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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 A finite group scheme over a field is a group scheme whose coordinate algebra is finite dimensional. The author reviews the structure of finite group schemes with particular interest in simple finite group schemes. A simple finite group scheme over an algebraically closed field is either the étale group scheme associated to a finite simple group or the height one infinitesimal group scheme associated to a simple restricted Lie algebra (with the latter case occurring only if the field has prime characteristic). The author then reviews the classifications of finite simple groups and simple restricted Lie algebras. Finally, the author considers the infinitesimal deformations of a finite simple group scheme or equivalently the second cohomology of the group scheme with coefficients in the adjoint representation (its Lie algebra). The state of knowledge for the various cases is summarized. In particular, previous results of the author for simple Lie algebras of Cartan type are presented showing that the cohomology is non-zero (i.e., the Lie algebra is non-rigid) in contrast to the case of Lie algebras of classical type where in most cases it is zero (i.e., the Lie algebra is rigid). finite group scheme; simple group; infinitesimal group scheme; restricted Lie algebra; infinitesimal deformations; Lie algebra cohomology Viviani, F.: Deformations of simple finite group schemes Group schemes, Modular Lie (super)algebras, Cohomology of Lie (super)algebras, Graded Lie (super)algebras, Finite simple groups and their classification Simple finite group schemes and their infinitesimal deformations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f:(\mathbb{C}^2 , 0) \rightarrow (\mathbb{C}, 0)\) be the germ of a holomorphic function with an isolated critical point at the origin and let \((C, 0)\) be the irreducible plane curve singularity defined by \(f = 0\). Let \(V_{f}\) be the Milnor fiber of the singularity and let \(h_{f}: V_{f} \rightarrow V_{f}\) be the monodromy transformation. Let \(\zeta_{f}(t)\) be the zeta function of \(h_{f}\), as defined by \textit{N. A'Campo} [Comment. Math. Helv. 50, 233-248 (1975; Zbl 0333.14008)]. In this paper the authors prove that \(\zeta_{f}(t)\) is exactly the Poincaré series of the ring of functions of \(C\). plane curve singularity; Poincaré series; zeta function; critical point; Milnor fiber Gusein-Zade, S., Delgado, F., Campillo, A.: On the monodromy of a plane curve singularity and the Poincaré series of its ring of functions. Funct. Anal. Appl. 33(1), 56-57 (1999) Singularities of curves, local rings, Local complex singularities, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities of differentiable mappings in differential topology, Singularities in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the monodromy of a plane curve singularity and the Poincaré series of its ring of 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 The paper under review gives a short tutorial introduction to the singularities arising from minimal model program. In doing this the author first recalls the main techniques and tools, such as Reid's canonical cover and discrepancy inequalities, that are used to study this singularities [for a more technical introduction see \textit{J. Kollár}, in: Algebraic geometry, Proc. Summer Res. Inst., Santa Cruz 1995, Proc. Symp. Pure Math. 62(pt. 1), 221--287 (1997; Zbl 0905.14002)]. Then he gives a treatment in dimension two and three and discusses some conjectures stated by Shokurov. minimal model program; canonical singularities; terminal singularities V. A. Iskovskikh, ''Singularities on minimal models of algebraic varieties,'' J. Math. Sci. (New York), 106 (2001). no. 5, 3269--3285. Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds Singularities on minimal models of algebraic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X_ g\) be the space of binary forms of degree \(g\), and let \(X_{p,g}\) be the subspace of forms having a root of multiplicity \(p\). \(X_ g\) can be identified with \(\text{Spec} R_ g\), where \(R_ g=\text{Sym} (S_ gV)\) for a fixed two-dimensional vector space \(V\) over \(\mathbb{C}\). Let \(J_{p,g}\) be the ideal of polynomials in \(R_ g\) vanishing on \(X_{p,g}\). For \(p=2\) it is well known that \(J_{p,g}\) is generated by one element of degree \(2g-2\), namely the discriminant. In this paper a formula for the dimensions of the graded pieces of \(J_{p,g}\) in the general case is derived. If \(g-1=(p-1) h+1\), \(0 \leq 1<p-1\), it is conjectured that \(J_{p,g}\) is generated by its elements of degrees \(2h\), \(2h+1\), and \(2h+2\). Hilbert functions of multiplicity ideals; binary forms Weyman, J.: On Hilbert functions of multiplicity ideals. J. algebra 161, 358-369 (1993) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Relevant commutative algebra, General ternary and quaternary quadratic forms; forms of more than two variables, General binary quadratic forms On the Hilbert functions of multiplicity ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert series of a polarized algebraic variety \((X, D)\) is a powerful invariant that, while it captures some features of the geometry of \((X, D)\) precisely, often cannot recover much information about its singular locus. This work explores the extent to which the Hilbert series of an orbifold del Pezzo surface fails to pin down its singular locus, which provides nonexistence results describing when there are no orbifold del Pezzo surfaces with a given Hilbert series, supplies bounds on the number of singularities on such surfaces, and has applications to the combinatorics of lattice polytopes in the toric case. del Pezzo surface; Hilbert series; mirror symmetry; orbifold singularity Fano varieties, Singularities of surfaces or higher-dimensional varieties, Graded rings Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 aim of this paper is to join the algebraic and the geometric information on \(A\), where \((A,m)\) is an excellent two-dimensional normal local ring, using the theory of the Hilbert functions and the theory of the resolution of singularities.
Inspired by \textit{T. Okuma} [Ill. J. Math. 61, No. 3--4, 259--273 (2017; Zbl 1402.14044)], the authors investigate the integrally closed m-primary ideals of elliptic singularities and of strongly elliptic singularities.
The authors present a result characterizes algebraically the strongly elliptic singularities. They show that there exist excellent two-dimensional normal local rings having no strongly elliptic ideals. Finally, the authors present necessary and sufficient conditions for the existence of strongly elliptic ideals in terms of the existence of certain cohomological cycles. When there exist, they present an effective geometric construction. Hilbert coefficients; elliptic ideals; elliptic singularities; reduction number; 2-dimensional normal domain Integral domains, Singularities of surfaces or higher-dimensional varieties, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Elliptic surfaces, elliptic or Calabi-Yau fibrations Normal Hilbert coefficients and elliptic ideals in normal two-dimensional 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 Jet schemes of monomial schemes are known to be equidimensional but are not reduced in general. We give a formula for the multiplicity along every component of the jet schemes of a simple normal crossing divisor, inspired by recent work of \textit{R. A. Goward} and \textit{K. E. Smith} [Commun. Algebra 34, No. 5, 1591--1598 (2006; Zbl 1120.14055)]. DOI: 10.1080/00927870701512168 Computational aspects in algebraic geometry, Computational aspects and applications of commutative rings The multiplicity of jet schemes of a simple normal crossing divisor | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 notion of blow-analytic equivalence was introduced by T.C. Kuo. The semi-algebraic counterpart is the blow-Nash equivalence. In the paper, the author considers simple Nash function germs (i.e., Nash germs belonging to the list of ADE singularities) and he proves that two germs of that type are blow-Nash equivalent if and only if they are analytically equivalent. Furthermore, in the paper, it is proved that a non-simple Nash germ cannot belong to the same blow-Nash class of a simple germ. blow-Nash equivalence; simple singularities; virtual Poincaré plynomial Fichou, G., Blow-Nash type of simple singularities, J. Math. Soc. Japan, 60, 2, 445-470, (2008) Singularities in algebraic geometry, Nash functions and manifolds, Topology of real algebraic varieties, Equisingularity (topological and analytic) Blow-Nash types of simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies fundamental groups related with log-terminal singularities.
The main theorem shows that the fundamental group of a normal variety \(X\) is preserved by a resolution of its singularities if there exists an effective \(\mathbb{Q}\)-divisor \(\Delta\) such that \((X, \Delta)\) is a KLT (Kawamata log-terminal) pair.
\textit{J. Kollár} [Invent. Math. 113, No. 1, 177--215 (1993; Zbl 0819.14006)] proved the above statement in some special cases (as \(\dim X=3\)), and proved the analogous for algebraic fundamental groups.
The second theorem of this paper, a slight generalization of a previous result of the author [J. Algebr. Geom. 10, No. 4, 713--724 (2001; Zbl 1096.14009)], shows that fundamental groups are preserved also by proper surjective morphisms \(f:X \rightarrow S\) of normal varieties with connected fibres such that there exists a \(\Delta\) with \((X,\Delta)\) KLT and \(-(K_X+\Delta)\) \(f\)-nef and \(f\)-big.
As a corollary of its results the author can show that the fundamental group is invariant under various fundamental operations in the minimal model program, as contractions of extremal rays, flips and pluricanonical morphisms of minimal varieties of general type. fundamental group; \(L^2\)-index theorem; extremal rays; flips; pluricanonical morphisms Coverings in algebraic geometry, Singularities in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects), Minimal model program (Mori theory, extremal rays) Local simple connectedness of resolutions of log-terminal singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A sandwiched singularity \(X\) of a surface is a normal surface singularity which dominates birationally a non singular surface \(S\). These singularities are obtained from the basis \(S\) by blowing-up a complete \(\mathfrak M\)-primary ideal \(I\) of the local ring \({\mathcal O}_{S,O}\), \(S\) is a complex regular analytic surface. The usual problem in this theory is to get all possible information from \(I\subset {\mathcal O}_{S,O}\) on the singularities of \(X\). Here the informations given by the author are the number of singularities on \(X\), their fundamental cycles and multiplicities.
The main theorem 3.5 gives, in the more general case where \({\mathcal O}_{S,O}\) is a local \(\mathbb C\)-algebra having a rational singularity, a bijection between the set of complete ideals of codimension 1 (as \(\mathbb C\)-vector space) contained in \(I\) and the set of points in the exceptional locus of the surface \(X=\text{Bl}_I(R)\).
In the section 4, the author applies his theorem to the case where \(S\) is regular and \(X\) a sandwiched singularity. To every exceptional point \(Q\in X\), there is the associated ideal \(I_Q\) of 3.5 and to \(J\) a complete \(\mathfrak M\)-primary ideal, the author associates the weighted cluster of base points of \(J\) (base points are closed points in \(X'\) where general elements of \(J\) go through, \(X'\to X\) is any sequence of blow-ups of closed points).
All this leads to an explicit formula in theorem 4.7, which gives the multiplicity of an exceptional point \(Q\in X\) in terms of the clusters given by the base points of \(I\) and of \(I_Q\). regular; complete ideals; sandwiched singularity; analytic surface Fernández-Sánchez, J.: On sandwiched singularities and complete ideals. J. pure appl. Algebra 185, 165-175 (2003) Singularities of surfaces or higher-dimensional varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Rational and birational maps, Complex surface and hypersurface singularities On sandwiched singularities and complete 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 From the abridged English version: Let \(\chi = (d + 1)\) \((d + 2)/2 - N\) with \(N\) and \(d\) natural numbers. Let \(H_N\) be the Hilbert scheme parametrizing subschemes of \(\mathbb{P}^2\) of length \(N\). Let \(W^i = W^i_N [d] = \{[X]\in H_N |h^0 (I_x(d)) \geq \chi^+ + i + 1\}\) where \(\chi^+\) is the positive part of \(\chi\). We call the \(W^i\) the Brill-Noether strata of \(H_N\).
The problem of determining the irreducible components of the \(W^i\) remains largely open because only the case \(W^0\) is known: for \(\chi \leq 0\) it is fairly easy to see that \(W^0\) has only one irreducible component, while a list of the irreducible components of \(W^0\) for \(\chi > 0\) has been given by \textit{M.-A. Coppo}. We will identify the irreducible components of \(W^1\) for \(\chi \leq 0\), i.e. for \(N > d\) \((d + 3)/2\). We are thus identifying the irreducible components of the locus of \([X] \in H_N\) for which \(X\) is contained in at least one pencil of curves of degree \(d\) in exactly the range of \((N,d)\) for which the general member of \(H_N\) is not contained in any curve of degree \(d\).
We describe our result. For \(0 < d' < d\) and \(0 \leq N' \leq N\) with \(N - N' \leq (d - d')^2\) we define the subscheme \(X(N,d, N',d')\) of \(\mathbb{P}^2\) as the union of \(N'\) generic points on the generic curve of degree \(d'\) and the generic part of degree \(N - N'\) of the complete intersection of two generic curves of degree \(d - d'\). We may similarly define \(X(N,d, 0,0)\) when \(N \leq d^2\). We write \(W(N,d, N',d')\) for the subvariety of the Hilbert scheme which is the closure of the point representing \(X(N,d, N',d')\). Our result is the following:
Theorem: Let \(N\) and \(d\) be integers with \(N > d(d + 3)/2\). The irreducible components of \(W^1\) are on the one hand \(W(N,d, 0,0)\) if \(N \leq d^2\), and on the other hand the \(W(N,d, N',d')\) for \(N'\) and \(d'\) satisfying
\(N' < N\), \(0 < d' < d\),
\(N' \geq d'(d + 3) - d'(d' - 1)/2\),
\((d - d') (d - d' + 3)/2 - 1 \leq N - N' \leq (d - d')^2\),
\(N' + d'(d' + 3)/2 + (d - d') (d - d' + 3) \geq d(d + 3)\). Hilbert scheme; Brill-Noether strata Hirschowitz, A.; Rahavandrainy, O.; Walter, C.: Quelques strates de brill -- Noether du schéma de Hilbert de P2. C. R. Math. acad. Sci. Paris sér. I 319, 589-594 (1994) Parametrization (Chow and Hilbert schemes) Some Brill-Noether strata of the Hilbert scheme of \(\mathbb{P}^ 2\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For the bounded derived category of coherent sheaves on a smooth projective variety, it is an interesting problem to study its group of autoequivalences. There are some cases where the group is completely understood, such as varieties with ample or anti-ample canonical bundles, abelian varieties, toric surfaces and \(K3\) surfaces of Picard rank 1. In other situations, one needs to construct enough examples of autoequivalences before one could understand the group completely. The main result in the paper under review is a new autoequivalence for the Hilbert scheme \(X^{[n]}\) of \(n\)-points on a complex smooth projective surface \(X\).
The author used two tools in the construction. The first tool is the derived McKay correspondence (aka Bridgeland-King-Reid-Haiman equivalence) between the derived category \(D^b(X^{[n]})\) and the \(\mathfrak{S}_n\)-equivariant derived category \(D^b_{\mathfrak{S}_n}(X^n)\) of the \(n\)-fold Cartesian product \(X^n\). The second tool is the so-called \(\mathbb{P}^n\)-functors. In short, each \(\mathbb{P}^n\)-functor with the derived category in question as the target leads to an autoequivalence via a standard procedure.
The author proved that the composition of the functor \(\mathrm{triv}: D^b(X) \to D^b_{\mathfrak{S}_n}(X)\) given by equipping every object with the trivial \(\mathfrak{S}_n\)-linearization and the push-forward functor \(\delta_*: D^b_{\mathfrak{S}_n}(X) \to D^b_{\mathfrak{S}_n}(X^n)\) along the diagonal embedding is a \(\mathbb{P}^{n-1}\)-functor, which yields an autoequivalence of \(D^b_{\mathfrak{S}_n}(X^n)\), hence an autoequivalence of \(D^b(X^{[n]})\) via the derived McKay correspondence. This autoequivalence generalizes the known EZ-spherical twist induced by a line bundle on the boundary of the Hilbert scheme when the canonical bundle of \(X\) is trivial and \(n=2\). It differs from the standard autoequivalences and the ones constructed via the universal sheaf on the Hilbert scheme when \(X\) is a \(K3\) surface by \textit{N. Addington} [``New derived symmetries of some hyperkähler varieties'', \url{arXiv:1112.0487}]. When \(X\) is an abelian surface, the construction leads to \(n^4\) orthogonal \(\mathbb{P}^{n-1}\)-objects on the generalized Kummer variety, which generalizes the 16 spherical objects on the Kummer surface given by the exceptional curves. Hilbert schemes; derived categories; autoequivalences Krug, A., Ploog, D., Sosna, P.: Derived categories of resolutions of cyclic quotient singularities. Q. J. Math. (2017). https://doi.org/10.1093/qmath/hax048/4675118 Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Parametrization (Chow and Hilbert schemes) On derived autoequivalences of Hilbert schemes and generalized Kummer 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 Given a smooth projective variety \(Z\), it is interesting to study the autoequivalences of the bounded derived category of coherent sheaves \(D^b(Z)\). The standard autoequivalences are the ones generated by automorphisms of \(Z\), the shift functor and tensor products with line bundles. When the canonical bundle of \(Z\) is ample or anti-ample, there are no other autoequivalences, due to a classical result of \textit{A. Bondal} and \textit{D. Orlov} [Compos. Math. 125, No. 3, 327--344 (2001; Zbl 0994.18007)]; but when the canonical bundle of \(Z\) is trivial, non-standard autoequivalences are expected but difficult to construct in general. Several closely related techniques have been developed in the construction of new autoequivalences, among which are spherical functors studied by \textit{R. Anno}, \textit{T. Logvinenko} [Adv. Math. 231, No. 3--4, 2069--2115 (2012; Zbl 1316.14033)] and \textit{R. Rouquier} [Invent. Math. 165, No. 2, 357--367 (2006; Zbl 1101.18006)], generalizing \textit{P. Seidel} and \textit{R. Thomas}' notion [Duke Math. J. 108, No. 1, 37--108 (2001; Zbl 1092.14025)] of spherical objects, as well as \(\mathbb{P}^n\)-functors studied by \textit{N. Addington} [``New derived symmetries of some hyperkaehler varieties'', \url{arXiv:1112.0487}], generalizing \textit{D. Huybrechts} and \textit{R. Thomas}' notion [Math. Res. Lett. 13, No. 1, 87--98 (2006; Zbl 1094.14012)] of \(\mathbb{P}^n\)-objects.
The paper under review applies these techniques to construct a new derived autoequivalence for the Hilbert scheme of points \(X^{[n]}\) on an Enriques surface \(X\). This autoequivalence is neither standard nor a twist around a spherical object.
The construction is given in two steps. In the first step, the authors proved that for any surface \(S\) with \(p_g=q=0\) (\(S\) can be chosen to be the Enriques surface \(X\) for instance), the Fourier-Mukai transform from \(D^b(S)\) to \(D^b(S^{[n]})\) whose kernel is the universal ideal sheaf is fully faithful. Indeed, they proved that the composition \(F^RF\), where \(F^R\) is the right adjoint of \(F\), is isomorphic to the identity functor. In the second step, the authors proved that for any smooth projective variety \(Y\) whose canonical bundle is \(2\)-torsion (\(Y\) can be chosen to be the Hilbert scheme of points \(X^{[n]}\) on an Enriques surface \(X\) for instance), assuming that \(\pi: \widetilde{Y} \to Y\) is the canonical cover, any admissible subcategory \(i: \mathcal{A} \to D^b(Y)\) induces a split spherical functor \(\pi^*i: \mathcal{A} \to D^b(\widetilde{Y})\), whose associated twist on \(D^b(\widetilde{Y})\) is equivariant, hence descends to an autoequivalence of \(D^b(Y)\).
