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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We develop, for finite groups, a certain kind of resolutions of quotient spaces \(\mathbb{C}^n/G\) called pluri-toric resolutions. The pluri-toric resolutions extend the use of toric geometry, from the study of quotients by abelian finite groups to the study of quotients by arbitrary finite groups. This is done in such a way that the combinatorical and stratifying nature of toric resolutions is inherited by the pluri-toric resolutions.
We show that some of the properties of a pluri-toric resolution of \(\mathbb{C}^n/G\) can be deduced directly from the data of the group \(G\). One of these results states that a pluri-toric resolution, of a quotient by a finite subgroup of \(SL(n,\mathbb{C})\) where \(n\) is 2 or 3, is always crepant. The pluri-toric resolutions also give a generalised degree one McKay correspondence, i.e. a correspondence between certain conjugacy classes in \(G\) and the exceptional prime divisors in a pluri-toric resolution of \(\mathbb{C}^n/G\).
The pluri-toric resolutions are constructed in two steps. In the first step we use a toric resolution of the quotient space \(\mathbb{C}^n/A\), where \(A\) is a maximal abelian subgroup of \(G\), to construct a partial resolution, called a mono-toric partial resolution, of \(\mathbb{C}^n/G\). In the second step we take a mono-toric partial resolution for each maximal abelian subgroup of \(G\) and patch these together to a pluri-toric resolution of \(\mathbb{C}^n/G\). Even if the construction addresses the general case we mainly focus on the cases when \(\mathbb{C}^n/G\) is a surface or a threefold.
Our treatment leaves a number of open questions. resolutions of quotient spaces; pluri-toric resolutions; McKay correspondence Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Equisingularity (topological and analytic), Homogeneous spaces and generalizations Pluri-toric resolutions of quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A germ \(f: (\mathbb{C}^{n+1},0) \to (\mathbb{C},0)\) of analytic functions is called an isolated hyperplane singularity of transversal type \(A_k\), if \(f(x,y_1,y_2, \dots, y_n)= x^kg(x, y_1,y_2, \dots, y_n)\) for some germ \(g:(\mathbb{C}^{n+1}, 0)\to (\mathbb{C},0)\) of analytic functions such that both \(g(x,y_1,y_2, \dots, y_n)\) and \(g(0,y_1,y_2, \dots, y_n)\) have isolated singularities at the origin.
An isolated hyperplane singularity \(f=x^kg\) of transversal type \(A_k\) is studied in this article. Among other results, it is shown that the Milnor fibre of \(f\) is homotopy equivalent to the wedge of a circle and \(k\mu+ \sigma\) copies of the \(n\)-sphere, where \(\mu\) is the Milnor number of \(g(0,y_1,y_2, \dots, y_n)\), and \(\sigma= \dim_C \mathbb{C} \{x,y_1,y_2, \dots, y_n\}/(kg+ xg_x,g_{y_1}, g_{y_2}, \dots, g_{y_n})\). Morse point; isolated hyperplane singularity; transversal type; Milnor fibre Complex surface and hypersurface singularities, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Milnor fibration; relations with knot theory, Families, moduli, classification: algebraic theory, Singularities in algebraic geometry Hyperplane singularities of analytic functions of several complex variables | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 article, a difficult problem in algebraic geometry is translated into a problem about polyhedral complexes. The latter problem is easy to solve, leading to an easy solution of the algebraic geometry problem. The problem in algebraic geometry is a problem relating to semistable reduction, which is an extension of \textit{H. Hironaka}'s resolution of singularities [Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)].
The main result of the paper is that if \(B\) is the affine \(n\)-space over the complex numbers \(\mathbb{C}\), and \(U_B\) is the natural open subscheme of \(B\) whose underlying complex variety is \((\mathbb{C}^\ast)^n\), then given a morphism \(f\) from \(X\) to \(B\) satisfying some reasonable conditions, there is a finite toric morphism from \((B',U_{B'})\) to \((B,U_B)\), and a toroidal modification from \(Y\) to \(X\times_B B'\), such that \(Y\rightarrow B'\) is nearly semistable. (Those unfamiliar with the terminology are referred to the paper for definitions.)
The paper shows that this is equivalent to the following result about triangulations of polyhedral complexes, namely, that if a polyhedral complex has a subcomplex which has a triangulation induced by a lifting function, this triangulation may be extended to a triangulation of the original complex without adding new vertices. Finally, this latter result is proven, which proves the original result from algebraic geometry. resolution of singularities; toroidal variety; polyhedral Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Simplicial sets and complexes in algebraic topology Extending triangulations and semistable reduction. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(Q\) be a quiver and \(\mathbf d\) a dimension vector. Assume that there exists a representation \(T\) of \(Q\) with dimension vector \(\mathbf d\) such that \(\text{Ext}_Q^1(T,T)=0\). Such a representation, if exists, is uniquely determined by \(\mathbf d\). Let \(S_1,\dots,S_r\) be the pairwise-nonisomorphic simple objects in the category \(T^\perp\), which consists of the representations \(M\) of \(Q\) such that \(\Hom_Q(T,M)=0=\text{Ext}_Q^1(T,M)\). The author studies the following problem: under which conditions there exists a representation \(X\) of \(Q\) such that the following conditions are satisfied: {\parindent=6mm\begin{itemize}\item[---] \(\Hom_Q(X,S_i)\neq 0\) for each \(i=1,\dots,r\);\item[---] if \(N\) is a representation of \(Q\) of dimension vector \(N\cdot\mathbf d\) for some \(N\geq 1\) such that \(\Hom_Q(N,S_i)\neq 0\) for each \(i=1,\dots,r\), then \(N\) is a degeneration of \(X\oplus T^{N-1}\), i.e.\ there exists a representation \(Z\) of \(Q\) and an exact sequence of the form \(0\to Z\to Z\oplus X\oplus T^{N-1}\oplus N\to 0\).
\end{itemize}} Among other things the author proves that such a representation exists if \(Q\) is a Dynkin quiver and \(\mathbf d\) is sufficiently big.
The author's motivation for studying this problem is geometric. More precisely, for \(Q\) and \(\mathbf d\) as above one defines a variety called the variety of representations of \(Q\) with dimension vector \(\mathbf d\). It is an affine space equipped with an action of a product \(\text{GL}(\mathbf d)\) of general linear groups. The orbits with respect to this action correspond naturally to the isomorphism classes of the representations of \(Q\) with dimension vector \(\mathbf d\). Moreover, there exists a representation \(T\) of \(Q\) with dimension vector \(\mathbf d\) such that \(\text{Ext}_Q^1(T,T)=0\) if and only if there exists an open orbit in \(\text{GL}(\mathbf d)\) (and, if this is the case, this open orbit corresponds to the isomorphism class of \(T\)). Next, the set of the isomorphism classes of the representations \(X\) with dimension vector \(\mathbf d\) such that \(\Hom_Q(X,S_i)\neq 0\) for each \(i=1,\dots,r\) correspond to the set of the common zeros of the non-constant \(\text{GL}(\mathbf d)\)-semi-invariants. Finally, a representation \(N\) is a degeneration of a representation \(M\) if and only if the orbit corresponding to the isomorphism class of \(N\) is contained in the orbit corresponding to the isomorphism class of \(M\). quivers; semi-invariants; null cones Representations of quivers and partially ordered sets, Group actions on affine varieties, Group actions on varieties or schemes (quotients) Quiver representations with an irreducible null cone. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 desingularization process of a Lagrangian curve (a projectivization of a conic Langrangian variety of a symplectic manifold of dimension four) is studied. It was proved before by the author that the blow-up of a contact manifold with centre at a Lagrangian submanifold has a canonical structure of a contact manifold with logarithmic poles along the exceptional divisor of the blow-up.
Here, a canonical desingularization process is described. It is proved that the graphs of the canonical desingularization of two Lagrangian curves are equal if and only if their generic plane projections are equisingular. curves equisingularity; space curves; Lagrangian curve; desingularization Neto ( O. ) .- Equisingularity and Legendrian curves , Bull. London Math. Soc. 33, p. 527 - 534 ( 2001 ). MR 1844549 | Zbl 1032.58028 Topological invariants on manifolds, Singularities in algebraic geometry, Equisingularity (topological and analytic) Equisingularity and Lagrangean 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 determine the singular locus of the moduli \(W=W_ 1\) of rational Weierstrass fibrations over \(\mathbb{P}^ 1_ \mathbb{C}\). The singular locus has 7 components of dimensions 5, 4, 3, 1, 0, 0 and 3. We also compute explicitly the general singularities which turn out to be cyclic quotient singularities. singular locus of moduli of rational Weierstrass fibration; quotient singularities; elliptic surface Singularities in algebraic geometry, Families, fibrations in algebraic geometry, Families, moduli of curves (algebraic), Singularities of curves, local rings The moduli of rational Weierstrass fibrations over \(\mathbb{P}^ 1\): Singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let $S$ be a Noetherian scheme and $f:X\rightarrow S$ a proper morphism. By SGA 4 XIV [Séminaire de géométrie algébrique du Bois-Marie 1963--1964. Théorie des topos et cohomologie étale des schémas (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat. Tome 3. Exposés IX à XIX. Springer, Cham (1973; Zbl 0245.00002)], for any constructible sheaf $\mathscr{F}$ of $\mathbb{Z}/n\mathbb{Z}$-modules on $X$, the sheaves of $\mathbb{Z}/n\mathbb{Z}$-modules $\mathtt{R}^{i}f_{\star}\mathscr{F}$ obtained by direct image (for the étale topology) are themselves constructible, that is, there is a stratification $\mathfrak{S}$ of $S$ on whose strata these sheaves are locally constant constructible. After previous work of N. Katz and G. Laumon, or L. Illusie, on the special case in which $S$ is generically of characteristic zero or the sheaves $\mathscr{F}$ are constant (with invertible torsion on $S$), here we study the dependency of $\mathfrak{S}$ on $\mathscr{F}$. We show that a natural 'uniform' tameness and constructibility condition satisfied by constant sheaves, which was introduced by O. Gabber, is stable under the functors $\mathtt{R}^{i}f_{\star}$. If $f$ is not proper, this result still holds assuming tameness at infinity, relative to $S$. We also prove the existence of uniform bounds on Betti numbers, in particular for the stalks of the sheaves $\mathtt{R}^{i}f_{\star}\mathbb{F}_{\ell}$, where $\ell$ ranges through all prime numbers invertible on $S$. étale cohomology; constructible sheaf; stratification; tameness; cohomological descent; alteration; ultraproduct Étale and other Grothendieck topologies and (co)homologies, Ramification problems in algebraic geometry, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Ultraproducts (number-theoretic aspects), Algebraic moduli problems, moduli of vector bundles Uniform constructability and tameness in étale cohomology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 Zariski equisingularity. We recall its definition, main properties, and a variety of applications in Algebraic Geometry and Singularity Theory. In the first part of this survey, we consider Zariski equisingular families of complex analytic or algebraic hypersurfaces. We also discuss how to construct Zariski equisingular deformations. In the second part, we present Zariski equisingularity of hypersurfaces along a nonsingular subvariety and its relation to other equisingularity conditions. We also discuss the canonical stratification of such hypersurfaces given by the dimensionality type. Equisingularity (topological and analytic), Singularities in algebraic geometry, Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry Algebro-geometric equisingularity of Zariski | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth complex projective variety, and let \(\text{CH}^k(X,1)\) be the higher Chow group. A higher Chow cycle is said to be decomposable if it lies in the image of the intersection product, i.e., the group of decomposable higher Chow cycles is
\[
\text{CH}^k_{\text{dec}}(X,1)= \text{Im} \{\text{CH}^1(X,1) \otimes\text{CH}^{k-1}(X,1) \to\text{CH}^k (X, 1)\},
\]
and the group of indecomposable cycles is the quotient
\[
\text{CH}^k_{ \text{ind}}(X,1;\mathbb{Q})= (\text{CH}^k (X,1)/ \text{CH}^k_{\text{dec}} (X,1))\otimes\mathbb{Q}.
\]
The first major result of this paper is the following: When \(X=E_1\times E_2\) is a sufficiently general product of two elliptic curves, then \(\text{CH}^k_{\text{ind}}(X,1; \mathbb{Q})\neq \{0\}\).
The second one, which has a similar flavor to Mumford's famous theorem on the kernel of the Albanese map on the Chow group of zero-cycles on a surface of positive genus: When \(X=E_1 \times E_2\times E_3\) is a sufficiently general product of three elliptic curves, the kernel of the cycle class map \(c_{3,1}:\text{CH}^3(X,1; \mathbb{Q})\to H^5_{\mathcal D}(X, \mathbb{Q}(k))\) into Deligne cohomology is uncountable. higher Chow cycle; product of two elliptic curves; product of three elliptic curves; Deligne cohomology Gordon B. and Lewis J.\ D., Indecomposable higher Chow cycles on products of elliptic curves, J. Alg. Geom. 8 (1999), 543-567. (Equivariant) Chow groups and rings; motives, Elliptic curves, Parametrization (Chow and Hilbert schemes), Algebraic cycles Indecomposable higher Chow cycles on products of elliptic 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 his famous paper ``Local Normal Forms of Functions'' [Invent. Math. 35, 87--109 (1976; Zbl 0336.57022)], \textit{V. I. Arnold} presented a classification of hypersurface singularities of moda\-lity smaller or equal to two over the complex numbers together with a singularity determinator (an algorithm to compute the normal form of a given singularity). Arnold gave also a corresponding classification over the real numbers.
The authors developed algorithms to determine the normal form of simple and unimodal singularities over the real numbers. This paper contains the splitting lemma and the case of simple singularities. The algorithms are implemented in the \textsc{Singular} library \texttt{realclassify.lib}. hypersurface singularities; algorithmic classification; real geometry Marais, M., Steenpaß, A.: The classification of real singularities using Singular Part I: splitting lemma and simple singularities. J. Symb. Comput. 68, 61--71 (2015) Singularities in algebraic geometry, Computational aspects of algebraic surfaces, Real algebraic and real-analytic geometry The classification of real singularities using \textsc{Singular}. I: Splitting lemma and simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The bounded derived category of coherent sheaves \(\text{D}^{\text{b}}(X)\) on a smooth projective variety \(X\) has been investigated from many angles in recent years, for example, because it appears in Kontsevich's homological mirror symmetry conjecture or because of the fact that sometimes varieties have equivalent derived categories without being isomorphic. In noncommutative algebraic geometry (NCAG) one now thinks of any (reasonable) triangulated category \(\mathcal{T}\) as the category of sheaves on some noncommutative space. Hence, it is an interesting question whether concepts from geometry make sense in this setting.
In the paper under review the author studies the NCAG-analogues of line bundles, namely two-sided tilting complexes. By definition, if \(A\) is a ring, then a complex of \(A\)-bimodules is a two-sided tilting complex \(\sigma\) if it induces an autoequivalence of the unbounded derived category of \(A\)-modules \(D(\text{Mod}-A)\). If \(A\) is noetherian and has finite global dimension, one can equivalently take the bounded derived category of finitely generated \(A\)-modules \(\text{D}^{\text{b}}(\text{mod}-A)\).
The author defines when a \(\sigma\) as above is (very) ample and it is shown in Section 3 that, if \(\sigma\) is very ample, then the tensor algebra \(T=T_A(H^0(\sigma))\) of \(H^0(\sigma)\) is a graded connected coherent ring over \(A\). The first main result of the paper is that if \(A\) is a finite dimensional \(k\)-algebra of finite global dimension and \(\sigma\) a very ample tilting comlex, then there is an equivalence \(D(\text{mod}-A)\cong D(\text{qcoh}T)\), where \(\text{qcoh}T\) is an abelian category considered as the category of coherent sheaves on \(\text{proj}T\).
The next main result can be viewed as an analogue of \textit{A. Bondal} and \textit{D. Orlov}'s description of the group of autoequivalences of \(\text{D}^{\text{b}}(X)\) if the canonical bundle of \(X\) is ample or anti-ample [Compos.\ Math.\ 125, No.\ 3, 327--344 (2001; Zbl 0994.18007)]. Namely, the author proves that if \(A\) is a finite dimensional \(k\)-algebra of finite global dimension and if \(\omega:=\Hom_k(A,k)[-d]\) is ample or anti-ample for some integer \(d\) (in this case the \(k\)-algebra \(A\) is called Fano of dimension \(d\)), then any \(k\)-linear autoequivalence \(F\) of \(\text{D}^{\text{b}}(\text{mod}-A)\) is isomorphic to \(-\otimes^L_A \tau\) for a two-sided tilting complex \(\tau\).
Finally, in Section 5 the author shows that some representation theoretic properties of a path algebra \(kQ\) of a connected finite acyclic quiver \(Q\) can be captured by the ``canonical bundle'' \(\Hom_k(kQ,k)[-1]\) (the name stems from the fact that Serre functors can be considered as shifted canonical bundles). In particular, \(Q\) is of infinite representation type if and only if \(kQ\) is a Fano algebra. noncommutative algebraic geometry; triangulated categories; tilting complexes; Serre functors; quiver algebras Minamoto, H, Ampleness of two-sided tilting complexes, Int. Math. Res. Not. IMRN, 2012, 67-101, (2012) Noncommutative algebraic geometry, Representations of quivers and partially ordered sets, Derived categories, triangulated categories Ampleness of two-sided tilting complexes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((A,\mathfrak m_A,k)\) be a local Artinian algebra, where \(k \cong A/\mathfrak m_A\) is a field. The \textit{Hilbert function} of \(A\) is the function \(H_A(i) = \dim_{k}\mathfrak{m}_A^{i}/\mathfrak{m}_A^{i+1}\). The \textit{socle degree} of \(A\) is the largest integer such that \(H_A(d) \neq 0\). A result of Iarrobino says that if \(H_A(d-i) = H_A(i) \) for all \(i\) then the \textit{associated graded algebra} \(\text{gr }A = \bigoplus_{i \geq 0} \mathfrak{m}^{i}_A/\mathfrak{m}^{i+1}_{A}\) is Gorenstein. The motivating problem here is to understand local Artinian Gorenstein algebras with fixed Hilbert function up to isomorphism. The author reproves a result of Elias and Rossi that such an algebra with Hilbert function \((1,n,n,1)\) or \((1,n,\binom{n+1}{2},n,2)\) is isomorphic to its associated graded algebra. He then proves a similar result for the Hilbert functions \((1,2,3,3,2,1)\) and \((1,2,2,2,\dots,2,1)\), at least for general such algebras with this Hilbert function. He analyzes the canonically graded case from the point of view of the dual socle generators, showing that this set is irreducible, but in general is neither open nor closed. He shows that there are eleven isomorphism types for algebras with Hilbert function \((1,3,3,3,1)\). Finally, he classifies algebras with Hilbert function \((1,2,2,2,1,1,1)\), obtaining \(|k|+1\) isomorphism types. At the end of the introduction he proposes some natural problems for researchers in this area. Artinian; Gorenstein; local Artinian Gorenstein algebras; Hilbert function Jelisiejew, J.: Classifying Local Artinian Gorenstein Algebras. Collect. Math. \textbf{68}(1), 101-127 (2017) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Commutative rings of differential operators and their modules, Parametrization (Chow and Hilbert schemes) Classifying local Artinian Gorenstein 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 \(V\) be a finite dimensional complex vector space. The author is interested in the Chow variety \(\mathrm{Ch}_d(V)\) of polynomials of degree \(d\), which decompose completely into linear factors. This variety is contained in \(\mathbb{P}S^dV\) and study of its equations is a subject of interest in algebraic geometry and commutative algebra with a long tradition. The Chow variety \(\mathrm{Ch}_d(V)\) is invariant under the action of \(\mathrm{GL}(V)\) induced on the symmetric algebra of \(V\). Thus the ideal of \(\mathrm{Ch}_d(V)\) has a structure of a \(\mathrm{GL}(V)\)-module. The author is interested in the ideal of the set \(\mathrm{Ch}_d(V)\) (that is, he ignores the scheme structure). The main result of this article is Theorem 1.1 which identifies the \(\mathrm{GL}(V)\)-module structure of the graded part of the ideal of \(\mathrm{Ch}_d(V)\) in an explicit way. The methods are standard tools from the representation theory rendered by clever explicit calculations. Chow variety; \(\mathrm{GL}(V)\)-module; Brill's equations; Brill's map Y. Guan, Brill's equations as a \(G L(V)\)-module, arXiv:1508.02293 (2015). Special varieties, Projective techniques in algebraic geometry, Parametrization (Chow and Hilbert schemes) Brill's equations as a \(\mathrm{GL}(V)\)-module | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies the apolar algebra of a complex cubic form \(f\) in 6 variables, which corresponds to a cubic hypersurface \(F\) in a projective space \(\mathbb P^5\). Assuming that \(F\) is not a cone, then the apolar algebra of \(f\) determines a Gorenstein subscheme \(G\) of length \(14\), supported at the origin of an affine space \(\mathbb A^6\). The author studies the location of subschemes \(G\) arising from cubic forms in the scheme \(\mathcal{H}\) parametering Gorenstein subschemes of length \(14\). It is known that \(\mathcal{H}\) consists of two components: the component \(\mathcal{H}_{\mathrm{gen}}\) containing smooth schemes (i.e, sets of distinct points), which has dimension \(84\), and another \(78\)-dimensional component \(\mathcal{H}_{1661}\), corresponding to schemes concentrated at the origin. The author denotes with \(\mathcal{H}_{1661}^{\mathrm{gr}}\) the closed subvariety of \(\mathcal{H}_{1661}\) corresponding to schemes invariant under scalar multiplication.
The main result of the paper shows a connection between the geometry of the decompositions of cubic forms \(f\) in 6 variables and the locus \(\mathcal{H}_{1661}^{\mathrm{gr}}\). Indeed, the author shows that \(\mathcal{H}_{1661}^{\mathrm{gr}}\) is canonically isomorphic to the open scheme parametrizing cubic forms in \(6\) variables which do not correspond to cones. In other words, the apolar subschemes of forms \(f\) not corresponding to cones are in \(1:1\) correspondence with points of \(\mathcal{H}_{1661}^{\mathrm{gr}}\). The intersection \(\mathcal{H}_{\mathrm{gen}}\cap \mathcal{H}_{1661}^{\mathrm{gr}}\) is a divisor of \(\mathcal{H}_{1661}^{gr}\) which, in the isomorphism, corresponds to the \textit{Iliev-Ranestad divisor} \(D\) defined in [\textit{A. Iliev} and \textit{K. Ranestad}, Trans. Am. Math. Soc. 353, No. 4, 1455--1468 (2001; Zbl 0966.14027)]: \(D\) is the divisor of cubic forms \(f\) such that the variety \(\mathrm{VSP}(f,10)\), which parameterizes the Waring decompositions of \(f\) with \(10\) summands, corresponds to the Hilbert scheme of two points on a \(K3\) surface.
