \}\) where \(A\subseteq K\) is finite, \(R_{\nu}\) the valuation ring of \(\nu\). Together with the Zariski-topology induced by \(\mathcal X\), the set \(RZ(X,x)\) is referred to as the Riemann-Zariski-space of \(X\) at \(x\).
In the simplest case \(\dim( X) =1\), the points of \(RZ(X,x)\) correspond to the local analytic branches of \(X\) at \(x\). The article under review is dedicated to study the relation between \(RZ(X,x)\) and the geometry of \(X\) at \(x\) in general, where additional results are obtained with emphasis to the surface case.
In contrast to \(RZ(X,x)\), a different space of valuations is obtained from the analytification \(X^{\mathrm{an}}\) in the sense of Berkovich geometry: \(L (X,x)\) is the subspace of points in \(X^{\mathrm{an}}\) which specialize to \(x\) except the trivial one. The quotient \(NL(X,x)\) of \(L (X,x)\) is given by identification of points which define equivalent valuations; it is said to be the normalized non-Archimedian link of \(x\) in \(X\).
The author studies the topology of both spaces: There exists a canonical continuous surjective map \( RZ(X,x) \to NL(X,x)\). For a normal surface singularity, it is the largest Hausdorff quotient of \(RZ(X,x)\). This second statement may fail in higher dimensions.
The main result in the regular case is:
Theorem A. Let \(x\in X\), \(y\in Y\) be regular closed points of algebraic varieties over \(k\). The following conditions are equivalent:
(1) \(RZ(X,x)\) and \(RZ(Y,y)\) are homeomorphic.
(2) \(NL(X,x)\) and \(NL(Y,y)\) are homeomorphic.
(3) \(\dim(X) =\dim(Y)\).
Using resolution of singularities, the following corollary is obtained:
Assume \(\operatorname{char} (k)=0\) and let \(x\in X\) be a regular closed point. Then for any open subset \(U\subseteq RZ(X,x)\) there exists \(V\subseteq U\) such that \(V\) is homeomorphic to \(RZ(X,x)\). A topological space with this property is said to be \textit{self-homeomorphic}. This notion resembles in a way self-similarity of fractals.
From now on the case of normal algebraic surfaces \(X\) is studied. For a good resolution of \((X,x)\), the dual \(\Gamma\) of its graph is uniquely determined by \((X,x)\) up to the following equivalence: Two graphs \(\Gamma\), \(\Gamma'\) are said to be equivalent if \(\operatorname{Core}(\Gamma)\) and \(\operatorname{Core}(\Gamma')\) have the same topological realization.
Here the main result is:
Theorem B. Let \((X,x)\) and \((Y,y)\) be normal algebraic surface singularities over \(k\) with associated graphs \(\Gamma_{(X,x)}\) and \(\Gamma_{(Y,y)}\). Then the following conditions are equivalent:
(1) \(RZ(X,x)\) and \(RZ(Y,y)\) are homeomorphic.
(2) \(NL(X,x)\) and \(NL(Y,y)\) are homeomorphic.
(3) \(\Gamma_{(X,x)}\) and \(\Gamma_{(Y,y)}\) are equivalent.
The theorem implies that for a rational surface singularity \((X,x)\) the valuation spaces \(RZ(X,x)\) and \(NL(X,x)\) respectively, are homeomorphic to the to the corresponding spaces of the pointed affine plane. The converse statement does not hold as shown by the example \((Y,y)\) of a germ of a cone over an elliptic curve: The associated graph of the minimal embedded resolution is a tree as well as for a rational \((X,x)\) and thus has an empty core. Riemann-Zariski space; normalized non-Archimedean link; valuative tree; dual graph Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Rigid analytic geometry Topology of spaces of valuations and geometry 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 an algebraic variety defined over an algebraically closed field \(K\). There are two homeomorphic varieties associated to \(X\), the weak normalization \(X^*\) of \(X\) and the Lipschitz saturation \(\tilde X\) of \(X\) such that if \(X^{\#}\) is the normalization of \(X\) one has the following decomposition of the normalization morphism
\[
\pi: X^{\#} \to X: X^{\#} \to X^* \to \tilde X \to X.
\]
If \(X\) is weakly normal (i.e \(X^*=X)\) then it is Lipschitz saturated (i.e. \(\tilde X=X)\). The converse of this assertion is false.
The main result of this paper is the following: if \(X\) is obtained from a nonsingular projective variety by means of a linear projection from a center in general position, then \(X^*=\tilde X=X\). The main tool to prove this is to compare the weak normalization and the Lipschitz saturation using the double point scheme of the projection morphism. weak normalization; Lipschitz saturation; linear projection W. Adkins,Weak normality and Lipshitz saturation for ordinary singularities, Compositio Mathematika,51 (1984), pp. 149--157. Birational geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Weak normality and Lipschitz saturation for ordinary singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mu_n(\mathbb{C})\) be the group scheme of \(n\)--th roots of unity and \(\widehat{\mu}=\lim\limits_\leftarrow \mu_n(\mathbb{C})\). Let Var\(_{\mathbb{C},\widehat{\mu}}\) be the category of algebraic \(\mathbb{C}\)-varieties endowed with a \(\widehat{\mu}\)-action. The Grothendieck group \(K_0(\text{Var}_{\mathbb{C},\widehat{\mu}})\) becomes a ring with respect to the cartesian product. Let \(\mathcal{M}^{\widehat{\mu}}_{\mathbb{C}}=K_0(\text{Var}_{\mathbb{C},\widehat{\mu}})[\mathbb{L}^{-1}], \mathbb{L}=[\mathbb{A}^1_{\mathbb{C}}]\). The motivic Milnor fibre is an object in \(\mathcal{M}^{\widehat{\mu}}_{\mathbb{C}}\). Using the extended simplified resolution graph a fomula for the motivic Milnor fibre of an arbitrary complex plane curve singularity is given. The formula allows to express the spectrum of the singularity in terms of the spectra of certain quasi homogeneous singularities. plane curve singularity; Newton polyhedron; resolution of singularity; extended resolution graph; arc spaces; motivic integration; motivic zeta function; motivic Milnor fiber Quy-Thuong Lł.: Motivic milnor fibers of plane curve singularities (2017). arXiv:1703.04820 Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Arcs and motivic integration, Singularities of curves, local rings, Toric varieties, Newton polyhedra, Okounkov bodies, Milnor fibration; relations with knot theory Motivic Milnor fibers of plane curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\) be a commutative ring and \(M\) a finitely generated \(R\)--module, let \(A=\text{End}_R(M)\). If \(R\) is normal Gorenstein, \(M\) is reflexive and \(A\) is maximal Cohen--Macaulay with finite global dimension. \(A\) is a so--called crepant resolution of \(R\). Endomorphism rings of finite global dimension over commutative rings are studied. They occur as non-commutative crepant resolutions. crepant resolutions; rational singularities; endomorphism rings Dao, H; Faber, E; Ingalls, C, Noncommutative (crepant) desingularisations and the global spectrum of commutative rings, Algebras Represent. Theory, 18, 633-664, (2015) Singularities in algebraic geometry, Noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Noncommutative (crepant) desingularizations and the global spectrum of commutative 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 This article is an expanded version of the results announced in C. R. Acad. Sci., Paris, Sér. I 296, 855-858 (1983; Zbl 0527.32009). rational singularities; resolution; deformation; Cohen-Macaulay singularity Andreatta, M.; Silva, A., On weakly rational singularities in complex analytic geometry, Ann. Mat. Pura Appl., 136, 65-76, (1984) Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Deformations of singularities, Deformations of complex singularities; vanishing cycles, Complex singularities, Singularities in algebraic geometry On weakly rational singularities in complex analytic geometry | 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 deals with properties of the 3-dimensional singularities given by the equations \(A_ n\), \(D_ n\), \(E_ n\) (in Arnold's list) over an algebraically closed field of characteristic \(\neq 2\). For the canonical exceptional loci E (configurations of surfaces), embedded into the desingularization X, the vanishing property \(H^ i(E,{\mathcal O}_ E)=0 (i>0)\) is shown, if we choose the multiplicities of the components of E such that we obtain a negative ''symmetric'' embedding. An obvious consequence is the vanishing of the higher derived images of the structural sheaf with respect to the resolution morphism. exceptional divisor; canonical resolution; 3-dimensional singularities; desingularization Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Cohomological properties of the canonical exceptional loci of the 3- dimensional Arnold-singularities \(A_ n\), \(D_ n\) and \(E_ n\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field, and denote by \(\mathbb{M}(d)\) the algebra of \(d\times d\) matrices with coefficients in \(k\). For any (associative finitely generated unital) \(k\)-algebra \(A\) the set \(\text{mod}_{A}(d)\) of \(A\)-module structures on \(k^{d}\) is an affine variety, and can be viewed as a closed subset of \( \mathbb{M}(d)) ^{t}\) for some \(t\) in a natural way. The \(\text{GL}(d)\)-orbits in \(\text{mod}_{A}(d)\) (where \(\text{GL}(d)\) acts via conjugation) correspond to isomorphism classes of \(d\)-dimensional \(A\)-module, For a given \(A\)-module \(M\) its orbit will be denoted \(\mathcal{O}_{M},\) and the Zariski closure will be denoted \(\mathcal{\bar{O}}_{M}.\)
The work under review investigates the finite dimensional \(A\)-modules \(M\) such that \(\mathcal{\bar{O}}_{M}\) is a regular variety. The primary result is that, for a given \(A\)-module \(M,\) the orbit closure \(\mathcal{\bar{O}}_{M}\) is a regular variety if and only if \(B:=A/\text{Ann}(M)\) is hereditary and \(\text{Ext}_{B}^{1}( M,M) =0,\) where \(\text{Ann}(M)\) is the annihilator of \(M\). As \(\text{mod}_{B}(d)\) is a closed \(\text{GL}(d)\)-subvariety of \(\text{mod}_{A}(d)\) which contains \(\mathcal{\bar{O}}_{M},\) \(d=\dim_{k}M,\) and \(M\) is faithful as a \(B\)-module we see that the above result is equivalent to the following. Let \(M\) be a module over a finite dimensional algebra \(B\). Then \(\mathcal{\bar{O}}_{M}\) is a regular variety if and only if \(B\) is hereditary and \(\text{Ext}_{B}^{1}(M) =0.\)
The proof of main result exploits the relationship of the above to representations of quivers. Indeed, \(B\) is Morita-equivalent to a quotient of a finite quiver algebra \(kQ\) by an admissible ideal \(I\). Under this equivalence \(B\) is hereditary if and only if \(I\) is trivial. A faithful \(B\)-module corresponds to a representation \(N\) in \(\text{rep}_{Q}(\mathbf{d}) \) for some vector \(\mathbf{d}\) of natural numbers such that \(\text{Ann}(N) =I,\) where \(\text{rep}_{Q}(\mathbf{d})\) is the vector space of representations of \(Q\) of dimension \(\mathbf{d}.\) It turns out that \(\mathcal{\bar{O}}_{M}\) is regular if and only if \(\mathcal{\bar{O}}_{N}\) is, and the main theorem reduces to the following. Let \(N\) be a representation in \(\text{rep}_{Q}(\mathbf{d}) \) such that \(\text{Ann}(N)\) is an admissible ideal in \(kQ\) and \(\mathcal{\bar{O}}_{N}\) is regular. Then \(\text{Ann}(N)\) is trivial and \(\mathcal{\bar{O}}_{N}=\text{rep}_{Q}(\mathbf{d}).\) Most of the paper is devoted to the proof of this result. orbit closure; module varieties; representations of quivers Loc, NQ; Zwara, G, Regular orbit closures in module varieties, Osaka J. Math., 44, 945-954, (2007) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets Regular orbit closures in module 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 A summary of this thesis was presented by the author in Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 185- 192 (1983; Zbl 0527.14032). algebroid surface; resolution of singularity; quasi-ordinary singularity; normal double point Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Special surfaces, Singularities in algebraic geometry, Families, moduli, classification: algebraic theory On the structure of singularities of immersed algebroid surfaces. (Sobre la estructura de las singularidades de las superficies algebroides sumergidas). (Thesis, Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Algebra y Fundamentos) | 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 and a valuation \(\nu \) on \(K\), \(\kappa_\nu \) the residual field and \(\Gamma_\nu \) the group of the valuation \(\nu \). Let \(\overline \Gamma \) be a totally ordered group containing \(\Gamma_\nu \). Now we consider valuations \(\mu \) on the polynomial ring \(K[x]\) such that the restriction of \(\mu \) to \(K\) coincide with \(\nu .\) The author says that \(\mu \) is well specified if we have the following Abhyankar's equality
\[
\text{dim.alg}_K K(x)= \text{dim.alg}_{\kappa_\nu} \kappa_\mu+ \text{rank.rat}\,\Gamma_\mu/\Gamma_\nu =1.
\]
For any valuation \(\mu \) on the polynomial ring \(K[x]\), and \(\gamma \in \overline \Gamma \), let \(P_\gamma =\{ f\in K[x] \mid \mu (f)\geq \gamma \}, \) \( P_\gamma^+ =\{ f\in K[x] \mid \mu (f)> \gamma \}. \) The graded algebra associated to \(\mu \) is defined by \(\text{gr}_\mu K[x]:=\bigoplus_{\gamma \in \overline \Gamma}P_\gamma/P_\gamma^+\).
The main result in this paper is the following: i) If the valuation \(\mu \) is not well specified then \(\text{gr}_\mu K[x]=\overline {G_0}\), where \(\overline {G_0}\) is a simple graded algebra, that is any non zero homogeneous element has an inverse. ii) If the valuation \(\mu \) is well specified, then \(\text{gr}_\mu K[x]=\overline {G_0}[T]\), where \(\overline {G_0}\) is a simple graded algebra.
As a consequence the author obtains a generalization of a result of MacLane. valuation; graded algebra; local uniformization; resolution of singularities Vaquié, M.: Algèbre graduée associée à une valuation de K[x]. Adv. stud. Pure math. 46, 259-271 (2007) General valuation theory for fields, Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Graded algebra associated to a valuation of \(K[x]\) | 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 considers the following question: ``How bad can the deformation space of an object be?'' It is shown that usually the deformation space may be as bad as possible according to the philosophy ``There is no geometric possibility so horrible that it cannot be found generically on some component of some Hilbert scheme'' [cf. \textit{J. Harris, I. Morrison}, ``Moduli of Curves'', Grad. Texts Math. 187, Springer (1998; Zbl 0913.14005)]. By definition Murphy's law holds for a moduli space if every singularity type of finite type over \(\mathbb Z\) appears on that moduli space. It is proved that a lot of the well understood moduli spaces satisfy Murphy's law. For instance the Hilbert scheme of nonsingular curves in projective space, the Hilbert scheme of surfaces in \(\mathbb P^4\), the versal deformation spaces of isolated normal Cohen--Macaulay threefold singularities. deformation; moduli space; Murphy's law Vakil, R., Murphy\(###\)s law in algebraic geometry: badly-behaved deformation spaces, Invent. Math., 164, 569-590, (2006) Local deformation theory, Artin approximation, etc., Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory, Plane and space curves, Deformations of singularities, Fine and coarse moduli spaces, Singularities in algebraic geometry Murphy's law in algebraic geometry: Badly-behaved deformation 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 A Fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field. This technique in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence, simple unified proofs are obtained for formulas of Poincaré polynomials of toric hyper-Kähler varieties (recovering results of Bielawski-Dancer and Hausel-Sturmfels), Poincaré polynomials of Hilbert schemes of points and twisted Atiyah-Drinfeld-Hitchin-Manin (ADHM) spaces of instantons on \(\mathbb C^2\) (recovering results of Nakajima-Yoshioka), and Poincaré polynomials of all Nakajima quiver varieties. As an application, a proof of a conjecture of Kac on the number of absolutely indecomposable representations of a quiver is announced. quiver varieties; Weyl-Kac character formula Tamás Hausel, Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform, Proc. Natl. Acad. Sci. USA 103 (2006), no. 16, 6120 -- 6124. Momentum maps; symplectic reduction, Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A blow-up of an ideal is called a prime blow-up if its exceptional set is a unique irreducible divisor. Recall the following result from the work of \textit{S. Izumi} [Publ. Res. Inst. Math. Sci. 21, 719--735 (1985; Zbl 0587.32016)]:
Let \(X\) be an affine normal variety, \(E_i, E_j\) be two irreducible exceptional divisors over (a partial resolution of) \(X\) both corresponding to a point \(x\), then there exists a positive number \(c_{ij}\) such that \(v_{E_j}(f)\geq c_{ij} v_{E_i}(f)\) for every regular function \(f\) on \(X\), where \(v_{E_j}\) is the valuation defined by \(E_j\).
The author of the paper under review says that \(f\) is an extremal function for \(c_{ij}\) if we have \(v_{E_j}(f)= c_{ij} v_{E_i}(f)\) and that the constant \(c_{ij}\) is attained. The main purpose of the paper is to prove the equivalence between the two conditions:
(A) For an irreducible exceptional divisor \(E_0\) over \(X\), the constant \(c_{j0}\) for some exceptional divisor \( E_j\) .