Some other related interesting results are also studied in the paper: the authors showed that an exceptional sequence on a surface \(S\) can be used to construct an exceptional sequence on the Hilbert scheme \(S^{[n]}\); they also showed that the so-called truncated ideal functor can be a good alternative for the Fourier-Mukai transform with the universal ideal sheaf as the kernel for constructing admissible subcategories or \(\mathbb{P}^n\)-functors in some cases. derived categories; semi-orthogonal decompositions; Fourier-Mukai functors; Hilbert schemes of points on surfaces Krug, A; Sosna, P, On the derived category of the Hilbert scheme of points on an Enriques surface, Selecta Math. (N.S.), 21, 1339-1360, (2015) Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K3\) surfaces and Enriques surfaces On the derived category of the Hilbert scheme of points on an Enriques 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 Inspired by the influential work of \textit{A. Suslin} and \textit{V. Voevodsky} [Invent. Math. 123, No. 1, 61--94 (1996; Zbl 0896.55002)] and \textit{V. Voevodsky, A. Suslin} and \textit{E. M. Friedlander} [``Cycles, transfers, and motivic homology theories'', Ann. Math. Stud. 143 (2000; Zbl 1021.14006)], the article under review introduces a singular homology theory on schemes of finite type over a Dedekind domain and verifies several basic properties.
As an application to class field theory, the author constructs a reciprocity isomorphism from the zeroth integral singular homology and the abelianized modified tame fundamental group for an arithmetic scheme. algebraic cycles; class field theory; arithmetic schemes Schmidt A.: Singular homology of arithmetic schemes. Algebra Number Theory 1(2), 183--222 (2007) Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Class field theory, Motivic cohomology; motivic homotopy theory, Singular homology and cohomology theory Singular homology of arithmetic schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting variety of some parabolic subalgebras of \(M_{n}(\Bbbk)\) and to the punctual nested Hilbert scheme. By this method, we obtain a lower bound on the dimension of these moduli spaces. We characterize the cases where they are irreducible. In some reducible cases, we describe the irreducible components and their dimensions. Hilbert scheme; commuting variety; GIT; parabolic algebra; nilpotent orbit [4] Michael Bulois &aLaurent Evain, &Nested punctual Hilbert schemes and commuting varieties of parabolic subalgebras&#xhttp://arxiv.org/abs/1306.4838 Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Geometric invariant theory, Coadjoint orbits; nilpotent varieties Nested punctual Hilbert schemes and commuting varieties of parabolic subalgebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathcal I_{d,g,r}\) denote the Hilbert scheme parametrizing smooth curves of degree \(d\) and genus \(g\) in complex projective space \(\mathbb P^{r}_{\mathbb C}\).
\textit{L. Ein} proved \(\mathcal I_{d,g,r}\) is irreducible if \(d \geq \frac{(2r-2)g+(2r+3)}{r+2}\) [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 469--478 (1986; Zbl 0606.14003); Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], which proves Severi's claim of irreducibility for \(d \geq g+r\). He also gave examples showing that Severi's claim fails for \(r \geq 6\).
In the paper under review, the author expands the known irreducibility range in case \(r=5\), showing that \(\mathcal I_{d,g,r}\) is irreducible for \(d \geq \max\{\frac{11}{10}g+2,g+5\}\). The outline of the proof runs as follows. If \(V \subset \mathcal I_{d,g,r}\) is any irreducible component, then \(V\) is generically a fibre bundle over a closed subset of a component \(\mathcal G \subset \mathcal G^{r,d}\) with fibre \(\text{Aut}(\mathbb P^{r})\) and there is \(\alpha \geq r\) such that \(\mathcal G\) is generically a fibre bundle over a closed subset of an irreducible component \(\mathcal W \subset \mathcal W^{\alpha,d}\) with fibre \({\roman Gr}(r,\alpha)\), thus we arrive at the inequality
\[
(r+1)d-(r-3)(g-1) \leq \mathcal W + r^{2}+2r+(r+1)(\alpha-r)
\]
(the spaces \(\mathcal G^{r,d}\) and \(\mathcal W^{\alpha,d}\) are the global versions of \(G^{r,d}(C)\) and \(W^{\alpha,d}(C)\) from the book of \textit{E. Arbarello, M. Cornalba, P. A. Griffiths} and \textit{J. Harris} [Geometry of Algebraic Curves, Grundlehren der Mathematischen Wissenschaften, 267. New York etc.: Springer-Verlag (1985; Zbl 0559.14017)]).
Now if the expected dimension \(\rho(d,g,r)=g-(r+1)(g-d+r)\) is positive, one constructs a unique irreducible component \(V \subset \mathcal I_{d,g,r}\) which dominates the moduli space \(\mathcal M_{g}\). If \(V_{1}\) were a different irreducible component, we would produce \(\mathcal W\) as above which does not dominate \(\mathcal M_{g}\), but then a series of estimates contradicts the inequality above. Hilbert scheme; Brill-Noether theory; linear series H. Iliev, On the irreducibility of the Hilbert scheme of curves in \(\mb{P}^5\), Comm. Algebra 36 (2008), no. 4, 1550--1564. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) On the irreducibility of the Hilbert scheme of curves in \(\mathbb P^{5}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We compute some generating series of integrals related to tautological bundles on Hilbert schemes of points on surfaces \(S^{[n]}\), including the intersection numbers of two Chern classes of tautological bundles, and the Euler characteristics of \(\Lambda_{-y}TS^{[n]}\). We also propose some related conjectures, including an equivariant version of Lehn's conjecture. Hilbert scheme; tautological sheaf; intersection number Wang, X.Y.; Chen, H.B.; Wang, Y.J., Solution structures of tensor complementarity problem, Front. Math. China, (2017) Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry Generating series of intersection numbers on Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((R,\mathfrak{m})\) be a two dimensional regular local ring with infinite residue field. The theory of complete (i.e., integrally closed) ideals in this setup was initiated by Zariski, with the initial motivation of studying under an algebraic perspective the problem of curves passing through a fixed set of points, with given multiplicities. Since then, several authors have worked on these and related aspects of the theory: for example, \textit{J. Lipman} [in: Algebraic geometry and commutative algebra, Vol. I, 203--231 (1988; Zbl 0693.13011)], \textit{S. D. Cutkosky} [Invent. Math. 98, No. 1, 59--74 (1989; Zbl 0715.13012)] and \textit{C. Huneke} [Publ., Math. Sci. Res. Inst. 15, 325--338 (1989; Zbl 0732.13007)], just to mention some. Let \(I\) be a complete simple residually rational \(\mathfrak{m}\)-primary ideal of \(R\). The main focus of this article is the study of the singular points of \(\Sigma = \text{Bl}_I(R)\), the blow-up of \(I\). The authors show that \(\Sigma\) has either one or two singular points, which in any case are rational singularities. The number of such singular points depends on whether the largest base point \(S_I\) of \(I\) is a free \(R\)-module or not. In addition, the authors compute the multiplicity of \(\Sigma\) at the singular point (respectively, points) in terms of certain non-simple ideal (respectively, ideals) adjacent to \(I\). blow-up; simple ideal; multiplicities Regular local rings, Singularities of surfaces or higher-dimensional varieties, Special surfaces The blow-up of a simple ideal | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 shows how the multigraded Hilbert scheme construction of \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] can be used to construct a quasi-projective scheme which parametrize left homogeneous ideals in the Weyl algebra having fixed Hilbert function. Fixing an integral domain \(k\) of characteristic zero, the Weyl algebra \(W=k \langle x_1, \dots, x_n, \partial_1, \dots, \partial_n \rangle\) has a \(k\)-basis consisting of the set \(\mathcal B = \{ x^\alpha \partial^\beta | \alpha, \beta \in \mathbb N^n \}\). If \(A\) is an abelian group, then any \(A\)-grading \(\mathbb N^n \to A\) on the polynomial ring \(S = k[x_1,\dots,x_n]\) extends to an \(A\)-grading \(\mathbb N^{2n} \to A\) on \(W\) by \(\deg (x^\alpha \partial^\beta) = \deg \alpha - \deg \beta\), which induces a decomposition \(W = \bigoplus_{a \in A} W_a\). Given a Hilbert function \(h:A \to \mathbb N\), the corresponding Hilbert functor \(H^h_W\) takes a \(k\)-algebra \(R\) to the set of homogeneous ideals \(I \subset R \otimes_k W\) such that \((R \otimes _k W_a)/I_a\) is a locally free \(R\)-module of rank \(h(a)\) for each \(a \in A\). The main theorem says that \(H^h_W\) is representable by a quasi-projective scheme over \(k\).
The strategy of the proof is similar to that of Haiman and Sturmfels [loc. cit.], but there are some new behaviors regarding monomials in the Weyl algebra \(W\) not seen in the polynomial ring \(S\). An obvious difference is that a product of monomials in \(W\) need not be a monomial. Another difference is that \(W\) has infinite antichains of monomial ideals, unlike the polynomial case: see work of \textit{D. MacLagan} [Proc. Am. Math. Soc. 129, 1609--1615 (2001; Zbl 0984.13013)]. Moreover, the natural extension of Gröbner basis theory for \(S\) to \(W\) does not work well, so the author considers the initial ideal of a left ideal in the associated graded algebra \(\text{gr} W\) and uses Gröbner basis theory for \(W\) developed by \textit{M. Saito} et al. [Gröbner deformations of hypergeometric differential equations. Berlin: Springer (2000; Zbl 0946.13021)]. The arguments are well presented along with examples showing the novel points. Hilbert schemes; Weyl algebras Parametrization (Chow and Hilbert schemes), Rings of differential operators (associative algebraic aspects) Multigraded Hilbert schemes parametrizing ideals in the Weyl algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(n\) be a positive integer. The Hilbert scheme of \(n\) points in the complex plane has a natural \((\mathbb C^*)\)-action induced by the action of the torus \(T:=(\mathbb C^*)^2\) on \(\mathbb C^2\). Let now \(T:=\left\{(t^a,t^b)\in T\,\,|\,\, t\in \mathbb C^* \right\}\), where \(\gcd(a,b)=1\) and \(a,b\geq 1\) be a one dimensional subtorus of \(T\). The set of fixed points under the action of such a subtorus has the structure of a smooth variety, denoted by \((\mathbb C^2)_{a,b}^{[n]}\).\newline In the paper under review, the author generalizes \textit{A. Iarrobino} and \textit{J. Yaméogo} [Commun. Algebra 31, No. 8, 3863--3916 (2003; Zbl 1048.14003)] by providing a formula for the class of the irreducible components of \((\mathbb C^2)_{1,k}^{[n]}\) in terms of polynomials in \(\mathbb L\), the class of \(\mathbb A_{\mathbb C}^1\) in the Grothendieck ring of complex quasi-projective varieties. Based on computer calculations, the author also makes a conjecture on a possible formula for the Grothendieck ring classes of the more general varieties \((\mathbb C^2)_{a,b}^{[n]}\).\newline Another result in the paper is an interesting relation between the classes of certain open strata of \((\mathbb C^2)^{[n]}\) and the \((q,t)\)-Catalan numbers.\newline Finally, using a well known quiver description of \((\mathbb C^2)^{[n]}\), the author provides sufficient conditions under which a \((1,k)\)-quasi-homogeneous Hilbert scheme of points is isomorphic to a homogeneous nested Hilbert scheme of points. The latter result generalizes \textit{J. Cheah} [Pac. J. Math. 183, No. 1, 39--90 (1998; Zbl 0904.14001)]. Hilbert schemes of points; torus action; \((q,t)\)-Catalan numbers; quiver varieties A. Buryak, ''The classes of the quasihomogeneous Hilbert schemes of points on the plane,'' Mosc. Math. J., 12:1, 21--36; http://arxiv.org/abs/1011.4459 . Parametrization (Chow and Hilbert schemes), Combinatorial aspects of partitions of integers, Representations of quivers and partially ordered sets The classes of the quasihomogeneous Hilbert schemes of points on the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a Noetherian integral scheme, and let \(\mathcal{M}\) be a coherent sheaf on \(X\); a modification \(f:\;Y\rightarrow X\) is called the \textit{blowup of \(X\) at \(\mathcal{M}\)} (denoted \(\mathrm {Bl}_{\mathcal{M}} (X)\)) if, on one hand, \(f\) is a flattening of \(\mathcal{M}\) (this means that the torsion free pullback \(f^{\star} (\mathcal{M})=f^*\mathcal{M}/\mathrm{torsion}\) is locally free) and, on the other hand, that any other flattening of \(\mathcal{M}\) factors through \(f\). The reader will note that, when \(\mathcal{M}\) is an ideal sheaf, \(\mathrm {Bl}_{\mathcal{M}} (X)\) is just the usual blowup with respect to an ideal.
From now on, suppose that \(X\) has prime characteristic \(p\), and that the absolute Frobenius \(F: X\rightarrow X\) is finite; for any integer \(e\geq 0\), one can define the \textit{\(e\)th F-blowup of \(X\)} to be \(\mathrm {Bl}_{F_*^e \mathcal{O}_X} (X):=\mathrm {FB}_e (X)\).
Hereafter, suppose that \((X,x)\) is a simple elliptic surface singularity in prime characteristic \(p\) with exceptional (elliptic) curve \(E\) on the minimal resolution \(\widetilde{X}\). Motivated by [Algebra Number Theory 7, No. 3, 733--763 (2013; Zbl 1303.14018)], the main goal of the paper under review is to understand the structure of \(\mathrm {FB}_e (X);\) it turns out that this structure depends on the intersection number \(-E^2\). Indeed, if \(-E^2\) is not a power of \(p\), then \(\mathrm {FB}_e (X)\cong\widetilde{X}\) (see Theorem 1.1). In this way, assume now that \(-E^2=p^n\) is a power of \(p\), let \(P_0\in E\) be the zero element of the group law, and pick an integer \(e\geq 0\) such that \(p^e\geq\max\{3,p^n\}\). One has to distinguish two cases (see Theorem 1.2).
On one hand, if \(E\) is ordinary (this means that, when \(q\) is a power of \(p\), the set of all \(q\)-torsion points on \(E\) has exactly \(q\) different points), then \(\mathrm {FB}_e (X)\) coincides with the blow up of \(\widetilde{X}\) at these points if \(n\geq 1\), and is the blow up at these points ruling out \(P_0\) if \(n=0\). On the other hand, if \(E\) is supersingular (this means that, when \(q\) is a power of \(p\), the set of all \(q\)-torsion points on \(E\) boils down to \(P_0\)), then \(\mathrm {FB}_e (X)\) coincides with the blow up of \(\widetilde{X}\) at an ideal defining a fat point at \(P_0\), where \(P_0\in\widetilde{X}\) is defined in local coordinates \(t,u\) by \((t,u^{p^e})\) if \(n\geq 1\), and is defined by \((t,u^{p^e-1})\) if \(n=1\). It is also worth noting that, in any case, \(\mathrm {FB}_e (X)\cong\widetilde{X}\) provided \(1\leq e<n\). F-blowup; Frobenius map; simple elliptic singularity; F-pure Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Structure of the F-blowups of simple 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 We study families \(V\) of curves in \(\mathbb{P}^2(\mathbb{C})\) of degree \(d\) having exactly \(r\) singular points of given topological or analytic types. We derive new sufficient conditions for \(V\) to be \(T\)-smooth (smooth of the expected dimension), respectively to be irreducible. For \(T\)-smoothness these conditions involve new invariants of curve singularities and are conjectured to be asymptotically proper, that is, optimal up to a constant factor; for curves with nodes and cusps these conditions are indeed optimal up to linear terms in \(d\). To obtain the results, we study the Castelnuovo function, prove the irreducibility of the Hilbert scheme of zero-dimensional schemes associated to a cluster of infinitely near points of the singularities and deduce new vanishing theorems for ideal sheaves of zero-dimensional schemes in \(\mathbb{P}^2\). Moreover, we give a series of examples of cuspidal curves where the family \(V\) is reducible, but where \(\pi_1(\mathbb{P}^2\setminus C)\) coincides (and is abelian) for all \(C\in V\). vanishing theorems; Castelnuovo function; singular plane curves; equisingular-families of algebraic curves Gert-Martin Greuel, Christoph Lossen & Eugenii Shustin, ``Castelnuovo function, zero-dimensional schemes and singular plane curves'', J. Algebr. Geom.9 (2000) no. 4, p. 663-710 Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Castelnuovo function, zero-dimensional schemes and singular 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 In order to calculate the multiplicity of an isolated rational curve \(C\) on a local complete intersection variety \(X\), i.e. the length of the local ring of the Hilbert scheme of \(X\) at \([C]\), it is important to study infinitesimal neighborhoods of the curve in \(X\). This is equivalent to infinitesimal extensions of \(\mathbb{P}^1\) by locally free sheaves. In this paper we study infinitesimal extensions of \(\mathbb{P}^1\), determine their structure and obtain upper and lower bounds for the length of the local rings of their Hilbert schemes at \([\mathbb{P}^1]\). multiplicity; length; rational curve; local complete intersection; locally free sheaves; symbolic power; Calabi-Yau 3-fold; minimal model program Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry Infinitesimal extensions of \(\mathbb P^1\) and their Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Lie algebra \({\mathfrak g}={\mathfrak{sl}}(z, \mathbb C)\oplus {\mathfrak {sl}}(z, \mathbb C)\) is used to construct the semi-universal deformation of simple elliptic singularities of type \(\widetilde{D}_5\). The nilpotent variety of the Lie algebra above is \(\mathcal N=\{\binom{a\;b}{c-a}\;| \;a^2+bc=0\}\times \{\binom{d\;e}{f-d}\;| \;d^2+ef=0\}\). The surface singularity \((\mathcal N\cap S, 0)\) is simple elliptic of type \(\widetilde{D}_5\) for a generic slice \(S\) at \(0\). If the slice is defined by \(c=d+e\) and \(f=a+b\) the construction is as follows. Let \(\mathfrak{h}=\{\binom{a\;0}{0-a}\}\oplus\{\binom{d\;0}{0-d}\}\) and consider the adjoint quotient \(\chi:\mathfrak{g}\to\mathfrak{h}/W\;,\;\chi(\binom{a\;b}{c-a}, \binom{d\;e}{f-e})=(-a^2-bc, -d^2-ef)\). \(W\cong\mathbb Z/2\mathbb Z \oplus\mathbb Z/2\mathbb Z\) is the Weyl group of \(\mathfrak{g}\). For \((\alpha, \beta)\in \mathbb C^2\) let \(f_{(\alpha,\beta)}(\binom{a\;b}{c-a},\;\binom{d\;e}{f-e}) = (-a^2-bc-\alpha e, -d^2-ef-\beta b)\) and the slice \(S_{(\gamma, \delta, \varepsilon)}\) be defined by \(c=d+e+\gamma, f=a+b+\delta e+\varepsilon\). Let \(S:=\mathbb C^2\times \mathbb C^3\times \mathfrak{h}/W\) and \(X=\{(A, \alpha, \beta, \gamma, \delta, \varepsilon, \lambda, \mu)\in \mathfrak{g}\times S\;| \;f_{(\alpha,\beta)}(A)=(\lambda, \mu), A\in S_{(\gamma, \delta, \varepsilon)}\}\). Let \(o:=(0,0,0,0,0,0,0)\in S\) and \(q:=(0,o)\in X\) then the morphism \((X,q)\to (S,o)\) is a semi-universal deformation of the simple elliptic singularity defined by the slice \(c=d+e, f=a+b\). simple elliptic singularity; semi-universal deformation Deformations of singularities, Deformations of complex singularities; vanishing cycles Semi-universal deformation spaces of some simple 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 Authors' abstract: We prove a closed formula for leading Gopakumar-Vafa BPS invariants of local Calabi-Yau geometries given by the canonical line bundles of toric Fano surfaces. It shares some similar features with Göttsche-Yau-Zaslow formula: Connection with Hilbert schemes, connection with quasimodular forms, and quadratic property after suitable transformation. In Part I of this paper we will present the case of projective plane, more general cases will be presented in Part II. Gopakumar-Vafa BPS invariant; Gromov-Witten invariants; local Calabi-Yau; Hilbert scheme; quasimodular form S. Guo, J. Zhou, Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. II, preprint. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects) Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. 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 \(C \subset \mathbb P^s\) be a non-degenerate rational complex curve of degree \(d>s\) with \(s \geq 3\) and at worst ordinary singularities. Letting \(f: \mathbb P^1 \to C\) be the normalization map, one obtains an exact sequence of vector bundles
\[
0 \to {\mathcal T}_{\mathbb P^1} \overset{df}{\to} f^* {\mathcal T}_{\mathbb P^s} f^* {\mathcal N}{_C} 0\tag{1}
\]
on \(\mathbb P^1\), where \({\mathcal N}_C\) is the normal sheaf of \(C\) and \({\mathcal T}_X = {\mathcal Hom}(\Omega^1_{X/\mathcal C}, {\mathcal O}_X)\). Writing \(\mathbb P^1 = \mathbb P V\) with \(V\) a 2-dimensional vector space with basis \(\{x,y\}\), the morphism \(f\) factors as the \(d\)-uple embedding \(\nu_d: \mathbb P V \hookrightarrow \mathbb P S^d V\) followed by projection from a linear subspace \(\mathbb P T \subset \mathbb P S^d V\). The main theorem of the paper gives an identification of the vector spaces \(H^0 f^* {\mathcal T}_{\mathbb P^s} (-d-2-k)\) and \(H^0 f^* {\mathcal N}_C (-d-2-k)\) with the respective vector spaces \(\text{ker} D \cap (S^k V \otimes T) \subset S^k V \otimes S^d V\) and \(\text{ker} D^2 \cap (S^k V \otimes T) \subset S^k V \otimes S^d V\), where \(D = \partial_x \otimes \partial_y - \partial_y \otimes \partial_x\) is the first order transvectant operator acting by derivation. The proof consists of careful algebra with tensors and builds on previous work of the authors [J. Pure Appl. Algebra 219, 1320--1335 (2015; Zbl 1305.14021)].