The author points out several consequences, both for the study of the Waring decompositions of cubic forms and for the geometry of the Gorenstein locus \(\mathcal{H}\). For instance, the author proves that the apolar scheme of forms of border rank at most \(9\) is smoothable, i.e. it lies in the intersection \(\mathcal{H}_{\mathrm{gen}}\cap \mathcal{H}_{1661}^{gr}\). This presents naturally the \(9\)-secant variety as a divisor of \(D\). The author also finds an effective description of the equation of \(D\), as the determinant of a \(120\times 120\) matrix. Another consequence of the description of \(\mathcal{H}_{1661}^{gr}\) is the proof that both the components of the Gorenstein locus \(\mathcal{H}\) are rational. Hilbert scheme; secant variety Parametrization (Chow and Hilbert schemes), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on \(\mathbb{A}^6\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Moduli spaces of stable quiver representations are important geometric invariant theory constructions as they play an principal role in classification problems using linear algebra data. In the case of fine quiver moduli spaces for acyclic quivers, i.e., smooth projective and rational varieties, several geometric properties of cohomological nature have been well-established by many mathematicians; their singular cohomology is algebraic, the Betti numbers and a certain counting polynomial can be determined explicitly, and their torus-equivariant Chow rings are tautologically generated and presented.
For each indivisible dimension vector for an acyclic quiver, the authors study a class of stabilities, i.e., the canonical chamber, for which the moduli space is a Fano variety, of fixed dimension, Picard rank, and index (Theorem 4.3, page 993). In the special case of toric quiver moduli spaces, the Fano property can also be derived in the context of Mori Dream Spaces.
The authors investigate the theory of Fano varieties and the theory of quiver moduli since it adds classes of arbitrarily high-dimensional Fano varieties to the existing classes of examples. In particular, deep properties of Fano varieties, such as the Mukai conjecture and its various generalizations, can be verified for this class, in the spirit of such a verification for toric varieties, horospherical varieties, and symmetric varieties. On the other hand, rigid properties and the refined invariants of Fano varieties facilitate the study of the geometry of these quiver moduli.
The authors provide the classical construction of quiver moduli spaces and their basic geometric properties, and then explain line bundles on them. By recalling the construction of tautological bundles under the assumption of ample stability, they determine their Picard group and identify one chamber of their ample cone. They establish the Fano property by performing a Chern class computation for the class of the tangent bundle.
The authors also identify all fine moduli spaces of ordered point configurations in projective spaces, such as Fano varieties, including a discussion of a subtle example related to the Segre cubic. They then prove that all fine Kronecker moduli spaces are Fano of nontrivial index and verify the Kobayashi-Ochiai theorem for them purely numerically. They also exhibit a class of Fano quiver moduli spaces of arbitrary Picard rank and index, which can also be interpreted as certain torus quotients of Grassmannians, for which the Mukai conjecture can then be verified. They finally parameterize all Fano toric quiver moduli, which allows them to exhibit particular examples of two- and three-dimensional Fano quiver moduli spaces. quiver representations; quiver moduli spaces; Fano varieties Representations of quivers and partially ordered sets, Algebraic moduli problems, moduli of vector bundles, Fano varieties Fano quiver moduli | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Du Bois singularities have the property that the natural map \(H^i(X^{\text{an}},\mathbb{C}) \to H^i(X^{\text{an}},\mathcal{O}_{X^{\text{an}}})\) is surjective; together with the requirement that the general hyperplane section is also Du Bois this essentially characterises them. Log canonical singularities are Du Bois. In [\textit{S. J. Kovács}, Kyoto J. Math. 51, No. 1, 47-69 (2011; Zbl 1218.14021)]. the notion was generalised to pairs \((X,\Sigma)\). The Authors call a singular point potentially Du Bois, if it has a Zariski open neighbourhood \(U\) containing a subvariety \(\Sigma_U\), such that \((U,\Sigma_U)\) is a Du Bois pair. They prove that the non Du Bois locus has codimension at least three, and that a normal variety \(X\) with \(K_X\) Cartier and potentially Du Bois singularities is log canonical and Du Bois. On the other hand, they develop in detail an example of a normal isolated 3-dimensional potentially Bu Bois singularity with \(K_X\) \(\mathbb{Q }\)-Cartier, that is not Bu Bois. singularities of the minimal model program; Du Bois pairs; log canonical singularities P. Graf, S.J. Kovács, Potentially Du Bois spaces. J. Singul. 8, 117--134 (2014) Singularities in algebraic geometry, Local complex singularities Potentially Du Bois spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(M = M(r,c_ 1, c_ 2)\) 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 note discusses results presented at the 2016 meeting `Workshop on Positivity and Valuations' at Centre de Recerca Matemàtica. Much of the content discussed below appears in [\textit{H. Blum}, Bull. Korean Math. Soc. 58, No. 1, 113--132 (2021; Zbl 1474.14009)] with further details. Singularities in algebraic geometry, Minimal model program (Mori theory, extremal rays), Divisors, linear systems, invertible sheaves Notes on divisors computing MLD's and LCT's | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0743.00011.]
Let \(k\) be an algebraically closed field, \(\text{char}(k)=0\), and \(f\in k[x,y]\), let \((X,k)\) be the canonical embedded resolution of the plane curve \(f=0\). Let \(E_ i\), \(i\in T\), be the irreducible components of \(h^{-1}(f^{-1}(\{0\}))\) (inverse image of \(f=0)\). To each \(i\in T\), associate \((N_ i,\nu_ i)\) where \(N_ i\) and \(\nu_ i\) are the multiplicity of \(E_ i\) in the divisor of \(f\circ h\) and \(h^*(dx\bigwedge dy)\) on \(X\). Fix one component \(E\) with its invariants \((N,\nu)\), denote by \(E_ 1,\dots,E_ k\) the other components of \(h^{-1}(f^{-1}(\{0\}))\) intersecting \(E\), then, you have: \((*)\;\sum^ k_{i=1}(\alpha_ i-1)+2=0\), where \(\alpha_ i=\nu_ i- {\nu\over N}N_ i\), \(1\leq i\leq k\).
In this conferences article, the author gives (without proofs) some generalizations of \((*)\) for hypersurfaces in \(\mathbb{A}_ n(k)\), \(3\leq n\). This problem is much deeper than in the case of plane curves:
(1) The components of \(h^{-1}(f^{-1}(\{0\}))\) may happen not to be linear projective spaces;
(2) If you blow-up a smooth variety inside several components of \(h^{- 1}(f^{-1}(\{0\}))\), you may happen not to separate them and you modify their Pic.
The author finds two formulas \(B_ 1\), \(B_ 2\) generalizing \((*)\) and a special one \(A\) for the components of \(h^{-1}(f^{-1}(\{0\}))\) created by blowing-ups of type 2. intersections; embedded resolution; multiplicity; Pic Embeddings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Picard groups Relations between numerical data of an embedded resolution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present article proves the threefold isolated singularity case of a famous conjecture relating two seemingly completely independent notions: log canonical and \(F\)-pure singularities.
To define \(F\)-pure singularities, we may work locally. An affine variety \(Y\) over a perfect field \(k'\) of characteristic \(p>0\) is \(F\)-pure if the \(\mathcal{O}_Y\)-homomorphism \(\mathcal{O}_Y \to F_* \mathcal{O}_Y\) splits, where \(F : Y \to Y\) is the absolute Frobenius morphism of \(Y\). A variety \(X\) over a field \(k\) of characteristic zero is called of \(F\)-pure type if for every model \(X_A\) of \(X\) over a finitely generated \(\mathbb{Z}\) algebra \(A\) in \(k\), for a dense set of prime ideals \(q \subseteq A\), the reduction \(X_{q}\) mod \(q\) is \(F\)-pure.
On the other hand, the definition of log canonical singularities in characteristic zero follows a different patter. If \(X\) is a variety over a characteristic zero field \(k\), and \(f : Z \to X\) is a log-resolution of singularities, then we may write uniquely \(K_Z \equiv_X E\) for some exceptional divisor \(E\). That is, the intersection of \(K_Z\) and \(E\) with every \(f\)-exceptional curve agrees. Then we say that \(X\) has log canonical singularities, if every coefficient of \(E\) is at least \(-1\).
The surprising conjecture relating the above two notions is that a singularity in characteristic zero is log canonical if and only if it is of \(F\)-pure type. The present article proves this conjecture for the isolated threefold case. The idea is to relate the action of the Forbenius on \(\mathcal{O}_{X_{q}}\) to the action on \(H^{\dim V}(V, \mathcal{O}_V)\), where \(V\) is a minimal stratum in a log resolution of the locus with discrepancy \(-1\). The isomorphism is via Hodge theory, using a previous article of the first author [\textit{O. Fujino}, J. Math. Sci., Tokyo 18, No. 3, 299--323 (2011; Zbl 1260.14006)]. log canonical singularities; \(F\)-pure singularities FT O.~Fujino and S.~Takagi, On the \(F\)-purity of isolated log canonical singularities, Compositio Math. \textbf 149 (2013), no. 9, 1495--1510. Singularities in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Minimal model program (Mori theory, extremal rays) On the \(F\)-purity of isolated log canonical 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 English translation of the Russian original in: Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 193, 10-38 (1991; Zbl 0762.14017); for part I of this paper see Math. USSR, Izv. 35, No. 3, 607-627 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 6, 1269-1290 (1989; Zbl 0722.14019). classification of quartics; singularity Degtyarëv, A.I.: Classification of Quartics Having a Nonsimple Singular Point. II. Topology of Manifolds and Varieties, Advances in Soviet Mathematics, vol. 18, pp. 23-54. American Mathematical Society, Providence (1994) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Comparison of PL-structures: classification, Hauptvermutung, Singularities of curves, local rings, Complex surface and hypersurface singularities Classification of quartics having a nonsimple singular point. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is known that varieties of general type have a finite automorphism group [\textit{H. Matsumura}, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 34, 151--155 (1963; Zbl 0134.16601)]. A celebrated result of Hurwitz bounds the order of the automorphism group of a curve \(C\) of general type by \(84(g(C)-1)\). More recently \textit{G. Xiao} [J. Algebr. Geom. 4, No. 4, 701--793 (1995; Zbl 0841.14011)] proved an analog of Hurwitz theorem for surfaces of general type. Note that the genus of a smooth curve can be interpreted in terms of volume of the canonical class. It is therefore natural to ask if it is possible to bound the order of the automorpshism group of a variety of general type, say \(X\), with a constant depending only on the dimension and \(\mathrm{vol}(X,K_X)\). This impressive paper answers completely to this question and proves the excistence of a universal constant c, depending only on the dimension, such that for any projective variety of general type \(X\) the dimension of the birational automorphism group has at most \(c\cdot \mathrm{vol}(X,K_X)\) elements. This generalization is highly non trivial and uses all the tools and techniques of Minimal Model Program, in particular there is a very subtle part on Descending Chain Conditions, that in the authors words could provide an affirmative answer to the more general Kollár's DCC conjectures [\textit{J. Kollár}, Contemp. Math. 162, 261--275 (1994; Zbl 0860.14014)]. The paper is very well written. automorphism; varieties of general type; volume Christopher D. Hacon, James McKernan & Chenyang Xu, ``On the birational automorphisms of varieties of general type'', Ann. Math.177 (2013) no. 3, p. 1077-1111 Automorphisms of surfaces and higher-dimensional varieties, Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry On the birational automorphisms of varieties of general 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 We use toric geometry to investigate the recently proposed relation between a set of D3 branes at a generalized conifold singularity and type IIA configurations of D4 branes stretched between a number of relatively rotated NS5 branes. In particular we investigate how various resolutions of the singularity corresponds to moving the NS branes and how Seiberg's duality is realized when two relatively rotated NS-branes are interchanged. Unge, R., Branes at generalized conifolds and toric geometry, JHEP, 02, 023, (1999) String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Branes at generalized conifolds and toric geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f,g \in \mathbb C \{x,y\}\) define two germs of analytic curves \((X,0), (Y,0)\) at the origin of \(\mathbb C^2\). Let \((X_t,0)\) be the germ of the curve defined by \(f+tg=0\). An algorithm is given to describe the singularities \((X_t,0)\) for all but finitely many \(t \in \mathbb C\). In particular the topological type is described in terms of \((X,0)\) and \((Y,0)\). equisingularity; topological type Alberich-Carramiñana, M., An algorithm for computing the singularity of the generic germ of a pencil of plane curves, Commun. Algebra, 32, 1637-1646, (2004) Singularities in algebraic geometry, Equisingularity (topological and analytic) An algorithm for computing the singularity of the generic Germ of a pencil of plane curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let $K$ be an algebraically closed field. A parametrization of a germ of a plane curve singularity is given by a pair $(x(t), y(t))$ of power series, $x(t)$, $y(t)\in K[\![t]\!]$. $\Gamma=\{\mathrm{ord}_t(h)\mid h\in K[\![x(t),y(t)]\!]\}$ is the semigroup of the parametrization. It is assumed that the parametrization is primitive, i.e., $\dim_K(K[\![t]\!]/K[\![x(t),y(t)]\!]) <\infty$. Furthermore, it is assumed that $\mathrm{ord}_t(x(t))=:n<\mathrm{ord}_t(y(t))=:m$ and that $n\nmid m$. \par Two parametrizations $(x(t),y(t))$ and $(\overline x(t),\overline y(t))$ are said to be $\mathcal A$-equivalent if there exist automorphisms $\psi: K[\![t]\!]\to K[\![t]\!]$ and $\varphi=(\varphi_1,\varphi_2): K[\![x,y]\!]\to K[\![x,y]\!]$ such that $(x(\psi(t)),y(\psi(t)))=(\varphi_1(\overline x(t),\overline y(t)),\varphi_2(\overline x(t),\overline y(t))))$. A parametrization $(x(t),y(t))$ is called \textit{simple} if there are only finitely many $\mathcal A$-equivalent classes in a deformation of $(x(t),y(t))$. \par \textit{J. W. Bruce} and \textit{T. J. Gaffney} [J. Lond. Math. Soc., II. Ser. 26, 465--474 (1982; Zbl 0575.58008)] classified the simple parametrized plane curve singularities over the complex numbers $\mathbb C$; \textit{K. Mehmood} and \textit{G. Pfister} [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 60(108), No. 4, 417--424 (2017; Zbl 1399.57008)] gave a different proof for this classification and gave also the classification of unimodal singularities. Furthermore, they extended the results to a classification over the real numbers $\mathbb R$. \par In the paper under review the authors give a similar classification in positive characteristic. A parametrization is not simple if $\Gamma$ has a minimal system of generators consisting of more than three elements. For the cases of simple parametrizations the authors describe $\Gamma$ by its generators and give normal forms for the parametrization. characteristic \(p\); parametrized plane curves; simple singularities Singularities of curves, local rings, Singularities in algebraic geometry, Plane and space curves Simple singularities of parametrized plane curves in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be an abelian variety of dimension \(g\) over a field \(k\) and let \(h(A) = \bigoplus^{2g}_{i = 0} h^ i(A)\) be the Chow motive of \(A\) constructed by \textit{C. Deninger} and \textit{J.-P. Murre} [J. Reine Angew. Math. 422, 201-219 (1991; Zbl 0745.14003)]. The author proves the motivic analog of the hard Lefschetz theorem: \(L^{g-i} : h^ i(A) @>\sim>> h^{2g-i} (A)(g-i)\) and of the Lefschetz decomposition theorem
\[
h^ i(A) = \bigoplus_ k L^ kP^{i-2k} (A).
\]
Here \(L\) is the Lefschetz operator associated to a symmetric polarization of \(A\), and \(P^ i(A)\) is a motivic analog of the primitive part in cohomology. The decomposition isomorphism was independently proven by \textit{A. J. Scholl} [in Motives, Proc. Res. Conf. Motives, Seattle 1991, Proc. Symp. Pure Math. 55, Part 1, 163-187 (1994)].
All these results can be also applied to abelian schemes. As an application the author reproves an earlier result of \textit{C. Soulé} [Math. Ann. 268, 317-345 (1984; Zbl 0573.14001)] on isomorphism of the Chow groups \(CH^ p (A,\mathbb{Q}) \to CH^{g-p} (A,\mathbb{Q})\), \(2p \leq g\), for any abelian variety \(A\) over a finite field \(k\). abelian variety; Chow motive; hard Lefschetz theorem; Lefschetz decomposition Künnemann, K., A Lefschetz decomposition for Chow motives of abelian schemes, Invent. Math., 113, 1, 85-102, (1993) Generalizations (algebraic spaces, stacks), Algebraic theory of abelian varieties, Parametrization (Chow and Hilbert schemes), Algebraic cycles A Lefschetz decomposition for Chow motives of abelian 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 central concept discussed in this research monograph is that of ``local moduli suite'' of an algebro-geometric object X. Roughly speaking this is a collection \(\{M_{\tau}\}\) of algebraic spaces each \(M_{\tau}\) prorepresenting the \(\tau\)-constant deformations of X occurring in the family \(\pi_{\tau}\) obtained by restricting an algebraisation \(\pi:\quad \tilde X\to H\) of the formal versal family of X to the ``\(\tau\)-constant stratum''.
An abstract theorem is proved first which guarantees the existence of the local moduli suite in the presence of certain axioms on the formal versal family. Results along this line were independently obtained by Palamodov and Saito. A conjecture of \textit{J. Wahl} [cf. Topology 20, 219-246 (1981; Zbl 0484.14012)] on the dimension of a smoothing component is then proved [this was independently proved by \textit{G.-M. Greuel} and \textit{E. Looijenga} in Duke Math. J. 52, 263-272 (1985; Zbl 0587.32038)].
Next the case of hypersurface singularities is investigated in detail (some modifications of the general setting of local moduli suites are here necessary). Here one is interested in the dimensions of the loci \(M_{\mu \tau}\) of all points in \(M_{\tau}\) corresponding to singularities with a given Milnor number \(\mu\).
Quite precise results are obtained in the case of weighted homogeneous plane curve singularities; in particular a coarse moduli space is proved to exist for all plane curve singularities with given semigroup \(\Gamma =<a_ 1,a_ 2>\), \((a_ 1,a_ 2)=1\) and minimal Tjurina number \(\tau\) and a computation is given for its dimension.
Part of these results are joint work of the authors with \textit{B. Martin}; they are a remarkable contribution to Zariski's program on the moduli problem for curve singularities. local moduli suite; formal versal family; hypersurface singularities; homogeneous plane curve singularities; coarse moduli space; Tjurina number O. A. Laudal and G. Pfister, ''Local moduli and singularities,'' In: Lecture Notes in Math., Vol. 1310, Springer, Berlin (1988). Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities in algebraic geometry, Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Fine and coarse moduli spaces, Singularities of surfaces or higher-dimensional varieties, Families, moduli of curves (algebraic) Local moduli and singularities. Appendix (by B. Martin and G. Pfister): An algorithm to compute the kernel of the Kodaira-Spencer map for an irreducible plane curve singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper classifies analytically isolated plane curve singularities defined by weighted homogeneous polynomials \(f(y,s)\), which are not topologically equivalent to homogeneous polynomials, in an elementary way. Around the proof of this analytic classification theorem, assuming that \(g(y,z)\) either satisfies the same property as the above \(f\) does or is homogeneous, then it can be proved easily that the weights of the above \(g\) determine the topological type of \(g\) and conversely. Hence this gives another easy proof for the topological classification theorem of quasihomogeneous singularities in \(\mathbb{C}^{2}\), which was already known. As an application, it can be shown that for a given \(h\), where \(h(w_{1}, \dots{},w_{n})\) is a quasihomogeneous holomorphic function with an isolated singularity at the origin or \(h(w_{1})=w_{1}^{p}\) with a positive integer \(p\), analytic types of isolated hypersurface singularities defined by \(f+h\) are easily classified with \(f\) above-mentioned. plane curve singularity; weighted homogeneous polynomial; classification theorem; isolated singularity Kang Chunghyuk: Analytic types of plane curve singularities defined by weighted homogeneous polynomials. Trans. AMS 352(9), 3995--4006 (2000) Equisingularity (topological and analytic), Global theory and resolution of singularities (algebro-geometric aspects) Analytic types of plane curve singularities defined by weighted homogeneous polynomials | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 an unicuspidal curve of degree \(d\) and genus \(g\). The main result in the paper is to prove that
\[
k-g \leq I_{jd+1-2k}- \frac{(d-j-2)(d-j-1)}{2} \leq k,
\]
for every \(-1 \leq j \leq d-2\) and every \(0 \leq k \leq g\), where \(I_m\) is the number of gaps of the semigroup of the singularity in \([m, + \infty)\). This result has been independently proved by \textit{M. Borodzik}, \textit{M. Hedden} and \textit{C. Livingston} [``Plane algebraic curves of arbitrary genus via Heegaard Floer homology'', Preprint, \\url{arXiv:1409.2111}] and extends another by \textit{M. Borodzik} and \textit{C. Livingston} [Forum Math. Sigma 2, Article ID e28, 23 p. (2014; Zbl 1325.14047)] which proves a conjecture by Bobadilla, Luengo, Melle-Hernández and Nèmethi [\textit{J. Fernández De Bobadilla} et al., Proc. Lond. Math. Soc. (3) 92, No. 1, 99--138 (2006; Zbl 1115.14021)]. A generalization for curves with more cusps is also given. As an interesting application, the authors prove that for \(C\) unicuspidal with \(g \leq 1\) and only a Puiseux pair, with singularity of type \((a,b)\), it holds that \(a+b=3d\) if \(d\) is large enough. They also show that, as a consequence, for all genera \(g\) such that \(g \cong 2 \pmod{5}\) or \(g \cong 4 \pmod{5}\), there are only finitely many unicuspidal curves of genus \(g\) up to equisingularity. unicuspidal curves; Heegaard Floer homology Bodnár, J.; Celoria, D.; Golla, M., Cuspidal curves and Heegaard Floer homology, Proc. Lond. Math. Soc., 112, 3, 512-548, (2016) Plane and space curves, Singularities in algebraic geometry, Knots and links in the 3-sphere, Floer homology, Singularities of curves, local rings Cuspidal curves and Heegaard Floer 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 \textit{Yu. G. Zarkhin}'s trick [Math. USSR, Izv. 8, 477--480 (1975; Zbl 0332.14016); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 471--474 (1974)] claims that for an abelian variety \(A\) over a field \(k\), \((A\times\hat{A})^4\) admits a principal polarization. This implies the Tate conjecture for abelian varieties over finite fields [\textit{J. Tate}, Invent. Math. 2, 134--144 (1966; Zbl 0147.20303)]. The aim of the paper under review is to study a version of Zarhin's trick for \(K3\) surfaces and to give new geometric and simple proofs of the Tate conjecture for divisors of \(K3\) surfaces.