(B) \(E_0\) can be the exceptional divisor of a prime blow-up of \(X\). resolution; exceptional divisor Ishii, S., \textit{extremal functions and prime blow-ups}, Comm. Algebra, 32, 819-827, (2004) Rational and birational maps, Global theory and resolution of singularities (algebro-geometric aspects), Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves Extremal functions and prime blow-ups | 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 two papers under review deal with the same problem: the classification of three-dimensional exceptional log canonical hypersurface singularities, which is a problem of birational geometry that arises in the log minimal model program. The first paper classifies the ``well-formed'' singularities and the second deals with the others. The first paper [Izv. Math. 66, 949--1034 (2002; Zbl 1076.14049)] is the main, more important one.
This classification is a positive answer, in the three-dimensional case, to the conjecture that there are only a finite number of types of \(d\)-dimensional exceptional log canonical hypersurface singularities, for any \(d \geq 3\).
The first paper also contains the solution to the above problem for \(d = 2\). First, let us describe this 2-dimensional classification, which is more familiar: a 2-dimensional log terminal singularity is exceptional if and only if it belongs to one of the types \(\mathbb E_6, \mathbb E_7, \mathbb E_8\); a 2-dimensional strictly log canonical singularity is exceptional if and only if it is either simply elliptic or it belongs to one of the types \(\widetilde{\mathbb D}_4, \widetilde{\mathbb E}_6, \widetilde{\mathbb E}_7, \widetilde{\mathbb E}_8\). (A description of 3-dimensional exceptional strictly log canonical hypersurface singularities is also given in the first paper).
The essential difference between the 2-dimensional and the \(d\)-dimensional case, \(d\geq 3\), is that almost every type of multidimensional singularity contains an infinite number of non-isomorphic singularities with the same resolution.
Reviewer's remark. The classification of the singularities, divided into 12 tables, is very detailed and very long (with more than 900 singularities); the list takes up about 51 of the 85 pages of the first paper. The author proves that there is a finite number of types of these singularities as a consequence of the finite nature of the classification list, but no mention is made of the classification into different types (as in the above 2-dimensional case). Be that as it may, it is easy to understand the effort that went into this involved classification: for instance, the author needed 17 definitions to classify the singularities in the first paper. log minimal model program; \(d\)-dimensional exceptional log canonical hypersurface singularities \(3\)-folds, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Hypersurfaces and algebraic geometry Classification of three-dimensional exceptional log-canonical hypersurface singularities. II. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We introduce a new class of algebras, called reconstruction algebras, and present some of their basic properties. These non-commutative rings dictate in every way the process of resolving the Cohen-Macaulay singularities \(\mathbb C^2/G\) where \(G=\frac{1}{r}(1,a)\leq\mathrm{GL}(2,\mathbb C)\).
This paper is organized as follows. In Section 2 we define the reconstruction algebra associated to a labelled Dynkin diagram of type \(A\) and describe some of its basic structure. In Section 3 we prove that it is isomorphic to the endomorphism ring of some Cohen-Macaulay modules. In Section 4 the minimal resolution of the singularity \(\mathbb C^2/\frac{1}{r}(1,a)\) is obtained via a certain moduli space of representations of the associated reconstruction algebra \(A_{r,a}\), and in Section 5 we produce a tilting bundle which gives us our derived equivalence. In Section 6 we prove that \(A_{r,a}\) is a prime ring and use this to show that the Azumaya locus of \(A_{r,a}\) coincides with the smooth locus of its centre \(\mathbb C[x,y]^{\frac{1}{r}(1,a)}\). This then gives a precise value for the global dimension of \(A_{r,a}\), which shows that the reconstruction algebra need not be homologically homogeneous. reconstruction algebras; Cohen-Macaulay singularities; labelled Dynkin diagrams; endomorphism rings of Cohen-Macaulay modules; resolutions of singularities; moduli spaces of representations; tilting bundles; derived equivalences; global dimension Wemyss, M, Reconstruction algebras of type \(A\), Trans. Am. Math. Soc., 363, 3101-3132, (2011) Rings arising from noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Cohen-Macaulay modules, Representations of quivers and partially ordered sets Reconstruction algebras of type \(A\). | 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 important paper the author describes and gives numerous applications of the algorithm which allows one to decide whether two 3-manifolds obtained by plumbing according to a graph of \(S^ 1\)-bundles over compact surfaces (possibly with boundary) are homeomorphic. Namely the manifolds associated with two graphs are the same if and only if there is a sequence of moves (which belong to 8 types) which transforms one graph into the other. Practical applications are based on the use of normal forms to which any graph can be reduced and such that manifolds corresponding to different graphs are distinct. As the author points out, this calculus is implicit in \textit{F. Waldhausen}'s work [Invent. Math. 3, 308-333; ibid. 4, 87-117 (1967; Zbl 0168.445)] on classification of graph manifolds. The author earlier used this calculus to define an integral invariant of plumbed homology spheres [Lect. Notes Math. 788, 125-144 (1980; Zbl 0436.57002)]. Closely related to the author's calculus is Bonahon and Siebenmann's census of oriented diffeomorphism types of manifolds arising from weighted trees. This technique is applied to the analysis of two types of 3-manifolds naturally appearing in algebraic geometry. The first type is the singularity links, i.e., the boundaries of regular neighbourhoods of isolated singularities of complex surfaces.
The main result is that with two exceptions the fundamental group of the singularity link determines the genera, normal bundles and intersection numbers of a minimal good resolution of the singularity (i.e., the resolution in which all intersections are normal crossings, no more than two curves intersect in one point and there are no \(CP^ 1\)'s with self-intersection \(-1\)). The exceptional singularity links which do not determine the resolution are the lens spaces \(L(p,q)\) and the torus bundles over circles with monodromy \(A\in SL_ 2({\mathbb{Z}})\) such that trace \(A\) is \(\geq 3\). Among other facts the author shows that the singularity links are irreducible 3-manifolds (Problem 3.20 in \textit{R. Kirby}'s list [Proc. Symp. Pure Math. 32, Part 2, 273-312 (1978; Zbl 0394.57002)]) and that the resolution of a singularity is star-shaped provided the singularity link is a Seifert manifold. Another interesting result is that a complex surface V is topologically the suspension of a closed 3-manifold if and only if it is homeomorphic to an Inoue surface [\textit{M. Inoue}, Complex Anal. algebr. Geom. 91-106 (1977; Zbl 0365.14011)].
The second type of 3-manifold arising from algebraic geometry is the link of families of curves. The latter are the manifolds of the form \(\pi^{- 1}(\partial D)\), where \(\pi: W\to D\) is an analytic map of a complex surface on the unit disk such that all fibres except the one over the origin are nonsingular complete algebraic curves. The main result is that the fundamental group of the link of a minimal good family of curves defines the numerical type of the family, i.e., genera, normal bundles and intersection numbers of components. plumbing of circle bundles over compact surfaces; 3-manifolds; classification of graph manifolds; boundaries of regular neighbourhoods of isolated singularities of complex surfaces; fundamental group of the singularity link; lens spaces; torus bundles over circles; Seifert manifold; Inoue surface; link of families of curves Neumann W., A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299-344. Topology of general 3-manifolds, Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Fundamental group, presentations, free differential calculus, Algebraic topology on manifolds and differential topology, \(3\)-folds, Special surfaces, Transcendental methods of algebraic geometry (complex-analytic aspects), Families, moduli of curves (analytic), Singularities in algebraic geometry, Coverings in algebraic geometry A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex 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 A \textit{Rees algebra} over an algebraic variety \(V\) (smooth over a field \(k\)) is a finitely generated subalgebra of \({\mathcal O}_{V}[T]\) (with \(T\) an indeterminate). They are important in the theory of resolution of singularities. Indeed, one may define the singular locus of a Rees algebra, and if Rees algebras can be resolved in a suitable ``canonical'' or ``algorithmic'' way (by means of a sequence monoidal transformations with well determined smooth centers), there is a formalism leading to embedded resolution of a subvariety \(X\) of a smooth ambient one \(V\).
There are indications, specially thanks to the efforts of O. Villamayor, that, working over perfect fields of positive characteristic, Rees algebras are natural objects to consider in resolution problems. As usual, to resolve Rees algebras at a crucial step one would like to use induction on the dimension of the base variety \(V\). In positive characteristic, as indicated by Giraud, Hironaka and others, it seems hopeless to use smooth hypersurfaces on \(V\) to define an auxiliary inductive algebra to implement the inductive procedure. Instead, Villamayor considers (locally defined) suitable ``generic'' projections \(\beta:V \to V'\), where \(\dim V' < \dim V\), and introduces a new Rees algebra \({\mathcal R}_{ {\mathcal G}, \beta}\) over \(V'\), called the \textit{elimination algebra} (defined using techniques inspired by classical elimination theory).
Trying to compare \(\mathcal G\) and its elimination algebra, the so-called \(\tau\)-invariant, introduced by Hironaka long ago, seems to be very important. If \(\mathcal G\) is a Rees algebra over \(V\) and \(x \in V\) is a closed point, \({\tau}_{{\mathcal G},x}\) is defined as the codimension of a certain linear subspace (related to the tangent cone \(C_{{\mathcal G},x}\)) of the tangent space \(T_{V,x}\).
In the reviewed paper, Benito studies properties of the \(\tau\)-invariant. Working over a perfect field \(k\) of any characteristic, her main results are:
(1) A new proof of the fact that if \(\mathcal G\) and \({\mathcal G}'\) are Rees algebras over the variety \(V\) and both have the same integral closure (in \({\mathcal O}_V[T]\)), then for every closed point \(x \in V\) we have \({\tau}_{{\mathcal G},x}= {\tau}_{{\mathcal G '},x} \) (a result previously obtained, with other methods, by H. Kawanoue).
(2) If \(\beta: V \to V'\) is a generic projection of algebraic varieties over \(k\), with \(\dim V' = \dim V -1\), \(x\) is a closed point of \(V\) and \(x'=\beta(x)\) and, moreover, \(\mathcal G\) is \textit{differential} (i.e., closed under application of differential operators) then \({\tau} _ { {\mathcal R}_{{\mathcal G},\beta} , x' } = {\tau}_{{\mathcal G},x} -1\).
(3) If, in (2), we drop the assumption that \(\mathcal G\) is differential, then we have an inequality \({\tau} _ { {\mathcal R}_{{\mathcal G},\beta} , x' } \leq {\tau}_{{\mathcal G},x} -1\).
Much of the the paper is devoted to a review of basic material on Rees and elimination algebras, necessary to give an understandable presentation of the main results. Singularity; positive characteristic; differential operator; Rees algebra; tau invariant Benito A.: The {\(\tau\)}-invariant and elimination. J. Algebra 324, 1903--1920 (2010) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Birational geometry, Singularities of surfaces or higher-dimensional varieties The \(\tau \)-invariant and elimination | 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 describe a combinatorial game that models the problem of resolution of singularities of algebraic varieties over a field of characteristic zero. By giving a winning strategy for this game, we give another proof of the existence of resolution. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Polynomials in real and complex fields: location of zeros (algebraic theorems), Applications of game theory, Combinatorial games A simplified game for 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 In J. Am. Math. Soc. 15, No. 3, 599--615 (2002; Zbl 0998.14009), \textit{M. Mustaţă} provides characterisations of KLT and LC pairs \((X, qY)\) with \(X\) smooth via dimension of jet schemes of \(Y\). The aim of the paper is to extend this result to the case \(X\) is normal and \(\mathbb{Q}\)-Gorenstein. KLT singularities; LC singularities Yasuda, T.: Dimensions of jet schemes of log singularities. Amer. J. Math. 125, No. 5, 1137-1145 (2003) Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Dimensions of jet schemes of log 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 note is a short survey of two topics: Archimedean zeta functions and Archimedean oscillatory integrals. We have tried to portray some of the history of the subject and some of its connections with similar devices in mathematics. We present some of the main results of the theory and at the end we discuss some generalizations of the classical objects. Archimedean zeta functions; Bernstein-Sato polynomials; oscillatory integrals; oscillatory integral operators; multilinear level set operators; D-modules Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Singular and oscillatory integrals (Calderón-Zygmund, etc.), Other Dirichlet series and zeta functions, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Asymptotic approximations, asymptotic expansions (steepest descent, etc.), Local complex singularities, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) Archimedean zeta functions and oscillatory integrals | 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 0563.00006.]
When X is a non-singular, quasi-projective variety defined over an algebraically closed field, the study of vector bundles on X (read: the algebraic K-theory of X) is intimately related to the Chow group of algebraic cycles on X modulo rational equivalence. This relationship is elegantly expressed by Bloch's formula: \(H^ p(X,K_ p)=CH^ p(X)\). The left-hand side of this formula makes sense even if X has singularities, and much effort has been devoted recently to finding the appropriate ''geometric'' object to generalize the right-hand side for singular X. This paper surveys results on this question, due mainly to Gillet, Levine, Pedrini, Srinivas, Weibel and the author. Among the topics touched on are varieties with isolated singularities, surfaces with arbitrary singularities, the kernel of the map induced by desingularization and the torsion in \(CH^ 2(X)\). algebraic cycles modulo rational equivalence; Bloch formula; vector bundles; algebraic K-theory; Chow group; singularities; desingularization A. Collino, Quillen's \(K\)-theory and algebraic cycles on singular varieties , Geometry today (Rome, 1984), Progr. Math., vol. 60, Birkhäuser Boston, Boston, MA, 1985, pp. 75-85. (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Quillen's K-theory and algebraic cycles on singular varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Brieskorn's (meanwhile classical) construction of the rational double points ADE from the subregular points in the nilpotent cone \({\mathcal N}\) of a Lie-algebra \({\mathfrak g}\), associated to a complex simple algebraic group, is reconsidered with a different proof. This is done using normality of \({\mathcal N}\) in subregular points [cf. \textit{P. Slodowy}, ``Simple singularities and simple algebraic groups'', Lect. Notes Math. 815 (1980; Zbl 0441.14002)] and a resolution due to T. Springer. Especially, in the description of the desingularization, the intersection matrix is calculated in a purely cohomological way.
For a different approach, cf. also \textit{F. Knop} [Invent. Math. 90, 579- 604 (1987; Zbl 0648.14002)]. subregular element; nilpotent cone; rational double points; ADE [H2] V. Hinich,On Brieskorn's theorem, Israel Journal of Mathematics76 (1991), 153--160. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities On Brieskorn's theorem | 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 quasi-projective variety over an algebraically closed field. When X is non singular there is a well known ``intersection theory'' with values in the graded Chow ring of X. Here we consider the question of extending this theory to the singular case. In particular: we show how to develop a ``Chow theory'', starting with a suitable cohomology theory on X (e.g.: singular cohomology, De Rham cohomology, étale cohomology). algebraic cycles; singular varieties; intersection cohomology; intersection theory; graded Chow ring; Chow theory; singular cohomology; De Rham cohomology; étale cohomology Algebraic cycles, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, de Rham cohomology and algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry Algebraic cycles on singular varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we characterize the representation type of an acyclic quiver by the properties of its associated quiver Grassmannians. This characterization utilizes and extends known results about singular quiver Grassmannians and cell decompositions into affine spaces.
While all quiver Grassmannians for indecomposable representations of quivers of finite representation type are smooth and admit cell decompositions, it turns out that all quiver Grassmannians for indecomposable representations of tame quivers admit cell decompositions, but some of these quiver Grassmannians are singular (even as varieties). A quiver is wild if and only if there exists a quiver Grassmannian with negative Euler characteristic. Grassmannians, Schubert varieties, flag manifolds, Representation type (finite, tame, wild, etc.) of associative algebras, Singularities in algebraic geometry, Representations of quivers and partially ordered sets Representation type via Euler characteristics and singularities of quiver Grassmannians | 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 0675.00006.]
Generalizing a construction of Hirzebruch, the authors construct quintic threefolds in \({\mathbb{P}}^ 4\) with many nodes \((=ordinary\) double points). The constructed examples are of the form \(F(x,y)-G(u,v)=0\quad (affine\quad equation)\) where F and G are of degree \( 5.\) E.g. if \(G(x,y)=(x+1)(y^ 2-(x-2)^ 2/3)(x^ 2+y^ 2-8/5)\) \((G=0\) defines a union of a circle and a triangle) then \(G(x,y)-G(u,v)=0\) has 118 nodes; if \(F_ c(x,y)=(x+c)(y^ 4-y^ 2(2x^ 2-2x+1)+(x^ 2+x-1)^ 2/5)\) \((F_ c=0\) defines a skew pentagon which is regular if \(c=)\) then \(F_{-2}(x,y)-F_{-2}(u,v)=0\) has 120 nodes. Hirzebruch used the regular pentagon \(F_{}\) and obtained 126 nodes which is still the record. It appears as a special member of a 1-parameter family of quintics where the general member has 120 nodes.