The are two interesting applications. The first deals with Hilbert schemes \({\mathcal H}_{\overline c}\) of non-degenerate rational curves in \(\mathbb P^s\) whose normal bundle splits as \(\oplus {\mathcal O} (c_i+d+2)\), where \(d\) is the degree and \(\overline c = (c_1, c_2, \dots, c_{s-1})\). For more than 30 years it has been known from work of \textit{D. Eisenbud} and \textit{A. van de Ven} [Math. Ann. 256, 453--463 (1981; Zbl 0443.14015); Invent. Math. 67, 89--100 (1982; Zbl 0492.14016)] and \textit{F. Ghione} and \textit{G. Sacchiero} [Manuscr. Math. 33, 111--128 (1980; Zbl 0496.14021)] that these Hilbert schemes are irreducible for \(s=3\) and for general splitting type by work of \textit{G. Sacchiero} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 26, 33--40 (1980; Zbl 0496.14020)]. The authors use their theorem to show that rational curves of degree \(d=11\) in \(\mathbb P^8\) whose normal bundle has splitting type \((2,2,1,1,0,0,0)\) has exactly two irreducible components of dimension \(98\), the first reducible example in the literature. Their second application is a characterization of the smooth rational curves contained in a rational normal scroll in terms of the splitting type of the restricted tangent bundle along with a computation of the splitting type of the normal bundle, thereby generalizing results of \textit{D. Eisenbud} and \textit{A. van de Ven} for \(s=3\) [Math. Ann. 256, 453--463 (1981; Zbl 0443.14015)]. rational curves; normal bundle; tangent bundle; Hilbert schemes Alzati, A; Re, R, Irreducible components of Hilbert schemes of rational curves with given normal bundle, Algebraic Geom., 4, 79-103, (2017) Projective techniques in algebraic geometry, Parametrization (Chow and Hilbert schemes), Special algebraic curves and curves of low genus Irreducible components of Hilbert schemes of rational curves with given normal bundle | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors describe how certain standard opens \(\mathrm{Hilb}^\beta\) of the Hilbert scheme \(\mathrm{Hilb}^N_{\mathbf A^n_A}\) parameterizing \(N\) points of \(\mathbf A^n_A\) are embedded into Grassmannians. These open subsets of the Hilbert scheme are the intersection of a corresponding open affine of the Grassmannian and a closed stratum determined by a Fitting ideal. As an application, the authors give a cover for the scheme parameterizing \(0\)-dimensional closed subschemes in \(\mathbb P^n_k\), \(k\) an algebraically closed field, that are flat, finite of relative rank \(n + 1\), and non-degenerate. The authors also show that the ideal generated by the commutator relations, often used in literature in the constructions of the local open schemes \(\mathrm{Hilb}^\beta\), equals the Fitting ideal arising from the graded, global situation. As an application, the authors give a new proof of the following result, already proved by \textit{K. Ranestad} and \textit{F.-O. Schreyer} [J. Reine Angew. Math. 525, 147--181 (2000; Zbl 1078.14506)]: the scheme \(\mathrm{VPS}^{n+1}_Z\) of length \(n+1\) subschemes in \(\mathbb P^n\) apolar to the annihilator scheme \(Z\) of a smooth quadratic surface is closed in the Grassmannian of rank \(n+1\) quotients of the vector space of two-forms on projective \(n\)-space. In the Appendix, the authors restate some of their results on ideals and Hilbert schemes for modules and Quot schemes. fitting ideals; Hilbert schemes; quot schemes; Grassmannian; strongly generated; apolarity Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Varieties of low degree, Projective and free modules and ideals in commutative rings, Grassmannians, Schubert varieties, flag manifolds, Modules, bimodules and ideals in associative algebras Explicit projective embeddings of standard opens of the Hilbert scheme of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \({\mathbb P}^n\) be the projective \(n\)-space over a field \(k\), let \(X\) be a closed subscheme of \({\mathbb P}^n\) of dimension \(d\) and codimension \(r > 0\), and let \(I_X \subset R := k[X_0,\dots ,X_n]\) be the homogeneous ideal associated to \(X\). Let \(D \geq 2\) be an integer and assume that \(I_X\) is generated by homogeneous polynomials of degree \(\leq D\). If \(\text{char}\, k = 0\) and \(X\) is smooth, \textit{A. Bertram, L. Ein} and \textit{R. Lazarsfeld} [J. Am. Math. Soc. 4, No. 3, 587--602 (1991; Zbl 0762.14012)] proved that the Castelnuovo-Mumford regularity of \(I_X\) satisfies the following inequality:
\[
\text{reg}(I_X) \leq r(D-1)+1.
\]
\textit{M. Chardin} and \textit{B. Ulrich} [Am J. Math. 124, No. 6, 1103--1124 (2002; Zbl 1029.14016)] showed that this inequality remains true in the case where \(X\) has isolated singularities.
In the paper under review, the author shows that, for \(k\) of arbitrary characteristic, one has:
\[
\text{reg}(I_X) \leq d\, !(r(D-1)-1)+1
\]
if the dimension \(\delta\) of the singular locus of \(X\) is \(\leq 1\), and that, for \(\delta \geq 2\):
\[
\text{reg}(I_X) \leq \lambda D^{(n-\delta )2^{\delta -2}}
\]
where \(\lambda\) is a constant depending on \(n\), \(d\) and \(\delta\).
The author reduces the proof of these inequalities to the case where \(X\) has, except at finitely many points, locally complete intersection rational singularities using an inductive argument introduced by \textit{G. Caviglia} and \textit{E. Sbarra} [Compos. Math. 141, No. 6, 1365--1373 (2005; Zbl 1100.13020)]. When \(X\) satisfies this additional hypothesis, the author uses the method developed by Chardin and Ulrich in the above mentioned paper. Castelnuovo-Mumford regularity; projective scheme; singular locus; locally complete intersection; rational singularity Fall, Amadou Lamine: Bornes pour la régularité de Castelnuovo-Mumford des schémas non lisses, Ann. inst. Fourier (Grenoble) 59, No. 3, 1015-1027 (2009) Polynomial rings and ideals; rings of integer-valued polynomials, Syzygies, resolutions, complexes and commutative rings, Singularities in algebraic geometry, Linkage, Vanishing theorems in algebraic geometry Bounds for the Castelnuovo-Mumford regularity of singular schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Lyashko-Looijenga mapping associates to a polynomial (i.e. its coefficients) the set of critical values. This map is holomorphic and maps real points to real points.
Let \(X \subset \mathbb{C}^ \mu\) be the subset of the base of the miniversal deformation of a simple singularity corresponding to polynomials with \(\mu\) distinct critical values. The restriction of the Lyashko-Looijenga map to \(X\) is a finite-sheeted covering. The multiplicity of the Lyashko- Looijenga covering restricted to a connected component of the complement in \(\mathbb{R}^ \mu\) is expressed in terms of invariants of this component.
For \(A_ \mu\)-singularities an algorithm to compute these invariants is described. simple singularities; miniversal deformation; Lyashko-Looijenga mapping Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Singularities in algebraic geometry, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Deformations of complex singularities; vanishing cycles, Coverings of curves, fundamental group, Local complex singularities On the real preimages of a real point under the Lyashko-Looijenga covering for simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper describes the relative Hilbert Chow morphism for a flat projective family \(\pi: X \to B\) of generically nonsingular curves which are at worst nodal over an arbitrary irreducible base \(B\). The relative Hilbert Chow morphism is the cycle map \(c_m: X^{[m]}_B \to X^{(m)}_B\), where \(X^{[m]}_B\) is the relative Hilbert scheme of \(m\) points and \(X^{(m)}_B\) is the relative symmetric product. The main theorem states that \(c_m\) is equivalent to the blowing up of the discriminant locus \(D^m \subset X^{(m)}_B\). The author works over the complex numbers and uses Serre's GAGA principle, constructing a local analytic model \(H\) for \(X^{[m]}_B\) and reverse engineering an ideal sheaf \(G\) in \(X^{(m)}_B\) to have syzygies so that the blow up at \(G\) maps to the pullback \(OH\) of \(H\) over the Cartesian product via a map \(\gamma\). A local analysis shows that \(\gamma\) is an isomorphism and that \(G\) defines the ordered diagonal, hence descends to the isomorphism claimed. This provides the details of a proof sketched in the author's earlier paper [in: Projective varieties with unexpected properties. A volume in memory of Giuseppe Veronese. Proceedings of the international conference ``Varieties with unexpected properties'', Siena, Italy, June 8--13, 2004. Berlin: Walter de Gruyter. 361--378 (2005; Zbl 1186.14027)].
In the second half of the paper the author uses the local model \(H\) to glean information about the singularity stratification of \(X^{[m]}_B\), specifically the structure of certain node polyscrolls he used earlier [Asian J. Math. 17, No. 2, 193--264 (2013; Zbl 1282.14097)] to develop an intersection calculus for the Hilbert scheme. This extends the intersection theory and enumerative geometry of a single smooth curve developed by \textit{I. G. Macdonald} [Topology 1, 319--343 (1962; Zbl 0121.38003)] to families with at worst nodal singularities, extending work of \textit{E. Cotterill} [Math. Z. 267, No. 3--4, 549--582 (2011; Zbl 1213.14064)]. nodal curves; relative Hilbert-Chow morphism; enumerative geometry; node scrolls Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Singularities of curves, local rings Structure of the cycle map for Hilbert schemes of families of nodal 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 types of simple singularities which can occur for functions from \(\mathbb R^3\) to \(\mathbb R\) consist of two infinite sequences and three special cases. The zero sets of some of these singularities are presented here. algebraic varieties; implicit surfaces; singularity theory Surfaces and higher-dimensional varieties, Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Graphical methods in numerical analysis, Computational aspects of algebraic surfaces Zero sets of simple singularities of functions on \(\mathbb R^3\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a projective \(K3\) surface and \(S^{[k]}\) be the Hilbert scheme of \(k\) points on \(S\), where \(k \geq 2\). The author proves that any dominant rational map from \(S^{[k]}\) that is not generically finite has a rationally connected image. As an application the author gives an alternative proof of \textit{C. Voisin}'s theorem that any symplectic involution of a projective \(K3\) surface \(S\) acts as the identity on \(\mathrm{CH}_0(S)\) [Doc. Math., J. DMV 17, 851--860 (2012; Zbl 1276.14012)]. Using theorems of Graber-Harris-Starr [\textit{T. Graber} et al., J. Am. Math. Soc. 16, No. 1, 57--67 (2003; Zbl 1092.14063)], the author reduces his main theorem to the equivalent formulation that the image of a dominant rational map from \(S^{[k]}\) that is not generically finite is either a point or a uniruled variety. punctual Hilbert schemes; \(K3\) surfaces; rational connectedness 9. H.-Y. Lin, Rational maps from punctual Hilbert schemes of K3 surfaces, preprint (2013); arXiv:1311.0743. Rational and birational maps, \(K3\) surfaces and Enriques surfaces, Cycles and subschemes Rational maps from punctual Hilbert schemes of \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(H_{d,g}\) be the Hilbert scheme of locally Cohen-Macaulay space curve of degree \(d\) and genus \(g\). The genus \(g=(d-3)(d-4)/2\) is the biggest value for which the study of \(H_{d,g}\) is nontrivial. According to the techniques developed by \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184-185 (1990; Zbl 0717.14017)], in the present paper the author studies \(H_{d,g}\) in the case \(g=(d-3)(d-4)/2\). Hilbert scheme; locally Cohen-Macaulay space curve; degree; genus Aït Amrane, S, Sur le schéma de Hilbert des courbes de degré \(d\) et genre\((d-3)(d-4)/2\) de \(\mathbf{P}^3_k\), C. R. Acad. Sci. Paris Sér. I Math., 326, 851-856, (1998) Special algebraic curves and curves of low genus, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Parametrization (Chow and Hilbert schemes) On the Hilbert scheme of curves of degree \(d\) and genus \(\frac{(d-3)(d-4)}{2}\) | 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 We determine the automorphism group of the Hilbert scheme of two points on a generic projective \(K3\) surface of any polarization. We obtain in particular new examples of Hilbert schemes of points having non-natural non-symplectic automorphisms. The existence of these automorphisms depends on solutions of Pell's equation. Boissière, S., Cattaneo, A., Nieper-Wisskirchen, M., Sarti, A.: The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface (2014). arXiv:1410.8387 Parametrization (Chow and Hilbert schemes), Automorphisms of surfaces and higher-dimensional varieties, \(K3\) surfaces and Enriques surfaces The automorphism group of the Hilbert scheme of two points on a generic projective \(K3\) surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{D. S. Keeler} and the authors [Duke Math. J. 126, No. 3, 491--546 (2005; Zbl 1082.14003)] introduced the concept of naïve blowup of a scheme. The algebras studied in this work were obtained by blowing up in a single point, and they proved that these objects had properties quite unlike their commutative counterparts. In a companion paper [A class of noncommutative projective surfaces, in press], the authors use the concept of naïve blowups to classify a large class of noncommutative algebras. This classification requires one to naïvely blow up any suitably general zero-dimensional subscheme. The aim of the paper is therefore to study this more general case.
The underlying data for a naïve blowup is as follows. Fix an integral scheme \(X\), with automorphism \(\sigma\) and a \(\sigma\)-ample sheaf \(\mathcal{L}\). Let \(Z=Z_{\mathcal{I}}\subset X\) be a zero-dimensional subscheme, with defining ideal \(\mathcal I\subset\mathcal{O}_X\). The \textit{bimodule algebra} is given by \(\mathcal{R}=\mathcal{R}(X,Z,\mathcal{L},\sigma)=\mathcal{O}_X \oplus\mathcal{R}_1 \oplus\mathcal{R}_2 \oplus\cdots,\) where \(\mathcal{R}_n=\mathcal{L}_n \otimes_{\mathcal{O}_X}\mathcal{I}_n,\) for \(\mathcal{L}_n=\mathcal{L} \otimes\sigma^\ast\mathcal{L} \otimes\cdots \otimes(\sigma^{n-1})^\ast\mathcal{L}\), and \(\mathcal{I}_n=\mathcal{I}\cdot\sigma^\ast\mathcal{I} \cdots(\sigma^{n-1})^\ast\mathcal{I}.\) \textit{The naïve blowup algebra of \(X\) at \(Z\)} is then the algebra of sections
\[
R=R(X,Z,\mathcal{L},\sigma)= \text{H}^0(X,\mathcal{R})=k\oplus \text{H}^0(X,\mathcal{R}_1) \oplus\text{H}^0(X,\mathcal{R}_2) \oplus\cdots.