The \(K3\) version of Zarhins's trick consists of two steps. The first step is to construct big line bundles on moduli spaces of sheaves on \(K3\) surfaces. More precisely, for arbitrary positive integer \(d\) there exists a positive integer \(r\) such that, for infinitely many positive integers \(m\), if \((X,H)\) is a polarized \(K3\) surface of degree \(2md\) over a field \(k\), there exists a \(4\)-dimensional smooth projective moduli space \({\mathcal M}\) of stable sheaves on \(X\) together with a line bundle \(L\) on \({\mathcal M}\) such that \(c_1(L)^4 = r\) and the Beauville-Bogomolov form \(q(K)>0\) (Theorem~1.1 and Theorem~3.3). \(L\) or its dual is big and \({\mathcal M}\) is an analogue of \((A\times \hat{A})^4\) in the original version of Zarhin's trick.
The second step is to consider to what extent a birational version of Matsusaka's big theorem holds in this setting. The author proved that for arbitrary positive integers \(n\) and \(r\), the irreducible holomorphic sympletic varieties over \({\mathbb C}\) of dimension \(2n\) having the line bundles \(L\) with \(c_1(L)^{2n}=r\) and the Beauville-Bogomolov form \(q(L)>0\) are birationally bounded. Namely, there exists a projective morphism \({\mathcal X}\to S\) with a scheme \(S\) of finite type over \({\mathbb C}\) such that, for a complex point \(s\in S\), \({\mathcal X}_s\) is birational to such holomorphic symplatic variety \(X\) (Theorem~3.3 and Theorem~1.2). A weak version of this result over a finite field of characteristic \(\geq 5\) is also proved with a similar method, where the Kuga-Satake construction is used instead of the period map (Proposition~3.16).
As applications, the author gives new geometric and simple proofs of the Tate conjecture for \(K3 \)surfaces over fields of characteristic \(\geq 5\) or in any characteristic when the Picard number is \(\geq 2\). The latter contains the new result for characteristic~\(2\). holomorphic sympltic varieties; \(K3\) surfaces; Tate conjecture Charles, F., \textit{birational boundedness for holomorphic symplectic varieties, zarhin's trick for \textit{K}3 surfaces, and the Tate conjecture}, Ann. of Math. (2), 184, 487-526, (2016) \(K3\) surfaces and Enriques surfaces, Finite ground fields in algebraic geometry, Parametrization (Chow and Hilbert schemes) Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for \(K3\) surfaces, and the Tate conjecture | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 scheme (over an algebraically closed field of characteristic \(0\)), and \(\mathrm{Hilb}^{sc}(X)\) the Hilbert scheme of smooth connected curves in \(X\). For \(X={\mathbb P}^3\) or \(X\) a prime Fano \(3\)-fold it is known that \(\mathrm{Hilb}^{sc}(X)\) contains a generically non-reduced irreducible component (see the Introduction of the paper under review and references therein). Moreover, in this second case, the general element in this component is contained in a smooth element of the anticanonical linear system, which is a smooth \(K3\)-surface. The paper under review deals with the case of Enriques-Fano \(3\)-folds \(X \subset {\mathbb P}^N\), that is, normal projective \(3\)-folds whose general hyperplane section is an Enriques surface, and which are not a cone. In Theorem 1.1, sufficient conditions on \(X\) are provided to get the existence of a generically non-reduced component \(W\) of \(\mathrm{Hilb}^{sc}(X)\) whose general member is contained in an Enriques surface in \(X\). To be precise, the condition is the existence of a half pencil \(E\) (\(2E\) is an elliptic pencil) of anticanonical degree \(\geq 2\) on a smooth hyperplane section \(S \subset X\) and such that \(h^1 (E,N_{E/X}(E))=0\) (\(N_{E/X}\) stands for the normal bundle of \(E\) in \(X\)). The dimension of \(W\) and the linear class of the general curve in its surface are also provided. This result is a consequence of Theorem 1.2 where, under some hypotheses (see 1.2 for details), the (un)obstructedness of curves in smooth hyperplane sections of Enriques-Fano \(3\)-folds is studied and a computation of the dimension of \(\mathrm{Hilb}^{sc}(X)\) is provided. Hilbert scheme; obstruction; Enriques surface; Enriques-Fano threefold Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Formal methods and deformations in algebraic geometry Obstructions to deforming curves on an Enriques-Fano 3-fold | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 conjecture of \textit{M. Finkelberg} and \textit{A. Ionov} [Bull. Inst. Math., Acad. Sin. (N.S.) 13, No. 1, 31--42 (2018; Zbl 1397.05203)] is proved on the basis of a generalization of the Springer resolution and the Grauert-Riemenschneider vanishing theorem. As a corollary, it is proved that the coefficients of the multivariable version of Kostka functions introduced by Finkelberg and Ionov are nonnegative. Kostka-Shoji polynomials; cohomology vanishing; quivers; Lusztig convolution diagram Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Symmetric functions and generalizations, Representations of quivers and partially ordered sets, Vanishing theorems in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Higher cohomology vanishing of line bundles on generalized Springer resolution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors apply a \(\mathbb{Z}\)-algebra construction and good filtrations to study the representation theory of Cherednik algebras of type \(A_{n-1}\), denoted \(H_c\), and of its spherical subalgebras \(U_c=eH_ce\). This representation theory is related to \(\text{Hilb}(n)\) and to the resolution of singularities \(\tau:\text{Hilb}(n)\rightarrow\mathfrak h\oplus\mathfrak h^\ast/W.\)
It is assumed throughout the article that \(c\neq\frac 12+\mathbb{Z}\), and for simplicity that \(c\notin\mathbb{Q}_{\leq 0}.\) The aim of the article is to relate \(H_c\)- and \(U_c\)-modules to sheaves over the Hilbert scheme of points in the plane.
It is shown that finite-dimensional \(H_c\)- and \(U_c\)-modules form moduli for the sections of some sheaves on the Hilbert scheme of points on the plane. The Borel-Weil theorem describes the finite-dimensional simple modules of a complex semisimple Lie algebra \(\mathfrak g\) as sections of line bundles on the flag variety \(\mathfrak B=G/B,\) and some properties of \(U_c\) are similar to those of \(U(\mathfrak g)/P\) where \(P\) is a minimal primitive ideal in the enveloping algebra \(U(\mathfrak g),\) and it is the Beilinson-Bernstein equivalence of categories that provides the analogue for this approach.
Both \(H_c\) and \(U_c\) can be filtered by degree of differential operators with associated graded rings \(\text{gr}H_c\cong\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]\ast W\) and \(\text{gr}U_c\cong\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W\) where \(\mathfrak h\subset \mathbb{C}^n\) is the reflection representation of the \(n\)th symmetric group \(W\). The role of the Springer resolution is played by the crepant resolution \(\tau:\text{Hilb}(n)\rightarrow\mathfrak h\oplus\mathfrak h^\ast/W\), where \(\text{Hilb}(n)\) is the subvariety of \(\text{Hilb}^n\mathbb{C}^2\) consisting of those ideals of \(\mathbb{C}[x,y]\) of colength \(n\) whose supports have centre of mass at the origin.
The analogy with the Beilinson-Bernstein equivalence is then given by one of the main theorems of the article:
There exists a filtered \(\mathbb{Z}\)-algebra \(B\) such that
(1) \(U_c\)-mod, the category of finitely generated \(U_c\)-modules, is equivalent to \(B\)-{qgr}, the quotient category of finitely generated graded \(B\)-modules modulo those of finite length;
(2) gr\(B\), the associated graded ring of \(B\), is isomorphic to the \(\mathbb{Z}\)-algebra associated with the homogeneous coordinate ring \(\bigoplus_{k\geq 0}H^0(\text{Hilb}(n),\mathcal{L}^k)\) for a certain ample line bundle \(\mathcal{L}\) on \(\text{Hilb}(n)\).
This theorem shows that \(U_c\) can be regarded as a noncommutative deformation of a homogeneous coordinate ring of the Hilbert scheme \(\text{Hilb}(n)\), and it provides a recipe for passing from a left \(U_c\)-module \(M\) with a good filtration \(\Lambda\) to a coherent sheaf \(\Phi(M)=\Phi_{\Lambda}(M)\) on \(\text{Hilb}(n)\). This depends on the induced tensor product filtration. This also works for filtered \(H_c\)-modules \((N,\Lambda)\), resulting in a sheaf \(\widehat{\Phi}_{\Lambda}(N)\) also on \(\text{Hilb}(n)\). Under the hypotheses on \(c\), \(H_c\) and \(U_c\) are Morita equivalent. The basic technique of this article is to exploit the functors \(\Phi\) and \(\widehat{\Phi}\) to understand the representation theory of \(U_c\) and \(H_c\).
For \(\mathcal{O}_c\), the category of finitely generated \(H_c\)-modules on which \(\mathbb{C}[\mathfrak h^\ast]\) acts locally nilpotently, the analogues of Verma modules over a simple complex Lie algebra are the standard modules \(\Delta_c(\mu)\) where \(\mu\) is an irreducible representation of the \(n\)th symmetric group \(W\). For any good filtration \(\Lambda\) on \(e\Delta_c(\mu)\), the associated variety is independent of \(\mu\). On the other hand, the associated variety of \(\widetilde{e\Delta_c(\mu)}\) is a subvariety of \(\text{Hilb}(n)\) which does depend on \(\mu\). This result is stated by introducing the characteristic cycle, \(\text{\textbf{Ch}}(\Delta_c(\mu))\) which counts the irreducible components of the characteristic variety of \(\hat{\Phi}(\Delta_c(\mu))\) in \(\text{Hilb}(n)\) with multiplicities. Another main result of the article is the following:
Let \(\Delta_c(\mu)\) be the standard \(H_c\)-module corresponding to \(\mu\in \text{Irrep}(W)\). Then
\[
\text{\textbf{Ch}}(\Delta_c(\mu))=\sum_{\lambda}K_{\mu\lambda} [Z_\lambda]
\]
where \(K_{\mu\lambda}\) are Kostka numbers and the \(Z_\lambda\)'s are the irreducible components of \(Z=\tau^{-1}(\mathfrak h/W).\)
This introductory remarks gives just some of the important results of this article. It contains a lot more, not possible to give any meaning in such a short review as this. It should be said however that the definitions and theory needed to understand this theory is nicely and explicitly treated. Graded and filtered modules for \(\mathbb Z\)-algebras, rational Cherednik algebras and the Hilbert schemes mentioned above are both sections that are understandable without too much special knowledge. The two final sections treating the representations of \(H_c\), coherent sheaves on \(\text{Hilb}(n)\), finite-dimensional \(U_c\)-modules and their characteristic varieties is somewhat harder to completely understand. Kostka numbers; Procesi bundle; Verma modules; Morita equivalence I. Gordon, J. T. Stafford, Rational Cherednik algebras and Hilbert schemes. II. Representations and sheaves, Duke Math. J. 132 (2006), no. 1, 73--135. Parametrization (Chow and Hilbert schemes), Module categories in associative algebras, Modifications; resolution of singularities (complex-analytic aspects), Deformations of associative rings, Combinatorial aspects of representation theory Rational Cherednik algebras and Hilbert schemes. II: Representations and sheaves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Dans Ann. Math., II. Ser. 32, 485-511 (1931; Zbl 0001.40301), \textit{O. Zariski} étudie les revêtements de \(\mathbb{P}^ 2\) ramifiés le long d'une courbe \(C\) et d'une droite à l'infini \(L\) transverse à \(C\). Par définition, l'irrégularité \(q\) d'un tel revêtement \(\widehat{X}\) est la dimension du groupe de cohomologie \(H^ 1(Z,{\mathcal O}_ Z)\), où \(Z\) est une résolution des singularités de \(\widehat{X}\). Grâce à la théorie de Hodge, l'irrégularité est liée au premier nombre de Betti de la surface \(Z\) par l'égalité \(2q = b_ 1(Z) = \dim H^ 1(Z,\mathbb{C})\). Dans le cas où la courbe \(C\) a pour seules singularités des points doubles ordinaires et des points cuspidaux, Zariski calcule l'irrégularité du revêtement en fonction de la surabondance des systèmes linéaires des courbes de certains degrés fixés passant par les points cuspidaux de \(C\), c'est à dire en fonction des groupes de cohomologie \(H^ 1(\mathbb{P}^ 2,{\mathcal A}(n))\), où \(\mathcal A\) est l'idéal des fonctions holomorphes sur \(\mathbb{P}^ 2\) s'annulant aux points cuspidaux de \(C\).
Dans cet article nous nous proposons de généraliser ces résultats au cas d'une surface non singulière projective \(X\) et d'une courbe réduite \(C\) sur \(X\) ayant des singularités quelconques. Si nous notons \(\omega_ X\) le faisceau dualisant sur \(X\) et \({\mathcal A}_ \alpha\) le faisceau sur \(X\) constitué des fonctions holomorphes dont l'exposant de Hodge en chaque point singulier de \(C\) est strictement supérieur à \(\alpha\), nous avons les résultats:
Théorème 3.1. Soit \(H\) un diviseur très ample sur une surface non singulière projective \(X\), soit \(C\) une courbe réduite sur \(X\) appartenant au système linéaire \(| mH|\) et soit \(L\) une courbe non singulière appartenant à \(| H|\) transverse à \(C\); alors l'irrégularité \(q\) du revêtement cyclique de \(X\) de degré \(n\), ramifié le long de \(C \cup L\) est égale à:
\[
q = \sum \dim H^ 1(X,{\mathcal A}_ \alpha (\lceil pm/n\rceil H) \otimes \omega_ X)
\]
où \(\alpha = p/n - 1\), \(p = 0,1,\dots,n - 1\), et où pour tout nombre réel \(x\) on note \(\lceil x\rceil\) le plus petit entier supérieur ou égal à \(x\).
Théorème 3.6. Si \(\beta\) est le plus grand exposant strictement négatif appartenant au spectre d'une singularité de la courbe \(C\), le groupe \(H^ 1(X,{\mathcal A}_ \beta(sH) \bigotimes \omega_ X)\) est nul pour tout entier \(s\) tel que \(m \geq s > m(\beta + 1)\). irregularity; divisor; cyclic coverings; resolution of singularity; vanishing of first cohomology Vaquié, Michel, Irrégularité des revêtements cycliques des surfaces projectives non singulières, Amer. J. math., 114, 6, 1187-1199, (1992), MR 1198299 (94d:14015) Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Vanishing theorems in algebraic geometry Irregularity of cyclic coverings of nonsingular projective surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0718.00010.]
The paper deals with actions of a finite group of automorphisms of the projective space \(\mathbb{P}^ n\) on an algebraic variety \(X\subset\mathbb{P}^ n\). One takes the quotients of \(G\)-invariant cycles modulo ``averaged'' ones and then one considers the homotopy groups of these spaces. One shows that they are functorial invariants and satisfy the complex suspension property at all primes away from the order of the group. The paper reviews also some recent results on the structure of Chow schemes. actions on an algebraic variety; automorphisms; Chow schemes H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Algebraic cycles and group actions, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 261 -- 277. Algebraic cycles, Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes) Algebraic cycles and group actions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0559.00004.]
The paper studies the mixed Hodge structures on the cohomology groups H(X), \(H_ x(X)\) and IH(X) of an algebraic variety X with isolated singularities, and of a resolution \(f : \tilde X\to X.\) The results of the paper (purity theorems of IH(X), difficult Lefschetz theorem,...) were suggested by (and are consequences of) theorems on perverse sheaves by P. Deligne, O. Gabber, A. Beilinson and I. Bernstein, viewing the relation of the above cohomology groups with the sheaves of perverse cohomology \({\mathbb{R}}f_*{\mathbb{C}}_{\tilde X}\) and the perverse sheaf \(IC_ X^{\bullet}\). The author gives ''purely transcendental'' proofs of this results. He uses especially the filtred complex of de Rham \(\Omega_ X\) of X. He shows that the results on the vanishing of cohomology sheaves from \textit{F. Guillén}, the author and \textit{F. Puerta}: ''Théorie de Hodge via schémas cubiques'' (prepublication 1972)] lead to adequate variants for the decomposition theorem of Hodge, vanishing theorem of Kodaira, difficult theorem of Lefschetz for strongly pseudo-convex varieties (in the direction of Nakano and Grauert- Riemenschneider). Further the filtered de Rham complex of a rational singularity and an isolated hypersurface singularity is studied in detail. pseudo-convex varieties; mixed Hodge structures; purity theorems; perverse sheaves; vanishing theorem of Kodaira; theorem of Lefschetz; filtered de Rham complex of a rational singularity Navarro-Aznar V., Sur la théorie de Hodge des variétés algébriques à singularités isolées, Systèmes Différentiels et Singularités (Luminy 1983), Astérisque 130, Société Mathématique de France, Paris (1985), 272-307. Transcendental methods, Hodge theory (algebro-geometric aspects), de Rham cohomology and algebraic geometry, Singularities in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Singularities of surfaces or higher-dimensional varieties Sur la théorie de Hodge des variétés algébriques à singularités isolées. (On Hodge theory of algebraic varieties with isolated singularities) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We construct five families of 2D moduli spaces of parabolic Higgs bundles (respectively, local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve (respectively, twisted cotangent bundle). We show that the Hilbert scheme of \(m\) points of these surfaces is again a moduli space of parabolic Higgs bundles (respectively, local systems), confirming a conjecture of \textit{P. Boalch} [Publ. Math., Inst. Hautes Étud. Sci. 116, 1--68 (2012; Zbl 1270.34204)] in these cases and extending a result of \textit{A. Gorsky} et al. [Commun. Math. Phys. 222, No. 2, 299--318 (2001; Zbl 0985.81107)]. Using the McKay correspondence, we establish the autoduality conjecture for the derived categories of the moduli spaces of Higgs bundles under consideration. Gröchenig, M., Hilbert schemes as moduli of Higgs bundles and local systems, Int. math. res. not. IMRN, 2014, 23, 6523-6575, (2014) Relationships between algebraic curves and integrable systems, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, McKay correspondence Hilbert schemes as moduli of Higgs bundles and local systems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider the variety of \(X_d\) of punctual flags of length \(d\) in dimension 2, defined as the closure of the variety of complete curvilinear zero-dimensional subschemes of length \(\leq d\) with support at the fixed point on a non-singular algebraic surface; this closure is taken in the direct product of punctual Hilbert schemes.
It is known that for \(2\leq d\leq4\) the variety \(X_d\) is smooth and coincides with the projectivization of the rank 2 vector bundle over \(X_{d-1}\), described as the corresponding \(\mathcal{E}xt\)-sheaf. A similar bundle \(\mathcal E\) is also defined over \(X_4\). However, its projectivization \(\mathbb{P}(\mathcal E)\) is only birational isomorphic, but not isomorphic to \(X_5\). M. Gulbrandsen showed that \(X_5\) has an entire curve of singularities. In the present article, we give a precise description of a minimal birational transformation of \(X_5\) into \(\mathbb{P}(\mathcal E)\) and interpret this transformation and the singularities of \(X_5\) in terms of \(\mathcal{E}xt\)-sheaves. Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds, (Co)homology theory in algebraic geometry On the variety of complete punctual flags of length 5 in dimensions 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 A theorem of Hochster and Huneke says that if \(A\) is an equal characteristic regular domain, \(R\) is a module-finite and torsion-free extension of \(A\) and \(R \rightarrow S\) is any homomorphism from \(R\) to a regular ring \(S\) then for every \(A\)-module \(M\) and every \(i \geq 1\), the map \(\mathrm{Tor}_i^A (M,R) \rightarrow \mathrm{Tor}_i^A(M,S)\) vanishes. They conjectured that it also holds in mixed characteristic. This has become known as the \textit{vanishing conjecture for vanishing of Tor}, and it implies other well-known conjectures. Extending this, the author says that an excellent local domain \((S,\mathfrak n)\) \textit{satisfies the vanishing conditions for maps of Tor} if, for every \(A \rightarrow R \rightarrow S\) with \(A\) regular and \(A \rightarrow R\) a module-finite torsion-free extension, and every \(A\)-module \(M\), the map \(\mathrm{Tor}_i^A (M,R) \rightarrow \mathrm{Tor}_i^A(M,S)\) vanishes for every \(i \geq 1\). The main result of this paper is that in equal characteristic, rings that satisfy the vanishing conditions for maps of Tor are exactly \textit{derived splinters} in the sense of a paper of Bhatt. The author also shows that an equivalent condition is that for every regular local ring \(A\) with \(S = A/P\) and every module-finite torsion-free extension \(A \rightarrow B\) with \(Q \in \mathrm{Spec } B\) lying over \(P\), the map \(P \rightarrow Q\) splits as a map of \(A\)-modules. The author concludes with a corollary that characterizes rational singularities in terms of splittings in module-finite extensions. vanishing conjecture for maps of Tor; derived splinters; rational singularities 10.4171/JEMS/768 Homological conjectures (intersection theorems) in commutative ring theory, Singularities in algebraic geometry The vanishing conjecture for maps of Tor and derived splinters | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite subgroup \(\Gamma\subset\mathrm{SL}(2,\mathbb{C})\), we identify fine moduli spaces of certain cornered quiver algebras, defined in earlier work, with orbifold Quot schemes for the Kleinian orbifold \(\big[ \mathbb{C}^2\!/\Gamma\big]\). We also describe the reduced schemes underlying these Quot schemes as Nakajima quiver varieties for the framed McKay quiver of \(\Gamma\), taken at specific non-generic stability parameters. These schemes are therefore irreducible, normal and admit symplectic resolutions. Our results generalise our work [\textit{A. Craw} et al., Algebr. Geom. 8, No. 6, 680--704 (2021; Zbl 1494.16013)] on the Hilbert scheme of points on \(\mathbb{C}^2/\Gamma\); we present arguments that completely bypass the ADE classification. quot scheme; quiver variety; Kleinian orbifold; preprojective algebra; cornering Representations of quivers and partially ordered sets, Actions of groups on commutative rings; invariant theory, Algebraic moduli problems, moduli of vector bundles, McKay correspondence Quot schemes for Kleinian orbifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\mathcal F\) be a coherent sheaf on a projective space \({\mathbb P}_n\), with Hilbert polynomial \(P_{\mathcal F}(m) = (e/d!)m^d+\cdots \). \(d\) equals the dimension of \(\text{supp}{\mathcal F}\) and \(e\) is the multiplicity of \(\mathcal F\). The reduced Hilbert polynomial of \(\mathcal F\) is defined by \(p_{\mathcal F} := P_{\mathcal F}/e\). According to \textit{C.T. Simpson} [Inst. Hautes Études Sci. Publ. Math. 79, 47--129 (1994; Zbl 0891.14005)], \(\mathcal F\) is called semi-stable if it is pure (i.e., \(\text{supp}{\mathcal F}\) is equidimensional and the only associated points of \(\mathcal F\) are the generic points of the irreducible components of \(\text{supp}{\mathcal F}\)) and, for every non-zero subsheaf \({\mathcal F}^{\prime}\subset {\mathcal F}\), \(p_{{\mathcal F}^{\prime}}(m)\leq p_{\mathcal F}(m)\) for large \(m\). Simpson showed that, for every numerical polynomial \(P\), there exists a projective scheme \(M_P({\mathbb P}_n)\) which is a coarse moduli space for semi-stable sheaves on \({\mathbb P}_n\) with Hilbert polynomial \(P\). For \(P(m) = em+\chi \), the moduli spaces \(M_P({\mathbb P}_2)\) of semi-stable sheaves supported on plane curves were investigated by \textit{J. Le Potier} [Rev. Roumaine Math. Pures Appl. 38 (7--8), 635--678 (1993; Zbl 0815.14029)].
