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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For irreducible components \(E_i\), \(i=1,2,\dots,m\), of the exceptional locus \(E\) of the canonical resolution of the three-dimensional \(A_n\)-singularity \(X\), we compute the dimension of the cohomology groups H\(^0(O_{\mu E})\), where \(\mu E=\mu_1E_1+\cdots+\mu_nE_n\). 3-dimensional singularity; exceptional locus; canonical resolution Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies Dimension of the cohomology groups of the canonical resolution of some \(A_n\)-singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities ``The stringy Euler number and B-function of \textit{V. V. Batyrev} [in: Integrable systems and algebraic geometry. Proc. 41st Taniguchi Symp., Kobe 1997, Kyota 1997,World Scientific, 1--32 (1998; Zbl 0963.14015)] for log terminal singularities in dimension 2 can also be considered for a normal surface singularity with all log discrepancies nonzero in its minimal log resolution. Here we obtain a structure theorem for resolution graphs with respect to log discrepancies, implying that these stringy invariants can be defined in a natural way, even when some log discrepancies are zero, and more precisely for all normal surface singularities which are not log canonical. We also show that the stringy E-functions of log terminal surface singularities are polynomials (with rational powers) with non-negative coefficients, yielding well defined (rationally graded) stringy Hodge numbers.'' As applications some interesting examples of log terminal and weighted homogeneous singularities are discussed in detail. normal surface singularities; log-terminal surface singularities; log-canonical singularities; log discrepancy; stringy \(E\)-function; stringy Hodge numbers; Hirzebruch-Jung singularities; triangle singularities Veys, W.: Stringy invariants of normal surfaces. J. Alg. Geom. 13, 115--141 (2004) Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Topological properties in algebraic geometry Stringy invariants of normal 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 Log canonical singularities of a surface \(X\) have an index, i.e. a smallest positive integer \(I\) such that \(I\cdot K_X\) is Cartier. If \(\sigma:\widetilde{X}\to X\) is a good resolution of \(X\), \(E_i\) being the exceptional curves, \(I\) is obtained from the codiscrepancy \(\Delta\), where \(\sigma^*K_X= K_{\widetilde{X}}+ \Delta\) and \(\Delta=\sum \delta_iE_i\) and \(I=\min\{n\in \mathbb{N}_+\mid \forall i:n\cdot\delta_i\in \mathbb{Z}\}\). This paper studies log canonical and log terminal singularities, i.e. such singularities with \(\delta_i\leq 1\) for all \(i\) (\(\delta_i<1\), respectively), for a fixed number \(I\). There is a classification given in theorem A: The set of log terminal singularities consists of finitely many series (the number of series is explicitly given for each \(I\)) and in each series the singularities are indexed by natural numbers and ``behave very regular''. Essential for this investigation is a new numerical characterization of cyclic quotient singularities: The Hirzebruch-Jung continued fraction and the description of Riemenschneider, respectively, are compared with a triplet \((I,m,\alpha)\) involving the index \(I\), where \(n=I\cdot m\), \(q=m\cdot\alpha-1\) if the singularity is of type \((n,q)\). A series of cyclic quotient singularities is obtained simply in the form \((I,m,\alpha), (I,m+I,\alpha), (I,m+2I,\alpha),\dots\) -- The theorem provides an algorithm to obtain all log canonical singularities of given index \(I\). The author points out that his recent implementation on a computer proved useful to find log surfaces of prescribed index and Kodaira dimension 0.
In theorem B, the weights \(b_i:= E_i^2\) of the singularity are considered. They are shown to measure ``how singular'' is the singularity for the case of log terminal singularities, whereas this is not the case in general. -- As an application, a new bound for the maximal number of weights \(\neq 2\) of a log terminal singularity of index \(I\) is given. Further, ascending and descending chain conditions for maximum and minimum, respectively of the \(\delta_i\) are considered.
The author points out a partial overlap with a paper by \textit{A. Alexeev} [Duke Math. J. 69, No. 3, 527-545 (1993; Zbl 0791.14006)]. index of singularity; codiscrepancy; Cartier divisor; log terminal singularities; cyclic quotient singularities; Hirzebruch-Jung continued fraction; algorithm; log canonical singularities; weights Blache, R. (1994): Two aspects of log terminal surface singularities. Abh. Math. Sem. Univ. of Hamburg64, 59--87 Singularities of surfaces or higher-dimensional varieties, Divisors, linear systems, invertible sheaves, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Two aspects of log terminal 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 This article deals with a notion of higher Nash transform. Given an algebraic variety \(X\), of dimension \(d\), over an algebraically closed field \(k\), and an integer \(n \geq 0\), we may associate to a regular closed point \(x \in X\) a point \([x^{(n)}]\) of \(H_n : = \text{Hilb}_N (X)\), \(N= {{n+d}\choose{b}}\) (the Hilbert scheme of ``\(N\) points'' of \(X\)). The \(n\)-th Nash blowing-up of \(X\), denoted by \(\text{Nash}_n(X)\), is defined as the closure (in \(X \times H_n\)) of the set of the points \((x,[x^{(n)}]), x\) regular in \(X\). There is a natural projection \(\text{Nash}_{n}(X) \to X\), which is a proper, birational morphism, an isomorphism over the set of regular points of \(X\). Similarly, by taking the closure of the points \([x^{(n)}]\) (\(x\) regular in \(X\)) in \(H_n\) we get a variety \(\text{Nash}'_n(X)\).
In this paper properties of these constructions are studied. Some of the basic ones are:
(a) A natural morphism \(\text{Nash}_n(X) \to\text{Nash}'_n(X)\) is bijective and if \(\mathrm{ch}(k)\) (the characteristic of \(k\)) is zero, it is an isomorphism.
(b) \(\text{Nash}_1(X)\) is naturally isomorphic to the usual Nash blowing-up of \(X\) (limits of tangent spaces at regular points).
Other alternative constructions of \(\text{Nash}_n(X)\), using relative Hilbert schemes or Grassmamnians are presented. In general there is no easy connection between \(\text{Nash}_n(X)\) an \(\text{Nash}_{n+1}(X)\). A natural question is: given \(X\) as above, is there an index \(n\) such that \(\text{Nash}_n(X)\) is smooth? The author proves that this true for curves, if char(\(k\))=0. This is a consequence of a difficult theorem characterizing, for an irreducible ``algebroid '' curve \(X\) (i.e, \(X= {\text{Spec}}(R)\), \(R\) a complete one-dimensional \(k\)-algebra, \(\mathrm{ch}(k)=0\), residual field \(k\), without zero divisors) the normality of \(\text{Nash}_n(X)\) in terms of the monoid of values of the curve \(X\). (Since \(X\) is algebraizable, \(\text{Nash}_n(X)\) is defined). One also uses some more general results on separating branches via higher Nash blowing-ups, previously proved in the paper. When \(\mathrm{ch}(k)>0\), the author gives examples of curves \(X\) over \(k\) such that \(X=\text{Nash}_n(X)\) for all \(n\), so they cannot be desingularized using this process. The paper is well written and, spite of its technical nature, pleasant to read. Nash blowing-up; resolution of singularities; Hilbert scheme of points; numerical monoid of a branch. Yasuda, T., Higher Nash blowups, Compos. Math., 143, 1493-1510, (2007) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of curves, local rings, Modifications; resolution of singularities (complex-analytic aspects), Local deformation theory, Artin approximation, etc. Higher Nash blowups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 algebraic variety in characteristic zero. To an effective divisor on \(X\) Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igusa) zeta function, associated to a regular function on \(X\), which specializes to both the classical \(p\)-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant. This paper treats a generalization to singular varieties. Batyrev already considered such a `Kontsevich invariant' for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any \(\mathbb{Q}\)-Gorenstein variety \(X\) we associate a motivic zeta function and a `Kontsevich invariant' to effective \(\mathbb{Q}\)-Cartier divisors on \(X\) whose support contains the singular locus of \(X\). singularity invariant; topological zeta function; motivic zeta function Veys, W.: Zeta functions and ``Kontsevich invariants'' on singular varieties. Can. J. Math. 53(4), 834-865 (2001) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Zeta functions and ``Kontsevich invariants'' on singular varieties. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper collects a series of lectures given by H. Hauser at the Clay Summer School on Resolution of Singularities in 2012. It is a very good introduction to the theory of blowups and the resolution of singularities in characteristic zero. It is also a good way to discover the bad phenomena of the positive characteristic. The point of view is to study the resolution of singularities in a global way finding some invariants of the singularities. The author, with the help of S. Perlega and V. Roitner gives lot of examples and exercises with some help to solutions in the last chapter.