\]
If \(Z\) is the empty set, \(R\) is simply the twisted homogeneous coordinate ring \(B(X,Z,\mathcal{L},\sigma)\) which is essential in noncommutative projective geometry. If \(B=B(X,Z,\mathcal{L},\sigma)\) for a \(\sigma\)-ample invertibele sheaf \(\mathcal{L}\), then \(B\) has extremely pleasant properties, among others: (a) \(B\) is \textit{strongly noetherian} (b) \(qgr-B\cong \text{coh} X\) (c) When \(B\) is generated in degree one, the set of point modules for \(B\), both in \(gr-B\) and in \(qgr-B\), is parametrized by the scheme \(X\). (d) \(B\) has balanced dualizing complex.
In contrast, the naïve blowup algebra \(R(X,Z,\mathcal{L},\sigma)\) for \(Z\neq\emptyset\) has properties quite unlike those just mentioned. These properties are the main object of this article and form the main theorem. Its proof is the main body of the article, and the rest is concerned with applying this theorem to get a deeper understanding of the properties of \(\mathcal{R}\) and \(R\). First of all, the authors prove that the properties described by the main theorem is not exceptional. As soon as \(Y\) has at least one critically dense \(\sigma\)-orbit each noetherian connected graded (cg) subalgebra of \(k(Y)[t,t^{-1};\sigma]\) that is generated in degree one inherit the properties.
Given a naïve blowup algebra \(R=R(X,Z,\mathcal{L},\sigma)\). The \textit{maximal right torsion extension} of \(R\) is defined to be the ring \(T=T(R)=\{x\in Q(R):xR_{\geq n}\subseteq R\text{ for some } n\geq 0\}\). If \(R\) is the naïve blowup algebra at a single point then \(T(R)/R\) is always finite dimensional. In contrast, if one blows up at multiple points then \(T(R)/R\) can be infinite dimensional. This implies that naïve blowup algebras can have very nonsymmetric properties. To prove the main theorem, and to describe the maximal right torsion extension of a naïve blowup algebra, the authors pass to a larger class of algebras, called generalized naïve blowup algebras. The last main result of the article shows that the differences between the categories \(qgr-R\) and \(\text{coh}-X\) are quite subtle.
The article depends heavily on the article by Keeler, Stafford and Rogalski where the concept of naïve blowup algebras where introduced. The article is clearly important for the study of noncommutative geometry, and introduces a lot of new concepts useful for applications in this field. naive blowup; bounding function D. Rogalski and J. T. Stafford, Naïve noncommutative blowups at zero-dimensional schemes, J. Algebra 318 (2007), no. 2, 794 -- 833. Noncommutative algebraic geometry Naïve noncommutative blowups at zero-dimensional schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The stratification associated with the number of generators of the ideals of the punctual Hilbert scheme of points on the affine plane has been studied since the 1970s. In this paper, we present an elegant formula for the E-polynomials of these strata. Symmetric functions and generalizations, Polynomial rings and ideals; rings of integer-valued polynomials, Parametrization (Chow and Hilbert schemes) A note on the E-polynomials of a stratification of the Hilbert scheme of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The thesis under review may be viewed as a step forward in understanding the semi-universal deformation of complex analytic isolated complete intersection singularities. One main question is: given such a singularity, which singularities does it deform to ? The answer depends on a detailed knowledge of the geometry of the deformation, and the author does an exhaustive study for simple singularities on space curves. These were classified by \textit{M. Giusti} [Singularities, Summer. Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part I, 457-494 (1983; Zbl 0525.32006)].
The main result is that for the \(S_{\mu}, T_ 7, T_ 8\) and \(T_ 9\) singularities (Giusti's notation), the base space of the semi-universal deformation is isomorphic to a quotient X/W of a torus embedding X by a Weyl group W, such that the discriminant of the group action maps to the discriminant of the deformation. This parallels other known descriptions of the discriminant, for example as done by \textit{E. Brieskorn}, in Actes Congr. intern. Math. (1970), part 2, 279-284 (1981; Zbl 0223.22012) for simple hypersurface singularities [see also \textit{P. Slodowy}, ''Simple singularities and simple algebraic groups,'' Lect. Notes Math. 815 (1980; Zbl 0441.14002)].
The novelty is the introduction of torus embeddings which are constructed from extended Dynkin diagrams corresponding to generalized root systems. This construction, based on recent work by \textit{E. Looijenga} [Invent. Math. 61, 1-32 (1980; Zbl 0436.17005)], is done from a general point of view and should be applicable to other situations. The explicit study of the deformations uses classical algebraic geometry, e.g. results on families of hyperelliptic curves and del Pezzo surfaces. semi-universal deformation of complex analytic isolated complete; intersection singularities; simple singularities on space curves; torus embeddings; Dynkin diagrams; root systems; semi-universal deformation of complex analytic isolated complete intersection singularities Deformations of singularities, Singularities of curves, local rings, Deformations of complex singularities; vanishing cycles, Formal methods and deformations in algebraic geometry, Complete intersections Torus embeddings and deformations of simple singularities and space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{G. Kempf} [Ann. Math. (2) 98, 178--185 (1973; Zbl 0275.14023)] proved that the theta divisor of a smooth projective curve \(C\) has rational singularities. In this paper we estimate the dimensions of the jet schemes of the theta divisor and show that all these schemes are irreducible. In particular, we recover Kempf's theorem in this way. For general projective smooth curves, our method also gives a formula for the log canonical threshold of the pair \((\text{Pic}^d (C), W^r_d(C))\). Brill-Noether loci; log canonical threshold Arcs and motivic integration, Singularities in algebraic geometry, Special divisors on curves (gonality, Brill-Noether theory) Jet schemes and singularities of \(W^r_d(C)\) loci | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 infinite field, whose characteristics is neither \(2\) nor \(3\). The paper deals with smoothable Gorenstein \(K\)-points in a punctual Hilbert scheme, getting the following main results
\begin{itemize}
\item every \(K\)-point defined by local Gorenstein \(K\)-algebras with Hilbert function \((1,7,7,1)\) is smoothable;
\item the Hilbert scheme \(\mathrm{Hilb}_{16}^7\) has at least five irreducible components.
\end{itemize}
A new elementary component in \(\mathrm{Hilb}_{15}^7\) is found, starting from the study of \(\mathrm{Hilb}_{16}^7\), The problem is studied via properties double-generic initial ideals and of marked schemes. We remark that the considered problem is deeply related to the study of the irreducibility of the Gorenstein locus in a Hilbert scheme and, more in general, of the irreducibility of a Hilbert scheme, a relevant and open question. Gorenstein algebra; Hilbert scheme; strongly stable ideal; marked basis Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Smoothable Gorenstein points via marked schemes and double-generic initial 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 Here we use a paper of \textit{T. Mignon} [J. Algebr. Geom. 10, No.~2, 281--297 (2001; Zbl 0987.14019)] to extend Severi's theory of nodal plane curves to the case some (but not too many) of the prescribed singularities are ordinary multiple points with arbitrary multiplicity. Singularities of curves, local rings, Plane and space curves, Families, moduli of curves (algebraic) Plane curves with ordinary singularities and many nodes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 tautological bundle of the Hilbert scheme of points on a smooth projective surface is a central object of algebraic geometry and its application. Nevertheless, as is pointed out in the introduction of this paper, there are few studies on positivity of the tautological bundles. The paper under review gives an explicit criterion of tautological bundles on \(K\)-trivial surfaces to be big and nef. The strategy of derivation is the usage of the Segre integral formula established recently in [\textit{A. Marian} et al., J. Eur. Math. Soc. (JEMS) 24, No. 8, 2979--3015 (2022; Zbl 1495.14006)]. The main text begins with the study of \(K3\) surfaces in \S 2, and turns to other \(k\)-trivial surfaces in \S 3. The same strategy works for other smooth projective surfaces, including blowups of \(K3\), minimal surfaces of general type, and also for the punctual Quot schemes of curves, presented in \S 4. Hilbert scheme; quot scheme; tautological bundles Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Big and nef tautological vector bundles over the Hilbert scheme of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{F. Severi} claimed in the 1920s that the Hilbert scheme \({\mathcal H}_{d,g,r}\) of smooth irreducible non-degenerate curves \(C \subset \mathbb P^r\) of degree \(d\) and genus \(g\) is irreducible for \(d \geq g+r\) [Vorlesungen über algebraische Geometrie. Übersetzung von E. Löffler. Leipzig u. Berlin: B. G. Teubner (1921; JFM 48.0687.01)]. \textit{L. Ein} proved Severi's claim for \(r=3\) and \(r=4\) [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 469--478 (1986; Zbl 0606.14003); Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], but there are counterexamples of various authors for \(r \geq 6\).
It has been suggested by \textit{C. Ciliberto} and \textit{E. Sernesi} [in: Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co. 428--499 (1989; Zbl 0800.14002)] that Severi intended irreducibiity of the Hilbert scheme \({\mathcal H}^{\mathcal L}_{d,g,r} \subset {\mathcal H}_{d,g,r}\) of curves whose general member is linearly normal: indeed, the counterexamples above arise from families whose general member is not linearly normal.
Here the authors prove irreducibility for \(g+r-2 \leq d \leq g+r\) (the Hilbert scheme is empty for \(d > g+r\) by Riemann-Roch) and for \(d=g+r-3\) under the additional assumption that \(g \geq 2r+3\). This extends work of \textit{C. Keem} and \textit{Y.-H. Kim} [Arch. Math. 113, No. 4, 373--384 (2019; Zbl 1423.14028)]. Hilbert scheme; linear series; linearly normal curves Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) On the Hilbert scheme of linearly normal curves in \(\mathbb{P}^r\) of relatively high degree | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle of the threefold satisfies certain positivity conditions, we relate the DT invariants to Carlsson-Okounkov formulas for the ``twisted Euler number'' of the punctual Hilbert schemes of nonsingular surfaces, and conclude they have a modular property. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Parametrization (Chow and Hilbert schemes) Donaldson-Thomas invariants, linear systems and punctual Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(0\in C\) be a germ of a smooth curve and \(f_U:Y_U\rightarrow U=C\setminus \{0\}\) a family of smooth surfaces of general type over \(U\). Then \(f_U\) can be completed in a unique way to a family \(f:Y\rightarrow C\) such that \(\omega_{Y/C}^{[k]}\) is invertible and ample for some \(k>0\) and the central fiber \(X=f^{-1}(0)\) is a stable surface [see \textit{V. Alexeev}, Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 515--536 (2006; Zbl 1102.14023) and \textit{J. Kollár} and \textit{N. I. Shepherd-Barron}, Invent. Math. 91, No. 2, 299--338 (1988; Zbl 0642.14008)]. Thus the moduli space of surfaces of general type can be compactified by adding stable surfaces and it is important to know which stable surfaces are smoothable (this is related with the minimal model program). The purpose of this paper is to study the deformation theory of schemes with non-isolated singularities and to write some smoothability and nonsmoothability criteria. minimal model program; deformation theory; semi-log-canonical singularities; stable surface Tziolas N.: Smoothings of schemes with nonisolated singularities. Michigan Math. J. 59(1), 25--84 (2010) Formal methods and deformations in algebraic geometry, Families, moduli of curves (algebraic), Deformations of singularities, Fibrations, degenerations in algebraic geometry Smoothings of schemes with nonisolated 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 short text takes up a survey talk on recent results about the Riemann-Hilbert correspondence in case of irregular singularities in higher dimension. It explains the new features of Hodge theory in this wild context, where irregular singularities are present, and exemplifies applications to Landau-Ginzburg potentials. Structure of families (Picard-Lefschetz, monodromy, etc.), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain, Transcendental methods, Hodge theory (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects) Riemann-Hilbert correspondence, irregular singularities and Hodge theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 if \(X\subset {\mathbb P}^r\) is any 2-regular scheme (in the sense of Castelnuovo-Mumford) then \(X\) is small. This means that if \(L\) is a linear space and \(Y:= L\cap X\) is finite, then \(Y\) is linearly independent in the sense that the dimension of the linear span of \(Y\) is \(\deg Y+1\). The converse is true and well-known for finite schemes, but false in general. The main result of this paper is that the converse, ``small implies 2-regular'', is also true for reduced schemes (algebraic sets). This is proven by means of a delicate geometric analysis, leading to a complete classification: we show that the components of a small algebraic set are varieties of minimal degree, meeting in a particularly simple way. From the classification one can show that if \(X\subset {\mathbb P}^r\) is 2-regular, then so is \(X_{\text{red}}\), and so also is the projection of \(X\) from any point of~\(X\). Our results extend the Del Pezzo-Bertini classification of varieties of minimal degree, the characterization of these as the varieties of regularity 2 by \textit{D. Eisenbud} and \textit{S. Goto} [J. Algebra 88, No. 1, 89--133 (1984; Zbl 0531.13015)], and the construction of 2-regular square-free monomial ideals by \textit{R. Fröberg} [in: Topics in Algebra, Part 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw 1988, Banach Cent. Publ. 26, 57--70 (1990; Zbl 0741.13006)]. Eisenbud, D., Green, M., Hulek, K., Popescu, S.: Small schemes and varieties of minimal degree. Am. J. Math. 128(6), 1363--1389 (2006) Projective techniques in algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Small schemes and varieties of minimal degree | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 very beautiful connection was found by Brieskorn and Slodowy between simple singularities (rational double points) and simple Lie algebras. In the paper under review the author seeks an analogue of this connection in the class of 1-dimensional simple singularities which are complete intersection in \({\mathbb{C}}^ 3.\) These singularities are classified by \textit{M. Giusti} [Singularities, Summer Inst., Arcata/Calif. 1987, Proc. Symp. Pure Math. 40, Part 1, 457-494 (1983; Zbl 0525.32006)] and they are labelled \(S_{\mu} (\mu =5,6,7,...)\), \(T_ 7,T_ 8,T_ 9,U_ 7,U_ 8,U_ 9,W_ 9,W_{10},Z_ 9,Z_{10}\). The author associate to \(S_{\mu}^ a \)diagram called \(D=D_ k[*]\), \(k=\mu -1\) which is constructed from the Dynkin diagram \(D_ k\) by adding one distinguished vertex. They are connected as follows: The diagram determines a torus embedding \({\mathcal X}(D)\) and the parameter space of the semi-universal deformation of the singularity is identified with \({\mathcal X}(D)/W_ 2(D)\), where \(W_ 2(D)\) is an analogue of the Weyl group, and this identification respects the discriminants of the both spaces. A similar result is proved for \(T_{\mu}\) and \(E_{\mu}[*]\) \((\mu =6,7,8)\). extended Dynkin diagram; 1-dimensional simple singularities; complete intersection; torus embedding; semi-universal deformation K. Wirthmüller , Torus embeddings and deformations of simple singularities of space curves , Acta Math. 157 (1986), 159-241. Singularities of curves, local rings, Deformations of complex singularities; vanishing cycles, Deformations of singularities, Projective techniques in algebraic geometry, Embeddings in algebraic geometry Torus embeddings and deformation of simple singularities of space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper concerns the resolution of Gorenstein artinian quotients \(A\) of a polynomial ring \(K[x_1,\dots,x_n]\). For \(n=3\), the structure theorem of Buchsbaum and Eisenbud provides a bound for the Betti numbers of the quotient with preassigned Hilbert function. The authors discover a construction which, starting with some configurations of points in \({\mathbb P}^2\), gives examples of such quotients having maximal Betti numbers.
When the number of variables increases, few general results on the resolutions of quotients are known. The authors present here an extension of their previous construction to points in \({\mathbb P}^3\), which provides Gorenstein artinian quotients of \(K[x_1,\dots,x_4]\) having large Betti numbers. They conjecture that these examples have maximal Betti numbers, with respect to their Hilbert function. Gorenstein ideals of codimension 4; resolution of Gorenstein artinian quotients; polynomial ring; preassigned Hilbert function; Betti numbers Pegg, DT, Complement of the Hamiltonian, Phys. Rev. A, 58, 4307, (1998) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Polynomial rings and ideals; rings of integer-valued polynomials, Complete intersections The Hilbert function and the minimal free resolution of some Gorenstein ideals of codimension 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 In the theory of graded Gorenstein artinian quotients of \(S=k [x,y,z]\) (the codimension 3 Gorenstein \(k\)-algebra), where \(k\) is a field, there is no theorem analogous to that of Buchsbaum and Eisenbud for graded Gorenstein artinian quotients of \(k[x_1,\dots,x_n]\) when \(n\geq 4\). It is known that Gorenstein algebras achieving all the possible collections of graded Betti numbers could be constructed. In this paper the authors give a construction for Gorenstein ideals of codimension 4 which the authors believe have maximal graded Betti numbers for their Hilbert function. graded Gorenstein artinian quotients; codimension 4; maximal graded Betti numbers; Hilbert function Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Complete intersections The Hilbert function and the minimal free resolution of some Gorenstein ideals of codimension 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 A boundary singularity is a singularity of a function on a manifold with boundary. The relation between the characteristic polynomial of the mono\-dromy and the Poincaré series of the ambient hypersurface singularity for such a singularity is investigated. The simple boundary singularities arise from simple hypersuface singularities, and there is a generalization of the McKay correspondence for these cases. For seven of the 12 exceptional uni\-nodal boundary singularities a direct relation between the Poincaré series of the ambient singularity and the characteristic polynomial of the monodromy is given. boundary singularities; unimodal singularities; simple singularities; monodromy; McKay correspondence W. Ebeling and S. M. Gusein-Zade, On indices of 1-forms on determinantal singularities, Proceedings of the Steklov Institute of Mathematics 267 (2009), 113--124. Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local complex singularities Poincaré series and monodromy of the simple and unimodal boundary singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The classification by \textit{G. M. Greuel} and \textit{H. Kröning} in Math. Z. 203, No. 2, 339-354 (1990; Zbl 0715.14001) yields a list of equations of type \(A\), \(D\) and \(E\), which is reproduced in an annex to this paper. The author shows how to recognise these singularities from their weights.