In the paper under review, the authors study the moduli space \(M = M_P({\mathbb P}_3)\) for \(P(m) = 3m+1\). Since the coefficients of \(P\) are coprime, a semi-stable sheaf \(\mathcal F\) on \({\mathbb P}_3\) with Hilbert polynomial \(P\) is already stable and \(M\) is a fine moduli space. The authors show that either
(0) \({\mathcal F} = {\mathcal O}_C\) where \(C\) is a Cohen-Macaulay cubic curve in \({\mathbb P}_3\) (with Hilbert polynomial \(P\)), or
(1) \({\mathcal F} = {\mathcal H}om({\mathcal I},{\mathcal O}_C)\), where \(C\) is a cubic effective divisor on a plane \(H\subset {\mathbb P}_3\) and \({\mathcal I}\subset {\mathcal O}_C\) is the ideal sheaf of a point \(p\in C\).
Accordingly, \(M\) has two irreducible components \(M_0\) and \(M_1\). \(M_1\) is a fibration over the dual projective space \({\mathbb P}_3^{\ast}\) with fibre \(M_P({\mathbb P}_2)\) isomorphic to the universal family of plane cubics, and \(M_0\) is isomorphic to the closure of the space of twisted cubic curves in the Hilbert scheme \(\text{Hilb}_P({\mathbb P}_3)\) described by \textit{R. Piene} and \textit{M. Schlessinger} [Am. J. Math. 107, 761--774 (1985; Zbl 0589.14009)]. The intersection \(M_0\cap M_1\) parametrizes the sheaves of the form (1) with \(p\) a singular point of \(C\). \(M_0\), \(M_1\), and \(M_0\cap M_1\) are smooth, rational, of dimension 12, 13, and 11, respectively.
Using results of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [Ann. Math. (2) 130, 159--187 (1989; Zbl 0716.14002)], the authors also compute the Betti numbers of the Chow groups of \(M\). moduli space; twisted cubic curve Hans-Georg Freiermuth - Günther Trautmann, On the Moduli Scheme of Stable Sheaves Supported on Cubic Space Curves, Amer. J. Math., 126 (2004), pp. 363--393. Zbl1069.14012 MR2045505 Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes), Plane and space curves On the moduli scheme of stable sheaves supported on cubic 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 articles of this volume will be reviewed individually. Resolution; Singularities; Tribute; Textbook; Working week; Obergurgl (Austria); Dedication Hauser, H.; Lipman, J.; Oort, F.; Quirós, A.: Resolution of singularities. A research textbook in tribute to oscar Zariski papers from the working week. Prog. math. 181 (2000) Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Singularities in algebraic geometry, Complex singularities, Singularities of vector fields, topological aspects Resolution of singularities. A research textbook in tribute to Oscar Zariski. Based on the courses given at the working week in Obergurgl, Austria, September 7--14, 1997 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 complex variety \(X\), and a nonzero sheaf of ideals \(\mathcal{I}\) on it with zero locus \(Z\), the log canonical threshold of the pair \((X, Z)\) is an important local invariant of the singularities of \(Z\). In many cases one could assume \(X = \mathbb{C}^{d+1}\), and \(\mathcal{I}\) generated by a non-zero polynomial \(f \in \mathbb{C}[x_1, \dots x_{d+1}]\) with \(f(0) = 0\). Take a log resolution \(\pi: \widetilde{U} \rightarrow U\) of \(Z \cap U\) in a neighborhood \(U\) of 0, with \(E_i\) the irreducible components of \(\pi^{-1}(Z \cap U) = {\bigcup}_{i \in J}E_i\). If \(a_i = \mathrm{ord}_{E_i}(f \circ \pi)\) and \(b_i = \mathrm{ord}_{E_i} \det(\mathrm{Jac}(\pi))\) for all \(i \in J\), the log canonical threshold of \(f\) at 0 is defined as \(\mathrm{lct}_{0}f = \min_{i}\{\frac{b_{i}+1}{a_i}\}\). It is independent of the choice of the log resolution, and the same definition holds for a germ of complex analytic function. It is the reciprocal of Arnold multiplicity, and it evaluates how bad the singularity is. Intuitively, the worse the singularity, the higher the multiplicities \(a_i\) are, so the smaller the log canonical threshold is. It is related with other notions, for example, it could be computed in terms of the jet spaces of \(Z\) [\textit{M. Mustaţă}, J. Am. Math. Soc. 15, No. 3, 599--615 (2002; Zbl 0998.14009)] or, as the negative of the biggest root of Bernstein-Sato polynomial for \(f\). Also, it is the smallest \(\alpha > 0\) such that \(|f|^{-2\alpha}\) is not locally integrable.
In the article under review the log canonical threshold is computed for irreducible quasi ordinary hypersurface singularities. A germ \((Z, 0) \subset (\mathbb{C}^{d+1}, 0)\) of complex analytic hypersurface is quasi-ordinary if there is a finite morphism \(h: (Z, 0) \rightarrow (\mathcal{C}^d, 0)\) whose discriminant locus is contained in a simple normal crossings divisor. For such \((Z, 0)\) there exists an embedding in \((\mathcal{C}^{d+1}, 0)\), defined by the zero set \(Z(f)\) of a quasi-ordinary polynomial \(f \in \mathbb{C}\{x_1, \dots x_d \}[y]\). This means that \(f\) is an irreducible Weierstrass polynomial in \(y\) with discriminant \(\Delta_{f} = x^{a}.u\), where \(a \in \mathbb{Z}^d_{\geq 0}\), and \(u \in \mathbb{C}\{x_1, \dots, x_d\}\) is a unit in the ring of convergent power series. Moreover, in terms of coordinates \((x_1, \dots x_d, y)\) the morphisms \(h\) is the projection on the first \(d\) coordinates.
The geometry of such hypersurface germ could be described in terms of its characteristic exponents. These are uniquely defined vectors \(\lambda_1 \leq \dots \leq \lambda_s\) in \(\mathbb{Q}^d\), and they determine a nested sequence of characteristic lattices [\textit{Y.-N. Gau}, Mem. Am. Math. Soc. 388, 109--129 (1988; Zbl 0658.14004)]. The main theorem of the article computes \(\mathrm{lct}_{0}(f)\) in terms of the combinatorics of associated characteristic exponents for \(f\). The proof uses earlier result of the last two authors, based on the fact that the biggest pole of the local motivic zeta function \(Z_{\mathrm{mot}, f}(\mathbb{L}^{-s})_0\) is equal to \(-\mathrm{lct}_{0}(f)\) (see, for example, [\textit{L. H. Halle} and \textit{J. Nicaise}, Adv. Math. 227, No. 1, 610--653 (2011; Zbl 1230.11076)]). This result describes explicitly the possible poles of \(Z_{\mathrm{mot}, f}\), and from it and some combinatorial lemmas \(\mathrm{lct}_{0}(f)\) is obtained by discarding some of the candidates. As a corrolary is given a necessary and sufficient condition for the polynomial \(f\) to be log canocical in terms of its (this time unique) characteristic exponent. The calculations are demonstrated on particular examples. log canonical threshold; quasi-ordinary hypersurface; characteristic exponent Budur, N.; González-Pérez, P.D.; González-Villa, M., Log-canonical thresholds of quasi-ordinary hypersurface singularities, Proc. Amer. Math. Soc., 140, 4075-4083, (2012) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities Log canonical thresholds of quasi-ordinary 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 The paper deals with normal generic covers branched over \(\{x^n=y^m\}\), i.e., finite holomorphic maps \(\pi : S \rightarrow \mathbb{C}^2\) from a connected normal surface \(S\) to complex plane \(\mathbb{C}^2\), which is an analytic covering branched over a curve \(B=\{(x,y) \in \mathbb{C}^2 : x^n=y^m\}\), such that the fiber \(\pi^{-1}(p)\) for \(p \in B\setminus \{(0,0)\}\) is supported on \(\deg \pi -1\) points. The authors show that the germ of such a map \(\pi\) is equivalent to a germ of the branched covering defined on a surface \(X\) obtained by contracting a section \(C_0\) of a ruled surface \(\widetilde{X}\) on a smooth curve \(C\) and taking a quotient by an action of a finite cyclic group \(G\). Basing on the above they give the numerical characterization of all the smooth normal generic covers branched over \(\{x^n=y^m\}\) and state also the rationality criteria. branched coverings; normal surface singularities Complex surface and hypersurface singularities, Singularities in algebraic geometry, Coverings in algebraic geometry, Local complex singularities, Rational and ruled surfaces Ruled surfaces and generic coverings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors prove that the analytic intersection numbers (extended index of intersection) of \textit{P. Tworzewski} [see Ann. Pol. Math. 62, 177-191 (1995; Zbl 0911.32018)] and the Segre numbers of \textit{T. Gaffney} and \textit{R. Gassler} [J. Algebr. Geom. 8, 695-736 (1999; Zbl 0971.13021)], are generalized Samuel multiplicities, which have been introduced for an arbitrary ideal in an arbitrary local ring by \textit{R. Achilles} and \textit{M. Manaresi} [Math. Ann. 309, 573-591(1997; Zbl 0894.14005)]. improper intersection; intersection number; Segre number; generalized Samuel multiplicity R. Achilles and S. Rams, Intersection numbers, Segre numbers and generalized Samuel multiplicities, Arch. Math. (Basel) 77 (2001), 391-398. Multiplicity theory and related topics, Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Invariants of analytic local rings Intersection numbers, Segre numbers and generalized Samuel multiplicities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Using generalized power series and Newton's polygon method, the authors extend the notion of the local Łojasiewicz exponent as well as the Łojasiewicz exponent at infinity from \(\mathbb{C}\) to an algebraically closed field \(K\) of elements \(F=(f_1,\dots,f_m)\in K[[X,Y]]^m\) and \(F=(f_1,\dots,f_m)\in K[X,Y]^m\) respectively. They also study the basic properties of these two numbers. generalized power series; Łojasiewicz exponent; parametrization; Newton polygon method Brzostowski, S; Rodak, T, The łojasiewicz exponent over a field of arbitrary characteristic, Rev. Math. Complut., 28, 487-504, (2015) Formal power series rings, Singularities in algebraic geometry, Invariants of analytic local rings The Łojasiewicz exponent over a field of arbitrary characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field of zero characteristic finitely generated over a primitive subfield. Let \(f\) be a polynomial of degree at most \(d\) in \(n\) variables, with coefficients from \(k\), irreducible over an algebraic closure \( \bar{k} \). Then we construct an algebraic variety \(V\) nonsingular in codimension one and a finite birational isomorphism \(V \to Z(f)\), where \(Z(f)\) is the hypersurface of all common zeros of the polynomial \(f\) in the affine space. The running time of the algorithm for constructing \(V\) is polynomial in the size of the input. Computational aspects of higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Symbolic computation and algebraic computation Effective construction of an algebraic variety nonsingular in codimension one over a ground field of zero 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 In the paper under consideration a classification of bimodal germs of parametrized plane curves singularities of \( (\mathbb{C},0) \rightarrow (\mathbb{C}^{2},0) \) with respect to \( \mathcal{A} \) equivalence. \( \mathcal{A} = \mathcal{L} \times \mathcal{R} \) is a cross product of automorphysm groups of \( \mathcal{L} \) and \( \mathcal{R} \), where \( \mathcal{L} \) is the group automorphysm of \( \mathbb{C}[[x,y]] \) and \( \mathcal{R} \) is the group automorphysm of \( \mathbb{C}[[t]] \). The group \( \mathcal{A} \) acts on the space map of germs \( f: (\mathbb{C},0) \rightarrow (\mathbb{C}^{2},0) \) and two germs are \( \mathcal{A} \) equivalent if they lie in the same orbit under the action of \( \mathcal{A} \). The paper is illustrated by three tables of normal forms of simple, unimodal and bimodal germs. parametrized curves; bimodal singularities Singularities of curves, local rings, Singularities in algebraic geometry Bimodal singularities of parametrized plane curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors investigate simultaneous robust subspace recovery, i.e., given an \(m\)-tuple of data sets \(\mathcal{X}^j:=\{v^j_1, \ldots, v^j_n\} \subseteq \mathbb{R}^{d_j}\), where \(j \in [m]=\{1,2,\ldots, m \}\), investigate the existence of an effective way to find an \(m\)-tuple of subspaces \((T_1, \ldots, T_m)\) with \(T_j \leq \mathbb{R}^{d_j}\), with \(j \in [m]\), such that
\[
|\mathcal{I}_T|> \frac{\displaystyle{\sum_{j=1}^m} \dim T_j}{D} \cdot n,
\]
where \(\mathcal{I}_T = \{i \in [n] : v^j_i \in T_j \mbox{ for all } j \in [m]\}\) and \(D=\displaystyle{\sum_{j=1}^{m}} d_j\). One says that \(\mathcal{X}\) admits a lower-dimensional subspace structure if there exists a simultaneous robust subspace recovery solution for \(\mathcal{X}\).
When \(m=1\), this is a central problem in robust subspace recovery where the optimal balance between robustness and efficiency is the key. For \(m>1\), this problem is related to the Hilbert-Mumford numerical criterion for semistability of representations of quivers.
The authors prove the following (Theorem 2, page 214): let \(Q\) be a quiver without oriented cycles and let \((V, \sigma)\) be a quiver datum, defined over the rational numbers, such that \(\sigma \cdot \dim V=0\). Then there exist deterministic polynomial-time algorithms to check whether \(V\) is \(\sigma\)-semi-stable or not, and assuming that \(Q\) is a bipartite quiver (not necessarily complete) and \(\sigma\) is nowhere zero, there exists a deterministic polynomial time algorithm that constructs a subrepresentation \(W \leq V\) such that \(\text{disc}(V, \sigma)=\sigma \cdot \dim W\). In particular, if \(\sigma\cdot \dim W=0\), then \(V\) is \(\sigma\)-semi-stable. Otherwise, \(W\) is an optimal witness to \(V\) not being \(\sigma\)-semi-stable. capacity of completely positive operators; partitioned data sets; semi-stability of quiver representations; shrunk subspaces; simultaneous robust subspace recovery; Wong sequences Representations of quivers and partially ordered sets, Actions of groups on commutative rings; invariant theory, Geometric invariant theory Simultaneous robust subspace recovery and semi-stability of quiver representations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main aim of this paper is to characterize ideals \(I\) in the power series ring \(R = K [[x_1, \ldots, x_s]]\) that are finitely determined up to contact equivalence by proving that this is the case if and only if \(I\) is an isolated complete intersection singularity, provided \(\dim(R / I) > 0\) and \(K\) is an infinite field (of arbitrary characteristic). Here two ideals \(I\) and \(J\) are contact equivalent if the local \(K\)-algebras \(R / I\) and \(R / J\) are isomorphic. If \(I\) is minimally generated by \(a_1, \ldots, a_m\), we call \textit{I finitely contact determined} if it is contact equivalent to any ideal \(J\) that can be generated by \(b_1, \ldots, b_m\) with \(a_i - b_i \in \langle x_1, \ldots, x_s \rangle^k\) for some integer \(k\). We give also computable and semicontinuous determinacy bounds.
The above result is proved by considering left-right equivalence on the ring \(M_{m, n}\) of \(m \times n\) matrices \(A\) with entries in \(R\) and we show that the Fitting ideals of a finitely determined matrix in \(M_{m, n}\) have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in \(R\). Some results of this paper are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We include some open problems and a conjecture. singularities; finite determinacy; positive characteristic; algebraic group action; inseparable orbit action; specialization of power series; complete intersections Formal power series rings, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Local complex singularities Finite determinacy of matrices and ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider the orbifold curve that is a quotient of an elliptic curve \(\mathcal{E}\) by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov--Witten theory of the orbifold curve via the product of the Gromov--Witten theories of a point. This is done by employing mirror symmetry and certain results in FJRW theory. In particular, we present the particular Givental's action giving the CY/LG correspondence between the Gromov--Witten theory of the orbifold curve \(\mathcal{E}/\mathbb{Z}_{4}\) and FJRW theory of the pair defined by the polynomial \(x^{4}+y^{4}+z^{2}\) and the maximal group of diagonal symmetries. The methods we have developed can easily be applied to other finite quotients of an elliptic curve. Using Givental's action, we also recover this FJRW theory via the product of the Gromov--Witten theories of a point. Combined with the CY/LG action, we get a result in ``pure'' Gromov--Witten theory with the help of modern mirror symmetry conjectures. Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Singularities in algebraic geometry Givental-type reconstruction at a nonsemisimple point | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth integral scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families of cycles parameterized by curves, or by abelian varieties. In this article, we extend these results to arbitrary base fields. The strengthening of these results turns out to be a key step in our work elsewhere extending Murre's results on algebraic representatives for varieties over algebraically closed fields to arbitrary perfect fields. Algebraic cycles, Parametrization (Chow and Hilbert schemes), (Equivariant) Chow groups and rings; motives, Other nonalgebraically closed ground fields in algebraic geometry, Abelian varieties and schemes Parameter spaces for algebraic equivalence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We present an example of a real algebraic surface of degree 5 in \(\mathbb{P}^ 3\). The homology of the real part of a nonsingular model is of dimension 41 and generated entirely by algebraic cycles. The example is constructed by resolution of singularities of an algebraically singular surface homeomorphic to \(\mathbb{P}^ 2 (\mathbb{R})\). We study the behaviour of the topology during the resolution. Picard group; Picard number; real algebraic surface; homology; algebraic cycles; resolution of singularities Frédéric Mangolte, Une surface réelle de degré 5 dont l'homologie est entièrement engendrée par des cycles algébriques, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 4, 343 -- 346 (French, with English and French summaries). Singularities of surfaces or higher-dimensional varieties, Real algebraic sets, Classical real and complex (co)homology in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) A real algebraic surface of degree 5 whose homology is entirely generated by algebraic 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 \(f:X\to\text{Spec} R\) denote the smallest resolution for the local ring \(R={\mathcal O}_{Y,y}\) of a normal surface singularity \((Y,y)\), whose exceptional divisor \(E\) consists of intersecting transversally smooth curves \(E_1,\dots, E_s\). The singularity \((Y,y)\) is called Du Bois if the natural map \(R^1f_*{\mathcal O}_X \to H^1(E, {\mathcal O}_E)\) is an isomorphism. The author proves a slightly generalized version of \textit{J. M. Wahl}'s vanishing theorem [Invent. Math. 31, 17-41 (1975; Zbl 0314.14010)].
Namely if \((Y,y)\) is a Du Bois singularity with weighted dual graph \(\Gamma\) and \(\text{char} k=0\) or \(\text{char} k=p\) is greater than a certain bound given explicitly by \(\Gamma\), then \(H^1_E(X,\Omega^1_X (\log E))=0\), where \(\Omega^1_X (\log E)\) is the sheaf of differential 1-forms with log poles along \(E\). surface singularities; rational surfaces; exceptional divisors; resolution; vanishing theorem; Du Bois singularity; differential 1-forms Hara, N.: A characteristic p proof of wahl's vanishing theorem for rational surface singularities. Arch. math. (Basel) 73, 256-261 (1999) Vanishing theorems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure A characteristic \(p\) proof of Wahl's vanishing theorem for rational surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Robin Hartshorne's popular textbook ``Algebraic geometry'' [Graduate Texts in Mathematics, 52. New York - Heidelberg - Berlin: Springer-Verlag (1977; Zbl 0367.14001)], is well known to everyone who ever tried to get acquainted with the basic modern aspects of the subject during the last three decades. Due to its comprehensiveness, versatility, and expository mastery, Hartshorne's ``Algebraic Geometry'' is by far the most widely used introductory text (and reference book) in this discipline of contemporary mathematics. Generations of algebraic geometers have acquired a profound fundamental knowledge of the subject from R. Hartshorne's book ever since its first appearance, and the book itself has never been out of print so far.
Now, more than thirty years after the publishing of his standard text on modern algebraic geometry, R. Hartshorne obliges with another book in the field. Being more specialized, the present volume is to provide an introduction to the main ideas of deformation theory in algebraic geometry and to illustrate their use in a number of typical concrete situations. Actually, as the author points out in the preface, this book has an amazingly long history that began exactly thirty years ago. Namely, in the fall semester of 1979, R. Hartshorne taught a course on algebraic deformation theory at Berkeley, mainly with the goal to understand completely Grothendieck's approach to the local study of the Hilbert scheme using cohomological methods. The handwritten notes of this course circulated quietly for many years until D. Eisenbud urged the author to complete and publish them. Then, five years ago, R. Hartshorne expanded the old notes into a rough draft, which he used to teach a course in the spring of 2005. Finally, he rewrote those notes once more and, with the addition of numerous exercises, turned them into the book under review.
As the present text is intended to be of introductory nature, no effort has been made to develop the theory of deformations in full generality. Instead, the author has preferred to elaborate the basic ideas underlying the theory, without letting them get buried in too many technical details. Also, the approach has been kept as elementary as possible, thereby assuming only a basic familiarity with the concepts and methods of algebraic geometry as developed in the author's above-mentioned standard text.
In this vein, and very much to the benefit of the rather unexperienced reader, the author has not striven for stating results in their most general form, nor has he attempted to use the more recent state-of-the-art framework of Grothendieck topologies and algebraic stacks to its full extent. Overall, the purpose of this book is to explain the basic concepts and methods of deformation theory, to bring forth clearly their fundamental essence, to show how they work in various standard situations, and to provide some instructive examples and applications from the literature.
As for the contents, the book is divided into four chapters which, altogether comprise twenty-nine sections. The author's guiding principle is to focus on four standard situations:
(A) Deformations of subschemes of a fixed ambient scheme \(X\).
(B) Deformations of line bundles on a fixed scheme \(X\).
(C) Deformations of coherent sheaves on a fixed scheme \(X\).
(D) Deformations of abstract schemes,including the local study of deformations of singularities as well as the global study of deformations of non-singular varieties (global moduli).
For each of these particular situations, a number of typical problems is discussed, with the ultimate goal to establish a global parameter space classifying the isomorphism classes of the objects in question and, moreover, to describe its geometric properties.
However, in this introductory text, the technically involved proofs of the existence of these global classifying spaces are not provided, as the author's primary goal is rather to lay the foundations of the respective deformation theory that allow to describe the local structure of the (assumed) global parameter space.
Chapter 1 is titled ``First-Order Deformations'' and deals with algebraic deformations over the ring \(D:=k[t]/(t^2)\) of dual numbers associated to an algebraically closed field \(k\). Starting with the concept of Hilbert scheme as a deformation space of a closed subscheme of the projective space \(\mathbb{P}^n_k\), which serves as a model in the sequel, deformations over \(D\) are discussed for the particular situations (A), (B) and (C). Then, after an introduction to the cotangent complex and the \(T^i\) functors of Lichtenbaum and Schlessinger, deformations of abstract schemes (as in situation (D)) are explained, thereby using the infinitesimal lifting property with regard to non-singular varieties.