It is an interesting question whether there are small resolutions of such a quintic which are not projective algebraic. To decide this it is essential to know the divisors on V which pass through the nodes and which are not homologous to a multiple of the generic hyperplane section. The number of independent such divisors is called the defect. Although the authors cannot decide the projectivity of some small resolutions of their examples, they compute the defect by an interesting method based on computing the absolute value of the eigenvalues of Frobenius acting on étale cohomology and the Lefschetz fixed-point formula over a suitable finite field. quintic threefolds; nodes; small resolutions; defect; Frobenius; Lefschetz fixed-point formula B. van Geemen and J. Werner, Nodal quintics in \( {P^4}\), preprint. \(3\)-folds, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Projective techniques in algebraic geometry Nodal quintics 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 For complex algebraic surfaces, this paper provides the final step in the proof of the conjecture (due to John Nash, about 30 years ago) that the singularities of an algebraic variety can be resolved by a finite succession of proper birational maps \(\mu\) : \(\bar S\to S\), \(\mu^*\Omega_{S/k}/\)torsion locally free of rank dim(S) and \(\mu\) universal with respect to that property (so called ``Nash transformations''). It is known by \textit{A. Nobile} [Pac. J. Math. 60, 297-305 (1975; Zbl 0324.32012)] that (in characteristic 0) \(\mu\) is an isomorphism iff S is nonsingular, and in particular, for plane curve singularities the conjecture is true. \textit{G. González-Sprinberg} [Ann. Inst. Fourier 32, No.2, 111-178 (1982; Zbl 0469.14019)] proved for complex surfaces, that normalized Nash transformations resolve rational double points and cyclic quotient singularities. By a theorem of H. Hironaka, any surface singularity is transformed by a finite succession of Nash transformations into ``sandwiched singularities'', i.e. singularities of a surface which birationally dominates a nonsingular surface.
The main theorem of the article under review states that sandwiched singularities are resolved by normalized Nash transformations, thus completing the proof of the above mentioned conjecture for \(\dim(S)=2\), \(k={\mathbb{C}}.\)
Chapter II of the paper gives a classification of sandwiched singularities, a problem which is shown to be equivalent to the classification of plane curve singularities, complete ideals in 2- dimensional regular local rings or valuations with center in a regular 2- dimensional local ring [partially, this is an overview of results of the same author, cf. Am. J. Math. 112, No.1, 107-156 (1990; Zbl 0716.13003)].
In chapter III, the main theorem is proved in two steps: First of all, minimal singularities are considered, i.e. here: rational surface singularities with reduced fundamental cycle. If \((S,\xi)\) is such a singularity, \(\Gamma\) its dual graph and \(\mu:S'\to S\) the normalized Nash transformation, \(\Gamma '_ i\) the dual graphs of the singularities of \(S'\), then \(\#\{\)vertices of \(\Gamma'_ i\}\leq \#\{\)vertices of \(\Gamma\},\) i.e. such procedure terminates after finitely many steps. The proof is completed by showing that the problem can be reduced to minimal singularities, i.e. any sandwiched singularity is transformed into a minimal one after finitely many normalized Nash transformations. resolution of singularities; Nash transformations; surface singularity; classification of sandwiched singularities; classification of plane curve singularities Mark Spivakovsky, ``Sandwiched singularities and desingularization of surfaces by normalized Nash transformations'', Ann. Math.131 (1990) no. 3, p. 411-491 Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings Sandwiched singularities and desingularization of surfaces by normalized Nash transformations | 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 Starting from McKays observation on the description of (an essential part of) the representation theory of binary polyhedral groups \(\Gamma\) in terms of extended Coxeter-Dynkin-Witt diagrams \(\widetilde {\underline\Delta} (\Gamma)\) and working in the differential geometric framework of hyper-Kähler-quotients \textit{P. B. Kronheimer} [C. R. Acad. Sci., Paris, Sér. I 303, 53-55 (1986; Zbl 0591.53057) and J. Differ. Geom. 29, No. 3, 665-683 (1989; Zbl 0671.53045)] was able to give a new construction of the semiuniversal deformations of the Kleinian singularities \(X=\mathbb{C}^2/ \Gamma\) as well as of their simultaneous resolutions. As far as the deformations were concerned, he already gave a purely algebraic geometric formulation of his results in terms of representations of certain quivers naturally attached to the diagrams \(\widetilde {\underline\Delta} (\Gamma)\). By making use of the invariant-theoretic notion of ``linear modification'' (cf. section 6) and applying it to Kronheimer's quiver construction we show here how to obtain a purely algebraic geometric simultaneous resolution as well (section 7). On the way, we remind the reader of various facts about Kleinian singularities (section 1), McKay's observation (section 2), symplectic geometry (section 3), Kronheimer's work (section 4), and quivers (section 5). -- This article covers the main results of the doctoral dissertation by \textit{H. Cassens} (``Lineare Modifikation algebraischer Quotienten, Darstellungen des McKay-Köchers und Kleinsche Singularitäten'', Hamburg 1995; Zbl 0842.14036). linear modification; Kleinian singularities; simultaneous resolutions; quivers Cassens, H.; Slodowy, P., On Kleinian singularities and quivers, (Singularities. Singularities, Oberwolfach, 1996. Singularities. Singularities, Oberwolfach, 1996, Progr. Math., vol. 162, (1998), Birkhäuser: Birkhäuser Basel), 263-288 Singularities in algebraic geometry, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Global theory and resolution of singularities (algebro-geometric aspects) On Kleinian singularities and 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 [For the entire collection see Zbl 0517.00010.]
This work is a short version of a paper which will be published by the author and \textit{J. L. Verdier} in Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 409-449 (1983; Zbl 0538.14033). The results are given without proofs. Let G be a finite subgroup of SL(2,\({\mathbb{C}})\) and S the surface obtained as the a quotient \(S={\mathbb{C}}^ 2/G\), having the origin as double rational singularity. If \(q:\widetilde S\to S\) is a minimal resolution of the singularity, D the exceptional fiber of q and Irr(D) the union of the irreducible components of D, there are well-known results of M. Artin connecting the group G and the dual graph \(\Gamma\) of Irr(D). In this paper are considered the ring R(G) of the representations of G and the Grothendieck ring \(K(\widetilde S)\) and are announced the following results: (1) The rings R(G) and \(K({\mathbb{C}}^ 2,0)\) are isomorphic. - (2) There is a bijection \(\pi *R(G)\to K(\tilde S)\). - (3) If \(c\in R(G)\) is the canonical representation there is a ring homomorphism \(R(G)/(2-c)R(G)\to K(\tilde S)/K_ D(\tilde S)\). Grothendieck ring; representations of group of automorphisms; double rational singularity Singularities of surfaces or higher-dimensional varieties, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Homogeneous spaces and generalizations, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Representation of polyhedral groups and singularities of surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let consider a surface \(S\) with an isolated singular point \(O\) over a closed field of characteristic 0. \textit{J. F. Nash, jun.} proved in the sixties (see [Duke Math. J. 81, No.1, 31--38 (1995; Zbl 0880.14010)]) that the space of arcs passing trough \(O\) has a finite number of irreducible components, and he raised the question to know if this number coincides with the number of exceptional components of the minimal resolution of singularities of \(S\). This problem was considered to be difficult but recently there has been a lot of progress, and many partial answers were given by several authors (including the reviewer).
In the paper under review, the authors solve completely this problem. Their proof uses many tools developed by people trying to prove the Nash problem and some new ideas developed recently by the authors. One main point is the translation in terms of wedges by Monique Lejeune-Jalabert, and recent work done by the two authors [\textit{J. Fernández de Bobadilla}, Adv. Math. 230, No. 1, 131--176 (2012; Zbl 1248.14004); \textit{M. Pe Pereira}, J. Lond. Math. Soc., II. Ser. 87, No. 1, 177--203 (2013; Zbl 1272.32027)]. The proof is by contradiction, if the Nash's problem is not true for \(S\), there exists a convergent wedge \(\alpha \) with precise properties, \(\alpha \) can be viewed as a family of mappings \(\alpha_s :{\mathcal U}_s\rightarrow (S,O) \), so is natural to consider the lifts \(\tilde\alpha_s :{\mathcal U}_s\rightarrow \tilde S \), where \(\tilde S\) is the minimal resolution of singularities of \(S\). Their images \(Y_s\subset \tilde S\) converge to a limit divisor \(Y_0\), then by some computations on the Euler characteristics of \(Y_0\) they prove that this leads to a contradiction which finishes the proof. Nash's problem; surfaces; singularities; resolution of singularities; space of arcs; wedges; coins Fernández de Bobadilla, Javier; Pereira, María. Pe, The Nash problem for surfaces, Ann. of Math. (2), 176, 3, 2003-2029, (2012) Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) The Nash problem for surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be the germ of a normal surface singularity \((X,o)\) whose link \(M\) is a rational homology sphere, let \(\pi \colon \widetilde{X} \rightarrow X\) be a resolution of the singularity, and let \(K\) be the canonical class of \(\widetilde{X}.\) It is assumed that the exceptional divisor \(E = \pi^{-1}(o)\) has normal crossings.
The authors analyze the following hypothetical upper bound for the geometric genus of the singularity: \(sw_M(\sigma_{\text{ca}n})- {1\over 8}(K^2+ \#(E)) \geq p_g,\) where the first term is a rational number, the Seiberg-Witten invariant associated to the canonical \(\text{spin}^c\) structure of \(M,\) and \(\#(E)\) is the number of irreducible components of \(\pi^{-1}(o).\) In fact, \((K^2 + \#(E))\) does not depend on the choice of the resolution \(\pi.\) The authors further conjecture that for a \(\mathbb{Q}\)-Gorenstein singularity the inequality becomes equality, while for a smoothable Gorenstein singularity \(-sw_M(\sigma_{\text{can}}) = {1\over 8}\sigma(F).\) Here \(\sigma(F)\) is the signature of the Milnor fiber \(F\) of a smoothing.
It should be remarked when \(X\) is a suspension hypersurface singularity or Brieskorn-Hamm complete intersection with \(H_1(M,Z)=0\) then \({1\over 8}\sigma(F)\) is equal to the Casson invariant \(\lambda(M)\) of the link \(M\) [\textit{W. Neumann} and \textit{J. Wahl}, Comment. Math. Helv. 65, 58-78 (1990; Zbl 0704.57007)]. Making use of the Reidemeister-Turaev sign refined torsion theory for rational homology \(3\)-manifolds [\textit{V. G. Turaev}, Math. Res. Lett. 4, 679-695 (1997; Zbl 0891.57019)], properties of the Fourier transform, the generalized Fourier-Dedekind and Dedekind-Rademacher sums, and some other considerations the authors verify their conjectures by direct computations for several rational, cyclic quotient and other types of minimally elliptic singularities, for Brieskorn-Hamm complete intersections, etc. They underline that the obtained results extend and generalize earlier works of M. Artin, H. Laufer, S. S.-T. Yau, and many others [see, e.g. \textit{M. Artin}, Am. J. Math. 88, 129-136 (1966; Zbl 0142.18602), \textit{H. Laufer}, Am. J. Math. 99, 1257-1295 (1977; Zbl 0384.32003) and \textit{S. S.-T. Yau}, Trans. Am. Math. Soc. 257, 269-329 (1980; Zbl 0343.32009)]. surface singularities; \(Q\)-Gorenstein singularities; rational singularities; Brieskorn-Hamm complete intersections; geometric genus; Seiberg-Witten invariants of \(Q\)-homology spheres; Reidemeister-Turaev torsion; Casson-Walker invariant Némethi, András; Nicolaescu, Liviu I., Seiberg--Witten invariants and surface singularities, Geom. Topol., 6, 269-328, (2002) Complex surface and hypersurface singularities, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Applications of global analysis to structures on manifolds, Invariants of knots and \(3\)-manifolds, Global theory and resolution of singularities (algebro-geometric aspects), Milnor fibration; relations with knot theory, Knots and links in the 3-sphere Seiberg-Witten invariants and surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension. associated graded ring of a valuation; ramification; valuation 10.1080/00927872.2015.1065845 Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Global theory and resolution of singularities (algebro-geometric aspects) Ramification of valuations and local rings in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give a criterion for the existence of noncommutative crepant resolutions (NCCRs) for certain toric singularities. In particular, we recover \textit{N. Broomhead}'s result [Dimer models and Calabi-Yau algebras. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1237.14002)] that a three-dimensional toric Gorenstein singularity has an NCCR. Our result also yields the existence of an NCCR for a four-dimensional toric Gorenstein singularity, which is known to have no toric NCCR.
For part II, see [\textit{Š. Špenko} and \textit{M. Van den Bergh}, J. Noncommut. Geom. 14, No. 1, 73--103 (2020; Zbl 1454.13011)]. Noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Non-commutative crepant resolutions for some toric 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 Let \(X\) be a smooth complex variety. To a non-zero quasi-coherent sheaf of ideals \(\underline{\mathbf a}\) on \(X\) one can associate a sequence of ideals called the multiplier ideals of \(\underline{\mathbf a}\), which depend on a rational parameter. The behaviour of these ideals encodes in a subtle way the properties of the singularities of \(V(\underline{\mathbf a})\). Introduced first in the analytic context in the work of Demailly, Nadel, Siu and others, multiplier ideals have recently found surprising applications in algebraic geometry.
Here is the definition. Suppose that \(f:X'\to X\) is a log resolution of \((X,V(\overline {\mathbf a}))\), i.e., \(f\) is proper and birational, \(X'\) is smooth, and \(f^{-1}V (\underline{\mathbf a}) =D\) is a divisor with simple normal crossings. If \(K_{X'/X}\) is the relative canonical divisor of \(f\), the multiplier ideal of \(\underline {\mathbf a}\) with coefficient \(\alpha\in \mathbb{Q}_+\) is
\[
{\mathcal I}(X,\alpha \cdot\underline {\mathbf a})=f_*{\mathcal O}_{X'} \bigl( K_{X'/X} -[\alpha D]\bigr).
\]
Here \([\cdot]\) denotes the integral part function.
We prove that if \(\underline{\mathbf a},\underline{\mathbf b}\subseteq {\mathcal O}_X\) are non-zero sheaves of ideals on a complex smooth variety \(X\), then for every \(\gamma\in \mathbb{Q}_+\) we have the following relation between the multiplier ideals of \(\underline{\mathbf a}\), \(\underline{\mathbf b}\) and \(\underline {\mathbf a}+\underline {\mathbf b}\):
\[
{\mathcal I}(X,\gamma \cdot(\underline {\mathbf a}+ \underline {\mathbf b}))\subset \sum_{\alpha+ \beta=\gamma} {\mathcal I}(X,\gamma \cdot \underline {\mathbf a}) \cdot{\mathcal I}(X,\beta\cdot \underline{\mathbf b}).
\]
A similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals. We use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals. This is applied to compare the multiplier ideals associated to a scheme in different embeddings. multiplier ideals; log resolutions; monomial ideals Mustaţă, M., \textit{the multiplier ideals of a sum of ideals}, Trans. Amer. Math. Soc., 354, 205-217, (2002) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The multiplier ideals of a sum of 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 We give a combinatorial proof for a multivariable formula of the generating series of type D Young walls. Based on this we give a motivic refinement of a formula for the generating series of Euler characteristics of Hilbert schemes of points on the orbifold surface of type D. Hilbert scheme of points; Young walls; generating function Parametrization (Chow and Hilbert schemes), Combinatorial aspects of representation theory, Quantum groups (quantized enveloping algebras) and related deformations, Singularities in algebraic geometry Young walls and equivariant Hilbert schemes of points in type \(D\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an ambitious paper which lays the foundations of homological theory of filtration and applies them to study surface singularities. Here a filtration of a local ring (A,m) is a decreasing sequence of non- zero ideals \((F^ n| n\in {\mathbb{Z}})\) such that \(F^ n=A\quad (n\leq 0),\quad F^ iF^ j\subseteq F^{i+j},\quad \cap F^ n=(0),\) and such that the Rees ring \({\mathcal R}=\oplus_{n\geq 0}F^ nT^ n \) is finitely generated over \(A={\mathcal R}_ 0\). For the authors a good filtration is one whose associated graded ring \(G=\oplus F^ n/F^{n+1} \) has good properties (e.g. normal domain with isolated singularity), and from such filtration (which is in general not ideal-adic) they derive properties of A.
In chapter 1, {\S}1 to 3 are devoted to such subjects as the filtered blowing-up \(X=\Pr oj({\mathcal R})\), local cohomology, canonical modules, divisor class group, dualizing complexes. In {\S}4 a criterion for X to be normal with only rational singularities is given. The main result of {\S}5 is: if A is a normal 2-dimensional local ring and G is a domain with isolated singularity, then the (dual graph of the exceptional divisor of) resolution of singularity of Spec(A) is star-shaped.
In chapter 2, conversely, normal 2-dimensional singularities with star- shaped resolution are studied. Such a resolution defines a filtration on A, and its associated graded ring G is compared with another graded ring called Pinkham-Demazure construction. homological theory of filtration; surface singularities; Rees ring; good filtration; filtered blowing-up; local cohomology; canonical modules; divisor class group; isolated singularity; singularities with star-shaped resolution Tomari, M.; Watanabe, K., Filtered rings, filtered blowing-ups and normal two-dimensional singularities with ''star-shaped'' resolution, Publ. Res. Inst. Math. Sci., 25, 681-740, (1989) Deformations and infinitesimal methods in commutative ring theory, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Singularities of surfaces or higher-dimensional varieties, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Étale and flat extensions; Henselization; Artin approximation Filtered rings, filtered blowing-ups and normal two-dimensional singularities with ``star-shaped'' resolution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields of arbitrary characteristic. Furthermore, from the same perspective, we study a family of cluster algebras which are not of finite type and which arise from a star shaped quiver. cluster algebras; singularities; resolution of singularities; Dynkin diagrams; continuant polynomials Cluster algebras, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Classification of singularities of cluster algebras of finite type: the case of trivial 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 The articles of this volume will not be indexed individually.