Let \(X=\text{Spec} R\), with \(R\) a complete Cohen-Macaulay local \(k\)- algebra with algebraically closed residue field \(k\) of dimension \(n\geq 1\). Then \(X\) is of type \(A\), \(D\) or \(E\), if and only if \(X\) is semi- quasihomogeneous of some weight \(w\) \((=(w_ 1,\dots,w_ n))\) such that \(w_ 0+\cdots+w_ n<n/2\). For \(n>1\) the proof uses the characterisation of the \(ADE\) singularities as absolutely isolated double points, which is here established in positive characteristic for \(n>2\). \(ADE\) singularities; weights; positive characteristic M. Roczen, Recognition of simple singularities in positive characteristic, Math. Z., 210 (1992), 641-653. Singularities in algebraic geometry, Finite ground fields in algebraic geometry Recognition of simple singularities 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 Let k be a field and \(A=k[M]\) the monoid algebra where M is a cancellative, abelian monoid with unit element 0. The Poincaré series for A is \(P_ A(z)=\sum_{m\geq 0}\#(\dim_ kTor^ A_ m(k,k))z^ m.\quad Suppose\) M carries a simple, connected, standard, monoidal gradation, i.e. a monoid homomorphism \(d:\quad M\to N,\) such that \(d^{- 1}(0)=0\), \(d^{-1}(v)\) finite for all v and M generated by \(d^{- 1}(1)\). Then the Hilbert series for A with respect to d is \(H_ A(z)=\sum_{m\geq 0}(d^{-1}(m))z^ m.\quad In\) this article monoids M for which \(H_ A(z)P_ A(-z)=1\) are studied. The main means for the study is the characterization of monoidal homology due to Laudal, the monoidal version of the Möbius inversion formula due to Lallement and the study of Cohen-Macaulay finite posets due to Stanley, Baclawski etc. A corollary is a former result by the author and N. Manolache that Segre products of Veronese subrings of a polynomial ring satisfy the formula \(H_ A(z)P_ A(-z)=1\). algebraic singularity with linear resolution; Fröberg rings; monoid algebra; Poincaré series; Hilbert series; monoidal homology Homological methods in commutative ring theory, Polynomial rings and ideals; rings of integer-valued polynomials, Singularities in algebraic geometry Monoidal algebraic singularities with linear 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 The author announces the following result: Let \(i: Z\to Y\) be the inclusion of an irreducible complex subspace into a compact complex manifold. Assume that Z has ordinary singularities, i.e. the total space X of the normalization \(\nu\) : \(X\to Z\) is smooth and \(f=i\circ \nu: X\to Y\) is infinitesimally stable. Then the functors of a) locally trivial embedded deformations of Z in Y and of b) deformations of the map \(f: X\to Y\) have semiuniversal objects and are equivalent. analytic subvariety with ordinary singularities; locally trivial displacements; deformations S. Tsuboi: Locally trivial displacements of analytic subvarieties with ordinary singularities. Proc. Japan Acad., 61A, 270-273 (1985). Deformations of complex structures, Complex singularities, Modifications; resolution of singularities (complex-analytic aspects), Singularities of surfaces or higher-dimensional varieties, Structure of families (Picard-Lefschetz, monodromy, etc.) Locally trivial displacements of analytic subvarieties with ordinary singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0619.00007.]
This note gives an outline of the proof of the following. Let R be the local ring of a reduced plane curve singularity X. Then the torsion free rank 1 R-modules fall into finitely many 0-\ and 1-parameter families of isomorphism classes if and only if X is strictly unimodal (for contact equivalence), i.e. has modality \(\leq 2\) for right equivalence. - If X is not strictly unimodal, it deforms to at least one of \(J_{4,0}, X_{2, 0}, Z\) \(1_{2,0}\) or \(N_{16}\), and 2-parameter families can be explicitly constructed in these cases.
The remaining cases are settled by direct calculation. First it is shown that a torsion free rank 1\ R-module admits a special type of generating system (called short systems), and then isomorphism is computed in the context of these systems. plane curve singularity; unimodal Schappert, A. : A characterization of strict unimodular plane curve singularities , in: Singularities, Representation of Algebras, and Vector Bundles, Lambrecht 1985 (Eds.: Greuel, G.-M.; Trautmann, G.). Lecture Notes in Math., Vol. 1273, Springer, Berlin- Heidelberg-New York (1987) pp. 168-177. Singularities of curves, local rings, Singularities in algebraic geometry, Multiplicity theory and related topics A characterisation of strictly unimodular plane curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field, let \(\gamma\) be an arc of a \(k\)-variety \(X\), and assume that \(\gamma\) does not factor through the singular locus of \(X\). The Drinfeld-Grinberg-Kazhdan Theorem states that the formal neighborhood of \(\gamma\) in the arc scheme of \(X\) is isomorphic to the product of an infinite dimensional formal disc and a formal neighborhood in a finite type \(k\)-scheme. The latter formal neighborhood has been interpreted as a finite dimensional model of the formal neighborhood of \(\gamma\). The authors prove a generalization of the Drinfeld-Grinberg-Kazhdan theorem where \(X\) is replaced with a topologically finite type formal \(k[[T]]\)-scheme \(\mathcal{X}\), and they study consequences related to singularity theory.
In the case where \(\mathcal{X}\) is the completion of an affine \(k\)-variety, their proof gives an algorithm for computing the resulting finite dimensional model. In addition, they prove that these finite dimensional models are compatible with base change by separable extensions of \(k\), and they also prove a factoring statement involving the truncation morphisms from arc schemes to jet schemes.
The authors also present a proof, communicated to them by O. Gabber, of a cancellation theorem, which in particular allows them to define unique minimal finite dimensional models for an arc's formal neighborhood. The authors also use finite dimensional models to define an invariant they call the absolute nilpotency index of a singularity, and they prove that this is a formal invariant of the singularity. arc scheme; formal neighborhood Bourqui D., The Drinfeld--Grinberg--Kazhdan Theorem for Formal Schemes and Singularity Theory (2015) Arcs and motivic integration, Singularities in algebraic geometry The Drinfeld-Grinberg-Kazhdan theorem for formal schemes and singularity theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The classical Hilbert scheme introduced by \textit{A. Grothendieck} [Sém. Bourbaki 1960/1961, Exp. 221 (1961; Zbl 0236.14003)], parametrizes subschemes of \(\mathbb P^r_k, k\) a field, with a given Hilbert polynomial.
A toric variety is a variety parametrized by a finite number of monomials \({\mathbf t}^{a_i} = t_1^{a_1}\cdots t_d^{a_d}, i = 1,\ldots,n,\) in the polynomial ring \(k[t_1,\ldots,t_d],\) where \({\mathcal A} = \{a_1,\ldots,a_n\}\) denotes a subset of \(\mathbb N^d \setminus 0\) of \(n\) different vectors. Let \(S = k[x_1,\ldots, x_n]\) denote the polynomial ring in the variables \(x_1,\ldots,x_n\) of degree \(a_1,\ldots, a_n\) respectively. The toric ideal \(I_{\mathcal A}\) is the kernel of the natural map \(S \to k[t_1,\ldots, t_d], x_i \mapsto {\mathbf t}^{a_i}, i = 1,\ldots, n,\) which is a prime \(\mathbb N^d\)-graded ideal.
A homogeneous ideal \(M \subset S\) is called \(\mathcal A\)-graded if \(\dim_k (S/M)_b = 1\) if \(b \in \mathbb N \mathcal A\) and \(0\) otherwise. That is, \(S/M\) has the same multigraded Hilbert function as the toric ring \(S/I_{\mathcal A}.\) The authors construct the toric Hilbert scheme \(H_{\mathcal A}\) that parametrizes all ideals with the same multigraded Hilbert function as \(S/I_{\mathcal A},\) satisfying a universality property. It follows that there exists exactly one component containing the point \([I_{\mathcal A}].\) If char\((k)= 0,\) then this component is reduced and so the point \([I_{\mathcal A}]\) on \(H_{\mathcal A}\) is smooth.
Moreover, in the case of codim\((S/I_{\mathcal A}) = 2\) the authors prove the following additional results:
(1) The toric Hilbert scheme has one component. It is the closure of the orbit of the toric ideal under the torus action.
(2) The toric Hilbert scheme is 2-dimensional and smooth. Note that there is no restriction on the characteristic of the field \(k.\)
(3) \(H_{\mathcal A}\) is the toric variety of the Gröbner fan of \(I_{\mathcal A}.\)
In an unpublished paper, \textit{B. Sturmfels} started with a different construction in order to parametrize all ideals with the same Hilbert function as \(I_{\mathcal A}\) [``The geometry of \(\mathcal A\)-graded algebras'', preprint, \texttt{http://arxiv.org/abs/math.AG/94100032}] . toric variety; multigrading Peeva I., Stillman M., Toric Hilbert schemes, Duke Math. J., 2002, 111(3), 419--449 Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Toric varieties, Newton polyhedra, Okounkov bodies Toric Hilbert schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper proves a relative Lefschetz formula for some equivariant arithmetic schemes. More specifically, let \(X\) and \(Y\) be arithmetic schemes whose generic fibers are smooth. Assume that \(X\) admits an automorphism of order \(n\), and let \(f:X\to Y\) be the composition of an \(\mu_n\)-equivariant closed immersion \(X\hookrightarrow Z\) and an \(\mu_n\)-equivariant morphism \(Z\to Y\), where \(Z\) is a regular arithmetic scheme and the \(\mu_n\)-action on \(Y\) is trivial. The main theorem (Theorem 6.1) is the Lefschetz formula for such \(f\). This answers a conjecture by \textit{V. Maillot} and \textit{D. Rössler} [in: From probability to geometry II. Volume in honor of the 60th birthday of Jean-Michel Bismut. Paris. Astérisque 328, 237--253 (2009; Zbl 1232.14016)], and it is an Arakelov-geometry analog of \textit{R. W. Thomason} [Duke Math. J. 68, No. 3, 447--462 (1992; Zbl 0813.19002)] as well as a generalization of an earlier work of the author [J. Reine Angew. Math. 665, 207--235 (2012; Zbl 1314.14049)] to a singular case. The formula takes place in the equivariant arithmetic Grothendieck groups \(\widehat {G_0}\), which are defined with respect to fixed wave front sets. The bulk of the paper consists of developing this \(\widehat {G_0}\)-theory and of proving two key results: the arithmetic concentration theorem (Theorem 5.5) and the vanishing theorem (Theorem 6.3). The paper has a thorough history of Lefschetz formula in various settings in Section 1, and Section 2 recalls necessary differential-geometric facts, such as equivariant Chern-Weil theory, equivariant analytic torsion forms, equivariant Bott-Chern singular currents, and Bismut--Ma immersion formula. fixed point formula; singular arithmetic schemes; Arakelov geometry Riemann-Roch theorems, Arithmetic varieties and schemes; Arakelov theory; heights, Group actions on varieties or schemes (quotients), Index theory and related fixed-point theorems on manifolds, Determinants and determinant bundles, analytic torsion A Lefschetz fixed point formula for singular arithmetic schemes with smooth generic fibres | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a complex simple Lie group and \(\mathfrak{g}\) the corresponding Lie algebra. Let further \(S(\mathfrak{g})\) be the \(G\)-module of polynomial functions on \(\mathfrak{g}\) and \(\mathrm{Sing}(\mathfrak{g})\) be the closed algebraic cone of singular elements in \(\mathfrak{g}\). Let \(\mathcal{L}\) be the graded ideal defining \(\mathrm{Sing}(\mathfrak{g})\) and let \(2r\) be the dimension of the \(G\)-orbit of a regular element in \(\mathfrak{g}\). Then \(\mathcal{L}_k=0\) for all \(k<r\) and there exists a \(G\)-module \(M\subset \mathcal{L}_r\) such that \(x\in \mathrm{Sing}(\mathfrak{g})\) if and only if \(f(x)=0\) for all \(f\in M\). The main result of the paper under review describes the structure of this \(G\)-module \(M\) as follows: \(M\) is shown to be multiplicity free and its irreducible components are parameterized by the set of ideals in \(\Delta_+\) of cardinality \(\mathrm{rank}(\mathfrak{g})\) in a natural way (with an explicit description of the corresponding highest weights). Lie algebra; singular element; ideal; module; highest weight Kostant, B.; Wallach, N.: On the algebraic set of singular elements in a complex simple Lie algebra, Contemp. math. 557, 215-229 (2011) Lie algebras of Lie groups, Simple, semisimple, reductive (super)algebras, Group actions on varieties or schemes (quotients), General properties and structure of complex Lie groups On the algebraic set of singular elements in a complex simple Lie algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 obtain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) the Hirzebruch homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. This result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as on a formula for the generating series of the Hirzebruch homology characteristic classes of symmetric products. We apply the same methods for the calculation of generating series formulae for the Hirzebruch classes of the push-forwards of ``virtual motives'' of Hilbert schemes of a threefold. As corollaries, we obtain counterparts for the MacPherson (and Aluffi) Chern classes of Hilbert schemes of a smooth quasi-projective variety (resp. for threefolds). For a projective Calabi-Yau threefold, the latter yields a Chern class version of the dimension zero MNOP conjecture. Hilbert scheme; symmetric product; generating series; power structure; Pontrjagin ring; motivic exponentiation; characteristic classes S.\ E. Cappell, L. Maxim, T. Ohmoto, J. Sch\"nurmann and S. Yokura, Characteristic classes of Hilbert schemes of points via symmetric products, Geom. Topol. 17 (2013), no. 2, 1165-1198. Parametrization (Chow and Hilbert schemes), Representations of finite symmetric groups, Symmetric products and cyclic products in algebraic topology, Grothendieck groups, \(K\)-theory and commutative rings, Mixed Hodge theory of singular varieties (complex-analytic aspects) Characteristic classes of Hilbert schemes of points via symmetric products | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the notion of the Gröbner cell for the Hilbert scheme of points in the plane, as well as that of the punctual Hilbert scheme is comprehensively defined. An explicit parametrization of the Gröbner cells in terms of minors of a matrix is recalled. The main core of this paper shows that the decomposition of the Punctual Hilbert scheme into Grönber cells induces that of the compactified Jacobians of plane curve singularities. As an important application of this decomposition, the topological invariance of an analog of the compactified Jacobian and the corresponding motivic superpolynomial for families of singularities is concluded. Hilbert schemes; affine plane; Grothendieck-Deligne map; Gröbner cells; zeta functions; plane curve singularities Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Plane and space curves, Exact enumeration problems, generating functions, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Jacobians, Prym varieties, Hecke algebras and their representations, Combinatorial aspects of representation theory, Braid groups; Artin groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Gröbner cells of punctual Hilbert schemes in dimension two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an \(r\)-dimensional smooth complex quasiprojective variety and denote the Hilbert scheme parametrizing zero-dimensional subschemes of length \(n\) of \(X\) by \(\text{Hilb}^nX\). A nested Hilbert scheme on \(X\) is defined to be a scheme of the form
\[
Z_{\mathbf n}(X):=\{(Z_1, Z_2,\dots, Z_m) : Z_i \in \text{Hilb}^{n_i}X
\]
and \(Z_i\) is a subscheme of \(Z_j\) if \(i < j\},\) where the symbol \(\mathbf n\) is used as a shorthand for the \(m\)-tuple \((n_1, n_2,\dots,n_m)\). If \(({\mathcal U}_1, {\mathcal U}_2,\dots, {\mathcal U}_m)\) is the universal element over the nested Hilbert scheme \(Z_{\mathbf n}(X)\), we call the scheme \({\mathcal U}_1\times_{Z_{\mathbf n}(X)}{\mathcal U}_2 \times_{Z_{\mathbf n}(X)}\cdots \times_{Z_{\mathbf n}(X)} {\mathcal U}_m\) the universal family over \(Z_{\mathbf n}(X)\). \textit{J. Cheah} [J. Algebr. Geom. 5, No. 3, 479-511 (1996)], expressed the virtual Hodge polynomials of the smooth Hilbert schemes \(\text{Hilb}^n X\) in terms of that of \(X\). In the paper under review we indicate how the arguments of the cited paper can be modified to express the virtual Hodge polynomials of all the smooth nested Hilbert schemes (and those of their universal families when \(r\geq 2\)) in terms of that of \(X\). More generally, we obtain the virtual Hodge polynomials of the schemes
\[
\begin{cases} \text{Hilb}^nX,\\ Z_{n-1,n}(X),\\ {\mathcal F}_n(X), \\ {\mathcal F}_{n-1,n}(X),\\ \{(P,Z_1,Z_2)\in X\times \text{Hilb}^{n-1}X\times \text{Hilb}^n X: \\ \qquad P \text{ lies in the support of }Z_2, Z_1 \text{ is a subscheme of }Z_2\} \\ \{(P, Q, Z) \in X \times X\times \text{Hilb}^nX: P \text{ and }Q\text{ lie in the support of }Z\}\end{cases}\tag{1}
\]
in terms of the virtual Hodge polynomial of \(X\) and those of the reduced schemes
\[
\text{Hilb}^k (\mathbb{A}^r,0) = \{Z \in \text{Hilb}^k\mathbb{A}^r : Z\text{ is supported at the origin\}}
\]
and
\[
{\mathcal Z}_{k-1,k}(\mathbb{A}^r, 0)=\{(Z_1, Z_2) \in \text{Hilb}^{k-1}(\mathbb{A}^r,0)\times \text{Hilb}^k(\mathbb{A}^r,0): Z_1\text{ is a subscheme of }Z_2\}.
\]
When \(r= 2\) or \(n\geq 3\), the virtual Hodge polynomials of the spaces listed in (1) can be given purely in terms of that of \(X\). Note that when \(r\geq 2\), this includes all the smooth nested Hilbert schemes \(Z_{\mathbf n}(X)\) and their universal families. If \(r = 1\), the schemes \(Z_{\mathbf n}(X)\) are products of symmetric powers of \(X\) and their virtual Hodge polynomials are easily determined using the formula for the virtual Hodge polynomials of symmetric powers given in the paper cited above.