Chapter 2 turns to the more general case of higher-order deformations, that is, to deformations over arbitrary Artin rings, together with the according obstruction theories for the respective situations (A), (B), (C), and (D). Along the way, three special cases are treated in greater detail, namely Cohen-Macaulay subschemes of codimension 2, locally complete intersection schemes, and Gorenstein subschemes in codimension 3. Additional illustrating material concerns the obstruction theory for local rings, the classical bound on the dimension of the Hilbert scheme of projective space curves, and Mumford's ``pathological'' example of a family of non-singular projective space curves whose Hilbert scheme is generically non-reduced.
Chapter 3 is devoted to the study of formal moduli spaces. Starting from the explicit case of plane curve singularities, the author discusses the general problem in terms of functors of Artin rings, including Schlessinger's criterion for pro-representability as the crucial technical tool in this context. In the sequel, this formal apparatus is applied to each of the standard situations (A), (B), (C) and (D), along with numerous concrete examples and applications. Further material concerns a comparison of embedded and abstract deformations, with a special view toward general surfaces in \(\mathbb{P}^4_k\) of degree greater than 3 the problem of algebraization of formal moduli (à la M. Artin), and -- as a further application -- the question of lifting varieties from characteristic \(p> 0\) to characteristic \(0\).
Chapter 4 comes with the headline ``Global Questions''. Here the methods of infinitesimal and formal deformations from the previous chapters are applied to study global moduli problems. After introducing the notions of fine moduli space and coarse moduli space, the Hilbert functor and the Picard functor are described as examples of representable moduli functors.
The rest of this concluding chapter is devoted to the discussion of a number of concrete classical moduli spaces and their geometric properties. More precisely, the author illuminates in detail the global moduli spaces of rational and elliptic curves, Mumford's concept of modular families of curves of higher genus (together with the idea of stacks), the moduli space of stable vector bundles over a curve, and the notions of formally smoothable schemes and smoothable singularities. As an application of the general theory, the reader encounters here Mori's theorem on the existence of rational curves in non-singular varieties in characteristic \(p>0\) whose canonical divisor is not numerically effective, on the one hand, and instructive examples of non-smoothable singularities on the other.
Much more material is covered by the huge number of further-leading exercises complementing each section of the book. These exercises provide much more of the respective theories as well as a wealth of additional examples. Most of the exercises are quite challenging, but they are also well-structured and equipped with guiding remarks or hints.
The author's approach to deformation theory shows lucidly how far the subject can be treated with relatively elementary methods, without providing the technically complicated proofs of the existence of the main classifying schemes, and without using more advanced toolkits like geometric invariant theory, Artin's approximation theorems, simplicial complexes, differential graded algebras, fibered categories, stacks, or derived categories.
No doubt, this masterly written book gives an excellent first introduction to algebraic deformation theory, and a perfect motivation for further, more advanced reading likewise. It is the author's masterful style of expository writing that makes this text particularly valuable for seasoned graduate students and for future researchers in the field. The list of 177 references at the end of the book, which the author frequently refers to throughout the text, is another special feature of the volume under review.
As for complementary and parallel reading, the recent monograph ``Deformations of Algebraic Schemes'' by \textit{E. Sernesi} [Deformations of algebraic schemes. Grundlehren der Mathematischen Wissenschaften 334. Berlin: Springer (2006; Zbl 1102.14001)] might be quite instructive and useful,especially in view of the technical prerequisites and subtleties as well as for working some of the exercises. deformation theory; infinitesimal methods; formal neighborhoods; algebraic moduli problems; formal moduli; Hilbert schemes; moduli of curves; moduli of vector bundles R. Hartshorne, \textit{Deformation Theory} (Springer, Berlin, 2010), Grad. Texts Math. 257. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Families, fibrations in algebraic geometry, Infinitesimal methods in algebraic geometry, Formal methods and deformations in algebraic geometry, Local deformation theory, Artin approximation, etc., Deformations of singularities, Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes) Deformation 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 study certain \(\Delta\)-filtered modules for the Auslander algebra of \(k[T]/T^n\rtimes C_2\) where \(C_2\) is the cyclic group of order two. The motivation of this lies in the problem of describing the \(P\)-orbit structure for the action of a parabolic subgroup \(P\) of an orthogonal group. For any parabolic subgroup of an orthogonal group we construct a map from parabolic orbits to \(\Delta\)-filtered modules and show that in the case of the Richardson orbit, the resulting module has no self-extensions.
Let \(k\) be an algebraically closed of characteristic different from 2, and let \(G\) be a reductive algebraic group over \(k\), \(P\subset G\) a parabolic subgroup. Now \(P\) acts on its unipotent radical \(U\) by conjugation and on the nilradical \(\mathfrak n=\text{Lie\,}U\) by the adjoint action. By a fundamental result of \textit{R. W. Richardson} [Bull. Lond. Math. Soc. 6, 21-24 (1974; Zbl 0287.20036)], this action has an open dense orbit, the so-called `Richardson orbit' of \(P\). But in general, the number of orbits is not finite and it is a very hard problem to understand the orbit structure. The question of deciding whether \(P\) has a finite number of orbits in \(\mathfrak n\) has been asked by \textit{V. L. Popov} and \textit{G. Röhrle} [in Aust. Math. Soc. Lect. Ser. 9, 297-320 (1997; Zbl 0887.14020)].
If \(P\) is a parabolic subgroup of \(\text{SL}_N\), then there is an explicit description of the \(P\)-orbits in work of \textit{L. Hille} and \textit{G. Röhrle} [Transform. Groups 4, No. 1, 35-52 (1999; Zbl 0924.20035)], and \textit{T. Brüstle, L. Hille, C. M. Ringel}, and \textit{G. Röhrle} [Algebr. Represent. Theory 2, No. 3, 295-312 (1999; Zbl 0971.16007)], via a connection with a quasi-hereditary algebra, namely the Auslander algebra \(A_n\) of the truncated polynomial ring \(R_n:=k[T]/T^n\). They have shown that the \(P\)-orbits are in bijection with the isomorphism classes of certain \(\Delta\)-filtered modules of \(A_n\) with no self-extensions. This list has finitely many indecomposable modules, parametrized as \(\Delta(I)\) where \(I\) runs through the subsets of \(\{1,2,\dots,n\}\).
Our main goal in this paper is to establish an analogous correspondence between \(P\)-orbits for parabolic subgroups of the special orthogonal groups \(\text{SO}_N\) and certain \(\Delta\)-filtered modules for the Auslander algebra of \(k[T]/T^n\rtimes C_2\), the skew group ring of one considered by \textit{T. Brüstle} et al. [loc. cit.], where \(C_2\) is a cyclic group of order two. This article establishes the Auslander algebra of \(k[T]/T^n\rtimes C_2\) as the correct candidate for such a correspondence. There is one major difference as compared with \textit{T. Brüstle} et al. [loc. cit.]. In our situation, a list of all \(\Delta\)-filtered modules with no self extensions is difficult to obtain and may even be infinite. This must be expected however, as the construction of the Richardson elements for \(\text{SO}_N\) involves symmetric diagrams and hence gives rise to symmetric \(\Delta\)-dimension vectors. Here, we use signed sets \(I\), that is certain subsets of \(\{\pm 1,\pm 2,\dots,\pm n\}\). We associate to each signed set \(I\) another set \(J\) (a symmetric complement) and an extension \(E(I,J)\) that is \(\Delta\)-filtered with no self extensions and has the required symmetric \(\Delta\)-dimension vector. These extensions may then be used to construct a \(\Delta\)-filtered module that corresponds to the Richardson element, using the work of \textit{K. Baur} [J. Algebra 297, No. 1, 168-185 (2006; Zbl 1144.17004)] and \textit{K. Baur} and \textit{S. M. Goodwin} [Algebr. Represent. Theory 11, No. 3, 275-297 (2008; Zbl 1147.17011)] on orthogonal Lie algebras. connected reductive algebraic groups; orthogonal groups; parabolic subgroups; dense orbits; unipotent radical; Auslander algebras; truncated polynomial algebras; Richardson orbit; filtered modules; numbers of orbits Baur, K; Erdmann, K; Parker, A, {\(\Delta\)}-filtered modules and nilpotent orbits of a parabolic subgroup in \(O\)\_{}\{\(N\)\}, J. Pure Appl. Algebra, 215, 885-901, (2011) Representations of quivers and partially ordered sets, Lie algebras of linear algebraic groups, Linear algebraic groups over arbitrary fields, Classical groups (algebro-geometric aspects), Coadjoint orbits; nilpotent varieties \(\Delta\)-filtered modules and nilpotent orbits of a parabolic subgroup in \(\mathrm O_N\). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove that a foliation \({\mathcal F}\) of degree \(\neq 1\) on \(\mathbb{P}^2\) is completely determined by its singular subscheme \(\text{SingS}({\mathcal F})\) of \(\mathbb{P}^2\). We apply this result to obtain a similar characterization of \({\mathcal F}\) in terms of the configuration of base points associated to its singular scheme, in case every singularity of \({\mathcal F}\) has nontrivial linear part. Our main motivation comes from a well-known fact: in case a foliation \({\mathcal F}\) of degree \(r\geq 2\) on \(\mathbb{P}^n\) has only isolated singularities of multiplicity 1, then \({\mathcal F}\) is completely determined by its singular set \(\text{Sing} ({\mathcal F})\). plane foliation; singular subscheme Campillo, A., Olivares, J.: Plane foliations of degree different from one are determined by their singular scheme. Comptes Rendus de l'Académie de Sciences - Série I - Mathématiques 328, 877-882 (1999) Singularities of holomorphic vector fields and foliations, Singularities in algebraic geometry A plane foliation of degree different from 1 is determined by its singular scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective variety of dimension \(n\) over \(\mathbb C\).
The authors introduce a notion of slope-semistability with respect to a fixed class \(\alpha\in N_1(X)_{\mathbb R}\) of real-valued \(1\)-cycles modulo numerical equivalence. The corresponding condition on the \(\alpha\)-slopes of non-trivial subsheaves \(F\subset E\) reads \[ \mu_{\alpha}(F)=\frac{c_1(F)\cdot \alpha}{\mathrm{rank}(F)}<(\le)\frac{c_1(E)\cdot \alpha}{\mathrm{rank}(E)}=\mu_{\alpha}(E). \] This modification of the definition of semistability preserves the information about the ample divisor classes. The power map \[ p_{n-1}: \mathrm{Amp}(X)_{\mathbb R}\to N_1(X)_{\mathbb R},\quad \alpha\mapsto \alpha^{n-1}, \] turns out to be injective and gives a homeomorphism from \(\mathrm{Amp}(X)_{\mathbb R}\) to its open image \(\mathrm{Pos}(X)_{\mathbb R}\).
The authors show the boundedness of torsion-free sheaves with fixed Chern classes with respect to \(1\)-cycles lying in compact subsets of \(\mathrm{Pos}(X)_{\mathbb R}\). In the cases when (1) the rank of the torsion-free sheaves is at most two or (2) \(n\le 3\) or (3) the Picard rank of \(X\) is at most two, the boundedness is also shown for classes of \(1\)-cycles from convex hulls of powers of ample integral divisors on \(X\).
For fixed invariants and a fixed compact subset \(K\) of \(\mathrm{Pos}(X)_{\mathbb R}\) there exist finitely many rational walls. This defines a chamber structure on \(K\): in every chamber the notion of \(\alpha\)-semistability remains unchanged. However, it does not necessarily mean that the semistability notion changes at each wall.
In every chamber of \(K\) there exists a class \(A^{n-2}.B\) defined by some ample integral divisor classes \(A\) and \(B\). This means that in suffices to construct moduli spaces for sheaves that are slope-semistable with respect to multipolarizations \((H_1,\dots, H_{n-1})\), i.~e., with respect to \(\alpha=[H_1]\cdot[H_2]\cdot\dots\cdot [H_{n-1}]\) for some ample integral divisor classes.
Let \(\mathcal{M}^{\mu-s}(\Lambda)\) be the coarse moduli space of slope-stable vector bundles with respect to a multipolarization \((H_1,\dots, H_{n-1})\) with fixed topological invariants and a fixed determinant line bundle \(\Lambda\). Let \((\mathcal{M}^{\mu-s}(\Lambda))^{w\nu}\to \mathcal{M}^{\mu-s}(\Lambda)\) be its semi-normalization. This map is in particular a homeomorphism.
The authors show that there is a compactification \(\mathcal{M}^{\mu-ss}(\Lambda)\) of \((\mathcal{M}^{\mu-s}(\Lambda))^{w\nu}\) such that every \((H_1,\dots, H_{n-1})\)-slope-semistable torsion free sheaf defines a point in \(\mathcal{M}^{\mu-ss}(\Lambda)\). This refines the correspondence that sends a sheaf \(E\) to the pair consisting of the associated graded sheaf \(gr^{\mu}E\) and the supporting \((n-2)\)-cycle \(C_E\) of the quotient torsion sheaf \(Q_E=(gr^\mu E)^{**}/gr^\mu E)\).
For ample line bundles \(L_1,\dots, L_{j_0}\), for some line bundles \(B_1,\dots, B_{j_0}\) on \(X\), and for a non-zero vector of non-negative real numbers \(\sigma=(\sigma_1,\dots, \sigma_{j_0})\) a notion of multi-Gieseker (semi)stability is introduced. The corresponding condition for the non-trivial subsheaves \(F\subset E\) reads \[ \frac{\sum_j \sigma_j\chi(F\otimes B_j\otimes L^n_j)}{\mathrm{rank}(F)}<(\le)\frac{\sum_j \sigma_j\chi(E\otimes B_j\otimes L^n_j)}{\mathrm{rank}(E)}, \quad n \gg 0. \] For fixed \(\sigma\), \(L_1,\dots, L_{j_0}\), and \(B_1,\dots, B_{j_0}\) there exists a projective moduli space \(\mathcal{M}_{\sigma}\) of multi-Gieseker-semistable sheaves. Moreover, on a threefold, the space of values \(\mathbb R^{j_0}_{\ge 0}\setminus\{0\}\) for \(\sigma\) is cut into chambers by finite number of linear rational walls such that the semistability does not change inside each chamber.
For \(j_0=1\), the moduli spaces \(\mathcal M_{L_1}\) and \(\mathcal M_{L_2}\) corresponding to ample line bundles \(L_1\) and \(L_2\) are related by a finite number of Thaddeus flips. Under certain conditions, the authors show that these flips are through moduli spaces of multi-Gieseker-semistable sheaves.
A multi-Gieseker-semistable sheaf as above is also slope-semistabe with respect to the \(1\)-cycle \(\gamma=\sum_j\sigma_j c_1(L_j)^{n-1}\). As a new result, the authors demonstrate the existence of a natural morphism from the semi-normalization \((\mathcal{M}_\sigma(\Lambda))^{w\nu}\) to \(\mathcal{M}^{\mu-ss}(\Lambda)\) for rational \(\sigma\).
The paper under review consists of five sections. Section~1 is an introduction. Section~2 deals with boundedness for slope-semistability. The corresponding moduli space of Donaldson-Uhlenbeck type is discussed in Section~3. Section~4 deals with Gieseker and multi-Gieseker stability. A multi-Gieseker-to-Uhlenbeck morphism is constructed in Section~5. variation of Gieseker moduli spaces; moduli of quiver representations; sheaves on Kähler manifolds; wall-crossing; Donaldson-Uhlenbeck compactification Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Minimal model program (Mori theory, extremal rays), Complex-analytic moduli problems, Geometric invariant theory, Representations of quivers and partially ordered sets, Moduli problems for differential geometric structures Moduli of vector bundles on higher-dimensional base manifolds - construction and variation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(R=k[x_1,\ldots,x_n]\) be a polynomial ring of prime characteristic \(p\) with maximal ideal \(\mathfrak{m}=(x_1, \ldots, x_n)\). For any ideal \(\mathfrak{a}\) in \(R\) and \(t\) a non-negative real number, the authors define the generalized Frobenius power \(\mathfrak{a}^{[t]}\) which agrees with the Frobenius powers when \(t=p^e\) with \(e \in \mathbb{Z}\). They first define the Frobenius power for positive integers \(m=m_0+m_1p+\cdots m_rp^r\) to be \[\mathfrak{a}^{[m]}:=\mathfrak{a}^{m_0}(\mathfrak{a}^{m_1})^{[p]}\cdots (\mathfrak{a}^{m_r})^{[p^r]},\] then for rational numbers of the form \(m/p^e\) with \(\text{gcd}(m,p)=1\): \(\mathfrak{a}^{[m/p^e]}:=(\mathfrak{a}^{[m]})^{[1/p^e]}\). For a general non-negative \(t\), they define \(\mathfrak{a}^{[t]}=\bigcup_{k=1}^{\infty} \mathfrak{a}^{[t_k]}\) where \(t_k=m_k/p^{r_k}\) is a monotonically non-increasing sequence of positive rational numbers converging to \(t\). A non-negative real number \(\lambda\) is a critical exponent of an ideal \(\mathfrak{a}\) if \(\mathfrak{a}^{[\lambda]} \subsetneq \mathfrak{a}^{[\lambda-\epsilon]}\) for \(0 < \epsilon\leq \lambda\). The critical exponent of \(\mathfrak{a}\) with respect to \(\mathfrak{b}\) is \[\text{crit}(\mathfrak{a},\mathfrak{b})=\text{sup}\{t>0 \mid \mathfrak{a}^{[t]} \nsubseteq \mathfrak{b}\}= \text{min}\{t>0 \mid \mathfrak{a}^{[t]} \subseteq \mathfrak{b}\}.\] Some of the main goals of this paper are determining the critical exponents in the interval \([0,1]\) of powers of \(\mathfrak{m}\) and diagonal ideals \((x_1^{a_1}, \ldots, x_n^{a_n})\). As a preliminary step, they show that for every \(\mathfrak{m}\)-primary monomial ideal \(\mathfrak{a}\) the critical exponents are of the form \(\text{crit}(\mathfrak{a},(x_1^{u_1}, \ldots x_n^{u_n}))\) for some \((u_1, \ldots, u_n) \in \mathbb{N}^n\). They further show that the critical exponents \(\lambda \in [0,1]\) of \(\mathfrak{m}^d\) are of the form \[\lambda=\displaystyle\frac{k}{d}-\displaystyle\frac{[kp^s\% d]}{dp^s}\] where \([kp^e \% d] \) is the remainder of \(kp^e\) modulo \(d\) and \(s\) is the infinmum of the set of all \(e \geq 1\) where the remainders are less than \(n\). For each critical exponent \(\lambda\) as above \((\mathfrak{m}^d)^{[\lambda]}=\mathfrak{m}^{k-n+1}\). Using similar techniques they give expressions for the critical exponents of diagonal ideals. The many explicit examples given help to visualize their results for the critical exponents and the Frobenius powers for \(t \in [0,1]\).
They conclude with discussing the relationship between \(F\)-jumping numbers of a polynomial of the form \(f=\alpha_1x_1^{a_1}+\cdots \alpha_n x_n^{a_n} \in \mathfrak{m}^d\) with the critical exponents of monomial ideals. In particular, they show that if the \(\alpha_i\) are algebraically independent then \(\tau(f^t)=(\mathfrak{m}^d)^{[t]}\) for \(t \in (0,1)\). Whereas if the \(\alpha_i\) are nonzero, not necessarily algebraically independent and \(p\) doesn't divide \(a_i\) then \(\tau(f^t)=(x_1^{a_1}, \ldots,x_n^{a_n})^{[t]}\) for \(t \in (0,1)\). Frobenius power; \(F\)-jumping number; test ideal Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry Frobenius powers of some monomial ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The jump of the Milnor number of an isolated singularity \(f_0\) is the minimal non-zero difference between the Milnor numbers of \(f_0\) and one of its deformations \((f_s)\). We prove that for the singularities in the \(X_9\) singularity class their jumps are equal to 2. Milnor number; singularity; deformation of singularity Brzostowski, S.; Krasiński, T., The jump of the Milnor number in the \({X}_9\) singularity class, Cent. Eur. J. Math., 12, 429-435, (2014) Local complex singularities, Invariants of analytic local rings, Singularities in algebraic geometry, Deformations of singularities The jump of the Milnor number in the \(X_9\) singularity class | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors construct the constant cohomology ring of the projective plane. Let \(A^*(\mathbb{P}^2)\) be the Chow cohomology ring of the projective plane and \(A^*(\mathbb{P}^2) [[T]]\) the ring of formal power series in one variable. If \(\alpha,\beta\) and \(\delta\) are elements of \(A^*(\mathbb{P}^2)\), then one can introduce the notion of the quantum product of \(\alpha\) and \(\beta\) deformed by \(\delta\) [see \textit{K. Behrend} and \textit{Yu. Manin}, Duke Math. J. 85, No. 1, 1-60 (1996; Zbl 0872.14019), \textit{M. Kontsevich} and \textit{Yu. Manin}, Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020) and \textit{M. Kontsevich} in: The moduli space of curves, Proc. Conf., Texel 1994, Prog. Math. 129, 335-368 (1995; Zbl 0885.14028)]. Each product, parametrized by \(\delta\in A^*(\mathbb{P}^2)\) is defined on the ring \(A^*(\mathbb{P}^2) [[T]]\). This product encodes the characteristic numbers \(N_d=\) the number of rational plane curves of degree \(d\) through \(3d-1\) general points (see the paper by M. Kontsevich and Yu. Manin, cited above). In section 3 the authors review material concerning the quantum cohomology and also give a new proof that the quantum product is associative (proposition 3.2). -- The contact product is defined by the authors in section 5. It is defined on the ring \(A^*(\mathbb{P}^2) [[T]]\), but it is parametrized by the Chow ring \(A^*I\) of the incidence variety of points and lines in \(\mathbb{P}^2\). The main theorem of section 5 is that this contact product is commutative and associative. Similarly as the associativity of the quantum product gives a recursive formula for the characteristic numbers \(N_d\), the associativity of the contact product allows the authors to derive a formula for the characteristic numbers \(N_d(a,b,c)\). Here \(N_d(a,b,c)\) is the number of rational plane curves of degree \(d\) through \(a\) general points, tangent to \(b\) general lines and tangent to \(c\) general lines at a specified general point on each line, where \(a+b+2c=3d-1\). Chow ring; constant cohomology ring; ring of formal power series; quantum cohomology; contact product; number of rational plane curves [EK2]Ernström, L. andKennedy, G., Contact cohomology of the projective plane,Amer. J. Math. 121 (1999), 73--96. Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Contact cohomology of the projective plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities From the point of view of combinatorial convexity, all the structure of a toric variety can be encoded in a fan, i.e., a finite collection of strictly convex cones spanned by finitely many vectors with integral coordinates and satisfying very natural incidence relations. Now the obvious question is how to extract algebro-geometric or topological invariants of the associated toric variety from these data. A most striking example is given by the famous Jurkiewiez-Danilov theorem [see \textit{V. I. Danilov}, Russ. Math. Surv. 33, No. 2, 97-154 (1978); translation from Usp. Mat. Nauk 33, No. 2(200), 85-134 (1978; Zbl 0425.14013); theorem 10.8 and remark 10.9]: The integral cohomology ring of a smooth compact toric variety can be explicitly computed in terms of the associated fan. Allowing mild singularities, the analogous result holds with rational coefficients. For toric varieties with arbitrary singularities, but still in the compact case, there is a spectral sequence relating data of the fan to integral cohomology that has been investigated by \textit{Stephan Fischli} in his dissertation [``On toric varieties'' (Thesis, Univ. Bern 1992)]. It admits the explicit computation of some integral cohomology groups in low and in high degrees; in particular, it yields complete results up to dimension three.