Contents: \textit{Kyoji Saito}, \(\theta\)-invariant formulas for extended affine root systems and moduli spaces of elliptic singularities (Japanese) (p. 1-22); \textit{Shigeru Mukai}, On groups of automorphisms of K3 surfaces (Japanese) (p. 23-56); \textit{Masaaki Yoshida}, Differential equations of rank 3 on \({\mathbb{C}}P^ 2\) (Japanese) (p. 57-60); \textit{Iku Nakamura}, Infinitesimal deformations of cusp singularities (p. 61-62); \textit{Jiro Sekiguchi}, A note on the invariant holonomic system (p. 63- 70); \textit{Hiroyasu Tsuchihashi}, 3-dimensional singularities with resolutions whose exceptional sets are toric divisors (p. 71-94); \textit{Fumio Sakai}, Normal surfaces and intersection theory (p. 95-109); \textit{Nobuo Sasakura}, A Čech cohomological method of construction of holomorphic vector bundles (p. 110-144); \textit{Mitsuyoshi Kato}, Complex surfaces corresponding to weighted line configurations on complex projective planes (Japanese) (p. 145-160); \textit{Isao Naruki}, Straying of 3-curves and families of conic sections (Japanese) (p. 161-175); \textit{Kimio Watanabe} and \textit{Shihoko Ishii}, A classification and construction of purely elliptic singularities (Japanese) (p. 176-209); \textit{Mutsuo Oka}, On the resolution of hypersurface singularities (p. 210-256); \textit{Masataka Tomai}, On calculational formulas for geometric genera and elliptic singularities (Japanese) (p. 257-289); \textit{Toshisumi Fukui}, Certain ramified coverings of \(P_ 2({\mathbb{C}})\) (Japanese) (p. 290-304). Singularities; Varieties; Proceedings; Symposium; Kyoto; RIMS Singularities in algebraic geometry, Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Deformations of singularities, Deformations of complex singularities; vanishing cycles, Singularities of surfaces or higher-dimensional varieties, Complex singularities, Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces Recent results on singularities of varieties. Proceedings of a Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, March 15-17, 1984 | 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 is related to the one reviewed above [\textit{R. Narasimhan,} ibid. 89, 402-406 (1983)] and gives a characteristic free proof that the 2-codimensional part of equimultiple locus of a singular hypersurface is hyperplanar. hyperplanarity; complete intersection; equimultiple locus of a singular hypersurface S. B. Mulay, Equimultiplicity and hyperplanarity, Proc. Amer. Math. Soc. 89 (1983), no. 3, 407 -- 413. Singularities of surfaces or higher-dimensional varieties, Multiplicity theory and related topics, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Equimultiplicity and hyperplanarity | 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 space of smooth genus-0 curves in projective space has a natural smooth compactification: the moduli space of stable maps, which may be seen as the generalization of the classical space of complete conics. In arbitrary genus, no such natural smooth model is expected, as the space satisfies ``Murphy's Law'' [cf. \textit{R. Vakil}, Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)]. In genus 1, however, the situation remains beautiful. We give a natural smooth compactification of the space of elliptic curves in projective space, and describe some of its properties. This space is a blowup of the space of stable maps. It can be interpreted as a result of blowing up the most singular locus first, then the next most singular, and so on, but with a twist--these loci are often entire components of the moduli space. We give a number of applications in enumerative geometry and Gromov-Witten theory. For example, this space is used by the second author to prove physicists' predictions for genus-1 Gromov-Witten invariants of a quintic threefold. The proof that this construction indeed gives a desingularization will appear in a subsequent paper. Vakil R., Zinger A.: A natural smooth compactification of the space of elliptic curves in projective space. Electron. Res. Announc. Am. Math. Soc. 13, 53--59 (2007) Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) A natural smooth compactification of the space of elliptic curves in projective space | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We show that the topological decomposition theorem for a proper semismall map \(f:X\to Y\) implies a ``motivic'' decomposition theorem for the rational algebraic cycles of \(X\) and, in the case \(X\) is compact, for the Chow motive of \(X\). We work in the category of pure Chow motives over a base. Under suitable assumptions on the stratification, we also prove an explicit version of the motivic decomposition theorem and compute the Chow motives and groups in some examples, e.g. the nested Hilbert schemes of points of a surface. In an appendix with T. Mochizuki, we do the same for the parabolic Hilbert scheme of points on a surface. decomposition theorem Cataldo, M.A.A.; Migliorini, L., The Chow motive of semismall resolutions, Math. Res. Lett., 11, 151-170, (2004) (Equivariant) Chow groups and rings; motives, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects), Mixed Hodge theory of singular varieties (complex-analytic aspects) The Chow motive of semismall 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 Let \(k\) be an algebraically closed field, and let \(A\) be a \(k\)-algebra. For \(d\geq1,\) let mod\(_{A}\left( d\right) \) be the set of \(A\)-module structures on the \(d\)-dimensional vector space \(k^{d}.\) As each \(M\in\)mod\(_{A}\left( d\right) \) corresponds to a \(k\)-algebra map from \(A\) into the ring of \(d\times d\) matrices with entries in \(k\), one obtains an action of \(\text{GL}\left( d\right) \) on mod\(_{A}\left( d\right) \) through conjugation on the image of this map. Let \(\mathcal{O}_{M}\) be the orbit of \(M\in\)mod\(_{A}\left( d\right) ;\) then \(\mathcal{O}_{M}\) corresponds to the isomorphism class of \(M\) (as an \(A\)-module). In a previous paper [\textit{Nguyen Quang Loc} and \textit{G. Zwara}, Osaka J. Math. 44, No. 4, 945--954 (2007; Zbl 1141.14003)], the Zariski closure \(\mathcal{\bar{O}}_{M}\) has been studied, paying particular attention to whether \(\mathcal{\bar{O}}_{M}\) is a non-singular variety. In that work, \(\mathcal{\bar{O}}_{M}\) is shown to be a non-singular variety if and only if \(B\) is hereditary and Ext\(_{B}^{1}\left( M,M\right) \cong0.\)
By contrast, in this work the authors give necessary and sufficient conditions for when \(\mathcal{\bar{O}}_{M}\) is a singular local hypersurface in the case where char \(k=0.\;\)The conditions here are more complex than in the non-singular case and, like the work cited above, much of this result is described using quivers. For a given \(A\)-module \(M,\) let \(B=A/\)Ann\(\left( M\right) .\) Associated to \(B\) one has the Gabriel quiver \(\Gamma\) along with an admissible ideal \(I\subset k\Gamma\) (where \(k\Gamma\) is the path algebra of \(\Gamma\)) such that the categories of \(B\)-modules and \(k\Gamma/I\)-modules are equivalent. It shown that (1) if \(B\) is hereditary and Ext\(_{B}^{1}\left( M,M\right) \cong k\) then \(\mathcal{\bar{O}}_{M}\) is a singular local hypersurface. Furthermore, (2) \(\mathcal{\bar{O}}_{M}\) is a singular local hypersurface if Ext\(_{B}^{1}\left( M,M\right) =0\) and \(I=\langle\gamma ^{2}\rangle\) for \(\gamma\) some loop in \(\Gamma\) at a vertex \(i\) with the \(i^{\text{th}}\) component of the dimension vector associated to \(M\) is \(2\). Also, (3) if Ext\(_{B}^{1}\left( M,M\right) =0\) and \(I=\langle\rho\rangle,\) where \(\rho\) is a relation between two vertices with corresponding dimension vector components equal to \(1\), then \(\mathcal{\bar{O}}_{M}\) is a singular local hypersurface. Finally, if \(\mathcal{\bar{O}}_{M}\) is a singular local hypersurface then one of (1), (2), or (3) must hold.
The proof of the main result above follows from another result given about quiver representations. For \(Q\) a quiver and \(\mathbf{d}=\left( d_{i}\right) _{i\in Q_{0}}\) a dimension vector, let \(N\) be a representation such that Ann\(\left( N\right) \) is an admissible ideal in \(kQ.\) Necessary and sufficient conditions are given for when \(\mathcal{\bar{O}}_{N}\) is a singular hypersurface, and these conditions are very similar to the ones above. module varieties; orbit closures; hypersurfaces; quiver representations; singularity Loc, NQ; Zwara, G, Modules and quiver representations whose orbit closures are hypersurfaces, Colloq. Math., 134, 57-74, (2014) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets Modules and quiver representations whose orbit closures are 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 We prove that certain quiver varieties are irreducible and therefore are isomorphic to Hilbert schemes of points of the total spaces of the bundles \(\mathcal{O}_{\mathbb{P}^1}(-n)\) for \(n \ge 1\). quiver representations; Hilbert schemes of points 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, Parametrization (Chow and Hilbert schemes) On the irreducibility of some 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 \((\mathfrak R,\mathfrak m)\) be a local noetherian ring and let \(I\subseteq R\) be an ideal. By \(\mathfrak R(I)\) we denote the Rees-ring of \(I\), which is defined as \(\mathfrak R(I)=\oplus_{n\ge 0}I^n\). The form ring \(Gr(I)\) of \(I\) is defined as \(Gr(I)=\mathfrak R(I)/I\mathfrak R(I)=\oplus_{n\ge 0}I^n/I^{n+1}\). Recall that the morphism \(\pi: \operatorname{Proj}(\mathfrak R(I))\to \operatorname{Spec}(R)\) defines the blow-up of \(\operatorname{Spec}(R)\) at \(\operatorname{Spec}(R/I)\) and that \(\operatorname{Proj}(Gr(I))\) is the exceptional fiber of this blow-up. This geometric interpretation gives the motivation to calculate the local cohomology modules \(H^i_{\mathfrak m(I)}(\mathfrak R(I))\) and \(H^i_{\mathfrak m(I)}(Gr(I))\) (in a certain range of \(i)\), where \(\mathfrak m(I)\) is the homogeneous maximal ideal \(\mathfrak m\oplus I\oplus I^2\oplus\cdots\) of \(\mathfrak R(I)\) and where \(I\) belongs to a particular class of ideals. This continues a corresponding investigation, begun in the first part of the paper [ibid. 81, 29--57 (1983; Zbl 0475.14001)] .
In part II we are mainly concerned with a class of ideals \(I\), whose Rees-rings may be understood as so called symbolic Rees-rings. In particular, our ideals are normally torsion-free in some important cases. We also investigate the behaviour of our blow-up under Veronese transforms. This leads to a characterization of some special (Buchsbaum-)singularities.
Our results also give the local background for an improved version of Faltings' ``Macaulayfication'' [the author, Comment. Math. Helv. 58, 388--415 (1983; Zbl 0526.14035)]. blow-up; Macaulayfication; Buchsbaum singularities; form ring; local cohomology modules; symbolic Rees-rings; Veronese transforms Brodmann, M.: Local cohomology of Certain Rees Form-rings. J. Algebra, 86, 457--493 (1984) Local cohomology and algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Local cohomology of certain Rees- and form-rings. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author investigates the relation between basic numerical invariants associated to a normal surface singularity (V,0) (e.g. Milnor number, Milnor number of a generic hyperplane section and local Euler obstruction) and numerical invariants associated to a resolution dominating the Nash modification of the singularity (V,0). numerical invariants; normal surface singularity; Milnor number; Nash modification Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities, Singularities in algebraic geometry, Normal analytic spaces Modification de Nash et invariants numériques d'une surface normale. (Nash modification and numerical invariants of a normal surface) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The abstract: Let \(G\) be a finite subgroup of \(GL (3,\mathbb{C})\). Then \(G\) acts on \(\mathbb{C}^ 3\). It is well known that \(\mathbb{C}^ 3/G\) is Gorenstein if and only if \(G \subseteq SL (3,\mathbb{C})\). In chapter one, we sketch the classification of finite subgroups of \(SL(3,\mathbb{C})\). We include two more types \((J)\) and \((K)\) which were usually missed in the work of many mathematicians. In chapter 2, we give a general method to find invariant polynomials and their relations of finite subgroups of \(GL(3,\mathbb{C})\). The method is in practice substantially better than the classical method due to Noether. In chapter 3, we recall some properties of quotient varieties and prove that \(\mathbb{C}^ 3/G\) has isolated singularities if and only if \(G\) is abelian and 1 is not an eigenvalue of \(g\) in \(G\). We also apply the method in chapter 2 to find minimal generators of the ring of invariant polynomials as well as their relations. finite subgroups of \(SL (3,\mathbb{C})\); quotient varieties; isolated singularities; ring of invariant polynomials S. Yau and Y. Yu, \textit{Gorenstein quotient singularities in dimension three}, \textit{Mem. Amer. Math. Soc.}\textbf{505}, American Mathematical Society, Providence RI U.S.A., (1993). Singularities in algebraic geometry, Homogeneous spaces and generalizations, Local complex singularities, Linear algebraic groups over the reals, the complexes, the quaternions, Geometric invariant theory, Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects) Gorenstein quotient singularities in dimension three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let (X,x) be a germ of an isolated singularity of an n-dimensional analytic space. To investigate a normal isolated singularity (X,x) K. Watanabe introduced pluri-genera \(\{\delta_ m(X,x)\}_{m\in {\mathbb{N}}}\). For a normal isolated Gorenstein singularity (X,x), it is known that either \(\delta_ m(X,x)=0\) for any m, \(\delta_ m(X,x)=1\) for any m or \(\delta_ m(X,x)\) grows in order n as a function in m.
In this article, it is shown that for a normal isolated Gorenstein singularity (X,x), \(\delta_ m(X,x)\leq 1\) holds for every \(m\in {\mathbb{N}}\) if and only if \(H^ i(\tilde X,{\mathcal O}_{\tilde X})\cong H^ i(E,{\mathcal O}_ E)\) for any \(i>0\), where \(f:\tilde X\to X\) is a resolution of the singularity (X,x) with \(E=f^{-1}(x)_{red}\) simple normal crossings. The singularity with the second property is called a Du Bois singularity (Steenbrink). - Let E be as above and decompose it into irreducible components \(E_ i (i=1,2,...,r)\). Then it is also shown that a normal isolated Gorenstein singularity (X,x) is Du Bois if and only if the canonical divisor \(K_{\tilde X}=\sum^{r}_{i=1}m_ iE_ i\) satisfies \(m_ i\geq -1\) for every i. More precisely, \(''\delta_ m(X,x)=0\) for every m'' iff \(m_ i\geq 0\) for every i, and \(''\delta_ m(X,x)=1\) for every m'' iff \(m_ i\geq -1\) for every i and \(m_ i=-1\) for some i. In the former case, (X,x) is rational. We call (X,x) purely elliptic in the later case. - We classify the set of n-dimensional purely elliptic singularities into n-types by the Hodge structure of the exceptional divisors, and consider the configuration of the exceptional divisor of a certain resolution of a singularity of each type. Next, we construct purely elliptic singularities of each type of any dimension \(n\geq 2\), by means of blowing down. germ of an isolated singularity; Du Bois singularity; n-dimensional purely elliptic singularities; Hodge structure of the exceptional divisors S. Ishii, On Isolated Gorenstein Singularities. Math. Ann.270, 541--554 (1985). Singularities in algebraic geometry, Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects) On isolated Gorenstein 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 concerns the study of families of smooth curves lying on a surface singularity \((S,O)\). Let \(\pi:X\to S\) be a minimal desingularization of \((S,O)\). The authors prove that for any irreducible component \(E\) of \(\pi^{-1}(0)\) such that \(\text{ord}_E (mO_X)=1\) (where \(\text{ord}_E\) is the divisorial valuation and \(m\) is the maximal ideal corresponding to \(O\) in \(S)\) the family of smooth curves \({\mathcal L}_E\) is nonempty. In that case a smooth curve in \({\mathcal L}_E\) corresponds to a point in \(E\). The authors conclude the existence of smooth curves lying on sandwich singularities. They also study wedge morphisms, that are morphisms \(h:\text{Spec} (k[[u,v]]) \to(S,O)\) such that the image of \(h\) is Zariski dense in some analytically irreducible component of \((S,O)\). arcs; resolution of singularities; surface singularity; desingularization; smooth curves lying on sandwich singularities; wedge morphisms Gonzalez-Sprinberg, Gérard; Lejeune-Jalabert, Monique, Families of smooth curves on surface singularities and wedges, Ann. Polon. Math., 67, 2, 179-190, (1997) Singularities of surfaces or higher-dimensional varieties, Plane and space curves, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) Families of smooth curves on surface singularities and wedges | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(P\) be a normal singularity of multiplicity \(d = 2\) or \(3\) of a complex surface \(X\). It is well-known that \(X\) is locally an irreducible finite cover \(\pi : X \to Y\) of degree \(d\) over a smooth surface \(Y\), and the singularity \((X, P)\) can be resolved by the canonical resolution \(X_k \to X_{k-1} \to \dots \to X_0 = X\), which is the pullback of the embedded resolution of the corresponding singularity \(p = \pi (P)\) of the branch locus. Let \(F\) be the maximal ideal cycle of this resolution. We will prove that \(F\) has a unique decomposition \(F = Z_1 + \dots + Z_d\) with \(Z_1 \geq Z_2 \geq \dots \geq Z_d \geq 0\), where \(Z_i\) is a fundamental cycle or zero. We show that \(w = p_a (Z_1) + \dots + p_a (Z_d)\) is an invariant of \((X, P)\) that can also be computed from the multiplicity of the branch locus at \(p\). \((X, P)\) is a rational singularity iff all of the singular points in the canonical resolution satisfies \(w \leq d - 1\). In order to get the minimal resolution from the canonical one, we need to blow down some exceptional curves, the number of blowing-downs is exactly that of fundamental cycles \(Z\) in the canonical resolution satisfying \(p_a (Z) = 0\) and \(Z^2 = -1\). Jung's resolution; canonical resolution; fundamental cycle; surface singularity; triple cover Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) On surface singularities of multiplicity three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams [cf. Trans. Am. Math. Soc. 363, No. 6, 3101-3132 (2011; Zbl 1270.16022)]. This paper deals with dihedral groups \(G=\mathbb D_{n,q}\) for which all special CM modules have rank one, and we show that all but four of the relations on such a reconstruction algebra are given simply as the relations arising from a reconstruction algebra of type \(A\). As a corollary, the reconstruction algebra reduces the problem of explicitly understanding the minimal resolution (= G-Hilb) to the same level of difficulty as the toric case. reconstruction algebras; quivers with relations; noncommutative resolutions; CM-modules; surface singularities; Cohen-Macaulay singularities; labelled Dynkin diagrams; resolutions of singularities Wemyss, M., Reconstruction algebras of type \textit{D} (I), J. Algebra, 356, 158-194, (2012) Representations of quivers and partially ordered sets, Rings arising from noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Cohen-Macaulay modules, Syzygies, resolutions, complexes in associative algebras Reconstruction algebras of type \(D\). 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 If D is a holomorphic vector field on a nonsingular algebraic variety V, and \(\pi: W\to V\) is a blowing-up, there is a natural lift of D, or at least of the \({\mathcal O}_ V\)-module generated by D, to W. The desingularisation problem is that of proving the existence of a sequence of blow-ups such that the singularities of the lifted vector field are of a simple kind. For vector fields \(D=a\partial /\partial x+b\partial /\partial y\) on \(K^ 2\), the order \(\nu (D)=\min (ord(a),ord(b))\) may increase on blowing-up, so this is modified by considering vector fields tangent to the exceptional divisor to an `adapted order'. This monograph is devoted to the proof that in the 3-dimensional case (over an algebraically closed field of characteristic 0) one can define a sequence of blowings-up to reduce the adapted order to 1.