This paper is composed of thirteen chapters. The first chapter is an introduction to the problem of resolution of singularities studying examples. The second is a quick introduction of schemes, algebraic varieties and completion. The third talk about singularities, normal crossing and different kind of singularities in all characteristic. The fourth, fifth and sixth chapters give all the equivalent definition of blowups, the functorial properties of blowups and the transformation of ideal and varieties under blowups. The seventh chapter give the different notions of resolution of singularities. The eighth talk about different kind of invariants of singularities used in the proofs of resolution theorems. The ninth chapter give the definition of maximal contact and talk of is existence in characteristic zero. The tenth chapter introduce the coefficient ideals. The eleventh chapter only talk about the existence of resolution of singularities in characteristic zero and compute some examples. The twelfth chapter present some problems and phenomena which appear in positive characteristic, specially in the purely inseparable case. The last chapter is composed of the solutions of the examples and exercises given in all the others sections. resolution; singularities; blowups Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Multiplicity theory and related topics, Rational and birational maps Blowups and 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 Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient 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 \textit{J. Fernández de Bobadilla} [Proc. Lond. Math. Soc. (3) 92, No. 1, 99--138 (2006; Zbl 1115.14021)] formulated a conjecture on the local topological types of plane curve singularities of rational unicuspidal complex projective curves. They checked the conjecture for all cuspidal curves \(C\) whose compliment \(\mathbb{P}^2\setminus C\) has logarithmic dimension \(\overline\kappa\leq 1\), and also for several curves with \(\overline \kappa=2\), in fact, for all known rational unicuspidal curves. \textit{M. Borodzik} and \textit{C. Livingston} [Forum Math. Sigma 2, Article ID e28, 23 p. (2014; Zbl 1325.14047)] proved the conjecture for rational unicuspidal curves in the general case, thus obtaining a necessary condition on numerical invariants of local plane curve singularities occurring on rational unicuspidal curves.
Using this condition, \textit{T. Liu} [``On planar rational cuspidal curves'', PhD. Thesis, Massachusetts Institute of Technology (2014; Zbl 1358.14006)] gave a full list of possible local types with two Newton pairs. In the paper under review, it is proved that all but two types on the list are realizable.
An analogous list for rational unicuspidal curves with one Newton pair is given by \textit{J. Fernández de Bobadilla, I. Luengo, A. Melle-Hernández} and \textit{A. Némehti} [J.-P. Brasselet (ed.) et al., Real and complex singularities, São Carlos workshop 2004. Papers of the 8th workshop, Marseille, France, July 19--23 (2004; Zbl 1120.14019)]. rational cuspidal curve; Newton pair; plane curve singularity Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Plane and space curves Classification of rational unicuspidal curves with two Newton pairs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the review in Zbl 0602.14004. weighted graphs of exceptional divisors for desingularizations; log- terminal singularities Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Log-terminal singularities of algebraic surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the preview in Zbl 0541.14004. exceptional locus of a desingularization; rigid quotient singularities; deformation ESNAULT, H., VIEHWEG, E.: Two dimensional quotient singularities deform to quotient singularities. Math. Ann. 271, 439--419 (1985) Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Formal methods and deformations in algebraic geometry, Singularities in algebraic geometry Two dimensional quotient singularities deform to 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 The author presents some of the ideas of Abhyankar that play a central role in desingularization. He restricts to the case of surfaces in 3- space.
In the first part the author recalls the case of plane curves. To show that the multiplicity of a singular point can be reduced by a finite sequence of quadratic transformations Abhyankar introduced a numerical character \(e\) called `magnitude' which is a measure of the infinitely near equimultiple points, \(e \in \mathbb{N}\). If the strict transform \(C'\) has the same multiplicity as \(C\), \(e(C')\) is equal to \(e(C) - 1\). Then after a finite sequence of quadratic transformations we get a desingularization.
In the second part of the article the author studies the case of a surface \(S\) in a 3-space and he considers a \(d\)-fold point \(p\) on \(S\), with \(d>1\). Locally we can choose a Weierstrass coordinate system such that the singularity is defined by \(z^ d + \sum z^ if_ i (x,y)\), with \(f_ i\) a power series in \(x,y\), and in the characteristic zero case we can suppose \(f_ 1 = 0\). The author defines
\[
g(x,y) = \text{gcd} \{f_ i^{1/i}, 2 \leq i \leq d\}
\]
and the point \(p\) is ``pre- good'' if for some choice of coordinates we have \(g = x^ a y^ b\). A pre-good \(d\)-fold point \(p\) of \(S\) is not an isolated \(d\)-fold point, i.e. there is a \(d\)-fold curve of \(S\) passing through \(p\), it is not a singular point of any \(d\)-fold curve, at most two \(d\)-fold curves pass through \(p\) and if there are two, they meet transversally at \(p\). Then the author defines the numbers \(E(p) = \inf \{d/m \text{ ord} (f_ m/g^ m)\), \(2 \leq m \leq d\}\) and \(e(p) = \text{ord} (g) = a + b\), and the point \(p\) is ``good'' if it is pre-good with \(E(p) < d\). If \(p\) is a good point for \(S\), for any finite sequence \(p_ n \to \cdots \to p_ 2 \to p = p_ 1\) of monoidal transforms centered at \(d\)-fold curves, if \(p_ n\) is a \(d\)-fold point then it is pre-good. After at most \(e(p)\) monoidal transformations a good point will be transformed into points of lesser multiplicity.
If \(p\) is not a good point, after a quadratic transformation of center \(p\) there is at most one point \(p'\) which is a \(d\)-fold point of the new surface \(S'\). This point \(p'\) is ``closer'' to being good than \(p\). After a finite sequence of quadratic transformations \(p\) will be transformed into points that are either good \(d\)-fold points or have multiplicity less than \(d\).
Then, in the characteristic zero case, the surface \(S\) can be desingularized by a finite sequence of monoidal and quadratic transformations. If the characteristic is positive the essential new difficulty provides from bad \(d\)-fold points, with \(d\) a multiple of the characteristic. pre-good point; bad \(d\)-fold points; desingularization; surfaces in 3- space; multiplicity of a singular point; strict transform; good point; monoidal transforms; characteristic Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Abhyankar's work on desingularization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 key point in the resolution of singularities is the choice of the centers for the blowing-ups. In characteristic zero the center as a subset of the locus of highest multiciplity is determined by induction passing to a so-called hypersurface of maximal contact (eliminating one variable). In characteristic \(p>0\) this is not possible in general. The multiplicity of an embedded hypersurface at a point can be described in terms of differential operators as well as in terms of general projections defined at étale neighbourhoods of this point. Both approaches are related in the paper. Invariants of embedded hypersurfaces are studied in terms of differential operators, which express properties of the ramification of the morphism.