In the work presented here, we also use spectral sequences to determine (co-)homological data of a toric variety in terms of the associated fan, but in a much more general setting: We investigate the homology with closed supports and the cohomology with compact supports and with arbitrary (constant) coefficients for not necessarily compact toric varieties with arbitrary singularities. We do not consider toric varieties over arbitrary fields but restrict ourselves to the complex case:
The toric variety \(X_\Delta\) associated to a fan \(\Delta\) in a vector space \(\mathbb{R}^n\) consists of \((\mathbb{C}^*)^n\)-orbits \(O_\sigma\) corresponding to the cones \(\sigma\in\Delta\); in particular, the full-dimensional cones \(\sigma \in\Delta^n\) correspond to fixed points \(x_\sigma\). It turns out that the natural filtration of the toric variety \(X_\Delta\) induced by its orbit structure provides convergent (co-)homology spectral sequences. An explicit calculation of the associated \(E^2\)- or \(E_2\)-terms yields formulae -- in low and in high degrees \(\ell\) -- for the homology groups \(H_\ell^{\text{cld}} (X_\Delta;G)\), the cohomology groups \(H_c^\ell (X_\Delta; G)\), and the integral local homology groups \({\mathcal H}_{\ell, x_\sigma}\) in the fixed points \(x_\sigma\). All Betti numbers \(b_\ell^{\text{cld}} (X_\Delta)\) are computable. For integral coefficients, the \(E^2_{p,p}\)-terms are isomorphic to the Chow groups \(A_p (X_\Delta)\), i.e., we obtain a generalization of the theorem of Jurkiewicz-Danilov to the singular, noncompact case. Up to dimension four our method admits the computation of almost the whole homology \(H_\bullet^{\text{cld}} (X_\Delta; \mathbb{Z})\) and almost the whole cohomology \(H_c^\bullet (X_\Delta; \mathbb{Z})\); only the computation of the torsion part \(\text{Tor} H_4^{\text{cld}} (X_\Delta; \mathbb{Z}) =\text{Tor} H_c^5 (X_\Delta; \mathbb{Z})\) of a 4-dimensional toric variety \(X_\Delta\) may fail. non compact toric varieties with singularities; homology with closed supports; cohomology with compact supports A. Jordan, ''Homology and Cohomology of Toric Varieties,'' PhD Thesis (Univ. Konstanz, 1997), available at http://www.inf.uni-konstanz.de/Schriften/preprints-1998.html#057 Toric varieties, Newton polyhedra, Okounkov bodies, Classical real and complex (co)homology in algebraic geometry, Singularities in algebraic geometry, Homology and cohomology theories in algebraic topology Homology and cohomology of toric varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In his fundamental paper [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV: Les schemes de
Hilbert'', Sémin. Bourbaki, exp. 221 (1961; Zbl 0236.14003)], \textit{A. Grothendieck} introduced the so called Hilbert scheme, which parametrizes all projective subschemes of the projective space with fixed Hilbert polynomial. Problems naturally arising in the study of the Hilbert scheme are irreducibility and number of components, dimension and smoothness. For instance, one knows that if \(X\subset \mathbb{P}^r\) is a local complete intersection projective subscheme and \(h^1(X,\mathcal N_{X,\mathbb{P}^r})=0\) (\(\mathcal N_{X,\mathbb{P}^r}=\) normal bundle of \(X\) in \(\mathbb{P}^r\)), then \(X\) is unobstructed, i.e. the corresponding point \([X]\) in the Hilbert scheme is smooth, and in such case the local dimension at \([X]\) is \(h^0(X,\mathcal N_{X,\mathbb{P}^r})\). But, in general, a necessary and sufficient condition for a subscheme to be unobstructed is not known [see also \textit{D. Mumford}, Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)] and \textit{E. Sernesi} [``Topics on families of projective schemes'', Queen's Pap. Pure Appl. Math. 73 (1986)].
Continuing previous works by \textit{J. O. Kleppe} [``The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in \(3-\)space'', Preprint (part of thesis), Univ. of Oslo, March (1981)], \textit{G. Bolondi} [Arch. Math. 53, No. 3, 300--305 (1989; Zbl 0658.14005)], and \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184--185 (1990; Zbl 0717.14017)], in the paper under review the author exhibits sufficient conditions and necessary conditions for unobstructedness of space curves \(C\subset \mathbb{P}^3\) which satisfy \(_{0}{\text{Ext}}^2_R(M,M)=0\) (e.g. of diameter\((M)\leq 2\)), and computes the dimension of the Hilbert scheme \(H(d,g)\) at \([C]\) under the sufficient conditions. Here \(C\subset \mathbb{P}^3\) denotes an equidimensional, locally Cohen-Macaulay subscheme of dimension one, \(d\) and \(g\) the degree and the arithmetic genus of \(C\subset \mathbb{P}^3\), \(M=\bigoplus_{v}H^1(\mathbb{P}^3, \mathcal I_C(v))\) denotes the Hartshorne-Rao module of \(C\), \(R=k[x_0,x_1,x_2,x_3]\) the polynomial ring over an algebraically closed field \(k\) of characteristic zero, and diameter\((M):=\max\{v\,| \,H^1(\mathbb{P}^3, \mathcal I_C(v))\neq 0\}-\min\{v\,| \,H^1(\mathbb{P}^3, \mathcal I_C(v))\neq 0\}+1\) (when \(H^1(\mathbb{P}^3, \mathcal I_C(v))=0\) for all \(v\), i.e. when \(C\) is arithmetically Cohen-Macaulay, then by \textit{G. Ellingsrud} [Ann. Sci. Éc. Norm. Supér. (4) 8, 423--431 (1975; Zbl 0325.14002)] one already knows that \(C\) is unobstructed).
In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of \(C\) turns out to be equivalent to the vanishing of certain graded Betti numbers of the free minimal resolution of the ideal \(I=\bigoplus_{v}H^0(\mathbb{P}^3, \mathcal I_C(v))\subset R\) of \(C\). The author also gives a description of the number of irreducible components of \(H(d,g)\) which contain an obstructed diameter one curve, and shows that in the diameter one case every irreducible component is reduced. Hilbert scheme; space curve; Buchsbaum curve; unobstructedness; cup-product; graded Betti numbers; ghost terms; linkage; normal module; postulation Hilbert scheme Dan, A.: Non-reduced components of the Noether-Lefschetz locus. Preprint arXiv:1407.8491v2 Parametrization (Chow and Hilbert schemes), Plane and space curves, Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The Hilbert scheme of space curves of small diameter | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 studies the closedness of the singular sets of a finitely generated \(A\)-module \(M\): \(S^*_ k (M) = \{p \in \text{Spec} A : \text{depth} M_ p + \dim A/p \leq k\}\). The author improves Grothendieck-Dieudonné's result: ``If \(A\) is a homomorphic image of a biequidimensional regular local ring, then \(S^*_ k (M)\) is closed'' by substituting ``Gorenstein'' to ``regular''. The proof lies on the fact that a finite dimensional Gorenstein ring has a fundamental dualizing complex, in which all the nonzero maximal ideals occur in the last term: in fact, that is the shown property. closedness of the singular sets; Gorenstein ring; dualizing complex; Spec DOI: 10.1080/00927879308824821 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Complexes, Singularities in algebraic geometry, Ideals and multiplicative ideal theory in commutative rings On the singular sets of a module | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 the exterior algebra in \(n+1\) variables over a field \(K\). We study the Auslander-Reiten quiver of the category of linear \(R\)-modules, and of certain subcategories of the category of coherent sheaves over \(\mathbb P^n\). If \(n>1\), we prove that up to shift, all but one of the connected components of these Auslander-Reiten quivers are translation subquivers of a \(\mathbb Z A_\infty\)-type quiver. We also study locally free sheaves over the projective \(n\)-space \(\mathbb P^n\) for \(n>1\) and we show that each connected component contains at most one indecomposable locally free sheaf of rank less than \(n\). Finally, using results from the theory of finite-dimensional algebras, we construct a family of indecomposable locally free sheaves of arbitrary large ranks, where the ranks can be computed using the Chebyshev polynomials of the second kind. Koszul algebras; linear modules; exterior algebras; coherent sheaves; locally free sheaves Roberto Martínez-Villa and Dan Zacharia, Auslander-Reiten sequences, locally free sheaves and Chebysheff polynomials, Compos. Math. 142 (2006), no. 2, 397 -- 408. Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homological dimension in associative algebras, Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.), Representations of quivers and partially ordered sets Auslander-Reiten sequences, locally free sheaves and Chebysheff polynomials. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 basic result of this thesis is the following theorem: Let \(f:({\mathbb C}^{n+1},0)\rightarrow ({\mathbb C},0)\) be an isolated singularity with \(n\) even, and let \(\tilde{f}:X\rightarrow S=\bar{\Delta}\) be a morsification of \(f\) and \(D_{\tilde{f}}=\{ z_1,\ldots ,z_{\mu}\}\) its discriminant. Choose a base point \(s\in \partial \Delta\).
The following statements are equivalent: (i) The intersection matrix is semidefinite. (ii) For each simple loop in \(S\) with base point \(s\) and bypassing \(D_{\tilde{f}}\) the corresponding monodromy is quasiunipotent. (iii) Statement (ii) is true for all loops going around exactly two critical points. (iv) Each Dynkin diagram of \(f\) contains only lines with a weight of absolute value \(\leq 2\).
The same equivalence is true with ``definite'' instead of ``semidefinite'', ``finite'' instead of ``quasiunipotent'', and ``\(\leq 1\)'' instead of ``\(\leq 2\)''. isolated singularity; hyperbolic singularity; modality; Dynkin diagram; morsification Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities in algebraic geometry The braid group representation on intersection matrices and monodromy of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors prove folklore conjectures about rigidity in families of curves in \(\mathbb P^3_{\mathbb C}\). Let \(\mathbf{Hilb}^{dz+1-g} (\mathbb P^r)\) denote the Hilbert scheme of one dimensional closed subschemes in \(\mathbb P^r\) of degree \(d\) and arithmetic genus \(g\) and let \(H_{d,g,r} \subset \mathbf{Hilb}^{dz+g-1} (\mathbb P^r)\) denote the union of irreducible components whose general member is smooth, connected and non-degenerate. An irreducible component \(Z \subset H_{d,g,r}\) is \textit{rigid in moduli} if the image of the natural rational map \(Z \to {\mathcal M}_g\) consists of a single point. It is expected that \(H_{d,g,r}\) has no component rigid in moduli unless \(g=0\). Using varieties of linear series [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume I. Berlim: Springer (1985; Zbl 0559.14017)] and classic tools such as the Accola-Griffiths-Harris bound [\textit{J. Harris}, Curves in projective space. With the collaboration of David Eisenbud. Seminaire de Mathematiques Superieures, 85. Seminaire Scientifique OTAN (NATO Advanced Study Institute), Department de Mathematiques et de Statistique - Universite de Montreal. Montreal, Quebec, Canada: Les Presses de l'Universite de Montreal. (1982; Zbl 0511.14014)], the authors prove this expectation when \(r=3\). They deduce a related conjecture [\textit{J. Harris} and \textit{I. Morrison}, Moduli of curves. New York, NY: Springer (1998; Zbl 0913.14005)], which says that the only family of curves whose deformations are all induced from automorphisms of \(\mathbb P^3\) is the family of rational normal curves. They give partial results for restricted ranges on the triples \((d,g,r)\) with \(r > 3\). Hilbert schemes; algebraic curves; linear series; gonality Parametrization (Chow and Hilbert schemes), Divisors, linear systems, invertible sheaves, Special divisors on curves (gonality, Brill-Noether theory) Irreducibility and components rigid in moduli of the Hilbert scheme of smooth 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 the paper under review, the author studies the minimal value of global Tjurina numbers for line arrangements. Let \(S = \mathbb{C}[x,y,z]\) be the graded polynomial ring and let \(C : \, f=0\) be a reduced curve of degree \(d\) in the complex projective plane \(\mathbb{P}^{2}_{\mathbb{C}}\). The minimal degree of a Jacobian relation for \(f\) is the integer \(\mathrm{mdr}(f)\) defined to be the smallest integer \(m\geq 0\) such that there is a non-trivial relation
\[a\frac{\partial \, f}{\partial \, x} +b \frac{\partial \, f}{\partial \, y} + c\frac{\partial \, f}{\partial \, z} = 0\]
with \(a,b,c \in S_{m}\). For a curve \(C\) we denote by \(\tau(C)\) the global Tjurina number of \(C\) which is the sum of the local Tjurina numbers of the singular points of \(C\). If now \(C\) is an arrangement of lines, then the global Tjurina number coincides with the global Milnor number. Let \(\mathcal{A} : \, f = 0\) be an arrangement of \(d\) lines in the complex projective plane and \(r:=\mathrm{mdr}(f)\). Let us denote by \(m(\mathcal{A})\) the maximal multiplicity of a singular point in \(\mathcal{A}\) and let \(n(\mathcal{A})\) be the maximal multiplicity of a point in \(\mathcal{A} \setminus \{p\}\), where \(p\) is a point of multiplicity \(m(\mathcal{A})\).
The main result of the paper can be formulated as follows.
Main Result: Let \(\mathcal{A}: \, f=0\) be an arrangement of \(d\geq 4\) lines in \(\mathbb{P}^{2}\) which is not free. If we set \(r =\mathrm{mdr}(f)\geq 2\) and \(\tau_{\min}(d,r) = (d-1)(d-r-1)\), then the following hold:
\begin{itemize}
\item[i)] \(\tau(\mathcal{A}) \geq \tau_{\min}'(d,r) := \tau_{\min}(d,r) + \binom{r}{2} + \binom{n(\mathcal{A})}{2}+1\).
\item[ii)] If \(r\neq d - m(\mathcal{A})\), then the following possibly stronger inequality holds:
\[\tau(\mathcal{A}) \geq \tau_{\min}''(d,r) := \tau_{\min}(d,r) + \binom{r}{2} + \binom{m(\mathcal{A})}{2}+1.\]
\end{itemize}
Corollary. Let \(\mathcal{A} : \, f=0\) be an arrangement of \(d\) lines in \(\mathbb{P}^{2}\) which is not free such that \(r = \mathrm{mdr}(f) \geq 3\) and \(n(\mathcal{A}) \geq 3\). Then
\[\tau(\mathcal{A}) \geq \tau^{N}_{\min}(d,r) = \tau_{\min}(d,r) + \binom{r}{2} + 4 \geq \tau_{\min}(d,r) + 7.\] Milnor number; line arrangement; Tjurina number Plane and space curves, Singularities in algebraic geometry, Relations with arrangements of hyperplanes On the minimal value of global Tjurina numbers for line arrangements | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study the connection between the geometry of Hilbert schemes and integrable hierarchies. The main goal of the paper is to establish a direct link between equivariant cohomology rings of Hilbert schemes \(X^{[n]}\) of \(n\)-points on a quasi-projective surface \(X\) and integrable hierarchies as well as the correspondence with stationary Gromov-Witten theory. The authors study the case of \(X={\mathbb C}^2\) -- the affine plane. The action of \(T={\mathbb C}^*\) given by the formula \(t(w,z)=(tw,t^{-1}z)\) induces an action on the Hilbert scheme \(X^{[n]}\) with finitely many fixed points parametrized by partitions of \(n\) [cf. \textit{G. Ellingsrud} and \textit{S. A. Strømme}, Invent. Math. 87, 343--352 (1987; Zbl 0625.14002)].
\textit{E. Vasserot} [C. R. Acad. Sci., Paris, Sér. I, Math. 332, No. 1, 7--12 (2001; Zbl 0991.14001)] has shown that the construction of the Heisenberg algebra given in [\textit{H. Nakajima}, Lectures on Hilbert schemes of points on surfaces (1999; Zbl 0949.14001)] can be extended to the \(T\)-equivariant cohomology of Hilbert schemes. All information of the equivariant cohomology ring \(H^{*}_T(X^{[n]})\) is then encoded in \({\mathbb H}_n=H^{2n}_T(X^{[n]})\) and \({\mathbb H}_{X}=\bigoplus_{n\geq 0}{\mathbb H}_n\) becomes the bosonic Fock space of a Heisenberg algebra. The ring \({\mathbb H}_n\) was identified in [Zbl 0991.14001] with the class algebra of the symmetric group \(S_n\).
This allows the authors to establish the correspondence between the \(k\)-th equivariant Chern characters of the tautological rank \(n\) vector bundle and the \(k\)-th power sum of Jucys-Murphy elements. The authors introduce the moduli spaces \({\mathcal M}(m,n)\) , where \(m\in {\mathbb Z}, n\geq 0\). The equivariant cohomology ring of \({\mathcal M}(m,n)\) corresponds to a ring \({\mathbb H}_n^{(m)}\) and \(\bigoplus_{n,m}{\mathbb H}_n^{(m)}\) can be identified with the fermionic Fock space via the boson-fermion correspondence. The introduction of the spaces \({\mathcal M}(m,n)\) allows one to reduce the study of equivariant intersection theory on the Hilbert schemes to the study of intersection numbers of equivariant Chern characters in \({\mathcal M}(m,n)\). The intersection numbers of equivariant Chern characters are studied via the generating functions.
The authors consider three types of generating functions: the \(N\)-point function, the multipoint trace function and the \(\tau\)-function. The authors show that the first function is related to the \(N\)-point disconnected series of stationary Gromov-Witten invariants of \({\mathbb P}^1\), the second is related to the characters on the fermionic Fock space and the third is actually the \(\tau\)-function for the Toda hierarchy. W.-P. Li, Z. Qin, and W. Wang, ''Hilbert schemes, integrable hierarchies, and Gromov-Witten theory,'' Int. Math. Res. Not. 40 (2004), 2085--2104. Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Virasoro and related algebras Hilbert schemes, integrable hierarchies, and Gromov-Witten 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 introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit closure in a semisimple Lie algebra, composed with a linear projection. symplectic resolutions; Springer resolution; semisimple Lie algebra 10.1093/imrn/rnu067 Global theory of complex singularities; cohomological properties, Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects) Symplectic resolutions for conical symplectic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a commutative ring and let \(R\) be a commutative \(k-\)algebra. The aim of this paper is to define and discuss some connection morphisms between schemes associated to the representation theory of a (non necessarily commutative) \(R-\)algebra \(A. \) We focus on the scheme \(Rep_A^n/\!/GL_n\) of the \(n-\)dimensional representations of \(A, \) on the Hilbert scheme \(\text{Hilb}_A^n\) parameterizing the left ideals of codimension \(n\) of \(A\) and on the affine scheme Spec \(\Gamma_R^n(A)^{ab} \) of the abelianization of the divided powers of order \(n\) over \(A. \) We give a generalization of the Grothendieck-Deligne norm map from \(\text{Hilb}_A^n\) to Spec \(\Gamma_R^n(A)^{ab} \) which specializes to the Hilbert Chow morphism on the geometric points when \(A\) is commutative and \(k\) is an algebraically closed field. Describing the Hilbert scheme as the base of a principal bundle we shall factor this map through the moduli space \(Rep_A^n/\!/GL_n\) giving a nice description of this Hilbert-Chow morphism, and consequently proving that it is projective. Hilbert-Chow morphism; Hilbert schemes; linear representations; divided powers F. Galluzzi, F. Vaccarino, Hilbert-Chow morphism for non-commutative Hilbert schemes and moduli spaces of linear representations. \textit{Algebr. Represent. Theory}\textbf{13} (2010), 491-509. MR2660858 Zbl 1203.14004 Schemes and morphisms, Parametrization (Chow and Hilbert schemes), Representation theory of associative rings and algebras, Noncommutative algebraic geometry Hilbert-Chow morphism for non commutative Hilbert schemes and moduli spaces of linear representations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Holomorphic foliations with singularities are considered. In this context, a singularity, or a singular locus, can be described as follows: in a fiber bundle \(\{\) \(F\to E\to B\}\) (with appropriate structure), or in a sheaf over B (with appropriate structure), let a section X be given over \(B-S_ 0\), with \(\overline{B-S_ 0}=B\). Then, in general, the closure \(\bar X\) in the total space E is a section with singularity \(S=\bar X-X\) over \(S_ 0\). A foliation of a manifold M defines a section in a Grassmann bundle over M, and this leads to a treatment of foliations with singularities. The Nash and Grassmann graph constructions, e.g. the blow-up process, are studied by the author. A geometric view of the Baum- Bott residues is presented. The rationality conjecture of \textit{P. Baum} and \textit{R. Bott} [J. Diff. Geom. 7, 279-342 (1972; Zbl 0268.57011)] is reduced to the domain of vector bundles in case the singular holomorphic foliation is given by the image sheaf of a bundle morphism. If the rationality conjecture holds for vector bundles that produce singular holomorphic foliations then it holds for integrable image sheaves of bundle morphisms. This is applied to the blow-up process. Chern classes; Holomorphic foliations with singularities; residues; rationality conjecture; vector bundles; sheaves Local complex singularities, Complex singularities, Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Characteristic classes and numbers in differential topology, Foliations in differential topology; geometric theory Residues of singular holomorphic foliations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(V \subset \mathbb{P}^n(\overline{\mathbb F}_q)\) be a complete intersection defined over a finite field \(\mathbb{F}_q\) of dimension \(r\) and singular locus of dimension at most \(s\), and let \(\pi: V \dashrightarrow \mathbb{P}^{s + 1}(\overline{\mathbb{F}}_q)\) be a generic linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning \(\pi\), namely an explicit upper bound of the degree of a proper Zariski closed subset of \(\mathbb{P}^{s + 1}(\overline{\mathbb{F}}_q)\) which contains all the points defining singular fibers of \(\pi\). For this purpose we make use of the concept of polar variety associated with the set of exceptional points of \(\pi\). As a consequence, we obtain results of existence of smooth rational points of \(V\), that is, conditions on \(q\) which imply that \(V\) has a smooth \(\mathbb{F}_q\)-rational point. Finally, for \(s = r - 2\) and \(s = r - 3\) we estimate the number of \(\mathbb{F}_q\)-rational points and smooth \(\mathbb{F}_q\)-rational points of \(V\). varieties over finite fields; rational points; singular locus; Bertini smoothness theorem; polar varieties; multihomogeneous Bézout theorem; Deligne estimate; Hooley-Katz estimate [CMP12]A. Cafure, G. Matera, and M. Privitelli, Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field , arXiv:1209.4938 [math.AG] (2012). 178G. Matera et al. Varieties over finite and local fields, Rational points, Finite ground fields in algebraic geometry, Singularities in algebraic geometry, Projective techniques in algebraic geometry Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is a solid step towards the understanding of algebraic invariants given by quadratic forms that arise naturally in motivic homotopy theory. In topology, if $f:\mathbb{R}^n\to\mathbb{R}^n$ is a continuous map with an isolated zero at the origin, the classical local Brouwer degree is the integer given by the degree of the induced map between sufficiently small $(n-1)$-spheres. In algebraic geometry, there is a similar definition using motivic homotopy theory (Definition 11): if $k$ is a field and $f:\mathbb{A}^n_k\to\mathbb{A}^n_k$ is a morphism of schemes with an isolated zero at the origin, its local $\mathbb{A}^1$-Brouwer degree is the $\mathbb{A}^1$-degree (as defined by \textit{F. Morel} [in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 1035--1059 (2006; Zbl 1097.14014)] of the induced endomorphism of the motivic sphere $\mathbb{P}^{n}_k/\mathbb{P}^{n-1}_k$, which is an element in the Grothendieck-Witt group $GW(k)$. The main result of the paper states that this class agrees with the Eisenbud-Khimshiashvili-Levine class (EKL class) of $f$ (Definition 7), which is the class of the symmetric bilinear form over the local algebra defined by the distinguished socle element (Definition 1) in $GW(k)$. This result contains former results of Eisenbud-Levine, Khimshiashvili and Palamodov as particular cases by taking the signature or rank of this equality. The authors first prove the result in the case where $f$ has simple zero using the work of \textit{M. Hoyois} [Algebr. Geom. Topol. 14, No. 6, 3603--3658 (2014; Zbl 1351.14013)], and then deduce the general case by moving along fibers to have a regular value. \par From a computational point of view, there are well-behaved algorithms that compute the ELK class from the equations of $f$. The authors then give applications of the main result in enumerative geometry: given an isolated hypersurface singularity at the origin in $\mathbb{A}^n_k$, the arithmetic (or $\mathbb{A}^1$-)Milnor number is defined as the local $\mathbb{A}^1$-Brouwer degree of the gradient of its defining equation. This invariant gives arithmetic information that measures the bifurcation of the hypersurface at the nodes. \(\mathbb{A}^1\) degree; \(\mathbb{A}^1\) enumerative geometry; Eisenbud-Levine/Khimshiashvili signature formula; Milnor number Motivic cohomology; motivic homotopy theory, Singularities in algebraic geometry, Degree, winding number The class of Eisenbud-Khimshiashvili-Levine is the local \(\mathbb{A}^1\)-Brouwer 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 Vafa-Witten invariants introduced by Tanaka-Thomas are enumerative invariants of Higgs pairs on surfaces. In the stable case, the \(SU(r)\) invariants are defined as
\[
VW_{r, L, c_2}= \int_{ [ (\mathcal{N}^\perp_{r,L,c_2})^{\mathbb{C}^*}]^\text{vir} } \frac{1}{e(N^\text{vir})}.