The first chapter contains general hypotheses and definitions and includes the statement of the theorem. It is necessary at each stage to choose a permissible centre of blowing-up (a regular closed subscheme having normal crossings with E, such that the adapted order cannot increase on blowing-up) so that one has sufficient control of the resulting germs at all centres on the new exceptional locus: the problem can thus be viewed as a 2-person game. It is not difficult to express these orders, exceptional sets, etc. explicitly in terms of suitable local coordinates.
The remainder of the book constructs a winning strategy. The problem is divided into 4 main cases, then into a couple of dozen subcases; the strategy is first devised in the simplest case, then later cases are reduced to previous ones. It is not easy to discern any over-riding pattern, though various techniques from the resolution of singularities of varieties are used. three-dimensional vector fields; desingularisation problem Cano, F.: Desingularization Strategies for Three Dimensional Vector Fields. Lecture Notes in Mathematics, vol. 1259. Springer, Berlin (1987) Global theory and resolution of singularities (algebro-geometric aspects), Ordinary differential equations in the complex domain, Research exposition (monographs, survey articles) pertaining to algebraic geometry, 2-person games, Singularities in algebraic geometry, \(3\)-folds Desingularization strategies for three-dimensional vector fields | 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 First we recall the embedding line theorem (Abhyankar-Moh): Let \(f\in\mathbb{C}[x,y]\). If \(f(x,y)=0\) defines an affine line \(C\) in \(\mathbb{C}^ 2\), then there exists \(g\in\mathbb{C}[x,y]\) such that \(\mathbb{C}[x,y]=\mathbb{C}[f,g]\). The author gives a proof of the embedding line theorem using the deformation \(f(x,y)+\lambda x^ n\), and showing this is an equisingular family (where \(n\) is the total degree of \(f\) and \(f\) is written in a convenient form). Abhyankar-Moh theorem; embedding line theorem; deformation; equisingular family Lejeune-Jalabert, M.: Sur l'équivalence des courbes algébroïdes planes. Coefficients de Newton. Contribution à l'etude des singularités du poit du vue du polygone de Newton, Paris VII, Janvier 1973, Thèse d'Etat. See also in Travaux en Cours, 36 (edit. Lê Dũng Trãng) Introduction à la théorie des singularités I, 49-124 (1988) Singularities of curves, local rings, Formal methods and deformations in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Jacobians, Prym varieties, Equisingularity (topological and analytic), Singularities in algebraic geometry On equisingularity, analytical irreducibility and embedding line theorem | 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 note we give an example which shows that the Denef-Loeser zeta function (usually called topogical zeta function) associated to a germ of a complex hypersurface singularity is not a topological invariant of the singularity. The idea is the following. Consider two germs of plane curves singularities with the same integral Seifert form but with different topological type and which have different topological zeta functions. Make a double suspension of these singularities (we consider them in a 4-dimensional complex space). A theorem of M. Kervaire and J. Levine states that the topological type of these new hypersurface singularities is characterized by their integral Seifert form. Moreover the Seifert form of a suspension is equal (up to sign) to the original Seifert form. Hence these new singularities have the same topological type. By means of a double suspension formula we compute the Denef-Loeser zeta functions for the two \(3\)-dimensional singularities and we verify that they are not equal. singularities of hypersurface; topological zeta function; Denef-Loeser zeta function; topological invariant of the singularity; integral Seifert form Artal Bartolo, E.; Cassou-Noguès, Pi.; Luengo, I.; Melle Hernández, A., The denef-loeser zeta function is not a topological invariant, J. lond. math. soc. (2), 65, 1, 45-54, (2002) Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Complex surface and hypersurface singularities, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) The Denef-Loeser zeta function is not a topological invariant | 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 is motivated by the construction of 3-dimensional Calabi-Yau (CY) algebras from dimer models, which was in turn motivated by theoretical physics. Previous work by Broomhead, Mozgovoy and Reineke, and by Davison had shown that certain dimer models give rise to 3-dimensional CY algebras via a quiver with potential construction. These algebras, called toric orders, are meant to be thought of as ``noncommutative toric resolutions'' of certain toric varieties. The article under review generalizes the idea of a dimer model to that of a weighted quiver polyhedron, and generalizes the idea of a toric order to that of a cancellation algebra.
An algebra is called a cancellation algebra if it is the category algebra of a category in which every morphism is both epic and monic; this gives rise to a cancellation law in the associated category algebra.
A quiver polyhedron is a strongly connected quiver (i.e., every vertex lies on an oriented cycle) along with two disjoint sets of cycles in the quiver, \(Q_2^+\) and \(Q_2^-\). These sets of cycles are required to satisfy:
(PO) every arrow of \(Q\) appears in exactly one cycle of \(Q_2^+\) and appears in exactly one cycle of \(Q_2^-\);
(PM) at each fixed vertex, the incidence graph of the cycles and arrows that meet there is connected.
The author shows that this is essentially the same as drawing a strongly connected quiver on a compact, orientable surface, in such a way that erasing the quiver leaves simply connected pieces bounded by cycles. A positive weighting on a quiver polyhedron is an assignment of a positive integer \(E_c\) to each cycle \(c\) in \(Q_2^+\cup Q_2^-\) such that \(E_c|c|>2\), where \(|c|\) denotes the number of arrows in \(c\). This roughly corresponds to putting some orbifold singularities in the surface described above. Now there is a natural superpotential associated to a weighted quiver polyhedron, obtained by taking the difference of the positive and negative cycles but with some additional technicalities from the weighting. A positive grading on a weighted quiver polyhedron is an assignment of a positive real number \(R_a\) to each arrow such that \(R_cE_c\) is the same for every cycle in \(Q_2^+\) and \(Q_2^-\). The existence of such a grading means that the superpotential is homogeneous; the results of the paper do not depend on the exact choice of grading.
A group acting on a weighted quiver polyhedron without fixing any vertices corresponds to a cover morphism between orbifolds and Galois cover on the level of Jacobi algebras. The author proves in Theorem 5.8 and Lemma 5.9 that the property of being a cancellation algebra, and that of admitting a positive grading are compatible with the notion of Galois covers. This is important because it allows one to reduce to the case of trivial weighting, with two exceptions (Theorem 5.10).
The main theorem then states that if a positively graded cancellation algebra is 3-CY, it necessarily comes from a graded weighted quiver polyhedron (Theorem 6.1). This is used to show in Section 7 that if we restrict our cancellation algebras to the case of toric orders, the underlying manifold of the associated quiver polyhedron must be a torus. In other words, a toric order which is 3-CY must come from a dimer model on a torus (Theorem 7.7). noncommutative resolutions; quivers; dimer models; Calabi-Yau algebras; toric orders; weighted quiver polyhedra; orientable surfaces; cancellation algebras; Galois covers Bocklandt, R.: Calabi-Yau algebras and quiver polyhedra Representations of orders, lattices, algebras over commutative rings, Representations of quivers and partially ordered sets, Global theory and resolution of singularities (algebro-geometric aspects), Noncommutative algebraic geometry, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.), String and superstring theories; other extended objects (e.g., branes) in quantum field theory Calabi-Yau algebras and weighted quiver polyhedra. | 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 related to resolution of singularities. We present here examples which explain why many arguments and proofs work in special situations, say small dimension or zero characteristic, but fail in general. This exhibits in particular the delicacy of resolution of singularity for arbitrary excellent schemes. We shall concentrate here on the classical approach developed by Zariski, Abhyankar, Hironaka and several other mathematicians towards a constructive proof of resolution of singularities by a sequence of well chosen monoidal transformations. The basic idea is to construct sufficiently fine local invariants of singularities which determine the center of blowing up at each stage as the locus of points on the variety where the invariants take their maximal values and to measure the improvement the variety undergoes when passing to the blown up variety by comparing these and possibly further invariants before and after the blowup. -- At present there is no completely satisfying answer to this objective. One reason is the lack of conceptuality of the proposed and studied invariants, already apparent in characteristic 0, the other, which is partly a consequence of the first, that the known invariants sometimes behave badly under blowup in positive characteristic.
We shall set up a catalogue of cautions one has to be aware of when using standard resolution invariants or searching new ones. resolution of singularities Hauser, H.: Seventeen obstacles for resolution of singularities. The Brieskorn anniversary (1998) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Seventeen obstacles for 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 \(k\) be an algebraically closed field and let \(A\) be a finitely generated algebra. Let \(M_d\) be the \(k\)-scheme such that \(M_d(R)\) is the \(k\)-algebra of \(d\times d\)-matrices with coefficients in \(R\), for any commutative \(k\)-algebra \(R\), and for \(d\geq 1\), a positive integer. Then the affine scheme \(\text{mod}_A^d\) is represented by the functor \(\Hom_{k\text{-alg}}(A,M_d(-))\). The general linear group \(\text{GL}_d\) acts on \(\text{mod}_A^d\) by conjugation so that for any \(d\)-dimensional left \(A\)-module \(M\) the corresponding orbit \(O_M\) in \(\text{mod}_A^d(k)\) is well-defined.
The paper under review is devoted to the study of types of singularities in the closures of orbits in module schemes \(\text{mod}_A^d\). The type of singularity \((X,x)\) is defined as the equivalence class with respect to the following relation: two pointed schemes \((X,x)\) and \((Y,y)\) are (smoothly) equivalent if there are smooth morphisms \(f\colon Z\to X\) and \(g\colon Z\to Y\) and a point \(z\in Z\) such that \(f(z)=x\) and \(g(z)=y\) [\textit{W. Hesselink}, Trans. Am. Math. Soc. 222, 1-32 (1976; Zbl 0332.14017)]. Among other things the author proves the following. Let \(0\to U\to M\to U\to 0\) be an exact sequence of finite dimensional left \(A\)-modules such that the codimension of the orbit \(O_{U\oplus U}\) in \(\overline O_M\) is equal to 2. Then \(\text{Sing}(M\oplus U^{p-1},U^{p+1})\) for any \(p\geq 1\) is nothing but the type of singularity of the nilpotent \((p+1)\times(p+1)\) matrices of rank at most 1 at the zero matrix. The author also gives the reference [\textit{G. Bobiński} and \textit{G. Zwara}, Manuscr. Math. 105, No. 1, 103-109 (2001; Zbl 1031.16012)] where his results are used in order to prove the normality of orbit closures for quivers of Dynkin type \(A_n\). module schemes; orbit closures; smooth morphisms; Grothendieck groups; Dynkin quivers; representations of finite-dimensional algebras Zwara, G.: Smooth morphisms of module schemes. Proc. Lond. Math. Soc. (3) 84(3), 539--558 (2002) Representations of quivers and partially ordered sets, Finite rings and finite-dimensional associative algebras, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry Smooth morphisms of module schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a finitely generated associative algebra with identity over an algebraically closed field \(k\), and let mod\(_{A}\left( d\right) \) denote the variety of \(d\)-dimensional representations of \(A\). For a \(d\)-dimensional module \(M\) we let \(\mathcal{O}_{M}\) be its \(\text{GL}\left( d\right) \) -orbit (the action being conjugation) and \(\mathcal{\bar{O}}_{M}\) the Zariski closure of \(\mathcal{O}_{M}.\) Given \(N\) a degeneration of \(M\) we let Sing\(\left( M,N\right) \) be the class of pointed varieties which are smoothly equivalent to the pointed variety \(\left( \mathcal{\bar{O}} _{M},n\right) \) for \(n\) an arbitrary point of \(N\).
This paper is a study of Sing\(\left( M,N\right) \) in the case where \(N\) has codimension two. The case of codimension one was considered in [\textit{G. Zwara}, J. Algebra, 283, 821--848, (2005; Zbl 1112.16018)], where it was shown that Sing\(\left( M,N\right) =\)Reg, where Reg is the singularity given by the regular points of \(\mathcal{\bar{O}}_{M}.\) In the case considered here there are some conditions for which Sing\(\left( M,N\right) =\)Reg. For example, suppose \(M=M^{\prime }\oplus X\) and \( N=N^{\prime }\oplus X.\) If \(N^{\prime }\) is of codimension one in \(M^{\prime }\) then Sing\(\left( M,N\right) =\)Reg.
On the other hand, if \(N^{\prime }\) is of codimension two then Sing\(\left( M,N\right) =\)Sing\(\left( M^{\prime },N^{\prime }\right) ,\) and hence it suffices to consider modules which are disjoint, i.e. have no common direct summand. A second condition that gives regularity is if \(M\) and \(N\) are disjoint and \(N\) decomposes into at least three indecomposable modules. These are the first two major theorems in the paper.
It is not true, however, that we always get regularity in codimension two -- the Kleinian singularity occurs in an example where \(N\) is the sum of two simple \(A\)-modules when \(A=k\left[ \varepsilon \right] /\left( \varepsilon ^{2}\right) .\) However, the final major theorem proves that when \(M\) is a module over the path algebra of a Dynkin quiver then \(\mathcal{\bar{O}}_{M}\) is regular in codimension two. module varieties; orbit closures; types of singularities Zwara, G.: Orbit closures for representations of Dynkin quivers are regular in codimension two. J. Math. Soc. Jpn. 57(3), 859--880 (2005) Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Representations of associative Artinian rings, Representations of quivers and partially ordered sets Orbit closures for representations of Dynkin quivers are regular in codimension two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper gives a nice exposition of, and simplifies, the key aspects of Abhyankar's proof of embedded resolution of surfaces in positive characteristic. Note that Abhyankar's original proof is obtained by combining the results of 5 sources, see [\textit{S. S. Abhyankar}, Math. Ann. 153, 81--96 (1964; Zbl 0121.37901); Wiss. Abh. Arbeitsgemeinschaft Nordrhein-Westfalen 33, Festschr. Gedächtnisfeier Karl Weierstrass, 243--317 (1966; Zbl 0144.03104); Math. Ann. 170, 87--144 (1967; Zbl 0158.04101); Ann. Mat. Pura Appl., IV. Ser. 71, 25--59 (1966; Zbl 0158.04201)] and [Resolution of singularities of embedded algebraic surfaces. New York-London: Academic Press (1966; Zbl 0147.20504)]. Thus, the fact that this paper is mostly self contained and not too long, is quite valuable for researchers working in this field. The first section also gives a clean outline of the proof strategy and some discussion of the main difficulties. resolution; singularity; surface; positive characteristic; blow up; valuation Cutkosky, SD, A skeleton key to abhyankar's proof of embedded resolution of characteristic \(P\) surfaces, Asian J. Math., 15, 369-416, (2011) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Valuations and their generalizations for commutative rings A skeleton key to Abhyankar's proof of embedded resolution of characteristic P 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 the paper under review, the author finds a geometric boson-fermion correspondence using the Hilbert schemes of points on the complex affine plane and certain quiver varieties. The main idea is to use equivariant cohomology of these Hilbert schemes under torus actions and the localization formula.