A central result in multiplicity theory of hypersurfaces is a form elimination of one variable in the description of highest multiplicity locus. In characteristic \(0\) this form of elimination is achieved with the notion of Tschirnhausen polynomials introduced by Abhyankar. This is a key point in the proof of embedded desingularization. A characteristic free approach to this form of elimination is given. Given a \((d-1)\)-dimensional hypersurface in a smooth \(d\)-dimensional scheme. The approach based on projections on smooth \((d-1)\)-dimensional schemes. The behaviour of invariants related to this form of elimination is discussed. Villamayor, O.: Hypersurface singularities in positive characteristic. Adv. in Math. 213 (2007), no. 2, 687-733. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Global theory of complex singularities; cohomological properties Hypersurface singularities in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give an explicit description of the semigroup of values of a plane curve singularity with several branches in terms of the usual invariants of the equisingularity type in the sense of Zariski. The main tool is the set of elements called maximals, especially the absolute and the relative ones. First, we describe the semigroup in terms of the relative maximals and these ones in terms of the absolute maximals by means of a symmetry property which generalizes the well known property of symmetry for the singularities with only one branch. Then the absolute maximals are described in terms of the theory of maximal contact of higher genus developed by Lejeune. semigroup of values of a plane curve singularity with several branches; relative maximals; absolute maximals; maximal contact of higher genus; equisingularity type Félix Delgado de la Mata, ``The semigroup of values of a curve singularity with several branches'', Manuscr. Math.59 (1987) no. 3, p. 347-374 Singularities of curves, local rings, Riemann surfaces; Weierstrass points; gap sequences, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) The semigroup of values of a curve singularity with several branches | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author describes three related techniques to work efficiently with power series in the context of (embedded) resolution of singularities, where the key problem is the construction of local invariants for singularities and their control under blowup and localization. The techniques are intended to provide a general framework for producing such invariants and observing their transformation rules.
1. One considers local flags in a regular ambient space, and local coordinates subordinate to the flag. The author constructs canonically new local flags on any blowup with centre transversal to the given flag. Choosing local coordinates subordinate to the flags before and after the blowup allows one to define significant invariants and to read off easily their change under the blowup.
2. The analog of the classical Gauss decomposition is proved for the group \(G\) of automorphisms of the formal (or convergent) power series in the variables \(x_1,\dots,x_n\) (i.e. local automorphisms of affine \(n\)-space): any \(\varphi \in G\) is a product \(\varphi =bus\), where \(b\) is an upper triangular automorphism, \(u\) is a lower unipotent de Joncquière automorphism and \(s\) is a permutation. As the \` Borel automorphisms\'\ \(b\) are just those which stabilize the flag with subordinate coordinates \(x_1,\dots,x_n\), one can restrict to automorphisms \(us\) when constructing subordinate coordinates out of ordinary ones. This is shown to be crucial for controlling a flag invariant under blowup or localization.
3. Initial ideals of an ideal of power series (with respect to monomial orders on \(\mathbb N^n\)) are a frequently used tool to investigate singularities, in particular the coordinate independent \textsl{generic initial ideal}. Here the author proposes to order the set of initial ideals with respect to all coordinate choices of a given ideal (by the ordering on monomial ideals through the lexicographic order taken on their minimal monomial generator systems). Then the minimal initial ideal is just the generic one, but the maximal initial ideal contains much more information on the singularity. Its existence is however not obvious and is shown in the paper. The Gauss decomposition of \(G\) is then used to determine coordinates realizing this maximal ideal.
The author summarizes the most important resolution invariants in the literature and indicates that they are special cases or variants of his flag invariants or minimal/maximal initial ideal. It is also interesting that the constructions in the paper are characteristic independent, thereby providing a potential contribution to the still unsolved characteristic \(p\) case. resolution of singularities; flag; Gauss decomposition of formal automorphisms; initial ideal Herwig Hauser, Three power series techniques, Proc. London Math. Soc. (3) 89 (2004), no. 1, 1 -- 24. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings, Modifications; resolution of singularities (complex-analytic aspects) Three power series techniques. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities An old problem in Algebraic Geometry asks about the maximum number \(\mu(n)\) of isolated singular points on a projective surface \(F\) of degree \(n\), contained in \(\mathbb{P}^3=\mathbb{P}^3(\mathbb{C})\). The paper under review is a survey on this subject, written by one of its a foremost experts.
Although its statement is elementary, this problem is very hard. The value of \(\mu(n)\) has been found for \(n<6\). For \(n>6\) only upper bounds for the number of isolated singularities were obtained, but probably they (specially for \(n\) large) are not sharp. Many ingenious examples of surfaces of degree \(n>6\) with many nodes were found, but the number is probably \(<\mu(n)\).
The article contains interesting historical information, for instance on a failed attempt by Severi to get a simple bound, based on the study of certain moduli, which turned out to be incorrect. There is a discussion of the proof of \textit{A. Beauville} [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006)] of the formula \(\mu(5)=31\), using a method that (with some homological arguments) reduces the problem to one on ``binary codes'', i.e., finite dimensional vector spaces over \(\mathbb{Z}_2\). This method, with more labour, yields the formula \(\mu(6)=65\), and might work for higher values of \(n\). The author also explains the work of Miyaoka, who found interesting general bounds and studied the asymptotic behaviour of \(\mu(n)/n^3\).
In addition the author discusses numerous examples of surfaces with many nodes, and several other related topics.
The paper is beautifully written, although some parts are expressed in the language of classical Italian Geometry, which might be a little challenge for some modern readers. The article contains a very extensive bibliography, that ranges from a 1750 article by Cramer to very recent preprints not published yet. surfaces in projetive 3-space; isolated singularities; rational double points Gallarati, D, Superficie algebriche con molti punti singolari isolati, Bull. Math. Soc. Sci. Math. Roumanie, 55, 249-274, (2012) Special surfaces, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Computational aspects of algebraic surfaces, Hypersurfaces and algebraic geometry Algebraic surfaces with many isolated singular points (Superficie algebriche con molti punti singolari isolati) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 any field, \(\text{Hilb}^{p(z)}_{\mathbb{P}^n_k}\) the Hilbert scheme of subschemes of \(\mathbb{P}^n\) with Hilbert polynomial \(p(z)\). \textit{F. J. Macaulay} showed [Proc. Lond. Math. Soc. (2) 26, 531-555 (1927; JFM 53.0104.01)] that there exists a unique saturated lexicographic ideal \(L\) such that \(k[x_0, \dots,x_n]/L\) has Hilbert polynomial \(p(z)\). We show that the scheme corresponding to \(L\) is parametrized by a smooth point on the Hilbert scheme. In the process we calculate the dimension of the unique component through this point explicitly, and describe explicitly the subscheme corresponding to the general point of this component. smoothness; lexicographic point; Hilbert scheme; JFM 53.0104.01; Hilbert polynomial A. Reeves - M. Stillman, Smoothness of the lexicographic point. J. Algebraic Geom., 6 (2) (1997), pp. 235-246. Zbl0924.14004 MR1489114 Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Smoothness of the lexicographic 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 This article contains a survey of results on ``generic singularities''. Many of the results discussed were obtained by the author. No proofs are presented.
More precisely, generic singularities are those that appear when we consider a sufficiently general linear projection \(\pi\) of a smooth \(r\)-dimensional projective variety \(X \subset {\mathbb P}^n\) into a linear subspace \( V \) (\( \approx {\mathbb P}^{m}\)) of \( {\mathbb P}^n\), where \( r+1 \leq m \leq 2r\), and (letting \(Y=\pi (X)\)) we exclude points of \(Y=\pi (X)\) in a suitable lower dimensional subvariety. Expanding results of M. Noether, E. Lluis, etc, J. Roberts developed the basic theory of these singularities in the 1970's, see \textit{J. Roberts} [Trans. Am. Math. Soc. 212, 229--268 (1975; Zbl 0314.14003)]. After briefly reviewing this theory, Zaare-Nahandi discusses some of his contributions. For instance, using the notation above, if \(y=\pi(x), x \in X\), is an analytically irreducible generic singularity, he has obtained a very explicit description of the induced homomorphism \(\pi ^* : {\hat {\mathcal O}}_{V,y} \to {\hat {\mathcal O}}_{X,x}\) and of the \textit{local defining ideal} of the singularity \(y\), namely Ker(\(\pi ^*\)). The local defining ideal is expressed in terms of minors of an associated matrix \({\mathcal M}\) with coefficients in \({\hat {\mathcal O}}_{V,y}\). The ring \({\hat {\mathcal O}}_{V,y}\) is isomorphic to a power series ring, and specializing some of the variables the matrix \(\mathcal M\) induces a matrix \({\mathcal M}_0\) with interesting properties. For instance, the defining ideal becomes a square-free monomial ideal. A simplicial complex may be associated to it, some of its properties are studied. As an application, a formula for the depth of \({\mathcal O}_{Y,y}\) is obtained and a partial answer to a conjecture of Andreotti, Bombieri and Holme weak normality of certain points of \(Y\) is gotten.