\]
Here \(\mathcal{N}^\perp_{r,L,c_2}\) denotes the moduli space of Gieseker semistable Higgs pairs \((E,\phi)\) on a smooth polarized projective surface \(S\) with \(\mathrm{rk}(E)=r, \det E=L, c_2(E)=c_2\) and \(\mathrm{tr}(\phi)=0\). It has a symmetric obstruction theory equivariant with respect to the \(\mathbb{C}^*\) action scaling \(\phi\), and the RHS is defined by virtual localization. The same recipe can be used to define pair invariants
\[
P^\perp_{r, L, c_2}(n)= \int_{ [ (\mathcal{P}^\perp_{r,L,c_2})^{\mathbb{C}^*}]^\text{vir} } \frac{1}{e(N^\text{vir})}.
\]
For a polarization \(H\) of \(S\) and \(n \gg 0\), \(\mathcal{P}^\perp_{r,L,c_2}\) denotes the moduli space of isomorphism classes of Joyce-Song type pairs \(( (E, \phi), s)\) that are stable, where \((E,\phi)\) is a semistable Higgs pair, \(s \colon \mathcal{O}_S(-nH) \to E\) a nonzero section, and the pair is stable if \(s\) does not factor through any destabilizing sub Higgs pair. A Higgs pair \((E,\phi)\) corresponds to a torsion sheaf \(\mathcal{E}\) on the total space \(X=\mathbb{V}_S(K_S)\) via the spectral construction, and \(( (E, \phi), s)\) corresponds to a Joyce-Song pair \((\mathcal{E}, s)\) on \(X\). In the presence of strictly semistable Higgs pairs and when \(\deg K_S>0\), Vafa-Witten invariants remain conjectural in general, and their existence is formulated using the invariants \(P^\perp_{r, L, c_2}(n)\). For a more detailed summary, see the introduction of the paper by \textit{Y. Tanaka} and \textit{R. P. Thomas} [Pure Appl. Math. Q. 13, No. 3, 517--562 (2018; Zbl 1404.53036)]. \(K\)-theoretic refinement of numerical Vafa-Witten invariants are discussed by \textit{R. P. Thomas} [Commun. Math. Phys. 378, No. 2, 1451--1500 (2020; Zbl 1464.14060)].
For a surface \(S\) with \(p_g(S)>0\), Vafa-Witten invariants \(VW_{\alpha}\) for \(\alpha=(r,L,c_2)\) are conjecturally defined by
\[
P^\perp_\alpha(n) = (-1)^{\chi(\alpha(n))-1}\chi(\alpha(n))VW_{\alpha}.
\]
The \(K\)-theoretic version is
\[
P^\perp_\alpha(n,t) = (-1)^{\chi(\alpha(n))-1}[\chi(\alpha(n))]_tVW_{\alpha}(t).
\]
The paper under review concerns the contribution of vertical components in \((\mathcal{P}^\perp_{\alpha})^{\mathbb{C}^*}\) to \(P^\perp_\alpha(n)\) and \(P^\perp_\alpha(n,t)\). Results and techniques of this paper can be viewed as extension of the author's previous work [Geom. Topol. 24, No. 6, 2781--2828 (2020, Zbl 07305780)]. As the author showed, vertical components can be explicitly constructed from nested Hilbert schemes. It is proved that the contribution is of the form \((-1)^{\chi(\alpha(n))-1}\chi(\alpha(n))VW^{\text{vert}}_{\alpha}\) for certain number \(VW^{\text{vert}}_{\alpha}\) independent of \(n\), parallel result holds for \(K\)-theoretic invariants \(VW^{\text{vert}}_{\alpha}(t)\). It follows from a vanishing result of Thomas that when the rank of \(E\) is prime, \(VW_\alpha^{\text{vert}}\) (\(VW^{\text{vert}}_{\alpha}(t)\)) accounts for all contribution from the monopole branch to \(VW_\alpha\) (\(VW_\alpha(t)\)). The author further established that the generating functions for \(VW^{\text{vert}}_{\alpha}\) and \(VW^{\text{vert}}_{\alpha}(t)\) can be expressed in terms of universal Laurent series \(A, B, C_{ij}\) that appeared in the author's paper [loc.cit., Theorem A] and Seiberg-Witten invariants. Vafa-Witten invariants; Joyce-Song pairs; tautological integrals Algebraic moduli problems, moduli of vector bundles, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Parametrization (Chow and Hilbert schemes) Vertical Vafa-Witten invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0607.00005.]
The three-dimensional Brieskorn singularity is the singularity \((V,0)\subset ({\mathbb{C}}^ 4,0)\) defined by the equation \(f(z)=z_ 0^{a_ 0}+z_ 1^{a_ 1}+z_ 2^{a_ 2}+z_ 3^{a_ 3}=0\). Let \(\pi: \tilde V\to V\) be the torus resolution connected with the simplicial decomposition \(\Sigma^*\) of the dual Newton diagram \(\Gamma^*(f)\) [see the author in Compl. analytic singularities, Proc. Semin., Ibaraki/Jap. 1984, Adv. Stud. Pure Math. 8, 405-436 (1987; Zbl 0622.14012)]. Let P be the ray of the decomposition \(\Sigma^*\) defined by the vector of weights of the quasi-homogeneous function f(z) and \(E(P)\subset \pi^{-1}(0)\) be the corresponding exceptional divisor. The goal of the article in question is to study the geometry of the divisor E(P). According to the powers \(a=(a_ 0,...,a_ 3)\), integers d, \(r_ i, r_{ij}\), \(0\leq i,j\leq 3\), are defined where \(d=\gcd (a_ 0,...,a_ 3),\quad r_ i=\gcd \{a_ j;j\neq i\}/d.\) It is proved that if for two Brieskorn singularities the numbers d, \(r_ i, r_{ij}\) coincide, then their divisors E(P) are birationally equivalent. A necessary and sufficient condition on the powers \(a=(a_ 0,...,a_ 3)\) for the surface E(P) to be either rational or a K3-surface is obtained. There are 14 cases where E(P) is a rational surface and 22 cases where E(P) is a K3-surface. geometry of divisor; three-dimensional Brieskorn singularity; torus resolution; K3-surface; rational surface M. OKA, On the resolution of hypersurfaces singularities, Adv. Studies in Pure Math. 8 (1986), 405-^36 Zentralblatt MATH: Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Modifications; resolution of singularities (complex-analytic aspects) On the resolution of the three dimensional Brieskorn 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 0727.00022.]
The paper deals with homogeneous vector bundles over manifolds of the form \(X=G/P\) where \(G\) is a compact semisimple algebraic group and \(P\subset G\) is a parabolic subgroup. The study of homogeneous bundles reduces to the study of representations of the subgroup \(P\), and the authors show that, under certain assumptions, the category of finite- dimensional algebraic representations of \(P\) is equivalent to the category of finite-dimensional modules over a certain category \(A(P)\). More precisely, \(A(P)\) is given by a quiver with relations, the vertices are integral \({\mathfrak a}\)-dominant weights, where \({\mathfrak a}\) is a Levi subalgebra, there are arrows \((\lambda)\to(\lambda+\gamma)\) for every root \(\gamma\) of the nilpotent radical and the relations are obtained from the relations of the corresponding Chevalley generators. Moreover the case \(X=\mathbb{P}^ n\) is considered. In particular the authors describe explicitly all vector bundles on \(\mathbb{P}^ 2\) of rank less then 100 which are exceptional [compare \textit{A. N. Rudakov}, Math. USSR, Izv. 32, No. 1, 99-112 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 1, 100-112 (1988; Zbl 0661.14017)]. exceptional vector bundle; representations of a quiver; homogeneous bundles A. I. Bondal and M. M. Kapranov, ''Homogeneous bundles,'' In:Helices and Vector Bundles, Vol. 148, Lond. Math. Soc. L. N. S. (1990), pp. 45--55. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homogeneous spaces and generalizations, Representations of quivers and partially ordered sets Homogeneous bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(T\) be a tree with \(n\) vertices, and let \(\mathbb{K}\) be a field. Let \({\alpha}=(\alpha_{t})_{t\in T}\) be an assignment to each vertex of \(T\) a value in \(\mathbb{K}.\) One can then consider the affine scheme \(X_{T}({\alpha})\) given by the equations \(x_{t}x_{t}^{\prime}=1+\alpha_{t}\prod x_{s}\), where the product ranges over all \(s\) adjacent to \(t\). This paper is concerned with computing the number of points \(N_{T}({\alpha})\) on this variety in the case where \(\mathbb{K}\) has \(q\) elements, focusing on three types of trees. Specifically, the trees considered are Dynkin diagrams of type \(\mathbb{A} ,\mathbb{D},\) and \(\mathbb{E}\) in the case where each \(\alpha_{i}\) is invertible.
Suppose that \(T=\mathbb{A}_{n}\). If \(n\) is even then \(X_{\mathbb{A}_{n} }(\alpha_{1},\dots,\alpha_{n})\cong X_{\mathbb{A}_{n}}(1,\dots,1)\): this is proved by studying full domino tilings of the tree. From this it follows that \(N_{\mathbb{A}_{n}}=(q^{n+2}-1)/(q^{2}-1).\) On the other hand, if \(n\) is odd we have \(X_{\mathbb{A}_{n}}(\alpha_{1},\dots,\alpha_{n})\cong X_{\mathbb{A}_{n}}(\alpha,1,\dots,1)\) for some \(\alpha\) depending on the \(\alpha_{i}\) with \(i\) odd: this follows from the partial domino tiling of \(\mathbb{A}_{n}\) leaving the first vertex uncovered. If \(\alpha\neq(-1)^{(n+1)/2}\) then \(N_{\mathbb{A} _{n}}=(q^{(n+1)/2}-1)(q^{(n+3)/2-1})/(q^{2}-1).\) Write \(X_{\mathbb{A} _{n}}(\alpha)=X_{\mathbb{A}_{n}}(\alpha,1,\dots ,1)\) (here \(n\) may be even or odd). If \(n\) is odd, and \(\alpha=(-1)^{(n+1)/2}\) then \(X_{n}(\alpha)\) has a unique singular point, namely \(x_{2k+1}=x_{2k+1}^{\prime}=0,\) \(x_{2k} =x_{2k}^{\prime}=-(-1)^{(n/2+k)}\). In all other cases, \(X_{n}(\alpha)\) is smooth.
While the focus is on counting the number of points on the varieties, further results are obtained when \(T=\mathbb{A}_{n}.\) Let \(Z_{\mathbb{A}_{n}}\) (resp. \(Y_{\mathbb{A}_{n}}\)) be the union of all \(X_{\mathbb{A}_{n}}(\alpha)\) not necessarily invertible (resp. necessarily invertible). Then \(Z_{\mathbb{A}_{n}}\) has \(q^{n+1}\) points and \(Y_{\mathbb{A}_{n}}\) is a smooth open subset of \(Z_{\mathbb{A}_{n}}\;\)having \((q^{n+2}+(-1)^{n+1})/(q+1)\) points.\ For \(0\leq i\leq n+1\) the non-zero cohomology groups with compact support for \(Y_{\mathbb{A} _{n}}\) are \(H_{c}^{i+n+1}(Y_{\mathbb{A}_{n}})\cong \mathbb{Q}(i)\), the Tate Hodge structure of weight \(i\). Also, for \(n\) even \(H_{c}^{i+n}(X_{\mathbb{A}_{n}}(1))\cong\mathbb{Q}(i)\) for all even \(0\leq i\leq n.\)
Suppose now that \(T=\mathbb{D}_{n}.\) By partial domino tilings if \(n\) is even \(X_{\mathbb{D}_{n}}(\alpha_{1},\dots,\alpha_{n})\cong X_{\mathbb{D}_{n}}(\alpha,\beta,1,\dots,1)\) for some \(\alpha \),\(\beta\) which depend on the \(\alpha_{i};\) if \(n\) is odd then \(X_{\mathbb{D} _{n}}(\alpha_{1},\dots,\alpha_{n})\cong X_{\mathbb{D}_{n} }(\alpha,1,\dots,1)\). Formulas are given for \(N_{\mathbb{D} _{n}}\) -- there are six different cases to consider.
Finally, for the diagrams of type \(\mathbb{E}\) different formulas are obtained for the different exceptional cases. Using domino tilings and counting, we have \(X_{\mathbb{E}_{6}}({\alpha})\cong X_{\mathbb{E} _{6}}(1,\dots,1)\) has \(q^{6}+q^{4}+q^{3}+q^{2}+1\) points and \(X_{\mathbb{E}_{8}}({\alpha})\cong X_{\mathbb{E}_{8} }(1,\dots,1)\) has \(q^{8}+q^{6}+q^{5}+q^{4}+q^{3}+q^{2}+1\) points. As \(\mathbb{E}_{7}\) affords only a partial domino tiling we get \(X_{\mathbb{E}_{7}}({\alpha})\cong X_{\mathbb{E}_{7} }(1,\dots,1,\alpha)\) where \(\alpha\) is the value on the last vertex on the long branch. Provided \(\alpha\neq-1\) we have \(q^{7}+q^{5} -q^{2}-1,\) \(N_{\mathbb{E}_{7}}(1,\dots,-1)\) has \(q^{7} +2q^{5}+q^{3}-q^{2}-1\) points. cluster algebras; affine varieties; Dynkin diagrams; finite fields Rational points, Representations of quivers and partially ordered sets, Cluster algebras, Finite ground fields in algebraic geometry On the number of points over finite fields on varieties related to cluster 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 Fix a field \(k\) and consider the class of all varieties which are smooth over \(k\). If \(W @>\pi>> \text{Spec} (k)\) is within this class, we can attach to it a relative tangent bundle or, equivalently, a locally free sheaf \(\Omega^1_{W/k}\) of relative differentials. This tangent bundle, intrinsic to \(\pi\), plays a central role in algebraic geometry. For instance
(a) in studying the birational class of \(W\),
(b) in analyzing the singular locus of a closed embedded subscheme of \(W\) (e.g. jacobian ideals).
Essential for the development of some of these problems is the fact that the class is closed by blowing up regular centers, namely has the following property:
(P) Let \(W\) be smooth over \(k\), \(C\) a regular closed subscheme of \(W\), and \(W\leftarrow W_1\) the blow-up at \(C\). Then \(W_1\) is also in the class (is also smooth over the field \(k)\).
However, this property fails to hold if we consider now the class of smooth schemes over \(\mathbb{Z}\). In this work we replace \(\text{Spec} (k)\) by a Dedekind scheme of characteristic zero (for instance \(Y=\text{Spec} (\mathbb{Z}))\), and define a class of schemes over \(Y\) such that:
(1) the class includes the smooth schemes over \(Y\),
(2) to any \(W @>\pi>> Y\) in the class there is an intrinsically defined tangent bundle,
(3) the class is closed by blowing up convenient regular centers.
As an application, in \(\S 4\), we analyze the behaviour of jacobian ideals of embedded arithmetic schemes. sheaf of relative differentials; tangent bundle; birational class; singular locus; blowing up; smooth schemes; jacobian ideals of embedded arithmetic schemes Villamayor, O.: On smoothness and blowing ups of arithmetical schemes. Math. Z. 225, 317-332 (1997) Schemes and morphisms, Global theory and resolution of singularities (algebro-geometric aspects), Arithmetic problems in algebraic geometry; Diophantine geometry On smoothness and blowing ups of arithmetic schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For 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 Given an affine surface \(X\) with rational singularities and minimal resolution \(X^{\prime}\), the covering of the Artin component of the deformation space of \(X\) where simultaneous resolutions are achieved is Galois and the Galois group is the Weyl group \(W\) associated with the configuration of \((-2)\)-curves on \(X^{\prime}\). This gives the existence of actions of \(W\) on polynomial rings over \(\mathbb{Z}\) where the ring of invariants is also polynomial. In turn, this leads to a description of the integral cohomology rings of flag varieties of type \(\mathit{ADE}\) that extends the known description of the rational cohomology rings as rings of coinvariants for actions of \(W\). Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory, Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds, Homology and cohomology of homogeneous spaces of Lie groups Weyl group covers for Brieskorn's resolutions in all characteristics and the integral cohomology of \(G/P\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Finding procedures in Algebraic Geometry that can be implemented in parallel computing is non-trivial, because some of the most popular algorithms for computations in polynomial rings, as those based on the determination of a Gröbner basis, have an intrinsic sequential nature. The authors describe some algebraic geometric questions that can be handled with parallel computing. The main problem that they address is the construction of a test for detecting the smoothness of algebraic varieties \(X\), following Hironaka's desingularisation approach which is based on the study of hypersurfaces of local maximal contact. The test splits \(X\) in a finite union of affine charts \(X_i\)'s embedded, with low codimension, in complete intersection smooth varieties. Then the test applies Hironaka's termination criterion and the Jacobian criterion to detect the smoothness of each \(X_i\). The algorithm has been implemented under the workflow management system GPI-Space, which controls the execution of many copies of the computer algebra system SINGULAR, for the study of the charts \(X_i\)'s. The authors show that their method works in some cases of surfaces of general type, too hard to be handled by a sequential approach. parallel computing; smoothness test Computational aspects of algebraic surfaces, Singularities in algebraic geometry Towards massively parallel computations in algebraic geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a nonsingular complex projective variety and \(D\) a locally quasi-homogeneous free divisor in \(X\). In this paper, we study a numerical relation between the Chern class of the sheaf of logarithmic derivations on \(X\) with respect to \(D\), and the Chern-Schwartz-MacPherson class of the complement of \(D\) in \(X\). Our result confirms a conjectural formula for these classes, at least after pushforward to projective space; it proves the full form of the conjecture for locally quasi-homogeneous free divisors in \(\mathbb{P}^n\). The result generalizes several previously known results. For example, it recovers a formula of \textit{M. Mustaţǎ} and \textit{H. K. Schenck} [J. Algebra 241, No. 2, 699--719 (2001; Zbl 1047.14007)] for Chern classes for free hyperplane arrangements. Our main tools are Riemann-Roch and the logarithmic comparison theorem of Calderon-Moreno, \textit{F. J. Castro-Jiménez} et al. [Trans. Am. Math. Soc. 348, No. 8, 3037--3049 (1996; Zbl 0862.32021)]. As a subproduct of the main argument, we also obtain a schematic Bertini statement for locally quasi-homogeneous divisors. (Equivariant) Chow groups and rings; motives, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Chern classes of logarithmic vector fields for locally quasi-homogeneous free divisors | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We observe that there exists a Białynicki-Birula decomposition of the Hilbert scheme \(\text{Hilb}^P_n\) such that the cells are homeomorphic to Gröbner strata of homogeneous ideals with fixed initial ideal. Using such a decomposition, we show that \(\text{Hilb}^P_n\) is singular at a monomial scheme if the corresponding Gröbner stratum is singular at \(J\). Hilbert scheme; Białynicki-Birula decomposition; Gröbner bases Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials Computable Białynicki-Birula decomposition of 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 The purpose of this paper is to summarize the results that we have obtained recently on four dimensional \(\mathcal{N}=2\) superconformal field theories from the point of view of singularity theory. superconformal field theories; rational singularity; Gorenstein singularity; deformation Supersymmetric field theories in quantum mechanics, Singularities in algebraic geometry, Local complex singularities Recent results on \(4d\) \(\mathcal{N}=2\) SCFT 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 This is a continuation of previous work of \textit{A. S. Tikhomirov} [same conference, Aspects Math. E 25, 183-203 (1994; see the preceding review)] to which we refer for definitions and notations. The main result of the present paper is an explicit formula for the polynomial \(\delta_ 4\) of a nonsingular projective surface \(S\). According to the above mentioned article, the computation reduces to an enumerative problem: the determination of the number of 4-secant planes to a convenient embedding of \(S\) into \(\mathbb{P}^{10}\). Hilbert scheme; Segre class; standard vector bundle; number of 4-secant planes Tikhomirov, A.; Troshina, T., Top Segre class of a standard vector bundle e4 D on the Hilbert scheme hilb4S of a surface S, 205-226, (1994) Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Enumerative problems (combinatorial problems) in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Top Segre class of a standard vector bundle \({\mathcal E}^ 4_ D\) on the Hilbert scheme \(\text{Hilb}^ 4S\) of a surface \(S\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(I\) be a homogeneous ideal in \(R=\mathbb K[x_0,\dots ,x_n]\), such that \(R/I\) is an Artinian Gorenstein ring. A famous theorem of Macaulay says that in this instance \(I\) is the ideal of polynomial differential operators with constant coefficients that cancel the same homogeneous polynomial \(F\). A major question related to this result is to be able to describe \(F\) in terms of the ideal \(I\). In this note we give a partial answer to this question, by analyzing the case when \(I\) is the Artinian reduction of the ideal of a reduced (arithmetically) Gorenstein zero-dimensional scheme \(\varGamma\subset\mathbb P^n\). We obtain \(I\) from the coordinates of the points of \(\varGamma\). Artinian Gorenstein ring; Macaulay inverse system; zero-dimensional scheme Commutative rings of differential operators and their modules, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Parametrization (Chow and Hilbert schemes) Finding inverse systems from coordinates | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 topology of line singularities, which are complex hypersurface germs with non-isolated singularity given by a smooth curve. We describe the degeneration of its Milnor fiber to the singular hypersurface by means of a vanishing polyhedron in the Milnor fiber. As a milestone, we also study the topology of the degeneration of a complex isolated singularity hypersurface under a nonlocal point of view. vanishing polyhedron; line singularity; degeneration; Milnor fiber; polar curve Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Equisingularity (topological and analytic), Global theory of complex singularities; cohomological properties, Complex surface and hypersurface singularities Lê's polyhedron for line singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we study the tangent spaces of the smooth nested Hilbert scheme \(\mathrm{Hilb}^{n,n-1}(\mathbb A^2)\) of points in the plane, and give a general formula for computing the Euler characteristic of a \(\mathbb T^2\)-equivariant locally free sheaf on \(\mathrm{Hilb}^{n,n-1}(\mathbb A^2)\). Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables \(q\) and \(t\) with non-negative integer coefficients. We call this conjecturally positive polynomial as the ``nested \(q,t\)-Catalan series'', for it has many conjectural properties similar to that of the \(q,t\)-Catalan series. Atiyah-Bott Lefschetz formula; (nested) Hilbert scheme of points; tangent spaces; diagonal coinvariants Can, M.: Nested Hilbert schemes and the nested q,t-Catalan series Parametrization (Chow and Hilbert schemes), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Nested Hilbert schemes and the nested \(q,t\)-Catalan 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 Denote by \(E\) an Enriques surface, and by \(E^{[n]}\) for \(n\geq 1\), the Hilbert scheme of \(n\) points on \(E\).