In sections one and two, the author reviews the boson-fermion correspondence from representation theory, equivariant cohomology, and the localization formula. In section three, a geometric realization of the fermionic Fock space using the quiver varieties of H. Nakajima and others is described in detail. Section four is devoted to the Hilbert schemes \(X_n\) of points on the affine plane \(\mathbb{C}^2\), certain torus action on \(X_n\), the equivariant cohomology of \(X_n\), and the fixed points. In section five, the author recalled the geometric realization of bosonic Fock space following the results of E. Vasserot, H. Nakajima, and W.-P. Li, W. Wang and the reviewer. In section six, using the localization formula, the author states the geometric boson-fermion correspondence relating the Hilbert schemes and the quiver varieties. quiver varieties; Hilbert schemes A. Savage, ``A geometric boson-fermion correspondence'', C. R. Math. Rep. Acad. Sci. Canada28 (2006) no. 3, p. 65-84 Vertex operators; vertex operator algebras and related structures, Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets A geometric boson-fermion correspondence | 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 describe projective resolutions of \(d\) dimensional Cohen-Macaulay spaces \(X\) by means of a projection of \(X\) to a hypersurface in \(d+1\)-dimensional space. We show that for a certain class of projections, the resulting resolution is minimal. resolutions; Cohen-Macaulay spaces Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Germs of analytic sets, local parametrization Projective resolutions associated to projections | 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 0509.00008.]
Normally flat deformations of an isolated singularity may be characterized by the property that they admit a singular section such that the Hilbert Samuel function is constant along this section. Since such deformations cannot be smoothings, it turns out to be a very interesting problem to describe the singularities which appear generically in normally flat deformations. Good candidates should be cone singularities. The aim of this paper is to give an affirmative answer in the important special case of rational and minimally elliptic singularities. As an application, one obtains a necessary condition to smooth minimally elliptic singularities. - It is not difficult to show that rational and elliptic cone singularities are generic singularities in the sense above. The crucial point is to construct non-trivial normally flat deformations of a given rational or minimally elliptic singularity (V,p): Let M be the minimal resolution of (V,p). Then a deformation of M to which the fundamental cycle Z lifts blows down to a normally flat deformation of (V,p). So one has to check smoothability of Z via deformations of M. Using a method introduced by the author in Math. Ann. 247, 43-65 (1980; Zbl 0407.14014)], it remains to verify the existence of a reduced subcycle of Z which satisfies certain simple numerical conditions. As a first step, the author describes a general strategy how to construct such nice subcycle. However, the verification of this algorithmic procedure turns out to be no at all trivial and needs a lot of tedious and delicate combinatorial arguments. minimally elliptic singularities; constant Hilbert-Samuel function; rational singularity; desingularization; normally flat deformations; cone singularities; fundamental cycle; smoothability U. Karras : Normally flat deformations of rational and minimally elliptic singularities . Proc. of Symp. in Pure Math. 40(I) (1983) 619-639. Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Deformations of complex singularities; vanishing cycles, Complex singularities, Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves, Formal methods and deformations in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Normally flat deformations of rational and minimally elliptic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field. This work is a classification of the isomorphism classes of quiver representations \(M\) whose orbit closure \(\overline{\mathcal{O}}_{M}\) has \(k\)-dimension at most 4 and has an invariant point. There are eleven such classes.
More explicitly, let \(M\) be a \(\mathbf{d}\)-dimensional representation of a quiver \(Q\), where \(\mathbf{d}\) is a vector of natural numbers whose length is given by the number of vertices in \(Q\). For \(Q\) the quiver given by \(1\leftarrow2\), \(\mathbf{d=}( p,q) \) and \(M\) given by a rank one matrix the orbit closure in GL\(( \mathbf{d}) \) is denoted \(\mathcal{D}( p,q) .\) The dimensions of \(\mathcal{D}( 2,2) \) and \(\mathcal{D}( 2,3) \) are 3 and 4 respectively. Additionally, let \(\mathcal{D}( 2,2,2) \) denote the image of the multilinear map \(( k^{2}) ^{3}\to( k^{2}) ^{\otimes3}\) -- this is a normal toric variety of dimension 4. Also, define \(\mathcal{HD}^{[ r] }( p,q) \) to be the matrices in \(\mathcal{D}( p,q) \) of the form \([ m_{ij}] \) such that \(m_{11}+\cdots+m_{rr}=0.\) The dimension of \(\mathcal{HD}^{[ 2] }( 2,2) \) is 2, the dimension of \(\mathcal{HD}^{[ 2] }( 2,3) \) is 3, and \(\dim \mathcal{HD}^{[ 2] }( 2,4) =\dim\mathcal{HD}^{[ 2] }( 3,3) =\dim\mathcal{HD}^{[ 3] }( 3,3) =4.\)
In the case where each entry in \(\mathbf{d}\) is equal to \(1\) it is easy to construct orbit closures based on two small quivers, each with five vertices. These are denoted \(\mathcal{C}( 2,3) ,\) and \(\mathcal{C}( 2,2,2) ,\) and are both four dimensional. Finally, if \(\overline {\mathcal{O}}_{M}\) has dimension one then \(\overline{\mathcal{O}}_{M}\cong k.\) The main result of this paper is that every such orbit closure is isomorphic to one of the eleven given above. This also establishes that every such \(M\) with orbit closure dimension at most four is normal and Cohen-Macaulay. Representations of quivers; orbit closures Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets Classification of low-dimensional orbit closures in varieties of 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 Let \(Y\) be a complex projective variety of dimension \(n\) with isolated singularities. Then for any resolution of singularities \(\pi \colon X \rightarrow Y\) with the exceptional locus \(G = \pi^{-1}(\mathrm{Sing}(Y))\) there is a natural map \(H^{k-1}(G) \rightarrow H^k(Y; Y\setminus\mathrm{Sing}(Y))\). The authors prove that this map vanishes for \(k>n\). They also remark that, as a consequence, for varieties with isolated singularities it follows a complete and short proof of the Decomposition Theorem from [\textit{A. A. Beilinson} et al., Astérisque 100, 172 p. (1982; Zbl 0536.14011)]. projective varieties; isolated singularities; resolution of singularities; simple normal crossing divisors, derived category; intersection cohomology; decomposition theorem; mixed Hodge theory; Hodge-Riemann bilinear relations Singularities in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Global theory and resolution of singularities (algebro-geometric aspects), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Topological properties in algebraic geometry, Global theory of complex singularities; cohomological properties, Mixed Hodge theory of singular varieties (complex-analytic aspects), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Hodge theory in global analysis, Topological properties of mappings on manifolds On the topology of a resolution of isolated singularities. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities To a non-constant polynomial \(f \in \mathbb C[x_1,\dots,x_n]\) with \(f(0)=0\) one associates its motivic, Hodge and topological zeta function. These are singularity invariants of the hypersurface germ \(0 \in \{f=0\}\), specializing to each other in the sense motivic \(\to\) Hodge \(\to\) topological. They are rational functions in one variable, and can be given by a concrete expression in terms of and embedded resolution \(\pi\) of \(0 \in \{f=0\}\). The denominator of this expression yields that each irreducible component of the total inverse image by \(\pi\) of \(\{f=0\}\) induces a candidate pole of these zeta functions. Miraculously, a lot of these candidate poles seem to cancel. In [Manuscr. Math. 87, No.~4, 435--448 (1995; Zbl 0851.14012)] the reviewer provided a geometric characterization of the actual poles when \(n=2\). For arbitrary \(n\) this is a very difficult problem.
In this paper the author proves for the Hodge and motivic zeta functions an interesting sufficient condition in full generality. For instance before this work no general sufficient condition was known. It clarifies previously known \` strange examples\', and its proof is very elegant. We introduce now the necessary notation for the \` Hodge\'\ statement.
Denote the irreducible components of \(\pi^{-1}\{f=0\}\) by \((E_i)_{i \in T}\), and by \(N_i\) and \(\nu_i -1\) the multiplicities of \(E_i\) in the divisor of \(f\circ \pi\) and \(\pi^*(dx_1\wedge \cdots \wedge dx_n)\), respectively. Put also \(E_I^\circ := (\bigcap_{i \in I} E_i) \setminus (\bigcup_{k \notin I} E_k)\). Then the Hodge zeta function of \(f\) is
\[
Z_{\text{Hod}}(f;T):= \sum_{I\subset T} H(E_I^\circ \cap \pi^{-1}\{0\}) \prod_{i\in I}\frac{(uv-1)T^{N_i}}{(uv)^{\nu_i}-T^{N_i}} \in \mathbb Q(u,v)(T)
\]
where \(H(\cdot) \in \mathbb Z[u,v]\) is the Hodge polynomial of a variety (encoding its Hodge numbers). Fix an exceptional component \(E_j\) with \(\pi(E_j) = \{0\}\). Suppose for simplicity that we are in the generic situation where \(\nu_j/N_j \neq \nu_i/N_i\) for \(i\neq j\). Say \(E_1,\dots,E_r\) are the other components intersecting \(E_j\) and put \(\alpha_i := \nu_i - \frac{\nu_j}{N_j}N_i\) for \(i=1,\dots,r\). (These numbers appear in the residue of the candidate pole \(T=(uv)^{\nu_j/N_j}\).) Denote finally by \(\chi(\cdot)\) the usual topological Euler characteristic. Then \(T=(uv)^{\nu_j/N_j}\) is a pole of \(Z_{\text{Hod}}(f;T)\) if
\[
\sum_{I\subset Z} \frac{\chi((E_j \cap E_I)^\circ)}{\prod_{i\in I}\alpha_i} \neq 0,
\]
where \(Z\) runs over the \(i \in \{1,\dots,r\}\) for which \(\alpha_i \in \mathbb Z\). In particular when no \(\alpha_i \in \mathbb Z\) it is a pole if \(\chi(E_j^\circ) \neq 0\). Already this last case is quite nonobvious.
There is a more general statement for arbitrary candidate poles, everything is also valid for the motivic zeta function, and, instead of for a hypersurface, for an effective \(\mathbb Q\)-divisor \(D\) on a \(\mathbb Q\)-Gorenstein variety with singular locus in the support of \(D\). Hodge and motivic zeta functions; poles; resolution of singularities Rodrigues, B.: On the geometric determination of the poles of Hodge and motivic zeta functions. J. Reine Angew. Math. 578, 129--146 (2005) Singularities in algebraic geometry, Zeta functions and \(L\)-functions, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties On the geometric determination of the poles of Hodge and motivic zeta 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 We show that the orbit closure of a directing module is regular in codimension one. In particular, this result gives information about a distinguished irreducible component of a module variety.
Given a finite-dimensional \(k\)-algebra \(\Lambda\) and an element \(\mathbf d\) of the Grothendieck group \(K_0(\Lambda)\) of the category of \(\Lambda\)-modules, one defines the variety \(\text{mod}^{\mathbf d}_\Lambda(k)\) of \(\Lambda\)-modules of dimension vector \(\mathbf d\). A product \(\text{GL}_{\mathbf d}(k)\) of general linear groups acts on \(\text{mod}^{\mathbf d}_\Lambda(k)\) in such a way that the \(\text{GL}_{\mathbf d}(k)\)-orbits correspond to the isomorphism classes of \(\Lambda\)-modules of dimension vector \(\mathbf d\). In particular, the author has shown [J. Math. Soc. Japan 54, No. 3, 609-620 (2002; Zbl 1048.16004)] that, if \(\mathbf d\) is the dimension vector of a directing module and \(\Lambda\) is tame, then \(\text{mod}^{\mathbf d}_\Lambda(k)\) is normal if and only if it is irreducible. Here a \(\Lambda\)-module \(M\) is called directing if there is no sequence \((X_0,\dots,X_n)\) of indecomposable \(\Lambda\)-modules such that \(\Hom_\Lambda(X_{i-1},X_i)\neq 0\) for each \(i\in [1,n]\), there exists an \(i\in [1,n-1]\) such that the Auslander-Reiten translate of \(X_{i+1}\) equals \(X_{i-1}\), and \(X_0\) and \(X_n\) are direct summands of \(M\). In general, if \(M\) is a directing module, then the closure \(\overline{{\mathcal O}(M)}\) of the \(\text{GL}_{\mathbf d}(k)\)-orbit \(\mathcal O(M)\) of \(M\) is an irreducible component of \(\text{mod}^{\mathbf d}_\Lambda(k)\). Thus the above result naturally raises the question about properties of \(\overline{{\mathcal O}(M)}\).
The question about the properties of \(\overline{{\mathcal O}(M)}\) for a directing module \(M\) is a special case of another geometric problem investigated in the representation theory of finite-dimensional algebras, namely, the study of the properties of the orbit closures in the module varieties. In particular, \textit{G. Zwara} and the author proved [J. Algebra 298, No. 1, 120-133 (2006; Zbl 1131.16010)] (using, among other things, the results of the author and \textit{A. Skowroński} [J. Algebra 215, No. 2, 603-643 (1999; Zbl 0965.16009)] that, if \(M\) is an indecomposable directing module, then \(\overline{{\mathcal O}(M)}\) is a normal variety. Recall that the normal varieties are regular in codimension one; that is, the set of singular points is of codimension at least two. Thus, the following main result of the paper is the first step in order to generalize the above result about the closures of indecomposable directing modules over tame algebras to arbitrary directing modules over arbitrary algebras.
Main Theorem. If \(M\) is a directing module, then \(\overline{{\mathcal O}(M)}\) is regular in codimension one.
The paper is organized as follows. In Section 1 we recall the definitions of quivers and their representations. We also describe the properties of directing modules that are needed in the proof of our main result. In Section 2 we discuss interpretations of extension groups that are useful in geometric investigations. Next, in Section 3 we define the module schemes and some schemes connected with them. Finally, in Section 4 we prove the main result of the paper.
The main idea of the proof is the following. Let \(M\) be a directing module over an algebra \(\Lambda\). We first observe that each minimal degeneration \(N\) of \(M\) (that is, a module \(N\) whose orbit is maximal in \(\overline{{\mathcal O}(M)}\setminus\mathcal O(M)\)) is of the form \(N=U\oplus V\) for a short exact sequence
\[
\xi:0\to U\to M\to V\to 0.
\]
Now we use a connection between the tangent space \(T_N\text{mod}^{\mathbf d}_\Lambda(k)\) to the module variety at \(N\) and the first extension group. As a consequence, it follows that \(\text{Ext}^2_\Lambda(V,U)\) measures the difference between \(\dim\overline{{\mathcal O}(M)}\) and \(\dim_kT_N\text{mod}^{\mathbf d}_\Lambda(k)\). On the other hand, we show for a general minimal degeneration \(N\) of \(M\) that, if
\[
\xi_1:0\to U\to W_1\to U\to 0\quad\text{ and }\quad\xi_2:0\to V\to W_2\to V\to 0
\]
are short exact sequences, then \((\xi_1,\xi_2)\) corresponds to an element of \(T_N\overline{{\mathcal O}(M)}\) if and only if the sequences \(\xi_1\circ\xi\) and \(\xi\circ\xi_2\) determine the elements in \(\text{Ext}^2_\Lambda(V,U)\) that differ by the sign. We prove that the space of such pairs of sequences is of codimension \(\dim_k\text{Ext}^2_\Lambda(V,U)\) in \(\text{Ext}^1_\Lambda(U,U)\times\text{Ext}^1_\Lambda(V, V)\), and this will complete the proof. finite-dimensional algebras; categories of modules; actions of general linear groups; Auslander-Reiten translations; irreducible components; indecomposable directing modules; normal varieties; degenerations; short exact sequences; module varieties Bobiński, G., Orbit closures of directing modules are regular in codimension one, J. lond. math. soc. (2), 79, 211-224, (2009) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Group actions on affine varieties, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Orbit closures of directing modules are regular in codimension one. | 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.]