From the equations of virtual Hodge polynomials, we obtain for free analogous equations of virtual Poincaré polynomials and Euler characteristics. In fact, since the Euler characteristics of the spaces \(\text{Hilb}^k(\mathbb{A}^r,0)\) and \({\mathcal Z}_{k-1,k}(\mathbb{A}^r,0)\) can be expressed in terms of the numbers of certain higher dimensional partitions, the Euler characteristics of the schemes listed in (1) are expressible in terms of these numbers and the Euler characteristic of \(X\). If \(X\) is projective, we also obtain formulae giving the Hodge (resp. Poincaré) polynomials of the smooth nested Hilbert schemes in terms of that of \(X\). virtual Hodge polynomials; nested Hilbert schemes; Poincaré polynomials; Euler characteristic J. Cheah, ''The Virtual Hodge Polynomials of Nested Hilbert Schemes and Related Varieties,'' Math. Z. 227, 479--504 (1998). Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The virtual Hodge polynomials of nested Hilbert schemes and related 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 the paper under review, the author studies particular loci of the Hilbert scheme \(\mathcal{H}\mathrm{ilb}^{r}_{n}\) of \(r\) points in the affine space \(\mathbb{A}^n\). In a previous paper [J. Commut. Algebra 3, No. 3, 349--404 (2011; Zbl 1237.14012)], the author introduced the functor \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta}_{n,k}: (k\mathrm{-Alg}) \rightarrow (\mathrm{Sets})\) that associates to any algebra \(B\) over a ring \(k\) the set of reduced Gröbner bases in the ring \(B[x_1,\ldots,x_n]\) with respect to the lexicographic order with a given standard set \(\Delta\) of \(r\) monomials. He proved that this functor is representable and represented by a locally closed subscheme of \(\mathcal{H}\mathrm{ilb}^{r}_{n}\) called Gröbner stratum. In this paper, the author studies the subfunctor \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta,\text{ét}}_{n,k}\) of \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta}_{n,k}\) that considers only the reduced Gröbner bases of ideals defining reduced points. The main result of the paper is that \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta,\text{ét}}_{n,k}\) is representable and, in the case of a ring \(k\) such that \(\mathrm{Spec}\, k\) is irreducible, all the connected components of the representing scheme have the same dimension. Moreover, the number of connected components and their dimension are nicely described in terms of combinatorial properties of the standard set \(\Delta\). Hilbert scheme of points; Gröbner stratum; lexicographic order; reduced points Mathias Lederer (2014). Components of Gröbner strata in the Hilbert scheme of points. \textit{Proc. Lond. Math. Soc}. (3) \textbf{108}(1), 187-224. ISSN 0024-6115. URL http://dx.doi.org/10.1112/plms/pdt018. Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomials over commutative rings, Enumerative problems (combinatorial problems) in algebraic geometry Components of Gröbner strata in the Hilbert scheme of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0584.00014.]
Suppose (A,\({\mathfrak M})\) is the local ring of a reduced curve C at an ordinary singular \(point\quad p.\) Let G(A) denote the associated graded ring of A, and let \(\{a_ i\}\) be the Hilbert function of G(A). If G(A) is a reduced ring, then p is said to be ''embedded in its tangent cone''.
In section one, we show that if \(a_ i=\min \{e(A),\left( \begin{matrix} \nu ({\mathfrak M}_ i)+i\\ \nu ({\mathfrak M}_ i)\end{matrix} \right)\}\), where e(A) is the multiplicity of A, and \(\nu\) (\({\mathfrak M})\) is the number of generators of \({\mathfrak M}\), then p is embedded in its tangent cone. This always occurs if C is a plane curve. However, p is not, in general, embedded in its tangent cone.
We show (section three) that for n sufficiently large, \((*)\quad a_ n=e(A)+b_ n-b_{n+1}\) where \(\{b_ i\}\) is the Hilbert function of G(A)/nil G(A). Finally we use (*) to give examples of 1-dimensional rings with (temporarily) decreasing Hilbert function. ordinary singular point of reduced curve; Hilbert function Singularities of curves, local rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Multiplicity theory and related topics, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Ordinary singularities of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We compute the intersection number between two cycles \(A\) and \(B\) of complementary dimensions in the Hilbert scheme \(H\) parameterizing subschemes of given finite length \(n\) of a smooth projective surface \(S\). The \((n+1)\)-cycle \(A\) corresponds to the set of finite closed subschemes the support of which has cardinality 1. The \((n-1)\)-cycle \(B\) consists of the closed subschemes the support of which is one given point of the surface. Since \(B\) is contained in \(A\), indirect methods are needed. The intersection number is \(A.B=(-1)^{n-1}n\), answering a question by H. Nakajima. punctual Hilbert scheme; intersection numbers Ellingsrud G., Strømme S.A.: An intersection number for the punctual Hilbert scheme of a surface. Trans. Amer. Math. Soc. 350, 2547--2552 (1999) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Parametrization (Chow and Hilbert schemes) An intersection number for the punctual Hilbert scheme of a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives a upper bound and a lower bound for the invariant \(\delta(f)\) where \(f\) is an analytic function with an isolated critical point at the origin and \(\delta(f)\) is the maximal possible number of the Morse points with the same critical value, into which the singularity of \(f\) can be decomposed. decomposition of hypersurface singularities; generic hyperplane section; isolated critical point; Morse points Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Singularities in algebraic geometry Some estimates for a decomposition of hypersurface singularities into the simplest ones | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the structure constants of the class algebra \(R_{\mathbb Z}(\Gamma_n)\) of the wreath products \(\Gamma_n\) associated to an arbitrary finite group \(\Gamma\) with respect to the basis of conjugacy classes. We show that a suitable filtration on \(R_{\mathbb Z}(\Gamma_n)\) gives rise to the graded ring \({\mathcal G}_\Gamma(n)\) with non-negative integer structure constants independent of \(n\) (some of which are computed), which are then encoded in a Farahat-Higman ring \({\mathcal G}_\Gamma\). The real conjugacy classes of \(\Gamma\) come to play a distinguished role and are treated in detail in the case when \(\Gamma\) is a subgroup of \(\text{SL}_2({\mathbb C})\). The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface \(X\). wreath products; class algebras; Hilbert schemes Wang, W.: The Farahat-Higman ring of wreath products and Hilbert schemes. Adv. Math. 187, 417--446 (2004) Frobenius induction, Burnside and representation rings, Parametrization (Chow and Hilbert schemes), Classical real and complex (co)homology in algebraic geometry, Extensions, wreath products, and other compositions of groups The Farahat-Higman ring of wreath products and Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a homogeous polynomial \(f\) of degree \(d\) in \(n+1\) variables with coefficients in \(\mathbb{C}\), which defines a holomorphic function germ at the origin of \(\mathbb{C}^{n+1}\), its monodromy map \(T\) is defined by \(x_j\mapsto e^{2\pi i/d}x_j\), \(j=0\), \(\ldots\), \(n\). When \(f\) is an isolated homogeneous singularity, several invariants such as the Milnor number, the characteristic polynomials of \(T^\ast\), the signature and Hodge numbers of the Milnor fibre can be computed by classical topological and algebraic methods as well as via mixed Hodge structures. The authors investigate the monodromy characteristic polynomials \(\Delta _l(t)\) as well as the relation between the monodromy zeta function and the Hodge spectrum of the singularity. In the case \(n=2\), they give a description of \(\Delta _C(t)=\Delta _1(t)\) in terms of the multiplier ideal. homogeneous singularity; log-resolution; local system; multiplier ideal; finite abelian cover; Hodge spectrum; spectrum multiplicity; monodromy zeta function Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves, Multiplier ideals, Global theory of complex singularities; cohomological properties, Mixed Hodge theory of singular varieties (complex-analytic aspects) On complex homogeneous 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 introduces and studies an algebraic object, named the \textit{Poincaré series}, attached to the sequence of multiplier ideals of any given simple complete ideal on a complex surface. Let \(R\) be a local ring at a closed point of a smooth complex variety, and let \({\mathcal P} \subset R\) be a simple complete ideal. The corresponding valuation is extracted by a simple sequence of point blowing-ups \(\pi : X \to X_0 = \text{Spec}(R)\). This gives a log-resolution of \({\mathcal P}\). Writing \({\mathcal P}{\mathcal O}_X = {\mathcal O}_X(-D)\), for every \(\iota \geq 0\) one defines the multiplier ideal
\[
{\mathcal J}({\mathcal P}^\iota) := \pi_*{\mathcal O}_X(K_{X/X_0} - \lfloor \iota D \rfloor)
\]
where \(K_{X/X_0}\) is the relative canonical divisor and \(\lfloor \iota D \rfloor\) is the round-down, component by component, of the \({\mathbb R}\)-divisor \(\iota D\). The jumping numbers are the values of \(\iota\) where the multiplier ideal ``jumps''. They form a discrete set \({\mathcal H} \subset {\mathbb Q_+}\), and at each jump the corresponding quotient is finite dimensional over \({\mathbb C}\). The Poincaré series is thus defined to be
\[
P({\mathcal P}) := \sum_{\iota \in {\mathcal H}} \dim_{\mathbb C} \left(\frac{{\mathcal J}({\mathcal P}^{\iota-\epsilon})}{{\mathcal J}({\mathcal P}^\iota)}\right) t^\iota,
\]
where \(0 < \epsilon \ll 1\). This series is related to the Hodge spectrum of the singularity. More precisely, it follows by \textit{N. Budur} [Math. Ann. 327, No.2, 257--270 (2003; Zbl 1035.14010)] that the terms of degree less than one of \(P({\mathcal P})\) coincide with those of the Hodge spectrum of a general element of \({\mathcal P}\).
The main result of the paper is that the Poincaré series \(P({\mathcal P})\) is rational. Moreover, the series can be described by an explicit formula which depends on the particular exceptional divisors in the log-resolution that compute to the jumping numbers, in the spirit of the work of \textit{K. E. Smith} and \textit{H. M. Thompson} [Irrelevant exceptional divisors for curves on a smooth surface. In Algebra, geometry and their interactions. Contemporary Mathematics 448, 245--254 (2007; Zbl 1141.14004)]. Multiplier ideal; Simple complete ideal; Poincaré series Galindo, C., Monserrat, F.: The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface. Adv. Math. 225, 1046-1068 (2010) Singularities in algebraic geometry, Regular local rings The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth 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 This note is an introductory survey to the author's theory, the Heisenberg Lie algebra and Hilbert schemes, a kind of abstract for the author's earlier book [\textit{H. Nakajima}, ``Lectures on Hilbert schemes of points on surfaces'', Univ. Lect. Ser. (1999; Zbl 0949.14001)].
Two theorems are explained:
(1) The idea of the proof for the theorem about the cohomology group of Hilbert schemes and Heisenberg Lie algebras [cf. \textit{H. Nakajima}, Ann. Math., II. Ser. 145, No.~2, 379-388 (1997; Zbl 0915.14001)].
(2) The theorem about the generating function of the cohomology group of the symmetric product of curves \(S^n C\)
\[
\sum_{n=0}^\infty z^n [S^n C] = \exp \left( \sum_{i=1}^\infty \frac{z^i P_{[C]}[-i]} {(-i)^{i-1}i} \right)\cdot 1
\]
where \(P_{[C]}[k]\) is an operator corresponding to the cohomology class \([C]\), with its application. Hilbert schemes; Heisenberg algebras; vertex algebras Parametrization (Chow and Hilbert schemes), Vertex operators; vertex operator algebras and related structures, Homological methods in Lie (super)algebras Hilbert schemes of points on a curved surface, Heisenberg algebras, and vertex algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \({\mathcal H}= \text{Hilb}^d (S)\) be the Hilbert scheme parametrizing the zero-dimensional subschemes of finite length \(d\) of a projective, irreducible, smooth surface \(S\), over the complex field. The study of \({\mathcal H}\) is related to many questions of enumerative geometry. -- \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193-207 (1990; Zbl 0679.14007)] and \textit{L. Göttsche} and \textit{W. Soergel} [Math. Ann. 296, No. 2, 235-245 (1993; Zbl 0789.14002)] have found the rational homology \(H_n ({\mathcal H}, \mathbb{Q})\) of \({\mathcal H}\).
In the present paper the authors exhibit two bases for \(H_n ({\mathcal H}, \mathbb{Q})\), one of them described by nonreduced subschemes, and another one described by reduced schemes, i.e. by sets of \(d\) distinct points of \(S\). The technique consists in showing that the elements of the two candidate bases intersect in a triangular matrix with nonzero determinant. bases for rational homology group; Hilbert scheme; zero-dimensional subschemes; enumerative geometry Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Surfaces and higher-dimensional varieties, (Co)homology theory in algebraic geometry Bases of the homology spaces of the Hilbert scheme of points in an algebraic 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 In a previous paper [Commun. Math. Phys. 221, 293--304 (2001; Zbl 1066.14038)] the authors compared the symplectic structures on the moduli space of stable Higgs bundles on a Riemann surface \(X\) and on the Hilbert scheme of zero-dimensional subschemes in the cotangent bundle \(\Omega_X\). In the paper under review they generalize this result using parabolic triples. More precisely, Higgs bundles in the first sentence are replaced by Higgs bundles with parabolic structure over a finite set \(S\) and the cotangent bundle is replaced by \(\Omega_X (S)\). parabolic triples DOI: 10.1007/s00220-003-0897-2 Vector bundles on curves and their moduli, Parametrization (Chow and Hilbert schemes), Symplectic structures of moduli spaces Symplectic structures on moduli spaces of parabolic Higgs bundles and Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Soit \(X\) une courbe irréducible et projective sur un corps algébriquement clos \(k\) qui n'a qu'un seul point double ordinaire \(x\) comme singularité. Soit \(Y\) sa normalisation avec \(y_1,y_2\) les deux points de \(Y\) au-dessus de \(x\). Notons \(J\) la jacobienne de \(Y\), et \(J_{y_1+y_2}\) la jacobienne généralisée de \(Y\) par rapport à \(y_1+ y_2\). Les jacobiennes généralisées du type \(J_{y_1+y_2}\) sont intéressantes puisque par exemple elles engendrent le groupe \(\text{Ext}(J,\mathbb{G}_m)\) des classes d'isomorphisme des extensions de \(J\) par \(\mathbb{G}_m\). \(J_{y_1+y_2}\) correspond à un couple d'éléments de \(\text{Ext} (J,\mathbb{G}_m)\), l'un inverse de l'autre. Par l'isomorphisme naturel \(\text{Ext} (J, \mathbb{G}_m) \cong \text{Pic}^0(J) \cong \text{Pic}^0(Y)\), alors ce couple d'éléments correspond au couple \(\pm(y_1-y_2)\). Autrement dit on peut décrire la jacobienne \(\text{Pic}^0_{X/k} = J_{y_1 + y_2}\) de \(X\) à l'aide des deux points de la courbe normalisée \(Y\) qui sont au-dessus du point singulier de \(X\).
Le but de ce papier est de donner une description analogue de la jacobienne \(\text{Pic}^0_{X/k}\) d'une courbe \(X\) (éventuellement réductible) à singularités ordinaires sur un corps algébriquement clos \(k\). generalized Jacobian; ordinary curve singularities Zhang B. (1997). Sur les jacobiennes des courbes à singularités ordinaires. Manuscripta math. 92: 1--12 Jacobians, Prym varieties, Singularities of curves, local rings On the Jacobians of curves with ordinary singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let X be a complete algebraic variety over an algebraically closed field. Let \(X^{[k+1]}\) be the Hilbert scheme of 0-cycles on X of length \(k+1\). Let L be a line bundle on X. L is said to be k-very ample if the restriction map \(H^ 0(X,L)\to H^ 0(Z,{\mathcal O}_ Z(L))\) is onto for every O-cycle \(Z\in X^{[k+1]}\). E.g. O-very ample means spanned by global sections, 1-very ample means very ample. In the case of smooth surfaces this notion was recently introduced by Sommese and the reviewer. Recently, the k-very ampleness has been extensively studied by several authors. The main result of the paper is to show that if L is k-very ample, then L is \((k+1)\)-very ample if and only if the morphism \(X^{[k+1]}\to Grass(k+1,H^ 0(X,L)^*)\) is an embedding. This improves a previous result by Sommese and the reviewer. Hilbert scheme; line bundle; k-very ampleness Fabrizio Catanese and Lothar Gœttsche, \?-very-ample line bundles and embeddings of Hilbert schemes of 0-cycles, Manuscripta Math. 68 (1990), no. 3, 337 -- 341. Parametrization (Chow and Hilbert schemes), Divisors, linear systems, invertible sheaves, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] d-very-ample line bundles and embeddings of Hilbert schemes of 0-cycles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective surface on \({\mathbb C}\). Let \(S^m(X)\) be the \(m\)-th symmetric product and \(\omega_X\) the canonical bundle of \(X\). Let \(X^{[m]}\) denote the Hilbert scheme of subschemes of \(X\) of length \(m\). Let \(L\) and \(A\) be two line bundles on \(X\). The symmetrization of the \(m\)-fold tensor product of \(A\) gives a line bundle on \(S^m(X)\). Let \({\delta}^A\) (the determinant bundle for \(A\)) denote the pull back of this line bundle to \(X^{[m]}\) by the canonical (Hilbert-Chow) morphism from \(X^{[m]}\) to \(S^m(X)\). Let \(S \subset X^{[m]} \times X\) be the universal scheme of couples \((Z,x)\) with \(x\in Z\). Let \(p_1\) and \(p_2\) be the projections from \(S\) to \(X^{[m]}\) and \(X\) respectively and \(L^{[m]} = {p_1}_* {p_2}^*L\). The main result is the following.
Theorem: If \({\omega_X}^{-1} \otimes A\) and \({\omega _X}^{-1} \otimes A \otimes L\) are ample, then
(i) \(H^q(X^{[m]}, L^{[m]}\otimes {\delta ^A}) = 0\) for \(q > 0\);
(ii) \(H^0(X^{[m]}, L^{[m]} \otimes {\delta ^A}) \approx S^{m-1}(H^0(A)) \otimes H^0(L\otimes A)\).