The author works over an algebraically closed field but if the characterisitc is positive the embedding \(X \subset {\mathbb P}^n\) must satisfy some extra conditions. generic projections; generic singularities; local defining ideal Singularities in algebraic geometry, Local theory in algebraic geometry, Projective techniques in algebraic geometry, Determinantal varieties, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes Algebraic properties of generic 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. Grothendieck constructed the Hilbert scheme \(\text{Hilb}^n_X\) of \(n\) points on \(X\), for any quasi-projective scheme \(X\) on a noetherian base scheme \(S\). In the paper under review, the authors are interested in showing the existence of the Hilbert scheme of \(n\) points on \(\text{Spec}({\mathcal O}_{X,P})\), where \(P\) is a (non necessarily closed) point on such a scheme \(X\). A natural candidate would be \(\bigcap_{P\in U_\alpha}\text{Hilb}^n_{U_\alpha}\), where \(U_\alpha\) varies in the set of open subsets of \(X\) containing \(P\). But in general an infinite intersection of open subschemes of a scheme is not a scheme. It is a scheme if one takes only locally principal open subschemes.
The authors introduce and study the notion of generalized fraction rings and localized subschemes, of which \(\text{Spec}({\mathcal O}_{X,P})\) is a particular case. They prove that, if \(X\) is a scheme such that \(\text{Hilb}^n_X\) exists, then the functor of points of a localized scheme \({\mathcal S}^{-1}X\) is representable. As a particular case, they get the following result:
If \(X\rightarrow S\) is a projective morphism of Noetherian schemes and \(P\) is a point in \(X\), then the Hilbert scheme of \(n\) points on \(\text{Spec}({\mathcal O}_{X,P})\) exists and coincides with the intersection of the Hilbert schemes of \(n\) points of the open subschemes of \(X\) containing \(P\). localized schemes; determinants; fraction rings Parametrization (Chow and Hilbert schemes) Infinite intersections of open subschemes and the Hilbert scheme of points. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article can be seen as a sequel to the first author's article [J. Algebr. Geom. 14, No. 4, 761-787 (2005; Zbl 1120.14002)], where he calculates the total Chern class of the Hilbert schemes of points on the affine plane by proving a result on the existence of certain universal formulas expressing characteristic classes on the Hilbert schemes in term of Nakajima's creation operators. The purpose of this work is (at least) two-fold. First of all, we clarify the notion of ``universality'' of certain formulas about the cohomology of the Hilbert schemes by defining a universal algebra of creation operators. This helps us to reformulate and extend a lot of the first author's previous results in a very precise manner.
Secondly, we are able to extend the previously found results by showing how to calculate any characteristic class of the Hilbert scheme of points on the affine plane in terms of the creation operators. In particular, we have included the calculation of the total Segre class and the square root of the Todd class. Using these methods, we have also found a way to calculate any characteristic class of any tautological sheaf on the Hilbert scheme of points on the affine plane. This in fact gives another complete description of the ring structure of the cohomology spaces of the Hilbert schemes of points on the affine plane. Samuel Boissière and Marc A. Nieper-Wißkirchen, Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces, J. Algebra 315 (2007), no. 2, 924 -- 953. Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Cycles and subschemes Universal formulas for characteristic classes on the 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 Given a Koszul algebra of finite global dimension we define its higher zigzag algebra as a twisted trivial extension of the Koszul dual. If our original algebra is the path algebra of a tree-type quiver, this construction recovers the zigzag algebras of Huerfano-Khovanov. We study examples of higher zigzag algebras coming from Iyama's type A higher representation finite algebras, give their presentations by quivers and relations, and describe relations between spherical twists acting on their derived categories. We connect this to the McKay correspondence in higher dimensions: if \(G\) is a finite abelian subgroup of \(\mathrm{SL}_{d+1}\) then these relations occur between spherical twists for \(G\)-equivariant sheaves on affine \((d+1)\)-space. trivial extension; braid group action; spherical twist; quiver; derived category; Koszul algebra; cluster tilting; equivariant sheaves Quadratic and Koszul algebras, Representations of quivers and partially ordered sets, Derived categories, triangulated categories, Derived categories and associative algebras, ``Super'' (or ``skew'') structure, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Higher zigzag 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 In this paper, the author develops a Bernstein-Sato theory for arbitrary ideals in an \(F\)-finite regular ring of positive characteristic, generalizing work of \textit{M. Mustaţă} [J. Algebra 321, No. 1, 128--151 (2009; Zbl 1157.32012); ``Bernstein-Sato polynomials for general ideals vs. principal ideals'', Preprint, \url{arXiv:1906.03086}] and \textit{T. Bitoun} [Sel. Math., New Ser. 24, No. 4, 3501--3528 (2018; Zbl 1423.13048)] in the case of principal ideals.
Let \(R\) be an \(F\)-finite regular ring of positive characteristic and let \(\mathfrak{a}=(f_1,\dots,f_r)\) be an ideal of \(R\). Consider the local cohomology module \(H_{(f_1-t_1,\dots,f_r-t_r)}^r(R[t_1,\dots,t_r])\). It is well-known that this module has a natural module structure over \(D_{R[t_1,\dots,t_r]}\), the ring of differential operators of \(R[t_1,\dots,t_r]\). The author defines a (decreasing) \(V\)-filtration on \(D_{R[t_1,\dots,t_r]}\), and more precisely on each \(D^e_{R[t_1,\dots,t_r]}=End_{R[t_1,\dots,t_r]^{p^e}}(R[t_1,\dots,t_r])\) (it is well-known that the union of these \(D^e\) is \(D\)). Then the author studies the module \(N_{\mathfrak{a}}^e\) which is defined as the quotient of \(V^0D^e_{R[t_1,\dots,t_r]} \cdot \delta\) by \(V^1D^e_{R[t_1,\dots,t_r]} \cdot \delta\), where \(\delta\) is the element \((f_1-t_1)^{-1}\cdots (f_r-t_r)^{-1}\in H_{(f_1-t_1,\dots,f_r-t_r)}^r(R[t_1,\dots,t_r])\), as well as their limit \(N_{\mathfrak{a}}:=\varinjlim_eN_{\mathfrak{a}}^e\). The higher Euler-type operator \(s_{p^m}:=\sum_{|\underline{a}|=p^m}\partial_{\underline{t}}^{[\underline{a}]}\underline{t}^{\underline{a}}\) acts on \(N_{\mathfrak{a}}^e\) for each \(0\leq m\leq e-1\). The key result which relates \(N_{\mathfrak{a}}^e\) with \(F\)-invariants is Theorem 3.11, where it was shown that the multi-eigenspace of \(N_{\mathfrak{a}}^e\) with eigenvalue \(\alpha=(\alpha_0,\dots,\alpha_{e-1})\) under the action \((s_{p^0},\dots, s_{p^{e-1}})\) is a direct sum of the modules in the set \(\{D_R^e\cdot \mathfrak{a}^{|\alpha|+s{p^e}} / D_R^e\cdot \mathfrak{a}^{|\alpha|+s{p^e}+1}\}\) where \(s=0, 1, \dots, r-1\) and \(|\alpha|=\alpha_0+p\alpha_1+\cdots+p^{e-1}\alpha_{e-1}\), and that each such module occurs in the direct sum. Since it is known (re-proved in this paper) that \(D_R^e\cdot \mathfrak{a}= (C_R^e\cdot \mathfrak{a})^{[p^e]}(=I_e(\mathfrak{a})^{[p^e]})\), it follows that the generalized eigenspace with eigenvalue \(\alpha\) is nonzero precisely when \(I_e(\mathfrak{a}^{|\alpha|+s{p^e}})\neq I_e(\mathfrak{a}^{|\alpha|+s{p^e}+1})\) for some \(0\leq s\leq r-1\). The latter is closely related to \(F\)-invariants such as \(F\)-jumping numbers, thus the author is able to obtain a series of results (Theorem 4.7, Theorem 4.12). The author further introduced Bernstein-Sato roots of \(N_{\mathfrak{a}}\) and proved that they are negative rational numbers (Theorem 6.7), and that there is a connection of these Bernstein-Sato roots with the \(F\)-jumping numbers of \(\mathfrak{a}\) (Theorem 6.11). The results obtained in this article can be viewed as analogs of Bernstein-Sato theory, multiplier ideals, and jumping numbers in characteristic zero, it would be interesting to explore the connections among them. \(F\)-jumping numbers; Bernstein-Sato polynomials; test ideals Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Bernstein-Sato theory for arbitrary ideals 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 In this paper, we introduce a method for resolving singularities of Galois closure covers for 5-fold covers between smooth surfaces. Applying this method, we determine types of singular fibers of a family of Galois closure curves for plane sextic curves. Consequently,we obtain an explicit construction of smooth projective minimal surfaces of general type with positive indices obtained as families of Galois closure curves of smooth plane sextic curves. \(\mathcal{S}_5\)-covers; canonical resolution; surfaces of general type with positive indices; Galois closure curves; Galois points Global theory and resolution of singularities (algebro-geometric aspects), Coverings in algebraic geometry, Ramification problems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Surfaces of general type Galois closure covers for 5-fold covers between smooth surfaces and its application | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 an extended Dynkin quiver and \(Z(Q,d)\) the common zero locus of all non-constant homogeneous semi-invariant polynomial functions on the vector space parametrizing representations of \(Q\) of dimension vector \(d\). The main result of the paper is that \(Z(Q,d)\) is a complete intersection provided that the multiplicities of the summands in the so-called canonical decomposition of \(d\) are large enough (explicit bounds are given in the paper). This extends to arbitrary dimension vectors the earlier result of the same authors on the case of a prehomogeneous dimension vector [Comment. Math. Helv. 79, 350--361 (2004; Zbl 1063.14052)]. A notable new phenomenon in the case of a non-prehomogeneous dimension vector is that the number of irreducible components of \(Z(Q,d)\) can be arbitrarily large (except when \(Q\) is an oriented cycle). The proof builds on the description of algebras of semi-invariants of tame quivers by \textit{A. Skowronski} and \textit{J. Weyman} [Transform. Groups 5, No. 4, 361--402 (2000; Zbl 0986.16004)] and on the representation theoretic interpretation of semi-invariants of quivers due to \textit{A. Schofield} [J. Lond. Math. Soc. (2) 43, No. 3, 385--395 (1991; Zbl 0779.16005)]. semi-invariants; quivers; representations; cofree action; complete intersection Riedtmann, Ch., Zwara, G.: The zero set of semi-invariants for extended Dynkin quivers. Trans. Am. Math. Soc. \textbf{360}(12), 6251-6267 (2009i:14064) (2008) \textbf{(MR2434286)} Geometric invariant theory, Representations of quivers and partially ordered sets The zero set of semi-invariants for extended Dynkin quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review is a detail survey of results obtained by the author with collaborators over the last two decades in the study of the classical theory of topological equisingularity and related topics. In the appendix, written by G.-M. Greuel and G. Pfister, some computational aspects of the theory are discussed. The bibliography contains 156 selected references and involves 16 items with the author's name.
The author begins with a reminder of basic notions, standard methods and known results concerning the topology of hypersurface singularities and its main objects such as the Milnor fibration, Morse theory, the polar and resolution methods, and others. Then he discusses some open problems closely related to Zariski's multiplicity conjecture [\textit{O. Zariski}, Bull. Am. Math. Soc. 77, 481--491 (1971; Zbl 0236.14002)], to questions on topological triviality for families of isolated hypersurfaces singularities and properties of \(\mu\)-constant strata and families, and so on. Further, the author explains his own contribution to the theory of topological triviality for families of nonisolated singularities [\textit{J. Fernández de Bobadilla}, Adv. Math. 248, 1199--1253 (2013; Zbl 1284.32018); \textit{J. Fernández De Bobadilla} and \textit{M. Marco-Buzunáriz}, Comment. Math. Helv. 88, No. 2, 253--304 (2013; Zbl 1271.14046)] and formulates several questions and conjectures. Among other things, some useful interactions of the theory of equisingularity with Floer homology of the Milnor fibration and Lipschitz geometry are mentioned also [\textit{L. Birbrair} et al., Proc. Am. Math. Soc. 144, No. 3, 983--987 (2016; Zbl 1338.14008); \textit{A. Némethi}, Publ. Res. Inst. Math. Sci. 44, No. 2, 507--543 (2008; Zbl 1149.14029)].
In the appendix you can find the following interesting remark: ``the failure to find a counter example to Zariski's conjecture was the most important reason for the development of SINGULAR as it is now'' (see [\textit{W. Decker} et al., SINGULAR 4-1-3 -- A computer algebra system for polynomial computations. (2020), \url{http://www.singular.uni-kl.de}]. nonisolated hypersurface singularities; Milnor fibration; morsification; equisingularity; Zariski's multiplicity conjecture; topological triviality; Floer homology; lattice homology; low dimensional topology; plumbing 3-manifolds; simultaneous resolutions; \(\mu\)-constant families; isolated surface singularities; topological triviality; Lipschitz equisingularity; motivic integration; arc spaces; vanishing cycles; monodromy; vanishing folds; cobordism theorem; computer algebra system ``Singular'' Singularities in algebraic geometry, Equisingularity (topological and analytic), Complex surface and hypersurface singularities, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants Topological equisingularity: old problems from a new perspective (with an appendix by G.-M. Greuel and G. Pfister on \textsc{Singular}) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 an idea of ``formal function along a subspace'', a notion that is related to Grothendieck's ``formal completion along a subscheme'', but expressed in concrete analytic terms using ``smooth coordinate charts''. Formal functions are families of formal power series parametrized by functions from a given class, satisfying natural commutativity relations between formal differentiation and differentiation with respect to the parameters. One of our main results is that, in generic coordinates for a chart \(U\), the standard basis of a local ideal \({\mathcal I}_{X,a}\) of a closed subspace \(X\) of \(U\) (the standard basis is a special set of generators of \({\mathcal I}_{X,a})\) extends as formal functions to generators along the Samuel stratum \(S\) of \(a\). \((S\) is the subset of \(X\) of points of the same singularity-type as \(a\), as measured by the Hilbert-Samuel function.) The simplest example of this phenomenon is ``implicit differentiation'' in elementary analysis. Our result can be used to reduce resolution of singularities over an infinite field, in general, to a ``hypersurface case''. formal function along a subspace; implicit differentiation; formal power series; standard basis of a local ideal; resolution of singularities Edward Bierstone and Pierre D. Milman, Standard basis along a Samuel stratum, and implicit differentiation, The Arnoldfest (Toronto, ON, 1997) Fields Inst. Commun., vol. 24, Amer. Math. Soc., Providence, RI, 1999, pp. 81 -- 113. Formal neighborhoods in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Formal methods and deformations in algebraic geometry, Formal power series rings, Relevant commutative algebra Standard basis along a Samuel stratum, and implicit differentiation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The goal of the authors is to describe a detail procedure for the reduction of the multiplicity of a hypersurface singularity over an algebraically closed ground field of positive characteristic along a valuation under additional assumptions. One of them is a requirement that there exists a finite linear projection of the hypersurface singularity which is defectless. The authors underline that their main result (Theorem 7.1) follows also from the paper [\textit{J.