This paper gives a solution to the following combinatorial problem. Let \(\xi\) be a singular point of an algebraic variety X embedded in a regular variety Z. Suppose we are given a coordinate system \(t_ 1,...,t_ n\) on Z. Two people, A and B, are playing the following game. A chooses a coordinate subspace \(Y^ k\subset X\) such that X is normally flat along Y at \(\xi\). They let the commutative diagram: \(X'\hookrightarrow Z'\to^{p}Z,\quad X'\to^{\pi}X\hookrightarrow Z\) be the blowing-up with center Y. \(p^{-1}(\xi)\) is covered by n-k affine coordinate charts \((k=\dim Y)\) with coordinate systems given by appropriate quotients \(t_ j/t_ k\). The second player, B, then chooses one of the coordinate charts, such that the origin \(\xi\) '\(\in X'\). \(\xi\) is replaced by \(\xi\) ', and the procedure is repeated. A wins if after a finite number of steps \(\ell\), the Hilbert-Samuel function of \(X^{(\ell)}\) at \(\xi^{(\ell)}\) satisfies \(H_{X^{(\ell)},\xi^{(\ell)}}(\nu)<H_{H,\xi}(\nu)\) (the inequality is strict for almost all \(\nu \in {\mathbb{Z}}_+)\). The question is: does there exist a winning strategy for A? Now consider the Newton polyhedron \(\Delta\) of \(\xi\in X\) with respect to the given coordinate system. The above game can then be rephrased as a purely combinatorial game on the polyhedron. The answer is affirmative: this paper gives a winning strategy. Since in characteristic 0 it is possible to choose a priori a coordinate system on Z such that the ''worst'' singularities of the blown- up varieties are the said origins of the coordinate charts upstairs, this gives a proof of resolution in characteristic 0. In characteristic p one needs to allow player B to perform additional transformations on \(\Delta\). Hironaka has proposed another game to take care of this, but it turned out to be false. See the author's paper ''A counterexample to Hironaka's hard polyhedra game'' (Publ. Res. Inst. Math. Sci. 18, 1009- 1012 (1982; see the following review). Hironaka polyhedra game; resolution of singularities; Newton polyhedron; blowing-up; normal flatness; Hilbert-Samuel function Spivakovsky, M., ''A solution to Hironaka's polyhedra game'', to appear in the Safarevich volume, Birkhauser. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, 2-person games, Polytopes and polyhedra, Permutations, words, matrices A solution to Hironaka's polyhedra game | 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 an algebraic variety \(X\) over a field \(k\), its space of arcs \(X_\infty\) (introduced in [\textit{J. F. Nash jun.}, Duke Math. J. 81, No. 1, 31--38 (1995; Zbl 0880.14010)]) is intrisically associated with it, and it bears important information about the singularities of \(X\). A cylinder set is preimage of a constructible set in some jet scheme \(X_m\) by the truncation morphism \(\psi_m : X_\infty \to X_m\). The stable points of \(X_\infty\) are fat points of finite codimension which are the generic points of irreducible cylinders.
Given a stable point \(P\), it is important to understand the properties of the local ring \(\mathcal{O}_{X_\infty,P}\) and its completion \(\widehat{\mathcal{O}_{X_\infty,P}}\). For example it is known that the latter is Noetherian ([\textit{A. J. Reguera}, Compos. Math. 142, No. 1, 119--130 (2006; Zbl 1118.14004)]). A main goal of this article is to compute or evaluate the dimension of the local ring.
Let \(E\) be an exceptional divisor in a resolution of singularities \(\pi : Y \to X\), with \(k_E\) being the discrepancy of \(X\) with respect to \(E\). The Mather discrepancy for \(E\) ([\textit{T. de Fernex} et al., Publ. Res. Inst. Math. Sci. 44, No. 2, 425--448 (2008; Zbl 1162.14023)]) is defined as \(\hat{k}_E := \mathrm{ord}_E(\hat{K}_{Y/X})\).
Suppose \(X\) is a reduced scheme of finite type over field of characteristic 0, and \(\nu\) is a divisorial valuation on an irreducible component \(X_0 \subset X\) whose center lies in \(X_{\mathrm{sing}}\). Take a resolution of singularities \(\pi : Y \to X\) such that the center of \(\nu\) on \(Y\) is a divisor \(E\). It defines a cylinder set \(N_{eE}\) of the arcs with contact \(e \geq 1\) with \(E\), and so its generic point is a stable point \(P_{eE} \in X_\infty\). In general, for any stable point \(Q\), \(\dim\mathcal{O}_{X_\infty,Q} \leq \operatorname{codim} X_\infty Z(Q)\) ([\textit{L. Ein} and \textit{M. Mustaţă}, Proc. Symp. Pure Math. 80, Pt. 2, 505--546 (2009; Zbl 1181.14019)]). In order to study \(\widehat{\mathcal{O}_{X_\infty,P_{eE}}}\), \(X_0\) is embedded in a complete intersection affine scheme \(X'\) of dimension \(\dim(X_0)\), which is an overweight deformation of an affine toric variety, associated to \(\nu\). Using this is proved that \(\operatorname{embdim}\mathcal{O}_{(X_\infty)_{\mathrm{red}},P_{eE}} = \operatorname{embdim}\widehat{\mathcal{O}_{(X_\infty),P_{eE}}} = e(\hat{k}_E + 1)\) (see also [\textit{A. J. Reguera}, J. Algebra 494, 40--76 (2018; Zbl 1388.13001)]; Am. J. Math. 131, No. 2, 313--350 (2009; Zbl 1188.14010)]). In other words, the embedding dimension of \(\mathcal{O}(X_\infty)_{\mathrm{red}},P_{eE}\) is the codimension of \(N_{eE}\) as a cylinder set. Moreover, it is described explicitly a minimal system of coordinates for \((\mathcal{O}_{(X_\infty)_{\mathrm{red}},P_{eE}})\), that is, a regular system of parameters with \(e(\hat{k}_E + 1)\) coordinates.
To obtain a lower bound for \(\dim \widehat{\mathcal{O}_{(X_\infty),P_{eE}}}\), \(X\) is embedded in a complete intersection scheme \(X'\) with the same Mather discrepancy, and with a divisor \(E'\) over \(X'\) which defines the same valuation \(\nu = \nu_{E'}\). The Mather-Jacobian log-discrepancy with respect to \(E\) is defined as \(a_{MG}(E,X) := \hat{k}_E -\nu(Jac_X)+1\). Then the ring \(\widehat{\mathcal{O}_{(X_\infty),P_{eE}}}\) could be replaced by \(\widehat{\mathcal{O}_{(X'_\infty),P'_{eE}}}\) and this permits to obtain a lower bound \(\dim \widehat{\mathcal{O}_{(X_\infty),P_{eE}}} \geq ea_{MG}(E, X)\). In particular, when \(X\) is normal and complete intersection, \(\dim \mathcal{O}_{(X_\infty)_{\mathrm{red}},P_{eE}} \geq e(k_E+1)\) holds. If moreover \(\operatorname{codim}_{X_\infty} P_E = 1\), it follows that the discrepancy \(k_E \leq 0\). space of arcs; divisorial valuations; embedding dimension; Mather discrepancy; stable point H. Mourtada and A. Reguera-Lopez, Mather discrepancy as an embedded dimension in the space of arcs, preprint (2016). Arcs and motivic integration, Singularities in algebraic geometry, Valuations and their generalizations for commutative rings, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) Mather discrepancy as an embedding dimension in the space of arcs | 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 set \(K[\![\mathbf x]\!]=K[\![x_1,\ldots,x_d]\!]\). Let \(f(\mathbf x,z)=z^n+f_1(\mathbf x)z^{n-1}+\cdots+f_n(\mathbf x)\in K[\![\mathbf x]\!][z]\) be an irreducible Weierstrass polynomial; \(f\) is said to be \textit{quasi-ordinary} if its discriminant as a polynomial in \(z\) is a monomial up to multiplication by a unit in \(K[\![\mathbf x]\!]\).
The roots of such a polynomial sit in \(K[\![x_1^{1/n},\ldots,x_d^{1/n}]\!]\) [\textit{S. Abhyankar}, Am. J. Math. 77, 575--592 (1955; Zbl 0064.27501); \textit{H. W. E. Jung}, J. Reine Angew. Math. 133, 289--314 (1908; JFM 39.0493.01); \textit{I. Luengo}, J. Algebra 85, 399--409 (1983; Zbl 0528.13019); \textit{K. Kiyek} and \textit{J. L. Vicente}, Arch. Math. 83, No. 2, 123--134 (2004; Zbl 1085.13009); \textit{A. Parusiński} and \textit{G. Rond}, J. Algebra 365, 29--41 (2012; Zbl 1268.13008)].
Let \(\zeta^{(i)}\), \(i\in\{1,\ldots,n\}\), be the roots of \(f\). Now \(\zeta^{(i)}-\zeta^{(j)}=\mathbf x^{\lambda_{ij}}\cdot \epsilon _{ij}\), \(\epsilon_{ij}\) being a unit in \(K[\![x^{1/n}_1,\ldots,x_d^{1/n}]\!]\); the exponents \(\lambda_{ij}\) are well ordered with respect to the product ordering \(\leq_{\text{poly}}\), hence can be written as \(\lambda_1<_{\text{poly}} \lambda_2<_{\text{poly}}\cdots<_{\text{poly}}\lambda_g\); they are called the \textit{characteristic exponents}. The semigroup associated to the singularity \((f=0)\) is determined by these exponents.
For \(c\), \(d\in \mathbb{Z}_+\), let \(W: \mathbb{Z}^d_{\geq0}\to \mathbb{Q}^c_{\geq0}\) be a map which is the restriction of a linear map \(\mathbb{Q}^d\to \mathbb{Q}^c\); if \(c=d\), let \(W_0\) be the identity, i.e., \(W_0(\mathbf a)=\mathbf a\) for all \(\mathbf a\in \mathbb{Q}^c\). Using \(W\), the authors assign a weight to the monomials of \(K[\![\mathbf x]\!]\): to the monomial \(x_i\) they assign the weight \(W(e_i)\in\mathbb{Q}^c_{\geq0}\) of the \(i\)-th unit vector in \(\mathbb{Z}^d\).
The authors define a polyhedron \(\Delta^W(f,\mathbf x, z)\subset \mathbb{R}^c_{\geq0}\), called the associated polyhedron for \(f(\mathbf x,z)\) with respect to the inclusion \(K[\![\mathbf x]\!]\to K[\![\mathbf x]\!][z]/(f)\) and the weight function \(W\). It is closely connected to Hironaka's characteristic polyhedron [\textit{H. Hironaka}, J. Kyoto Univ. 7, 251--293 (1967; Zbl 0159.50502)]. Now consider the irreducible hypersurface singularity of dimension \(d\) defined by \(f\) and the projection to the affine space defined by \(K[\![\mathbf x]\!]\). Using this polyhedron and successive embeddings of the singularity \((f=0)\) in higher-dimensional affine spaces, the authors construct an invariant which detects whether the singularity is quasi-ordinary. In this case this invariant determines also the semigroup of the singularity. quasi-ordinary singularities; characteristic polyhedron; overweight deformation Singularities in algebraic geometry, Local complex singularities, Formal power series rings, Global theory and resolution of singularities (algebro-geometric aspects) A polyhedral characterization of quasi-ordinary singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the author continues the study of Gorenstein three dimensional singularities coming from finite group quotients. The article of \textit{S. S.-T. Yau} and \textit{Y. Yu} [``Gorenstein quotient singularities in dimension three'', Mem. Am. Math. Soc. 505 (1993; Zbl 0799.14001)] describes the action of \(G\subset SL(3, \mathbb{C})\), a finite group, on \(\mathbb{C}^3\) and classifies the Gorenstein quotients which arise out of this procedure, by classifying the finite groups of the necessary type. The author's work is towards settling the following conjecture:
Let \(G\) be a finite subgroup of \(SL(3, \mathbb{C})\) acting on \(\mathbb{C}^3\). Then there exists a resolution of singularities \(\sigma: \widetilde X\to \mathbb{C}^3/G\) with \(\omega_{\widetilde X} = {\mathcal O}_{\widetilde X}\) and \(\chi (\widetilde X)= \#\)\{conjugacy classes of \(G\}\).
This is apparently of some interest to the physicists. Various cases of this conjecture had been proved earlier and the author gives a summary of what was known. The author had settled the case of trihedral groups [\textit{Y. Ito}, Proc. Japan Acad., Ser. A 70, No. 5, 131-136 (1994; Zbl 0831.14006) and Int. J. Math. 6, No. 1, 33-43 (1995; Zbl 0831.14005)]. In the present article more cases are settled from types (B) and (D), as opposed to the trihedral groups which are of type (C), in the classification table of Yau and Yu. The author also mentions a later preprint of \textit{S.-S. Roan} [Inst. Math., Acad. Sinica, preprint R940606-1 (1994)] where the rest of the cases which were not covered earlier are treated, thereby proving the conjecture in full. action of group on complex 3-space; Gorenstein three dimensional singularities; finite group quotients; resolution of singularities Ito, Y.: Gorenstein quotient singularities of monomial type in dimension three. J. math. Sci. univ. Tokyo 2, 419-440 (1995) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Modifications; resolution of singularities (complex-analytic aspects), Low codimension problems in algebraic geometry, Homogeneous spaces and generalizations, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects) Gorenstein quotient singularities of monomial type in dimension three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an expository article, where the author presents an informal introduction to the theory of resolution of singularities of algebraic varieties (in characteristic zero) and the closely related question of principalization of ideals. The approach is constructive, or algorithmic.
As is common in this type of work, one considers instead a similar problem for more technical auxiliary objects. In the present article, these are the so-called singular mobiles. These are systems \((W, \mathcal I,c,D,E)\), where \(W\) is a regular variety, \(\mathcal I\) a coherent sheaf of ideals on \(W\), \(c\) a positive integer and \(D=D_n, \dots, D_1\), \(E=E_n, \dots, E_1\) strings of normal crossing divisors satisfying certain conditions. These mobiles are useful to formalize a critical inductive step in the resolution process. A full presentation of this theory is given in \textit{S. Encinas} and \textit{H. Hauser} [Comment. Math. Helv. 77, 821--845 (2002; Zbl 1059.14022)].
The article under review includes introductory examples, a discussion of the basic concepts, blowing-ups, transforms, order and other invariants, and the notion of mobiles, as well as many examples. A large bibliography on the subject follows. This paper, together with \textit{H. Hauser} [Bull. Am. Math. Soc. 40, 323--403 (2003; Zbl 1030.14007)], is a good introduction to the subject. singularity; resolution; blowing-up; mobile Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Desingularization of ideals and 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 paper defines and studies rational Puiseux expansions of a plane curve. It is proven that rational Puiseux expansions easily give the classical Puiseux expansions and that they give more information than the classical ones (they give results of arithmetical nature about the curve in addition). One major result is the easy determination of the residual field of the places of a curve over a non-algebraically closed field.
By describing a variant of Newton's algorithm for computing rational Puiseux expansions their existence is proven. This algorithm is cheaper than the classical one. rational Puiseux expansions of a plane curve; Newton's algorithm for computing rational Puiseux expansions Duval D 1989 Rational puiseux expansions \textit{Compos. Math.}70 119--54 Singularities in algebraic geometry, Software, source code, etc. for problems pertaining to field theory, General field theory, Software, source code, etc. for problems pertaining to algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Rational Puiseux expansions | 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 Generalized Enriques diagrams are combinatorial data associated with constellations of infinitely near points and proximity relations. Classically they were introduced to deal with linear systems of curves with base conditions. We present a survey on some aspects and new results on this diagrams, examples and applications to relative characteristic cones and Zariski's complete ideal theory. infinitely near points; characteristic cones; complete ideals; toric varieties Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Generalized Enriques diagrams and characteristic cones | 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 the algebraic closure of the finite field with \(q\) elements and let \(\mathcal Q\) be a quiver with the underlying graph of Dynkin type \(A_n\). The group \(G_d=\text{GL}_{d_1}(F)\times\cdots\times \text{GL}_{d_n}\) acts by conjugation on \(E_d=\bigoplus_{i\to j\in{\mathcal Q}}\Hom_F(F^{d_i},F^{d_j})\). In the paper under review the authors describe the \(G_d\)-orbits \(\mathcal O\) with the property that the orbit closure \(\overline{\mathcal O}\) (in the Zariski topology) is rationally smooth. The approach is to consider the corresponding quantized enveloping algebra and to study the action of the bar involution on PBW bases. Then the authors use Ringel's Hall algebra approach to quantized enveloping algebras and Auslander-Reiten quivers, and describe the commutation relations between root vectors. As a result they obtain explicit formulas for the multiplication of an element of PBW bases adapted to a quiver with a root vector as well as recursive formulas to study the bar involution on PBW bases. As a consequence the authors derive that if the orbit closure is rationally smooth, then it is smooth.