As a corollary, it is shown that the conclusion of the theorem holds under the hypothesis \(H^q(X, A) = H^q(X, A\otimes L)= 0\) for \(q>0\). These results can be used to give examples supporting Le Potier's strange duality conjecture about moduli of semistable rank \(2\) torsionfree sheaves on a projective plane. Hilbert scheme of a surface; vanishing theorem; Le Potier's strange duality conjecture; cohomology Danila, G, Sur la cohomologie d'un fibré tautologique sur le schéma de Hilbert d'une surface, J. Algebraic Geom., 10, 247-280, (2001) Étale and other Grothendieck topologies and (co)homologies, Parametrization (Chow and Hilbert schemes), Vanishing theorems in algebraic geometry On the cohomology of a tautological fibre bundle on the Hilbert scheme of a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper concerns asymptotic connectedness of the Hilbert scheme of space curves. It is shown that any Buchsbaum curve can be connected to an extremal curve in the Hilbert scheme \(H_{d,g}\). As a corollary, one concludes that any curve having the same postulation and Rao function as a Buchsbaum curve can be connected to an extremal curve. Parametrization (Chow and Hilbert schemes), Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Asymptotic connectedness of the Hilbert scheme of space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities 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 \({\mathcal N}^0_{d,g}\) be the number of irreducible component of the Hilbert scheme parametrizing smooth arithmetically Cohen-Macaulay curves in the projective space \(\mathbb{P}^3\), of degree \(d\) and genus \(g\). The author constructs families of couples \((d,g)\) such that \({\mathcal N}^0_{d,g}\) increases at least exponentially. space curve; Hilbert scheme; arithmetically Cohen-Macaulay curves Parametrization (Chow and Hilbert schemes), Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the number of components of the Hilbert scheme of ACM space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Nonstandard mathematics furnishes a remarkable connection between analytic and algebraic geometry. We describe this interplay for the most basic notions like complex spaces/algebraic schemes, generic points, differential forms etc. We obtain -- by this point of view -- in particular new results on the prime spectrum of a Stein algebra. complex spaces; nonstandard schemes; generic points; analytic Nullstellensatz; Stein algebras; internal polynomials Khalfallah, Adel; Kosarew, Siegmund, Complex spaces and nonstandard schemes, J. Log. Anal., 2, Paper 9, 60 pp., (2010) Complex spaces, Nonstandard models in mathematics, Schemes and morphisms, Nonstandard analysis, Stein spaces, Differential forms in global analysis Complex spaces and nonstandard schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives ''a necessary and sufficient condition for d pieces of hypersurface to be contained in an algebraic hypersurface of degree d''. J. A. Wood, A simple criterion for local hypersurfaces to be algebraic, Duke Math. J. 51 (1984), 235-237. Zbl0584.14021 MR744296 Surfaces and higher-dimensional varieties, Projective techniques in algebraic geometry, Relevant commutative algebra A simple criterion for local hypersurfaces to be algebraic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C\) be a smooth curve with distinguished point \(0 \in C\). A \textit{simple degeneration} is a flat morphism \(f:X \to C\) with \(X\) a smooth algebraic space such that \(f\) is smooth outside \(X_0 = f^{-1} (0)\), \(X_0\) has normal crossing singularities, and the singular locus \(D \subset X_0\) is smooth. It is \textit{strict} if the irreducible components of \(X_0\) are also smooth. Given an orientation on the dual graph \(\Gamma (X_0)\) of \(X_0\) for a strict simple degeneration, \textit{J. Li} [J. Differ. Geom. 57, No. 3, 509--578 (2001; Zbl 1076.14540)] constructed an \textit{expanded degeneration} \(X[n] \to C[n]\), a \(G[n]\)-equivariant morphism in which \(C[n]/G[n] \cong C\) (\(G[n]\) is the \(n\)-torus \((\mathbb G_m)^n\)). \textit{M. G. Gulbrandsen} et al. used expanded degenerations to study degenerations of Hilbert schemes of \(n\) points on varying fibers \(X_t\), constructing a degeneration \(I^n_{X/C} \to C\) which coincides with the relative Hilbert scheme \(\text{Hilb}^n (X/C) \to C\) over \(C - \{0\}\), but differing at the central fiber. \(I^n_{X/C}\) is constructed as a GIT quotient of \(\text{Hilb}^n (X[n]/C[n])\) by the \(G[n]\)-action [Doc. Math. 24, 421--472 (2019; Zbl 1423.14068)]. If \(X\) is a scheme, then \(X[n]\) is a scheme if and only if \(\Gamma (X_0)\) has no loops and \(X \to C\) is projective if and only if \(\Gamma (X_0)\) has no directed cycles.
The authors study \(I^n_{X/C} \to C\) when \(\dim X_t \leq 2\) and \(\Gamma (X_0)\) is bipartite. They prove that \(I^n_{X/C}\) is normal with finite quotient singularities and \(\mathbb Q\)-factorial, the special fiber \((I^n_{X/C})_0\) is reduced, and the pair \((I^n_{X/C}, (I^n_{X/C})_0)\) is divisorial log terminal in the sense of the minimal model program. The crux of the proof is to show that the divisorial log terminal property passes from \(\text{Hilb}^n (X[n]/C[n])^{ss}\) to \(I^n_{X/C}\) under the GIT quotient. They also prove that if \(X \to C\) is a good minimal divisorial log terminal model, then so is \(I^n_{X/C} \to C\). With the same hypothesis, the authors prove that the dual complex for \(X \to C\) is a graph \(\Gamma\) and the dual complex \(\mathcal D ((I^n_{X/C})_0)\) is isomorphic to the \(n\)th symmetric product \(\text{Sym}^n (\Gamma)\) as a \(\Delta\)-complex. In particular, some results of \textit{M. V. Brown} and \textit{E. Mazzon} [Compos. Math. 155, No. 7, 1259--1300 (2019; Zbl 1440.14131)] on the essential skeleton of \(\text{Hilb}^n X\) in terms of the essential skeleton of \(X\) for K3 surfaces are recovered. In the motivating case where \(X \to C\) is a projective type II degeneration of K3 surfaces, they prove that the stack \(I^n_{X/C}\) is proper and semi-stable over \(C\); moreover, if \(K_{X/C}\) is trivial, then \(I^n_{X/C}\) carries an everywhere non-degenerate relative logarithmic \(2\)-form.
In the closing section, the authors compare their results with earlier work of \textit{Y. Nagai} [Math. Z. 258, 407--426 (2008; Zbl 1140.14008)] in the special case \(n=2\). There \textit{Y. Nagai} constructs a different degeneration \(H^2_{X/C} \to C\), even if \(\Gamma (X_0)\) is not bipartite; in case \(\Gamma (X_0)\) is bipartite, the authors relate \(H^2_{X/C} \to C\) to \(I^2_{X/C}\) by explicit birational maps. Recent work of \textit{Y. Nagai} constructs the Hilbert scheme degeneration \(I^n_{X/C} \to C\) using toric methods, describing the local structure of the singularities in \(\text{Sym}^n (X/C)\). He also gives an explicit \(\mathbb Q\)-factorial terminalization \(Y^{(n)} \to \text{Sym}^n (X/C)\) and isomorphism \(I^n_{X/C} \to \text{Sym}^n (X/C)\) [Math. Z. 289, 1143--1168 (2018; Zbl 1423.14069)]. strict simple degenerations; geometric Invariant theory; Hilbert schemes; good minimal dlt models; type II degenerations of \(K3\) surfaces Fibrations, degenerations in algebraic geometry, Parametrization (Chow and Hilbert schemes), \(K3\) surfaces and Enriques surfaces, Geometric invariant theory The geometry of degenerations of Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this note we attempt to generalize a few statements drawn from the 3-dimensional McKay correspondence to the case of a cyclic group not in \(SL(3,\mathbb{C})\). We construct a smooth, discrepant resolution of the cyclic, terminal quotient singularity of type \({1\over r}(1,1,r-1)\), which turns out to be isomorphic to Nakamura's \(G\)-Hilbert scheme. Moreover we explicitly describe the tautological bundles and use them to construct a dual basis to the integral cohomology of the resolution. McKay correspondence; terminal singularities Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients) Cohomology of the \(G\)-Hilbert scheme for \(\frac 1r(1,1,r-1)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(n>d\) be positive integers and let \({\mathcal A}=\{a_1,\dots,a_n\}\) be a subset of \({\mathbb N}^d\setminus \{(0,\dots, 0)\}\) such that the matrix with columns \(a_i\) has rank \(d\). Put \(S=k[x_1,\dots,x_n]\), where \(k\) is a field and where this polynomial ring is generated by variables \(x_1,\dots,x_n\) in \({\mathbb N}^d\)-degrees \(a_1,\dots,a_n\), respectively. The toric ideal \(I_{\mathcal A}\) is the kernel of the homomorphism \(S\rightarrow k[t_1,\dots,t_d]\) that maps \(x_i\) to \(t_1^{a_{i1}}\cdots t_d^{a_{id}}\) for \(1\leq i\leq n\). A homogeneous ideal \(M\) is called \({\mathcal A}\)-graded if the quotient \(S/M\) has the same multigraded Hilbert function as the toric ring \(S/I_{\mathcal A}\).
In another article [``Toric Hilbert schemes'', Duke Math. J., (to appear)], the authors introduce the toric Hilbert scheme \({\mathcal H}_{\mathcal A}\), which parametrizes \({\mathcal A}\)-graded ideals, and show that this scheme can be covered by finitely many open sets centered at monomial ideals. Here, the authors find local equations for this toric Hilbert scheme by determining the local ring at such a monomial ideal. The proof uses a reduction process based on Mora's tangent cone algorithm.
In the final section, the authors obtain a criterion for an ideal to be \({\mathcal A}\)-graded. This leads to an algorithm for computing all \({\mathcal A}\)-graded monomial ideals whose radical is the Stanley-Reiser ideal of a given triangulation of \({\mathcal A}\). This also gives a singly exponential bound on the degree up to which one must check to conclude that an ideal is \({\mathcal A}\)-graded, proving a conjecture of Sturmfels. toric ideal; toric Hilbert scheme; \({\mathcal A}\)-graded ideal; Stanley-Reiser ideal; conjecture of Sturmfels; tangent cone algorithm Irena Peeva and Mike Stillman, Local equations for the toric Hilbert scheme, Adv. in Appl. Math. 25 (2000), no. 4, 307-321. Parametrization (Chow and Hilbert schemes), Effectivity, complexity and computational aspects of algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Local equations for the toric Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper introduces two invariants of codimension 2 liaison classes and gives applications to bounding the dimensions of Hilbert scheme components. Let \(X \subset \mathbb P^n\) be a locally Cohen-Macaulay (lCM) subscheme of codimension 2. If \(R\) denotes the coordinate ring of \(\mathbb P^n\), then the saturated ideal \(I_X\) has a graded \(R\)-resolution
\[
0 \to \bigoplus_{i=1}^{r_{n+2}} R(-n_{n+2,i}) \to \dots \to \bigoplus_{i=1}^{r_{2}} R(-n_{2,i}) \to \bigoplus_{i=1}^{r_{1}} R(-n_{1,i}) \to I_X \to 0
\]
of length \(n\), from which the author defines functions
\[
\delta_X^m (v) = \sum_{j=1}^{n} \sum_{i=1}^{r_{j}} (-1)^{j+1} h^m({\mathcal I}_X (n_{j,i} + v))
\]
for \(0 \leq m \leq n\) and \(v \in \mathbb Z\). Also associated to \(X\) are its postulation \(\gamma\) and Rao functions \(\rho_i (l) = \dim H^i ({\mathcal I}_X (l))\) for \(0 < i < n-1\) and \(l \in \mathbb Z\). Setting \(\rho = (\rho_1, \rho_2, \dots \rho_{n-2})\), \(H_{\gamma,\rho}\) is the subset of the Hilbert scheme consisting of \(Y \subset \mathbb P^n\) with postulation \(\gamma\) and Rao functions \(\rho\). The remarkable fact proved is that the two integers
\[
\mathrm{obsumext}(X) = 1+\delta^{n-1}_X (-n-1) - \dim_{(X)} H_{\gamma,\rho}
\]
\[
\mathrm{sumext}(X) = 1+\delta^{n-1}_X (-n-1) - \dim \mathrm{Ext}^1_R (I_X,I_X)_{\rho}
\]
depend only on the biliaison class of \(X\). Moreover there is the inequality \(\mathrm{sumext}(X) \leq \mathrm{obsumext}(X)\) with equality if and only if \(H_{\gamma,\rho}\) is smooth at \(X\). If \(X\) is ACM, then \(\mathrm{sumext}(X) = \mathrm{obsumext}(X)=0\), recovering the well known smoothness result of \textit{G. Ellingsrud} [Ann. Sci. Éc. Norm. Supér. (4) 8, 423--431 (1975; Zbl 0325.14002)].
In low dimensions, the author uses these invariants to give lower bounds on Hilbert scheme components. If \(H(d,g)\) is the Hilbert scheme of lCM curves in \(\mathbb P^3\) and \(X \in H(d,g)\), then \(\dim_{(X)} H(d,g) \geq 4d + \delta^2 (0) - \delta^1 (0)\), which sometimes improves on the usual bound \(4d\), for example when \(X\) has large exceptionality. The lower bound for Hilbert scheme components of surfaces in \(\mathbb P^4\) is more complicated, but is used to produce examples of obstructed surfaces. From results of \textit{G. Horrocks} [Proc. Lond. Math. Soc., III. Ser. 14, 689--713 (1964; Zbl 0126.16801)] and \textit{P. Rao} [Math. Ann. 258, 169--173 (1981; Zbl 0493.14009)], a surface \(X \subset \mathbb P^4\) gives rise to two Rao modules \(M_1, M_2\) and \(b \in _0\mathrm{Ext}^2(M_2,M_1)\) and \textit{G. Bolondi} has shown explicitly how this creates a bijection between even liaison classes of surfaces and isomorphism classes of such triples modulo twist [in: Classification of algebraic varieties, Contemp. Math. 162, 49--63 (1994; Zbl 0823.14036)]. Bolondi's description is used and refined to prove that the natural map \(H_{\gamma, \rho} \to V_{\rho}\), where \(V_{\rho}\) is the set of isomorphism classes of such triples, is smooth. This generalizes the corresponding theorem for space curves of [\textit{M. Martin-Deschamps} and \textit{D. Perrin}, Sur la classification des courbes gauches. Centre National de la Recherche Scientifique. Astérisque, 184--185 (1990; Zbl 0717.14017)]. liaison; Hilbert scheme; codimension two subvarieties; unobstructedness Linkage, Parametrization (Chow and Hilbert schemes), Low codimension problems in algebraic geometry, Syzygies, resolutions, complexes and commutative rings Liaison invariants and the Hilbert scheme of codimension 2 subschemes in \(\mathbb{P}^{n+2}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\) be a Cohen-Macaulay local ring. Then \(R\) is said to be of finite Cohen-Macaulay type if there are only finitely many isomorphism classes of indecomposable (finite) Cohen-Macaulay \(R\)-modules. We refer to \textit{Y. Yoshino}'s book ``Cohen-Macaulay modules over Cohen-Macaulay rings'', Lond. Math. Lect. Note Ser. 146 (1990; Zbl 0745.13003) for a comprehensive list of publications on this subject.
Let \(R\) ba a one-dimensional, reduced, complete, equicharacteristic local ring with coefficient field \(k\). Assume that the residue fields \(K\) of the normalization of \(R\) are separable over \(k\). The author classifies those \(R\)'s which have finite Cohen-Macaulay type or equivalently which admit only finitely many non-isomorphic indecomposable torsionfree \(R\)- modules. (Actually in the case considered here, the condition for \(R\) to have finite CM type is equivalent to \([K:k]\leq 3\).) -- The proof uses the fact that \(R\) has finite CM type if and only if the (``special'') artinian pair \((R/c,\tilde R/c)\) has finite representation type where \(\tilde R\) is the normalization of \(R\) and \(c\) the conductor of \(R\) in \(\tilde R\) [see \textit{Yu. A. Drodz} and \textit{A. V. Rojter}, Izv. Akad. Nauk. SSSR, Ser. Mat. 31, 783-798 (1967; Zbl 0164.041)] and consists mainly of a classification of the special artinian pairs of finite representation type. Cohen-Macaulay local ring; finite Cohen-Macaulay type Wiegand, R.: Curve singularities of finite Cohen--Macaulay type. Ark. mat. 29, 339-357 (1991) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities of curves, local rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Curve singularities of finite Cohen-Macaulay type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a contractible Stein threefold representing the germ of an isolated canonical threefold singularity \(x\in X\). Assume that \(\varphi: Y\to X\) is a crepant partial resolution of \(X\) and that \(Y\) has only terminal analytic \(\mathbb Q\)-factorial singularities. The author is interested in the Betti numbers \(b_i(Y)=\dim H^i(Y,\mathbb Q)\) of \(Y\). These numbers are related to the rank \(\rho(X)\) of the analytic divisor class group of \(X\) in \(x\) and to the number \(c(X)\) of crepant valuations of \(X\). It is shown that \(b_1(Y)=b_5(Y)=b_6(Y)=0\), \(b_2=\rho(X)+c(X)\), \(b_4(Y)=c(X)\).
The calculation of \(b_3(Y)\) is more elaborate: \(b_3(Y)=\sum_{i=1}^{c(X)} b_3(E_i)\), where \((E_i)_{1\leq i\leq c(X)}\) is a system of nonsingular projective representatives of the crepant valuations of \(X\). For the special case where \((X,x)\) is the germ of an isolated canonical nondegenerate hypersurface singularity \((f=0)\) in \(\mathbb C^4\) the author gets a combinatorial formula for \(b_3(Y)\) in terms of the Newton diagram of the defining equation. The Betti numbers \(b_i(Y)\) are explicitly calculated for the examples \(f=x^2+y^3+z^6+t^n\), \(n\geq 6\), and \(f=x^3+y^4+z^4+t^4\).
Moreover the author presents an example of a threedimensional space, a blowup of the hypersurface singularity defined by \(x^3+x^2z+y^2z+z^4+t^5=0\), which is globally analytically \(\mathbb Q\)-factorial but not locally analytically \(\mathbb Q\)-factorial. canonical singularity; crepant valuation; Newton diagram Caibăr, Mirel, Minimal models of canonical 3-fold singularities and their Betti numbers, Int. Math. Res. Not., 26, 1563-1581, (2005) Minimal model program (Mori theory, extremal rays), \(3\)-folds, Complex surface and hypersurface singularities, Singularities in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Minimal models of canonical 3-fold singularities and their Betti numbers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0619.00007.]
Let \(A=\mathbb{C}[[ X_1,\ldots,X_ n]]\) and let \(G\) be a finite group acting linearly on \(A\) with ring of invariants \(R=A^ G\). It has been proved by Auslander and Reiten that \(R\) has a finitely generated Grothendieck group \(G(R)\) and that the reduced Grothendieck group \(\widetilde G(R)\) is finite, at least when \(G\) acts freely on the linear space \(L = \sum_{i=1} \mathbb{C} X_ i\) of 1-forms. In the present note, which is essentially completely self-contained, the authors show that \(\widetilde G(R)\) is finite if \(G\) is just assumed to be abelian. Moreover, the reduced Grothendieck group of all simple hypersurface singularities is explicitly calculated.
For the second part of the paper (the explicit calculation of \(\widetilde G(R)\)), the authors make use of the fact that for any two-dimensional normal local domain \((R,m,k)\) with \([k]=0\), \(\widetilde G(R) = Cl(R)\), the class group of \(R\). Indeed, this result implies that if \(G\) acts faithfully and linearly on \(A=\mathbb{C}[[ X,Y]]\), then \(\widetilde G(A^ G)=(G/H)^*\), where \(H\) is the subgroup of \(G\), which is generated by the pseudoreflections on \(G\). The explicit description of \(\widetilde G(R)\) referred to above, makes use of this as well as some results of Knörrer, which say that (i) if \(f\in \mathbb{C}[[ X_1,\ldots,X_ n]]\), then the stable \(AR\)-quivers of \(\mathbb{C}[[ X_1,\ldots,X_ n]]/(f)\) and \(\mathbb{C}[[ X_1,\ldots,X_ n,Y,Z]]/(f+Y^ 2+Z^ 2)\) coincide, and (ii) simple hypersurface singularities are of finite Cohen-Macaulay representation type. Indeed, in htis case the stable \(AR\)-quivers of \(M(R)\) determine \(G(R)\), so one only has to consider the Grothendieck groups of simple hypersurface singularities in dimension 1 and 2.