-C. San Saturnino}, J. Algebra 481, 91--119 (2017; Zbl 1370.13005)]. It should be remarked that the paper under review contains an interesting brief historical background concerning the development of the classical theories of local uniformization and resolution of singularities originated by O. Zariski, a series of fruitful ideas, comments and useful references related to these topics. local uniformization; hypersurface singularities; resolution of singularities; defect; multiplicity; valuations; Perron transforms; Zariski's reduction of singularities; defectless projection Global theory and resolution of singularities (algebro-geometric aspects), General valuation theory for fields, Valuations, completions, formal power series and related constructions (associative rings and algebras) Defect and local uniformization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 continues previous work, constructing and studying models of real subspace arrangements. Like the models of De Concini and Procesi for complex subspace arrangements, these are constructed in order to obtain information about invariants of the complement of the union of the subspaces. They are derived from combinatorial structures, called ``building sets'', obtained from a finite partially ordered set of subspaces which is determined by the arrangement. There are usually many building sets associated with an arrangement, each giving rise to a model. This paper studies two real structures associated with the model. The first is essentially the real part of the De Concini-Procesi complex model. The second, which arises when the arrangement is a Coxeter arrangement, is an ``extension'' of the model to a convex set, the face lattice of which generalizes in a way the ``Kapranov permutoassociahedron''. models of subspace arrangements; building sets; Coxeter arrangements; permutoassociahedron G. Gaiffi, ''Real structures of models of arrangements,'' Int. Math. Res. Not., iss. 64, pp. 3439-3467, 2004. Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Global theory and resolution of singularities (algebro-geometric aspects), Relations with arrangements of hyperplanes Real structures of models of 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 We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a godron (called also cusp of Gauss): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these properties and a projective invariant that we associate to each godron we present all possible local configurations of the tangent plane, the self-intersection line, the cuspidal edge and the flecnodal curve at a generic swallowtail in \(\mathbb{R}^3\). We present some global results, for instance: In a hyperbolic disc of a generic smooth surface, the flecnodal curve has an odd number of transverse self-intersections (hence at least one self-intersection). geometry of surfaces; tangential singularities; swallowtail; parabolic curve; flecnodal curve; cusp of Gauss; godron; wave front; Legendrian singularities R. Uribe-Vargas, ''A Projective Invariant for Swallowtails and Godrons, and Global Theorems on the Flecnodal Curve,'' Moscow Math. J. 6, 731--768 (2006). Catastrophe theory, Deformation of singularities, Singularities in algebraic geometry, Complex surface and hypersurface singularities, Projective differential geometry, Affine differential geometry, Surfaces in Euclidean and related spaces, Symplectic geometry, contact geometry, Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics A projective invariant for swallowtails and godrons, and global theorems on the flecnodal curve | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We present several conjectures on multiple \(q\)-zeta values and on the role which they play in certain problems of enumerative geometry. multiple \(q\)-zeta value; \(q\)-deformation; Hilbert scheme; CW/DT correspondence Okounkov, A, Hilbert schemes and multiple \(q\)-zeta values, Funct. Anal. Appl., 48, 138-144, (2014) Parametrization (Chow and Hilbert schemes), Binomial coefficients; factorials; \(q\)-identities, Enumerative problems (combinatorial problems) in algebraic geometry, Multiple Dirichlet series and zeta functions and multizeta values Hilbert schemes and multiple \(q\)-zeta values | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(f\) define a complex projective hypersurface with isolated singularities and let \(M(f)\) be its Milnor algebra. Denote by \(AR(f)\) the graded module of all relations between the partial derivatives of \(f\), and by \(KR(f)\) the module of Koszul relations. The main result of the paper is a characterisation of the fact that all singularities of the hypersurface \(\{f=0\}\) are weighted homogeneous. This is the case if and only if the projection of \(AR(f)_m\) upon its first factor, considered as element in the ring defined by the saturation of the Jacobian ideal, has as kernel precisely \(KR(f)_m\), for all \(m\). Here the coordinates have to be chosen such that the coordinate hyperplane \(\{x_0=0\}\) is transversal to the hyperplane. Several corollaries are derived.
As byproduct, the authors show that one can easily modify a basis for \(AR(f)_m\), computed with a computer algebra system, into one consisting of a basis for \(KR(f)_m\), extended by some elements, which give a basis for the quotient module, of essential relations. projective hypersurfaces; weighted homogeneous singularities; syzygies; Koszul relations A. Dimca and G. Sticlaru, Syzygies of Jacobian ideals and weighted homogeneous singularities, J. Symbolic Comput. 74 (2016), 627-634. Hypersurfaces and algebraic geometry, Complex surface and hypersurface singularities, Singularities in algebraic geometry, Syzygies, resolutions, complexes and commutative rings Syzygies of Jacobian ideals and weighted homogeneous singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f\in {\mathbb{Z}}[X_ 1,X_ 2,X_ 3]\) be a primitive quadratic form and \(M(f)=\Pr oj({\mathbb{Z}}[X_ 1,X_ 2,X_ 3]/(f)).\) The paper contains proofs of two results about resolution of singularities of the conic bundle surfaces M(f) over Spec \({\mathbb{Z}}\). The first says that the normalization of M(f) is M(B(f)), where B(f) is a suitable quadratic form, whose equivalence class is uniquely determined by f. The second says that if M(f) is normal, then there exists a sequence of conic bundle surfaces \(M_ 0=M(f),M_ 1=M(f_ 1),...,M_ n=M(f_ n)\) such that \(M_{i+1}\) is an elementary transform of \(M_ i\) at the singular points (on \(M_ i)\) in one of the fibers of \(M_ i\) for \(i=0,1,...,n-1\) and \(M_ n\) is regular. These results are presented in more general context of conic bundle surfaces corresponding to lattices on quadratic spaces over Dedekind schemes. The proofs depend on the relations between lattices, conic bundle surfaces and orders in quaternion algebras developed in the first part of the paper [Math. Scand. 46, 183-208 (1980; Zbl 0505.14006)] and extended in the present part: M(f) is normal iff the corresponding order O(f) is Bass and O(B(f)) is the Bass closure of the Gorenstein order O(f). The chain of the \(M_ i's\) corresponds to a minimal chain of Bass orders \(O(f)\subset O(f_ 1)\subset...\subset O(f_ n)\) ending with a hereditary order. quadratic space; conic bundle surface; resolution of singularities; orders in quaternion algebras Singularities of surfaces or higher-dimensional varieties, General binary quadratic forms, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Polynomial rings and ideals; rings of integer-valued polynomials, Global theory and resolution of singularities (algebro-geometric aspects) Arithmetical quadratic surfaces of genus 0. 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 [For the entire collection see Zbl 0664.00004.]