The recent paper [\textit{P. Caldero} and \textit{R. Schiffler}, Ann. Inst. Fourier 54, No. 2, 265--275 (2004; Zbl 1126.17013)] contains the characterization of the rationally smooth orbit closures of representations of quivers of type \(A\), \(D\) or \(E\). Comparing the methods of both papers, the present paper has the advantage that the approach is very explicit and recursive, and may be used for computer programming. But it cannot easily be generalized to type \(D\) and \(E\). representations of quivers; varieties of representations; rational smoothness; quantum groups; quantized enveloping algebra Robert Bédard and Ralf Schiffler, Rational smoothness of varieties of representations for quivers of type \({A}\), preprint. Quantum groups (quantized enveloping algebras) and related deformations, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Singularities in algebraic geometry, Representations of quivers and partially ordered sets, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Rational smoothness of varieties of representations for quivers of type \(A\) | 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 Par les mots de l'auteur ``cette présentation introductive des quelques désingularisation plongées constitue une base experimentale pour l'étude des liens avec les courbes tracées sur la surface et les points fixes des courbes polaires qui interviennents dans la transformation de Nash.'' En effet une telle désingularisation est realisée pour des singularités rationnelles de surface de type \(A_ 2\), \(D_ 4\), \(E_ 6\) et \(E_ 8\) en faisant des éclatements de courbes ou de points dans l'espace ambiant. desingularisation; surface singularities Gonzalez-Sprinberg, G., Quelques descriptions de desingularisations plongees d surface, Prepublication de lnstitut Fourier no. 123. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects), Singularities of surfaces or higher-dimensional varieties Some descriptions of embedded desingularizations of surfaces. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs \((C, o)\) of complex analytic curves contained in a smooth complex analytic surface \(S\). The embedded topological type of such a pair \((S,C)\) is usually defined to be that of the oriented link obtained by intersecting \(C\) with a sufficiently small oriented Euclidean sphere centered at the point \(o\), defined once a system of local coordinates \((x, y)\) was chosen on the germ \((S, o)\). If one works more generally over an arbitrary algebraically closed field of characteristic zero, one speaks instead of the combinatorial type of \((S, C)\). One may define it by looking either at the Newton-Puiseux series associated to \(C\) relative to a generic local coordinate system \((x, y)\), or at the set of infinitely near points which have to be blown up in order to get the minimal embedded resolution of the germ \((C, o)\) or, thirdly, at the preimage of this germ by the resolution. Each point of view leads to a different encoding of the combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques diagram, or a weighted dual graph. The three trees contain the same information, which in the complex setting is equivalent to the knowledge of the embedded topological type. There are known algorithms for transforming one tree into another. In this paper we explain how a special type of two-dimensional simplicial complex called a lotus allows to think geometrically about the relations between the three types of trees. Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants of it. This lotus is constructed from the finite set of Newton polygons created during any process of resolution of \((C,o)\) by successive toric modifications. plane curve singularities; embedded topological type; Newton-Puiseux series; Eggers-Wall tree; Enriques diagram; Newton polygons. Singularities of curves, local rings, Toric varieties, Newton polyhedra, Okounkov bodies, Combinatorial aspects of tropical varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, Research exposition (monographs, survey articles) pertaining to algebraic geometry The combinatorics of plane curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{B. Angeniol} and \textit{F. Elzein}, La classe fondamentale relative d'un cycle (The relative fundamental class of a cycle) (pp. 7- 12); \textit{R.-O. Buchweitz} and \textit{G.-M. Greuel}, Le nombre de Milnor, équisingularité et déformations des courbes réduites (The Milnor number, equisingularity and deformations of reduced curves) (pp. 13-30); \textit{H. Esnault}, Sur l'identification de singularités apparaissant dans des groupes algébriques complexes (On the identification of singularities appearing in complex algebraic groups) (pp. 31-59); \textit{M. Giusti} and \textit{J. P. Henry}, Minorations de nombres de Milnor (Lower bounds of Milnor numbers) (pp. 61-74); \textit{R. Langevin}, Courbure, feuilletages et singularités algébriques [Curvature, foliations and algebraic singularities] (pp. 75-86); \textit{Lê Dũng Tráng}, Ensembles analytiques complexes avec lieu singulier de dimension un (d'aprés I. N. Iomdine) (Complex analytic sets with one-dimensional singular locus (after Y. Yomdin)) (pp. 87-95); \textit{M. Lejeune-Jalabert}, Le théorème \(``AF+BG''\) de Max Noether (The \(``AF+BG''\) theorem of Max Noether) (pp. 97-138); \textit{Z. Mebkhout}, Dualité de Poincaré (Poincaré duality) (pp. 139-182); \textit{C. Sabbah}, Fonctions de Morse sur une variété analytique complexe (Morse functions on a complex analytic manifold) (pp. 183-191); \textit{B. Teissier}, Polyèdre de Newton jacobien et équisingularité (Jacobian Newton polyhedra and equisingularity) (pp. 193-221). Milnor number, equisingularity; algebraic singularities; Poincaré duality; Newton polyhedra Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Local complex singularities, Complex singularities, Duality theorems for analytic spaces, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Séminaire sur les singularités. (Seminar on singularities). (Université Paris VII, 1976-1977) | 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 work of \textit{H. Hironaka} in Ann. Math., II. Ser. 79, 109-203 and 203-326 (1964; Zbl 0122.386) there is a deep analysis of the effect of blowing up along a permissible centre; locally this means that the ideal I is a permissible centre for the regular local ring R iff R/I is again regular and moreover the (R/I)-module \(G_ I(R)=\otimes_{n}I^ n/I^{n+1}\) is free (i.e. R is normally flat along I). The paper extends some results well known for permissible blowing ups to pairs (R,I) such that R is normally flat along I but R/I is not necessarily regular. For instance the author obtains some equivalent conditions for R/(x) to be normally flat along P/(x), P being a prime ideal in the regular local ring R of dimension \(d>1\). R/P is not supposed to be regular, but such a hypothesis is replaced by the so called ''isomultiplicity'', a useful tool introduced in the paper: all the elements of any minimal bases of P are required to belong to the same maximal power of \(M=\max imal\) ideal of R. normal flatness; strictly complete intersection; permissible blowing up; isomultiplicity Michela Brundu, Normal flatness and isomultiplicity, Rend. Sem. Mat. Univ. Politec. Torino 40 (1982), no. 1, 163 -- 172 (Italian, with English summary). Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Multiplicity theory and related topics, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Regular local rings Normal flatness and isomultiplicity | 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 various properties of numerical data of an embedded resolution of singularities for plane curves, which are inspired by a conjecture of \textit{K. H. Nguyen} [``Uniform rationality of Poincaré series of \(p\)-adic equivalence relations and Igusa's conjecture on exponential sums'', Preprint, \url{arXiv:1903.06738}] and motivated by a conjecture of [\textit{J. I. Igusa}, Lectures on forms of higher degree. Notes by S. Raghavan. Berlin, Heidelberg, New York: Springer-Verlag (1978; Zbl 0417.10015)] on exponential sums. Singularities in algebraic geometry, Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects) On the log canonical threshold and numerical data of a resolution in dimension 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 The main purpose of this article is to get a handle on determining how far a non-rational singularity is from being rational, or in other words, introduce a measure of the failure of a singularity being rational. S. Kovács, Irrational centers , Pure Appl. Math. Q. 7 (2011), 1495-1515. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Minimal model program (Mori theory, extremal rays) Irrational centers | 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 construct small desingularizations of moduli spaces of semistable quiver representations for indivisible dimension vectors using deformations of stabilities and a dimension estimate for nullcones. We apply this construction to several classes of GIT quotients. quiver moduli; GIT quotients; small desingularizations Reineke, Markus, Quiver moduli and small desingularizations of some GIT quotients.Representation theory---current trends and perspectives, EMS Ser. Congr. Rep., 613-635, (2017), Eur. Math. Soc., Zürich Representations of quivers and partially ordered sets, Algebraic moduli problems, moduli of vector bundles, Intersection homology and cohomology in algebraic topology, Global theory and resolution of singularities (algebro-geometric aspects) Quiver moduli and small desingularizations of some GIT quotients | 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 0614.00007.]
In characteristic 0, a rational double point is a \(quotient\quad X\) of \(k^ 2\) (k is the field) by a finite \(subgroup\quad G\) of SL(2,k). The McKay correspondence establishes a one-to-one correspondence between the irreducible representations of G and the vertices of the extended Dynkin diagram associated to the minimal desingularization of X. Moreover, it says how to read on the Dynkin diagram the tensor product of the standard representation \(G\subset SL(2,k)\) with any other one. We refer to it as the multiplicative structure. One may rephrase it in erms of irreducible reflexive modules on X, knowing that they are in one-to-one correspondence with the irreducible representations of G, and that \(G\subset SL(2,k)\) corresponds to the Kähler one \(forms\quad \Omega^ 1_ X\) (as a reflexive module).
In characteristic p\(>0\), where the group G no longer exists in general, the one-to-one correspondence between irreducible reflexive modules and vertices of the extended Dynkin diagram is still true (Gonzalez- Sprinberg, Verdier and Artin, Verdier). In the paper under review the authors complete the picture in characteristic \(p.\) Replacing \(\Omega^ 1_ X\) by \(\Omega\), the unique non trivial extension of the maximal ideal by \({\mathcal O}\) (which was also considered by other people, among them M. Auslander), they show that the multiplicative behavior remains true, except in few cases which are studied precisely. rational double point; McKay correspondence; Dynkin diagram; characteristic p Gonzalez-Sprinberg, G.; Verdier, J. -L.: Structure multiplicative de modules réflexifs sur LES points doubles rationnels. Travaux en cours 22, 79-100 (1987) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Structure multiplicative des modules réflexifs sur les points doubles rationnels. (Multiplicative structure of the reflexive modules on the rational double points) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0607.00005.]
Let \(f: V\to T\) be a deformation of a 2-dimensional isolated hypersurface singularity \((V_ 0,p_ 0)\). We assume that there is a section morphism \(s: T\to V\) such that \(f\circ s\) is the identity on T and that f is smooth everywhere outside the image s(T). For a point \(t\in T\), we denote \(V_ t=f^{-1}(t)\) and \(p_ t=s(t)\). The point \(0\in T\) is the base point.
The author shows that if \(\mu^*(V_ t,p_ t)\) is constant as a function of t, then f has a strong simultaneous resolution. This gives an affirmative answer to a question raised by B. Teissier. Since it has been known that the converse of this statement holds, the above two conditions turn out to be equivalent in dimension 2. The main tools are Hironaka's theory on the Whitney conditions and Neumann's theory on plumbing on 3- manifolds.
In the last part the author takes up the family \(z^ 7+x(y^ 9+x^{27})+tzy^ 8=0\) of surface singularities with the parameter t. It has constant Milnor number and constant multiplicity, but \(\mu^*(V_ t)\) is \textit{not} constant. Moreover, it has a weak simultaneous resolution, but is has \textit{no} flat simultaneous resolution. (In this article three different notions of simultaneous resolutions are discussed. A strong simultaneous resolution implies a weak simultaneous resolution. A strong one also implies a flat simultaneous resolution. However, a weak one and a flat one have no implication relation.)
The reference list contains some incorrect numbers. simultaneous resolution; surface singularities; Milnor number Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Complex singularities, Singularities in algebraic geometry, Local complex singularities Strong simultaneous resolution for surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a real analytic germ f: (\({\mathbb{R}}^ 2,0)\to ({\mathbb{R}},0)\), which is analytically irreductible over \({\mathbb{C}}\) and with singularities at 0 in \({\mathbb{C}}^ 2\), poles of the generalized function \(| f(x,y)|^ s\), smaller than the largest pole, are determined; they are expressed in terms of the characteristic sequence associated to the germ of the complex curve defined by f. Bernstein-Sato polynomial; real analytic germ; singularities; characteristic sequence Lichtin, B., Some Formulae for Poles of|f(x,y)| s ,Amer. J. of Math. vol.107 (1985), pgs. 139--162. Complex singularities, Hyperfunctions, Hyperfunctions, analytic functionals, Germs of analytic sets, local parametrization, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Some algebro-geometric formulae for poles of \(| f(x,y)| ^ s\) | 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 construct a transverse transport for a multivalued function of type \(f^{\lambda_ 1}_ 1 ... f^{\lambda_ p}_ p\) (\(\lambda_ i\) complex numbers) near the origin of \({\mathbb{C}}^ 2\). This transport is unique up to isotopy. We deduce the existence of \textit{regular} neighborhoods all of whose fibers are \(C^ \infty\)-diffeomorphic (analytically diffeomorphic in the quasihomogeneous case). We obtain a generalization of the monodromy operator. We also compute the vanishing homology of the generic fiber, giving a description of its natural graduation. singularities; holomorphic foliations; desingularization; Liouville function; vanishing cycles; monodromy Paul, E.: Cycles évanescents d'une fonction de Liouville de type \$f\_\{1\}\^\{\(\backslash\)lambda\_\{1\}\} \(\backslash\)cdots f\_\{p\}\^\{\(\backslash\)lambda\_\{p\}\}\$ . Ann. Inst. Fourier 45(1), 31--63 (1995) Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Singularities in algebraic geometry, Singularities of holomorphic vector fields and foliations, Ordinary differential equations in the complex domain, Global theory and resolution of singularities (algebro-geometric aspects) Cycles évanescents d'une fonction de Liouville de type \(f^{\lambda_ 1}_ 1 \dots f^{\lambda_ p}_ p\) . (Vanishing cycles for a Liouville function of type \(f^{\lambda_ 1}_ 1 \dots f^{\lambda_ p}_ p\) ) | 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 \(X_ 0\) be a projective algebraic variety with isolated singularities and \(X\to X_ 0\) a resolution of the singularities of \(X_ 0\) with a normal crossings exceptional divisor E. Let \(E^ 0=X\) and let \(E^{(r)}\) be the disjoint union of the r-fold intersections of the irreducible components of E. The authors show that there is a formula that expresses the middle perversity intersection homology (mpih) Betti numbers of \(X_ 0\) as a linear function of the Betti numbers of \(X=E^ 0,E^ 1,E^ 2,..\). and which depends on \(X_ 0\), not on the chosen resolution.
This generalizes what was known about the mpih Betti numbers of a projective cone [see \textit{M. Goresky} and \textit{R. MacPherson}, Algebraic Geometry, Proc. int. Conf., La Rabida/Spain 1981, Lect. Notes Math. 961, 119-129 (1982; Zbl 0525.14010)]. middle perversity intersection homology; mpih Betti numbers Jonathan Fine and Prabhakar Rao, On intersection homology at isolated singularities, Algebras Groups Geom. 5 (1988), no. 4, 329 -- 340. (Co)homology theory in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Classical real and complex (co)homology in algebraic geometry, Complex singularities On intersection homology at isolated 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 present a survey of some aspects and new results on configurations, i.e., disjoint unions of constellations of infinitely near points, local and global theory, with some applications and results on generalized Enriques diagrams, singular foliations, and linear systems defined by clusters. infinitely near points; proximity relations; Enriques diagrams; clusters; toric varieties Campillo, A., Gonzalez-Sprinberg, G., Monserrat, F.: Configurations of infinitely near points. Sao Paulo J. Math. 3(1), 113-158 (2009) Divisors, linear systems, invertible sheaves, Toric varieties, Newton polyhedra, Okounkov bodies, Parametrization (Chow and Hilbert schemes), Infinitesimal methods in algebraic geometry, Singularities in algebraic geometry, Singularities of curves, local rings Configurations of infinitely near points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In contrast to the characteristic-zero situation, the residual order of an ideal may increase in positive characteristic under permissible blowups at points of the exceptional divisor where the order of the ideal has remained constant. The specific situations where this happens are described explicitly. resolution; singularities; positive characteristic; blowup Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Polynomials in real and complex fields: location of zeros (algebraic theorems) Characterizing the increase of the residual order under blowup in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We show that the types of singularities of Schubert varieties in the flag varieties \(\text{Flag}_n\), \(n \in {\mathbb N}\), are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type~\({\mathbb A}\). Similarly, we prove that the types of singularities of Schubert varieties in products of Grassmannians \(\text{Grass}(n,a) \times \text{Grass} (n,b)\), \(a,b,n \in {\mathbb N}\), \(a,b \leq n\), are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type~\({\mathbb D}\). We also show that the orbit closures in representation varieties of Dynkin quivers of type~\({\mathbb D}\) are normal and Cohen-Macaulay varieties. singularities of Schubert varieties; flag varieties; Dynkin quivers; Cohen-Macaulay varieties Bobiński, Grzegorz; Zwara, Grzegorz, Schubert varieties and representations of Dynkin quivers, Colloq. Math., 94, 2, 285-309, (2002) Grassmannians, Schubert varieties, flag manifolds, Representations of quivers and partially ordered sets, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Schubert varieties and representations of Dynkin 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 The author studies some special families of plane curve singularities. He considers a family \(C_t\) of (germs) of plane algebraic curves with two prescribed singular points, say, at \(x_t\) and \(y_t\). The author asks what can occur if for some \(t\to t_0\) the points \(x_t\) and \(y_t\) merge. Such degeneration is called a \textit{collision} of singular points. The problem of studying collisions is a special case of degenerations of singular points of plane curve, but it doesn't seem less difficult because of that.
Although the author does not prove any essentially new general results in that direction, the paper can well serve as a of precise definitions. In particular there are precise (even if very algebraic) definitions of what a generic collision is. There are very interesting examples in the article, too. Their computation is mostly algebraic. equisingular families; invariants of plane curve singularities; collisions of singularities; deformations of singularities Kerner, D.: On the collisions of singular points of complex algebraic plane curves. arXiv:0708.1228 Singularities of curves, local rings, Computational aspects of algebraic curves, Deformations of singularities, Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry On the collisions of singular points of complex algebraic 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 The so-called Enriques diagram is the graph naturally associated to a constellation of infinitely near points to an algebraic regular variety, which is also provided with the binary relation associated to proximity on blown-up points of the constellation. In the present work we characterize combinatorically the Enriques diagrams which are toric (i.e. associated to equivariant constellations with respect to an algebraic torus action) and we get the minimum dimension of a toric constellation which induces a given toric Enriques diagram. Secondly we characterize combinatorically the linear proximity relation in the toric case and by this method we prove that the characteristic cone of a toric constellation is regular (if) and only if it is inducible by a two dimensional constellation; this gives an inverse statement to a classical result of Zariski. proximity; characteristic cone Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies, Singularities in algebraic geometry On Enriques diagrams and toric constellation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the behaviour of the degree of the Fulton-Johnson class of a complete intersection under a blow-up with a smooth center under the assumption that the strict transform is again a complete intersection. Our formula is a generalization of the genus formula for singular curves in smooth surfaces. Euler characteristic; complete intersection; singularities Cynk, S., Complex and Differential Geometry, 8, Euler characteristic of a complete intersection, 99-114, (2011), Springer, Heidelberg Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Euler characteristic of a complete intersection | 0 |
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