Contrary to the case \(n=1\), the first result is to show that the small deformations of \(E^{[n]}\) for \(n\geq 2\) are induced from those of its universal covering space, by computing the dimensions of these deformation spaces.
In the second result, an equivalent condition for an automorphism \(f\) of \(E^{[n]}\) to be natural is given, namely, there exists an automorphism \(g\) of \(E\) such that \(f\) is naturally induced by \(g\). This is analogous to the case of the Hilbert scheme of \(n\) points on \(K3\) surfaces, that is, the automorphism \(f\) is natural if and only if it preserves the exceptional divisor of the Hilbert-Chow morphism of \(E^{[n]}\).
Contrary to the case of \(n=1\) (where the universal covering space being a \(K3\) surface), it is proved that there exists exactly one Enriques surface type quotient for the universal covering space \(X\) of \(E^{[n]}\). The proof is done by a classification of all involutions of \(X\) acting identically on \(H^2(X,\,\mathbb{C})\). Calabi-Yau manifld; Enriques surfaces; Hilbert scheme Calabi-Yau manifolds (algebro-geometric aspects), \(K3\) surfaces and Enriques surfaces, Parametrization (Chow and Hilbert schemes) Universal covering Calabi-Yau manifolds of the Hilbert schemes of \(n\) points of Enriques 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 Generalizing the construction of \textit{I. N. Bernstein} and \textit{S. I. Gel'fand} [Funkts. Anal. Prilozh. 3, No. 1, 84--85 (1969; Zbl 0208.15201)] the author introduces a zeta function of an algebraic variety defined over a finite extension of the field of \(p\)-adic numbers and proves that it is a rational function of several complex variables. As an application of this result, she obtains a new proof of rationality of the Poincaré series for any algebraic variety (cf. \textit{D. Meuser} [Math. Ann. 256, 303--310 (1981; Zbl 0471.12014)] and \textit{J. Denef} [Invent. Math. 77, 1--23 (1984; Zbl 0537.12011)] for previous results in this direction) and describes a finite set containing each of the poles of this series. local zeta-functions; singularities; Poincaré series; poles Deshommes, B.: Critères de rationalité et application à la série génératrice d'un système d'équations à coefficients dans un corps local. J. number theory 22, 75-114 (1986) Zeta functions and \(L\)-functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Global theory and resolution of singularities (algebro-geometric aspects) Rationality criteria and application to the generating series of a system of equations with coefficients in a local field | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 compact complex manifold \(X\) is called rigid if it has no nontrivial deformations, and it is said to be infinitesimally rigid if \(H^1(X,\Theta_X) = 0\), where \(\Theta_X\) is the sheaf of holomorphic vector fields on \(X\).
It is well known, that infinitesimal rigidity implies rigidity and the only rigid curve is \(\mathbb{P}^1\). In [\textit{I. Bauer} and \textit{F. Catanese}, Adv. Math. 333, 620--669 (2018; Zbl 1407.14003)] it is shown that rigid compact complex surfaces have Kodaira dimension \(-\infty\) or 2. In higher dimensions rigid manifolds are much more frequent and it is nowadays known, that for all \(n \geq 3\) and \(\kappa\in \{-\infty,0,\ldots,n\}\) there exists an infinitesimally rigid \(n\)-dimensional compact complex manifold of Kodaira dimension \(\kappa\).
A way to construct the infinitesimally rigid examples is to consider finite quotients of smooth compact complex manifolds \(X\) with respect to a (infinitesimally) rigid holomorphic group action of \(G\), i.e. \(G\) is a finite group acting holomorphically on \(X\) and \(H^1(X,\Theta_X)^G = 0\).
In the paper under review the authors consider \(X\) to be the product of elliptic curves and \(G\) a finite group acting \textit{diagonally} on the product. They firstly consider \textit{free} actions, and show that there are only four finite groups, that may carry a free rigid action on a product of elliptic curves. These four groups are called \textit{exceptional groups} and are exactly those groups admitting a rigid action on an elliptic curve such that the translation part is not uniquely determined (Proposition 4.4).
It is then show that actually only for 2 groups and dimension \(\geq 4\) such an action can be free, more precisely, the first main result of the paper is
{Theorem 1.4.} Let \(G\) be a finite group which admits an infinitesimally rigid free diagonal action on a product of elliptic curves \(E_1 \times \cdots \times E_n\). Then: \(n\geq 4\), \(E_1\cong \cdots \cong E_n\) is the Fermat elliptic curve, \(E_i\) is the Fermat elliptic curve for each \(1 \leq i \leq n\), and \(G\) is either \(\mathbb{Z}_2^3\) or the Heisenberg group \(He(3)\) of order 27.
The authors then consider non-free actions and for the not exceptional finite groups a strong structure result is proven:
{Theorem 1.5.} Assume that \(G\) is not exceptional and admits a rigid diagonal action on a product of elliptic curves \(E_1 \times \cdots \times E_n\). Then the elliptic curves are isomorphic \(E_1\cong \cdots \cong E_n=:E\), and the quotient is isomorphic to \(X_{n,d} := E^n/\mathbb{Z}_d\), where \(Z_d\) acts by multiplication with \(\exp\left(\frac{2\pi i }d\right) \cdot Id\). Here either \(d = 3, 6\) and \(E\) is the Fermat elliptic curve, or \(d = 4\) and
\(E\) is the harmonic elliptic curve.
Finally they give a complete classification of rigid diagonal actions of the two exceptional groups \(\mathbb{Z}_2^3\) and \(He(3)\) in dimensions 3 and the free ones in dimension 4:
{Theorem 1.7.}
\begin{itemize}
\item[1)] There is exactly one isomorphism class of quotient manifolds \(E^4/\mathbb{Z}_3^2\) resp. \(E^4/He(3)\) obtained by a rigid free and diagonal action. They have non isomorphic fundamental groups.
\item[2)] For each exceptional group \(\mathbb{Z}_2^3\) and \(He(3)\) there are exactly four isomorphism classes of (singular) quotients \(X_i := E^3/\mathbb{Z}_2^3\) and \(Y_i := E^3/ He(3)\) obtained by a rigid diagonal G-action:
\begin{itemize}
\item[(i)] \(X_4\) and \(Y_4\) are isomorphic to Beauville's Calabi-Yau threefold \(X_{3,3}\).
\item[(ii)] \(X_3\) and \(Y_3\) are also Calabi-Yau, uniformized by \(X_{3,3}\) and admit crepant resolutions, which are rigid.
\item[(iii)] \(X_2\) and \(X_3\), resp. \(Y_2\) and \(Y_3\), are diffeomorphic but not biholomorphic.
\item[(iv)] The eight threefolds \(X_i\), \(Y_j\) form five distinct topological classes.
\end{itemize}
\end{itemize} rigid complex manifolds; deformation theory; quotient singularities; hyperelliptic manifolds; crystallographic groups Local deformation theory, Artin approximation, etc., Calabi-Yau manifolds (algebro-geometric aspects), Deformations of special (e.g., CR) structures, Group actions on varieties or schemes (quotients), Abelian varieties and schemes, Families, moduli, classification: algebraic theory, \(n\)-folds (\(n>4\)), Singularities in algebraic geometry, Deformations of complex structures, Other geometric groups, including crystallographic groups Towards a classification of rigid product quotient varieties of Kodaira dimension 0 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a very interesting paper on Lipschitz equisingularity problems. The author states Sullivan's questions concerning Lipschitz equisingularity and its solution via Lipschitz stratifications, both in the complex and real cases. The paper also contains a solution to a question of M. Gromov, relations between the notions of Zariski's equisingularity and Lipschitz equisingularity and existence of moduli for the relation of Lipschitz equivalence of functions. Lipschitz equisingularity; Zariski equisingularity; Lipschitz equivalence of functions Critical points of functions and mappings on manifolds, Singularities in algebraic geometry, Equisingularity (topological and analytic) Lipschitz equisingularity problems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(f\) and \(g\) be reduced homogeneous polynomials in separate sets of variables. The article establishes a simple formula that relates the eigenspace decomposition of the monodromy operator acting on the cohomology of the Milnor fiber at the origin of \(fg\) to that of the singularities determined by \(f\), respectively \(g\). Hyper-surface singularities; Milnor fibration; Milnor fiber; monodromy; homogeneous singularities. D. Tapp, Picard-Lefschetz monodromy of products , J. Pure Appl. Algebra 212 (2008), 2314-2319. Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Picard-Lefschetz monodromy of 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 Neumann and Wahl introduced the notion of splice-quotient singularities, which is a broad generalization of quasihomogeneous singularities with rational homology sphere links. They proved the End Curve Theorem, that characterizes splice-quotient singularities, applying a theory of numerical semigroups. In this paper, the author gives a different proof of the End Curve Theorem, using combinatorics of monomial cycles and basic ring theory. surface singularity; splice-quotient singularity; rational homology sphere; splice type singularity; universal abelian cover Okuma, Another proof of the end curve theorem for normal surface singularities, J. Math. Soc. Japan 62 pp 1-- (2010) Complex surface and hypersurface singularities, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Another proof of the end curve theorem for normal surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V_{s,t}\) be the germ at the origin of the curve defined by
\[
f_{s,t}:=x^5+sx^4y+tx^3y^2+y^5=0\;,\;s,t\in\mathbb{C}\;.
\]
The biholomorphically equivalence problem is discussed for the family \(V_{s,t}\). It is proved that \(V_{s_1, t_1}\) and \(V_{s_2, t_2}\) are biholomorphically equivalent if and only if \(j(s_1, t_1)=j(s_2, t_2)\), \(k(s_1, t_1)=k(s_2, t_2)\) and \(l(s_1, t_1)=l(s_2, t_2)\). Here \(j, k, l\) are relatively complicated invariants from classical invariant theory, for example
\[
j(s,t)=\frac{5(1440000\cdot 10!)^2(125-3st^2)^2}{256s^5-1600s^3t-27s^2t^4+2250st^2+108t^5+3125}.
\]
plane curve singularities; binary forms; invariant theory Singularities of curves, local rings, Singularities in algebraic geometry, Local complex singularities Application of classical invariant theory to biholomorphic classification of plane curve singularities, and associated binary forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X^{[n]}\) denote the Hilbert scheme of \(n\) points on a nonsingular projective variety \(X\).
An isomorphism \(g:X \to Y\) of varieties induces an isomorphism \(g^{[n]}: X^{[n]} \to Y^{[n]}\).
\textit{S. Boissière} defined an isomorphism \(\sigma: X^{[n]} \to Y^{[n]}\) to be \textit{natural} if \(\sigma = g^{[n]}\) for some isomorphism \(g:X \to Y\) [Can. J. Math. 64, No. 1, 3--23 (2012; Zbl 1276.14006)].
Using work of \textit{A. Beauville} [J. Differ. Geom. 18, 755--782 (1983; Zbl 0537.53056)], \textit{R. Zuffetti} recently exhibited isomorphisms \(X^{[2]} \cong Y^{[2]}\) that are not natural, with \(X\) and \(Y\) (non-isomorphic) \(K3\) surfaces [Rend. Semin. Mat., Univ. Politec. Torino 77, No. 1, 113--130 (2019; Zbl 1440.14183)].
Defining a variety \(X\) to be \textit{weak Fano} if \(\omega_X^\vee\) is nef and big, the authors prove that if \(X\) is a smooth projective surface that is weak Fano or of general type and \(n\) is an integer, then every automorphism of \(X^{[n]}\) is natural unless \(n=2\) and \(X = C \times D\) is a product of two curves. In the latter case, if \(C,D\) are both rational or both of genus \(g \geq 2\), then there is a unique nonnatural isomorphism of \(X = C \times D\) up to composition with natural automorphisms.
They also prove that every automorphism of \((\mathbb P^n)^{[2]}\) is natural.
The authors also show that if \(X,Y\) are smooth projective surfaces with \(Y\) weak Fano or of general type, then every isomorphism \(X^{[n]} \to Y^{[n]}\) is natural (they assume \(n \geq 3\) if \(Y\) is a product of curves). This can be thought of as an analog to a theorem of \textit{A. Bondal} and \textit{D. Orlov} [Compos. Math. 125, No. 3, 327--344 (2001; Zbl 0994.18007)] and extends results of \textit{T. Hayashi}, who proved this for rational surfaces whose Iitaka dimension of \(\omega_X^\vee\) is at least one [Geom. Dedicata 207, 395--407 (2020; Zbl 1444.14012)]. Hilbert scheme of points; natural isomorphisms; smooth surfaces; automorphisms Parametrization (Chow and Hilbert schemes), Automorphisms of surfaces and higher-dimensional varieties Automorphisms of Hilbert schemes of points on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A bimeromorphic morphism f: Y'\(\to Y\) between two normal Gorenstein surfaces is called canonical if \(f_*\omega_{Y'}\cong \omega_ Y\). It is shown that such a morphism can be factored into a finite number of simple ones. A conceptual proof can be found in a more general frame work in the author's recent article: ''Weil divisors on normal surfaces'', Duke Math. 51, 877-888 (1984). In the definition of the generalized blowing up of type III in page 31, we need the following correction. There is given a sequence of usual blowing ups: \(U_ n\to...\to U_ 1\to U.\) We continue this as follows: Since \(\Delta_ nE_ n=1\) and \(E_ n\not\subset \Delta_ n\), we can find an irreducible component C of \(\Delta_ n\) with coefficient 1 which meets \(E_ n\) in a point \(x_{n+1}\). Blow up at this point, then blow up at the point \(x_{n+2}\) where the strict transform of C meets the exceptional curve \(E_{n+1}\) over \(x_{n+1}\), and in this way we continue r times, so that we obtain an extra sequence: \(U'=U_{n+r}\to...\to U_ n.\) Write \(\Delta_{n+i}=\eta^*_{n+i}\Delta_{n+i-1}-E_{n+i}.\) We put \(\Delta '=\Delta_{n+r},\) \(E'=E_{n+r},\) then \(\omega_{\Delta '}\cong {\mathcal O}_{\Delta '}.\) Let \(\pi ': U'\to V'\) be the contraction of \(\Delta\) ' and the chain of (-2)-curves consisting of the strict transforms of \(E_ n,E_{n+1},...,E_{n+r-1}\). On V', there are a non- rational Gorenstein singularity y' and a rational double point y'' of type \(A_ r\). We define a blowing up of type III by contracting the curve \(E=\pi '(E')\) to y. factorization of canonical morphism between two normal Gorenstein surfaces; blowing up F. Sakai, Normal Gorenstein surfaces and blowing ups, Saitama Math. J.1 (1983), 29--35. Rational and birational maps, Global theory and resolution of singularities (algebro-geometric aspects), Special surfaces Normal Gorenstein surfaces and blowing ups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(Q=(Q_0,Q_1,t,h)\) be a quiver with set of vertices \(Q_0\), set of arrows \(Q_1\) and functions \(t,h\colon Q_1\to Q_0\) attaching to an arrow \(\alpha\) its tail \(t(\alpha)\) and head \(h(\alpha)\). A representation of \(Q\) of dimension vector \(\underline n=(n_i\mid i\in Q_0)\) is a collection of linear maps \(L=(L_\alpha\colon V_{t(\alpha)}\to V_{h(\alpha)}\mid\alpha\in Q_1)\), where \(V_i\) is an \(n_i\)-dimensional vector space associated with \(i\in Q_0\). Let the base field \(K\) be infinite and of any characteristic. Identifying \(V_i\) with the space \(K^{n_i}\) of column vectors, and \(L_\alpha\) with an \(n_{h(\alpha)}\times n_{t(\alpha)}\) matrix operating by left multiplication, the representation \(L\) corresponds to a point in the affine space \(R=R(Q,\underline n)=\bigoplus_{\alpha\in Q_1}K^{n_{h(\alpha)}\times n_{t(\alpha)}}\). The group \(\text{GL}(\underline n)=\prod_{i\in Q_0}\text{GL}_{n_i}(K)\) acts on \(R\) such that \(g\cdot L=(g_{h(\alpha)}L_\alpha g_{t(\alpha)}^{-1}\mid\alpha\in Q_1)\) for any \(g\in\text{GL}(\underline n)\).
The purpose of the paper under review is to describe the algebra of \(\text{GL}(\underline n)\)-semi-invariants (i.e., the algebra of \(\text{SL}(\underline n)\)-invariants) of the quiver \(Q\). There is a canonical way to blow up an arbitrary quiver to obtain a bipartite quiver, i.e., a quiver for which any vertex is either a source or a sink. The crucial observation of the authors is that this construction corresponds to forming an associated fibre bundle. Algebraically this gives induction of modules over algebraic groups. Applying Frobenius reciprocity, the authors conclude that the semi-invariants of the original quiver can be obtained from those of the enlarged bipartite quiver.
The authors define generating semi-invariants of bipartite quivers as partial polarizations of determinants of block matrices. The proof is very natural and is based on a translation of the first fundamental theorem of classical invariant theory. As an application the authors obtain the generators of the algebra of invariants of \(d\)-tuples of \(n\times n\) matrices under simultaneous conjugation in the case of a field of any characteristic [see \textit{S. Donkin}, Comment. Math. Helv. 69, No. 1, 137-141 (1994; Zbl 0816.16015)].
As the authors mention in the paper, similar results in the case of characteristic 0 have been independently obtained by \textit{A. Schofield} and \textit{M. Van den Bergh} [Indag. Math., New Ser. 12, No. 1, 125-138 (2001; Zbl 1004.16012)]. matrix invariants; representations of quivers; semi-invariants of quivers; algebras of semi-invariants; fibre bundles; bipartite quivers Mátyás Domokos and, A. N. zubkov. semi-invariants of quivers as determinants, Transformation Groups, 6, 9-24, (2001) Trace rings and invariant theory (associative rings and algebras), Representations of quivers and partially ordered sets, Geometric invariant theory, Vector and tensor algebra, theory of invariants, Actions of groups and semigroups; invariant theory (associative rings and algebras), Representation theory for linear algebraic groups Semi-invariants of quivers as determinants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper we will show that if \(V \subset \mathbb{P}^ r_ \mathbb{C}\) is an irreducible nondegenerate projective variety of dimension \(n\) and degree \(d\), not contained in any irreducible subvariety of \(\mathbb{P}^ r_ \mathbb{C}\) of dimension \(n + 1\) and degree \(< s\), and if \(d\) is large with respect to \(s\), then the genus of \(V\) (arithmetic if \(n = 1\); geometric if \(n \geq 2)\) is bounded above by a number \(G = d^{n+1}/(n + 1)! s^ n + O(d^ n)\) which depends only on \(n,r,d\) and \(s\). We will see that this upper bound \(G\) is sharp (at least in some cases): generalized Castelnuovo varieties are varieties whose genus is \(G\).
In the case of nondegenerate curves (i.e. \(n = 1\), \(s = r - 1)\) this result is well known since the end of the last century and at the beginning of the 1980's it has been proved for \(n = 1\), \(r = 3\), \(s \geq 2\), \(d \gg s\) and for \(n = 1\), \(r \geq 3\), \(r - 1 \leq s \leq 2r - 3\), \(d \gg s\); the general result has been only recently proved for \(n = 1\), \(r \geq 3\), \(s \geq r - 1\) and \(d \gg s\) [cf. \textit{L. Chiantini}, \textit{C. Ciliberto} and the author, Duke Math. J. 70, No. 2, 229-245 (1993; Zbl 0799.14011)]. The case \(n \geq 1\) has been analyzed by \textit{J. Harris} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 35-68 (1981; Zbl 0467.14005)] for \(s = r - n\) (in this case the varieties which achieve the bound \(G\) are called ``Castelnuovo varieties'') and in 1990 by \textit{U. Nagel} and \textit{W. Vogel} [in: Topics in Algebra, Part 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw 1988, Banach Cent. Publ. 26, Part 2, 163-183 (1990; Zbl 0735.14032)] for codimension two arithmetically Cohen-Macaulay subvarieties \(V\) of \(\mathbb{P}^ r\) [in this case, when \(d \gg s\), our bound is more precise than the bound obtained by U. Nagel and W. Vogel (loc. cit.)]. bounded genus; generalized Castelnuovo varieties DOI: 10.1007/BF02567861 Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory Generalized Castelnuovo varieties | 0 |
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