The authors present a result from their forthcoming paper [``El diagrama de Enriques de una curva algebroide plana e irreducible''] which gives a complete system of invariants (called the Enriques diagram) of an equisingularity class of an algebroid plane curve. resolution of singularities; equisingularity class; algebroid plane curve Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings Enriques diagram of a plane algebroid branch and the classification of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this very well-written article, the authors establish several results relating finite schemes and secant varieties over arbitrary fields. Some of these results were previously known only over the complex numbers. The paper is in part expository and contains background material on scheme theory, apolarity theory, Castelnuovo-Mumford regularity, Hilbert schemes, and secant varieties. Let \(\mathbb K\) be a field and \(R\) be a finite scheme over \(\mathbb K\). One of the main objectives is to study the \textit{smoothability} of \(R\) both as an abstract scheme and as an embedded scheme in some algebraic variety \(X\). The condition of smoothability can be easily seen over an algebraically closed field: a finite scheme \(R\) is smoothable if and only if it is a flat limit of distinct points. Theorem 1.1 gives the equivalence between the abstract smoothability and the embedded smoothability in some algebraic variety \(X\), whenever \(X\) is smooth. Moreover, smoothability over \(\mathbb K\) is equivalent to smoothability in the algebraic closure of \(\mathbb K\) (Proposition 1.2). Let \(\mathbb K\) be an algebraically closed field. Let \(X\) be an algebraic variety \(\mathbb K\) and let \(r\) be an integer. Condition \((\star)\) holds if every finite \textit{Gorenstein} subscheme over \(\mathbb K\) of \(X\) of degree at most \(r\) is smoothable in \(X\). One of the main results is Theorem 1.7. This relates the scheme theoretic condition above with the possibility of giving \textit{set-theoretic equations} for secants of sufficiently high Veronese embeddings of \(X\), by determinantal equations from vector bundles on \(X\). If condition \((\star)\) does not hold, then those equations are not enough to cut them. Interestingly, the locus of determinantal equations from vector bundles contain more general loci than secants: the \textit{cactus varieties}. This containment is the ultimate reason for the failure of present methods to give good enough lower bounds on tensor ranks. smoothable; secant varieties; finite Gorenstein scheme; cactus variety; Veronese reembedding; Hilbert scheme Buczyński, J.; Jelisiejew, J., Finite schemes and secant varieties over arbitrary characteristic, Differential Geom. Appl., 55, 13-67, (2017) Determinantal varieties, Local deformation theory, Artin approximation, etc., Parametrization (Chow and Hilbert schemes), Schemes and morphisms, Homogeneous spaces and generalizations Finite schemes and secant varieties over 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 This papers deals with cell decompositions and algebraicity of cohomology for quiver Grassmannians; in particular, the authors prove that the cohomology ring of a quiver Grassmanian associated with a rigid quiver representation satisfies that there is no odd cohomology and the cycle map is an isomorphism. Also, they proved that the corresponding Chow ring admits explicit generators defined over any field. Moreover, they establish the polynomial point count property, after that they consider only quiver to finite or affine type and show that a quiver Grassmanian associated with an indecomposable representation admits a cellular decomposition. Finally, as a particular result, the authors establish a cellular decomposition for quiver Grassmannians associated with representations with rigid regular part. quiver Grassmannians; cellular decomposition; property (S); cluster algebras Representations of quivers and partially ordered sets, Cluster algebras, Derived categories, triangulated categories, Grassmannians, Schubert varieties, flag manifolds, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Cell decompositions and algebraicity of cohomology for quiver Grassmannians | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The problem of resolution of singularities and its solution in various contexts can be traced back to I. Newton and B. Riemann. This paper is an attempt to give a survey of the subject starting with Newton till the modern times, as well as to discuss some of the main open problems that remain to be solved. The main topics covered are the early days of resolution (fields of characteristic zero and dimension up to three), Zariski's approach via valuations, Hironaka's celebrated result in characteristic zero and all dimensions and its subsequent strengthenings and simplifications, existing results in positive characteristic (mostly up to dimension three), de Jong's approach via semi-stable reduction, Nash and higher Nash blowing up, as well as reduction of singularities of vector fields and foliations. In many places, we have tried to summarize the main ideas of proofs of various results without getting too much into technical details. Global theory and resolution of singularities (algebro-geometric aspects), Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Resolution of singularities: an introduction | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper we use some properties of spherical blowing-up to give an alternative and more geometric proof of Gau-Lipman Theorem [\textit{Y. N. Gau} and \textit{J. Lipman}, Invent. Math. 73, 165--188 (1983; Zbl 0498.32003)] about the differentiable invariance of the multiplicity of complex analytic sets. Moreover, we also provide a generalization of the Ephraim-Trotman Theorem [\textit{R. Ephraim}, Duke Math. J. 43, 797--803 (1976; Zbl 0352.14017); \textit{D. Trotman}, in: Real analytic and algebraic singularities. Harlow: Longman. 215--221 (1998; Zbl 0921.32013)]. Zariski multiplicty problem; Gau-Lipman theorem; Ephraim-Trotman theorem Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Singularities in algebraic geometry A proof of the differentiable invariance of the multiplicity using spherical blowing-up | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0632.00005.]
The contribution of this paper is to show how to parametrize a cubic surface given its implicit equation. This capability makes cubic surfaces unique.
One parametrization is presented which works for any cubic surface, but which involves the use of a square root. Other parametrizations are presented which involve only rational polynomials but which do not apply to every cubic surface. - \(Section\quad 2\) in this paper reviews the development of piecewise algebraic surfaces. - \(Section\quad 3\) discusses the existence of \(27\quad straight\quad lines\) which occur on every nonsingular cubic surface, and which are used in the parametrization. - \(Section\quad 4\) discusses several methods of parametrizing a cubic surface. rational polynomials; parametrize a cubic surface Sederberg, T W; Snively, J, Parametrization of cubic algebraic surfaces, 299-2, (1987), New York Software, source code, etc. for problems pertaining to algebraic geometry, Parametrization (Chow and Hilbert schemes), Surfaces and higher-dimensional varieties Parametrization of cubic algebraic surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors present a new approach to the study of multiplier ideals in a regular two dimensional ring \((R,m)\). This allows to unify the definition on ideals, filtration of ideals or plurisubharmonic functions. This work is based on the introduction of the space \({\mathcal V}\) of all \(\mathbb R\cup\{ +\infty \} \)-valued valuations of \(R\) centered at the maximal ideal \(m\) and normalized by \(\nu(m)=1\). \({\mathcal V}\) is a tree and encodes in a natural way all possible blow-ups of \(R\) centered in \(m\). The points of \({\mathcal V}\) that are not ends, form a subtree \({\mathcal V}_{qm}\). Multiplier ideals are defined in terms of some functions on \({\mathcal V}_{qm}\), called tree potentials. Among the applications are a formula for the singularity exponent of a plurisubharmonic function in terms of Kiselman numbers, and a proof of a conjecture by Demailly and Kollar. They give a new proof of the fact obtained by Lipman and Watanabe that any integrally closed ideal is a multiplier ideal. regular local ring; dimension two; blow up C. Favre and M. Jonsson, \textit{Valuations and multiplier ideals}, J. Amer. Math. Soc. 18(2005), no. 3, 655--684. Singularities in algebraic geometry, Lelong numbers, Regular local rings Valuations and multiplier ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((R, \mathfrak{m})\) be a \(d\)-dimensional commutative Noetherian local ring of prime characteristic \(p\). The main result of the paper is a new lower \textit{I. M. Aberbach} and \textit{F. Enescu} [Mich. Math. J. 57, 1--16 (2008; Zbl 1222.13005)], the authors of the paper introduce new ideas and also obtain other interesting results on Hilbert-Kunz multiplicity. For example, the paper presents a new inequality relating the Hilbert-Kunz multiplicity, Hilbert-Samuel multiplicity and the F-signature of the ring. Moreover, under some conditions on the ring and assuming an algebraically closed residue field, the authors prove that for an integrally closed \(\mathfrak{m}\)-primary ideal \(I\), \(\text{e}_{HK}(I) \geq \text{length}(R/I) + \text{e}_{HK}(R)-1\). \textit{I. M. Aberbach} and the reviewer have also refined their earlier work and improved the originally obtained lower bound in a new work [``New estimates of Hilbert-Kunz multiplicities for local rings of fixed dimension'', to appear in Nagoya Math. J.; \url{arXiv:1101.5078}], but the paper under review produces a better bound in important cases. The bound obtain in this paper is still weaker than the lower bound conjectured by \textit{K.-I. Watanabe} and \textit{K.-I. Yoshida} in [Nagoya Math. J. 177, 47--75 (2005; Zbl 1076.13009)]. Hilbert-Kunz multiplicity; F-signature; Watanabe-Yoshida conjecture Celikbas, O.; Dao, H.; Huneke, C.; Zhang, Y., Bounds on the Hilbert-Kunz multiplicity, Nagoya Math. J., 205, 149-165, (2012) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Integral closure of commutative rings and ideals, Multiplicity theory and related topics, Singularities in algebraic geometry Bounds on the Hilbert-Kunz multiplicity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 volume at hand is the first part of a profound introduction to algebraic geometry in its modern setting, that is, in A. Grothendieck's revolutionary conceptual framework of algebraic schemes. In these days, about fifty years after Grothendieck's refoundation of algebraic geometry in the language of schemes, this approach is well-established as both a fundamental cornerstone and an indispensable toolkit for the current research in various fields of mathematics, including geometry, number theory, complex analysis, theoretical physics, and their many modern applications.