Note: As pointed out by the authors, Auslander and Reiten have proved independently that \(\widetilde G(R)\) is finite for any action of a finite, not necessarily abelian group. Moreover, the \(AR\)-quivers of simple hypersurface singularities have been calculated by Dieterich and Wiedemann in dimension~1 and by Auslander in dimension~2. ring of invariants; reduced Grothendieck group; simple hypersurface singularities; stable AR-quivers; exponential-type operators; uniform approximation Jürgen Herzog and Herbert Sanders, The Grothendieck group of invariant rings and of simple hypersurface singularities, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 134 -- 149. Singularities of surfaces or higher-dimensional varieties, Geometric invariant theory, Grothendieck groups (category-theoretic aspects), Grothendieck groups, \(K\)-theory and commutative rings, Representation theory of associative rings and algebras, Group actions on varieties or schemes (quotients) The Grothendieck group of invariant rings and of simple hypersurface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0619.00007.]
It is shown that the obstruction space \(T^ 2_ X\) for deformations of a two-dimensional cyclic quotient singularity (X,0) that is not a hypersurface singularity has dimension \(e(e+2)\) where e denotes the embedding dimension of (X,0). The proof uses ``Wahl's conjecture'' on the dimension of smoothing components which was proven by \textit{G.-M. Greuel} and \textit{E. Looijenga} [Duke Math. J. 52, 263-272 (1985; Zbl 0587.32038)] and \textit{O. Laudal, G. Pfister}. This formula allows to express \(\dim (T^ 2_ X)\) in terms of invariants of a suitable quasihomogeneous curve on X. These invariants can be computed using invariant theory for cyclic subgroups of GL(2,\({\mathbb{C}})\). dimension of obstruction space for deformations; cyclic quotient singularity; Wahl's conjecture Christophersen, J. , Monomial curves and obstructions on cyclic quotient singularities, in 'Singularities, representations of algebras and vector bundles, Lambrecht 1985 ,' Lecture Notes in Mathematics 1273, Springer Verlag, Berlin-Heidelberg- New York. Deformations of singularities, Deformations of complex singularities; vanishing cycles Monomial curves and obstructions on cyclic quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a survey paper where the author gives characterizations of various types of singularities in spherical varieties in terms of combinatorial invariants: locally factorial and \(\mathbb Q\)-factorial varieties, Gorenstein and \(\mathbb Q\)-Gorenstein varieties, terminal, canonical and Kawamata log terminal singularities. spherical varieties; singularities; minimal model program Compactifications; symmetric and spherical varieties, Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays) A survey on the singularities of spherical 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 \(I\subset S=\mathbb{K}[x_1,\dots,x_n]\) be a homogeneous prime ideal of a standard graded polynomial ring with
coefficients in a field. In the article under review, the authors pose the following conjecture: if \(\operatorname{Proj} S/I\) is nonsingular and \(\prec\) is a monomial order for which \(\operatorname{in}_\prec (I)\) is square-free, then
\(S/\operatorname{in}_\prec(I)\) is Cohen-Macaulay.
The authors start by showing that if this conjecture holds then, for any \(I\) satisfying the assumptions,
\(S/\operatorname{in}_\prec(I)\) has negative \(a\)-invariant (Proposition 2.4).
Then they show that if \(\prec\) is a degree reverse lexicographic order then the conjecture
holds. (Theorem 3.3 and Corollary 3.4).
In connection, the authors define the notion of a Gröbner-smoothable simplicial complex over a field
\(\mathbb{K}\). Such is a simplicial complex \(\Delta\), on \(n\) vertices, for which there exists a homogeneous ideal
\(I\subset S\) and monomial order \(\prec\) with \(\operatorname{Proj} S/I\) smooth
and \(I_\Delta\) (the Stanley-Reisner ideal of \(\Delta\)) equal to \(\operatorname{in}_\prec(I)\).
In section 4, the authors study the two assumptions of the conjecture in the light of the notion of Gröbner-smoothable
simplicial complexes. The results obtained provide conditions on \(\Delta\) for it to not be Gröbner-smoothable. acyclic Cohen-Macaulay simplicial complexes; reduced monomial schemes Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Combinatorial aspects of commutative algebra, Combinatorial aspects of simplicial complexes, Deformations of singularities, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Rings with straightening laws, Hodge algebras, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes Singularities and radical initial 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 [For the entire collection see Zbl 0527.00002.]
The following result is proved: A noetherian subalgebra A of a \({\mathbb{C}}\)- algebra of finite type is finitely generated if and only if the Gel'fand topology on Spec max A is locally compact. The Gel'fand topology is the weakest topology such that for every \(f\in A\) the function Spec max \(A\to {\mathbb{C}}\), \(m\mapsto residue\quad of\quad f\quad in\quad A/{\mathfrak m}={\mathbb{C}},\) is continuous. Actually there is a slightly more general result in terms of Spec A. spectrum; finite generation of subalgebra; algebra of finite type Relevant commutative algebra, Commutative rings and modules of finite generation or presentation; number of generators On the algebraization of some complex 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 \(M = M(r,c_ 1, c_ 2)\) be the moduli space for stable sheaves on \(\mathbb{P}^ 2\) with fixed rank and Chern classes. Assume that \(\text{gcd}(r,c_ 1,(c_ 1 - 1)/2 c_ 2) = 1\) and that \(0 < c_ 1 < r - 1\). Then \(M\) is a projective non-singular variety with universal family \(E\) of vector bundles on \(P^ 2 \times M\). It is proved that the Chow ring of \(M\) is generated by the Chern classes of the bundles \(R^ 1 p_{M^*} E(-j)\) for \(j = 1,2,3\), that numerical and rational equivalence coincide on \(M\), that over \(\mathbb{C}\) the cycle map of the Chow ring into integer cohomologies is an isomorphism. In particular, the Chow ring is a free \(\mathbb{Z}\)-module and there are no odd-dimensional cohomologies. moduli space for stable sheaves; Chern classes; cycle map; Chow ring Geir Ellingsrud and Stein Arild Strømme, Towards the Chow ring of the Hilbert scheme of \?², J. Reine Angew. Math. 441 (1993), 33 -- 44. Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles Towards the Chow ring of the Hilbert scheme of \(\mathbb{P}^ 2\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an announcement on the investigation of four-dimensional terminal cyclic quotient singularities, which are not Gorenstein ones. (For simplicity, the main attention is paid to the quotients with respect to cyclic groups of prime order.) Computations of all these singularities for the primes \(<1000\) using a computer, has led the authors to a structural theorem concerning these singularities (which is only a little more complicated than the known structural theorem of dimension three). As computations of the authors show, one has not to expect that, for a four-dimensional terminal singularity \(T\) a general element of the system \(| -K_ T|\) has, as a rule, only canonical singularities (this fact holds for every terminal singularity in dimension three). The authors succeed in discovering only six terminal singularities T of dimension four for which the general term of \(| -2K_ T|\) has singularities worse than the canonical ones. The authors suppose that the same fact is true in the general case, as well. four-dimensional terminal cyclic quotient singularities; computations S. Mori, D. R. Morrison and I. Morrison, On four-dimensional terminal quotient singularities, Math. Comp. 51 (1988), 769--786. JSTOR: \(4\)-folds, Computational aspects of higher-dimensional varieties, Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Software, source code, etc. for problems pertaining to algebraic geometry On four-dimensional terminal quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a survey on recent results of the author on Noetherian reduced subalgebras of algebras of finite type over a field \(k\). It is known that their corresponding schemes are generically algebraic, i.e. they contain open dense subsets which are algebraic over \(k\). The author discusses the following topics in the survey:
1) Generic algebraization of closed subschemes of generically algebraic schemes,
2) Local algebraization of generically algebraic schemes at an arbitrary point,
3) Obstructions to the finite generation of Noetherian algebras,
4) Local geometric structure of Noetherian algebras with height-one maximal ideals.
The main result is an obstruction to the finite generation of subalgebras of dimension \(> 1\) in terms of the existence of height-one maximal ideals. Noetherian algebra; finite generation; algebraization; height-one maximal ideals Relevant commutative algebra, Commutative Noetherian rings and modules Schemes generically algebraic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review constructs a categorification of the maximal commutative subalgebra of a type \(A\) Hecke algebra, spanned by the Jones-Wenzl projectors to irreducible subrepresentations of the regular representation. This categorification is geometric in nature and is constructed via a monoidal functor from the symmetric monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The latter monoidal category is a standard categorification of the Hecke algebra.
The adjoint of this functor allows one to relate the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme.
The general picture the authors produce leads to a number of conjectures. For example, one of these conjectures is that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. Several other conjecture can be found in the introduction. Hilbert schemes; Khovanov-Rozansky homology; Soergel bimodules Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Parametrization (Chow and Hilbert schemes), Hecke algebras and their representations, Braid groups; Artin groups Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(X\) is a non-degenerate, irreducible, embedded projective variety over an algebraically closed field \(k\) corresponding to a homogeneous prime ideal \(P \subseteq S = k[x_1,\dots,x_n]\), the Eisenbud-Goto conjecture predicts an upper bound for the regularity of \(X\): \(\hbox{reg}(X) \leq \deg(X) - \hbox{codim}(X) + 1\). This was open for many years, with special cases proved. Thus it was a bit of a shock when Peeva and the second author gave counterexamples to this conjecture, producing irreducible projective varieties with regularity much larger than their degrees. The two main new ideas in their work were so-called Rees-like algebras and step-by-step homogenization. All of the varieties thus produced are singular, and the current paper studies the singularities and their geometry. The authors compute the codimension of the singular locus of a Rees-like algebra over a polynomial ring, and then show that the step-by-step process can decrease the codimension of this singular locus. Thus the authors introduce prime standardization, as an alternative to step-by-step homogenization that preserves the codimension of the singular locus. They then look at the regularity of certain smooth hyperplane sections of Rees-like algebras and show that they all satisfy the Eisenbud-Goto conjecture. They also give a characterization of Rees-like algebras of Cohen-Macaulay ideals, and more generally they characterize when Rees-like algebras are seminormal, weakly normal, and in the case of positive characteristic, F-split. Rees-like algebra; regularity; seminormal; F-split; singular locus; prime standardization; step-by-step homogenization Syzygies, resolutions, complexes and commutative rings, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Singularities of surfaces or higher-dimensional varieties, Homological dimension and commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials Singularities of Rees-like algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((A,\mathfrak{m})\) be a two--dimensional excellent normal local domain containing an algebraically closed field \(K\). Let \(f:X\to \mathrm{Spec} (A)\) be a resolution of singularity, then \(p_g(A)=l_A(H^1(X, \mathcal{O}_X))\), the geometric genus of \(A\), is independent on the resolution. Let \(I\subset A\) be an integrally closed \(\mathfrak{m}\)-primary ideal and assume that \(Z\) is an anti-nef cycle on \(X\) such that \(I\mathcal{O}_X=\mathcal{O}_X (-Z)\) and \(I=I_Z:= H^0(X, \mathcal{O}_X(-Z))\). If \(l_A(H^1(X, \mathcal{O}_X(-Z)))=p_g(A)\) then \(Z\) is called a \(p_g\)-cycle and \(I\) is called a \(p_g\)-ideal. The core of \(I\) is the intersection of all reductions of \(I\).
It is proved that in case of \(\mathrm{char}(K)\neq 2\) \(A\) is a rational singularity, i.e. \(p_g(A)=0\), if and only if for any integrally closed \(\mathfrak{m}\)-primary ideals \(I'\subset I\), we have core \((I')\subset \mathrm{core } (I)\).
An \(\mathfrak{m}\)-primary ideal \(I\) is said to be good if \(I^2=QI\) and \(I=Q:I\) for some minimal reduction \(Q\) of \(I\). It is proved that \(A\) admits a good \(p_g\)-ideal. core of ideals; good ideal; \(p_g\)-cycle; \(p_g\)-ideal; surface singularity; rational singularity T. Okuma, K-i. Watanabe, and K. Yoshida, A characterization of two-dimensional rational singularities via Core of ideals, available from arXiv:1511.01553. Singularities in algebraic geometry, Integral closure of commutative rings and ideals, Singularities of surfaces or higher-dimensional varieties, Ideals and multiplicative ideal theory in commutative rings, Multiplicity theory and related topics A characterization of two-dimensional rational singularities via core 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 Motivated by work of \textit{S. Gusein-Zade} et al. [Mosc. Math. J. 10, No. 3, 593--602 (2010; Zbl 1206.14014)], we study a specific generating series of arm and leg statistics on partitions, which is known to compute the Poincaré polynomials of \(\mathbb{Z}_3\)-equivariant Hilbert schemes of points in the plane, where \(\mathbb{Z}_3\) acts diagonally. This generating series has a conjectural product formula, a proof of which has remained elusive over the last ten years. We introduce a new combinatorial correspondence between partitions of \(n\) and \(\{1,2\}\)-compositions of \(n\), which behaves well with respect to the statistic in question. As an application, we use this correspondence to compute the highest Betti numbers of the \(\mathbb{Z}_3\)-equivariant Hilbert schemes. Parametrization (Chow and Hilbert schemes), Combinatorial aspects of partitions of integers, Combinatorial identities, bijective combinatorics, Enumerative problems (combinatorial problems) in algebraic geometry Topology of \(\mathbb{Z}_3\)-equivariant Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathcal{O}/\mathcal{J}\) denote the tautological rank \(n\) bundle over the Hilbert scheme of \(n\) points on \(\mathbb{C}^2\). The author studies the \(\mathbb{C}^{*}\)-equivariant integrals over the Hilbert scheme of points on \(\mathbb{C}^2 \) of the form
\[
F(k_1,\dots,k_N)=\sum_{n=0}^{\infty}q^{n}\int \mathrm{ch}_{k_1}(\mathcal{O}/\mathcal{J})\cdots \mathrm{ch}_{k_N}(\mathcal{O}/\mathcal{J})\cdot e(T_{\beta}\mathrm{Hilb}^n \mathbb{C}^2),
\]
where \(\mathbb{C}^{*}\) acts on \(\mathbb{C}^2\) by the formula \(z\cdot (x,y)=(zx,z^{-1}y)\), \(\mathrm{ch}\) denotes the Chern character and \(T_{\beta}\mathrm{Hilb}^n \mathbb{C}^2 \) denotes the equivariantly twisted tangent bundle to the Hilbert scheme. The main result of the paper states that \(F(k_1,\dots,k_N)\) is a quasimodular form in \(q\).
These integrals generalize the so-called Nekrasov partition function appearing in mathematical physics, which was computed explicitly by N. Nekrasov and A. Okounkov.
Although the Hilbert scheme is not compact, the integrals can be defined using the equivariant localization at isolated fixed points of the torus action. The author computes explicitly the contributions of the fixed points, what allows him to obtain the combinatorial formula for \(F(k_1,\dots,k_N)\) as an infinite sum over all partitions. He then identifies the direct sum of equivariant cohomology of \(\mathrm{Hilb}^n\mathbb{C}^2\) with the Fock space, and shows that the operators of cup products with \(\mathrm{ch}_{k_N}(\mathcal{O}/\mathcal{J})\) and \(e(T_{\beta}\mathrm{Hilb}^n \mathbb{C}^2)\) can be expressed in terms of certain vertex operators \(\Gamma_{\pm(x)}\) on this space. This approach allows him to present \(F(k_1,\dots,k_N)\) as a contour integral in 2\(N\)-dimensional space and prove the quasimodularity of this function. Hilbert schemes; moduli of sheaves; vertex operators; quasimodular forms; Nekrasov partition function DOI: 10.1016/j.aim.2011.10.003 Parametrization (Chow and Hilbert schemes), Vertex operators; vertex operator algebras and related structures, Infinite-dimensional Lie (super)algebras Vertex operators and quasimodularity of Chern numbers on the Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For an ample line bundle on an abelian or \(K3\) surface, minimal with respect to the polarization, the relative Hilbert scheme of points on the complete linear system is known to be smooth. We give an explicit expression in quasi-Jacobi forms for the \(\mathcal X_{-y}\) genus of the restriction of the Hilbert scheme to a general linear subsystem. This generalizes a result of \textit{T. Kawai} and \textit{K. Yoshioka} [Adv. Theor. Math. Phys. 4, No. 2, 397--485 (2000; Zbl 1013.81043)] for the complete linear system on the \(K3\) surface, a result of Maulik, Pandharipande, and Thomas [\textit{D. Maulik} et al., J. Topol. 3, No. 4, 937--996 (2010; Zbl 1207.14058)] on the Euler characteristics of linear subsystems on the \(K3\) surface, and a conjecture of the authors. Göttsche, L.; Shende, V., The \(\chi _{-y}\)-genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces, Algebr. Geom., 2, 405-421, (2015) Parametrization (Chow and Hilbert schemes), \(K3\) surfaces and Enriques surfaces, Algebraic theory of abelian varieties The \(\mathcal X_{-y}\)-genera of relative Hilbert schemes for linear systems on abelian and \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a projective non-singular surface over \(\mathbb{C}\), \(H\) an ample line bundle on \(X\). Denote by \(M(H)\) the coarse moduli scheme of rank-2 \(H\)-stable sheaves with fixed Chern class \((c_{1},c_{2})\in NS(X)\times\mathbb{Z}\). Let \(H_{-}\) and \(H_{+}\) be ample line bundles on \(X\) separated by only one wall of type \((c_{1},c_{2})\). For a parameter \(a\in(0,1)\), one defines the \(a\)-stability of sheaves in such a way that \(a\)-stability of sheaves with fixed Chern class equals \(H_{-}\)-stability (respectively, \(H_{+}\)-stability) if \(a\) is sufficiently close to 0 (respectively, 1); there is a coarse moduli scheme \(M(a)\) of rank-2 \(a\)-stable sheaves with Chern classes \((c_{1},c_{2})\). Let \(a_{-}\) and \(a_{+}\in(0,1)\) be parameters which are separated by only one miniwall and assume \(M(a_{-})\) is non-singular. One constructs a desingularization of \(M(a_{+})\) by using \(M(a_{-})\) and wall-crossing methods. As an application, one studies whether singularities of \(M(a_{+})\) are terminal or not when \(X\) is ruled or elliptic. moduli scheme of stable sheaves on surfaces; singularities of moduli spaces; desingularization Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Algebraic moduli problems, moduli of vector bundles Desingularization and singularities of some moduli scheme of sheaves on a surface | 0 |
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