As the authors point out in the preface, the present textbook is to give a systematic and comprehensive introduction to the theory of schemes in its necessary generality, thereby providing the reader with a solid background for turning towards current research in algebraic geometry and its related areas within contemporary mathematics.
Actually, there is already a considerably large number of excellent textbooks on modern algebraic geometry explaining some basics of scheme theory as well, at least so to a reasonable extent. However, most of the great standard texts have their own special focus, use schemes as an appropriate, well-adapted language for certain parts of the exposition, and develop their general theory only as far as needed for particular purposes. In fact, the monumental pioneering work ``Elements of Algebraic Geometry'' (EGA I--IV) by A. Grothendieck and J. Dieudonné is still the only encyclopedic reference for scheme theory, but this multi-volume, highly abstract treatise is very far from being a suitable textbook for beginners in the field.
On the other hand, the volume under review is really written as a comprehensive textbook of introductory character, almost exclusively devoted to the theory of schemes, primarily geared toward graduate students, and thereby assuming only basic knowledge in abstract algebra and topology as prerequisites. Moreover, the authors have set a high value on carefully thought out didactic principles underlying the entire exposition, and that by developing the abstract, both conceptually and technically utmost demanding theory of schemes all through in very systematic, detailed, motivating, concrete and illustrating a manner.
As to the contents, the present volume comprises sixteen chapters and five supplementing appendices.
The introduction to the book briefly explains the main concern of both classical algebraic geometry and the theory of schemes, on the one hand, and provides then a practical guide for the use of the text, including some directing comments on the contents of the single chapters, on the other hand.
Chapter 1 begins with a discussion of affine algebraic varieties, prevarieties, and projective varieties as spaces with (algebraic) functions, that is, as both historical precursors and first important examples related to abstract algebraic schemes à la Grothendieck. Chapter 2 introduces the spectrum of a ring, its Zariski topology, the basics of sheaf theory, and finally affine schemes as locally ringed spaces forming the local building blocs for general schemes treated in the sequel.
Chapter 3 introduces the main objects of study of the book, namely schemes and their morphisms. Basic properties of schemes and morphisms in general, projective schemes, schemes associated to prevarieties, subschemes, and immersions of schemes are the principal topics discussed in this chapter.
The category of schemes and functors attached to schemes are more closely investigated in Chapter 4, with the focus on fiber products, base change properties, and the structure of fibers of a morphism.
In this context, other fundamental constructions such as inverse images and schematic intersections of subschemes, morphisms and products of projective spaces, the Segre embedding, and group schemes are treated along the way.
After the basics of scheme theory (as developed in Chapters 2--4), the more advanced part of the whole subject starts with Chapter 5, where schemes over a general ground field are studied, including the notion of dimension of a scheme of finite type over a field and, as a first application of the general theory developed so far, the intersection calculus for plane curves.
Chapter 6 is concerned with local properties of schemes, with particular emphasis on algebraic tangent spaces, smooth morphisms, regular schemes, and normal schemes.
Quasi-coherent module sheaves, their basic properties, and the fundamental constructions of these objects are described in Chapter 7, whereas the functorial viewpoint in scheme theory is further pursued in the subsequent Chapter 8. The latter part deals with representable functors in general, then with representable morphisms of schemes, Zariski sheaves, Zariski coverings of functors, and two important concrete examples: Grassmannians and Brauer-Severi schemes.
Chapter 9 turns to the notions of separated schemes and separated morphisms in its first part, while the second part is dedicated to the important topic of rational maps and function fields of schemes, thereby providing the first steps into birational geometry.
Generalizing the notion and properties of schemes locally of finite type over a field, Chapter 10 analyzes various finiteness conditions for arbitrary morphisms of schemes, constructible properties of schemes and quasi-coherent sheaves as well as the structure of schemes over inductive limits of rings. This material provides fundamental concepts and techniques for the further general study of schemes, and the basic reference for it is the comprehensive volume ``EGA IV'' by Grothendieck and Dieudonné from the 1960s.
Chapter 11 treats two other central topics in algebraic geometry, namely vector bundles on schemes and divisors. This is done for general schemes (over a base scheme), with particular emphasis on locally free sheaves, line bundles, Cartier divisors, Weil divisors. Picard groups, divisor class groups, and the relations between these objects. Along the way, flattening stratifications for quasi-coherent modules of finite type, torsors, and non-Abelian cohomology are also briefly touched upon.
The main objects studied in Chapter 12 are affine, finite and proper morphisms. Apart from Chevalley's and Serres criteria for characterizing affine morphisms, the concept of normalization of an integral scheme and Zariski's Main Theorem appear in the limelight of this section of the book. Next, in Chapter 13, projective schemes are studied in greater detail, together with their distinguished quasi-coherent module sheaves and their embeddings into projective space. This includes the discussion of ample and very ample line bundles, immersions into projective bundles, linear systems, (quasi-)projective morphisms, the study of blow-ups, a general version of Chow's Lemma, and an outlook to the resolution of singularities likewise. Chapter 14 describes flat morphisms, their important geometric properties, and their various characterizations, in particular the valuative criterion for flatness.
In this context, a large number of results obtained from the principle of faithfully flat descent for schemes, quasi-coherent modules, and torsors is derived, together with a concrete application to Brauer-Severi varieties (as introduced in Chapter 8). The next parts are then dedicated to a refined treatment of dimension theory, especially with regard to the variation of the dimension of the fibers of a flat morphism, on the one hand, and with respect to Cohen-Macaulay schemes on the other. This chapter ends with a first short glimpse of the idea of Hilbert schemes as parametrizing objects. Chapter 15 gives an application of the general theory developed so far to the important class of Noetherian schemes of dimension one, i.e., to absolute algebraic curves and their divisors, ending with an outlook to the Riemann-Roch Theorem for curves. Finally, several important classes of concrete schemes are exhibited in the concluding Chapter 16. A special didactic feature of the presentation given here is that these examples are discussed in parallel to the advancement of the theory in the main part of the book, with respective references to the related previous chapters. More precisely, determinantal varieties and schemes, the Clebsch cubic surface and its relation to a special Hilbert modular surface, some quotients of algebraic surfaces by cyclic groups, and Abelian varieties are used as examples to illustrate many of the concepts, methods, and results developed in the single chapters of the book, and that in very vivid and instructive a manner.
In addition, there are five appendices at the end of the book.
Appendix A recalls some basic notions and result from category theory, whereas Appendix B provides a collection of those relevant facts from commutative algebra as they are used in the course of the main text. Appendix G contains a list of properties of morphisms of schemes satisfying various permanence principles, and an overview of the many relations between different properties of morphisms of schemes is given in Appendix D. Finally, the authors recall (and partly refine) several definitions and properties concerning constructible subsets of schemes in Appendix E. Generally, these appendices have been added for the convenience of the reader, basically in order to keep the book as self-contained as possible, on the one hand, and to increase both its lucidity and its utility as a reference book on the other.
Another outstanding feature of the present primer of basic scheme theory is given by the exceptionally large number of illustrating examples, supplementing remarks, and accompanying exercises permeating the entire text. In fact, each chapter concludes with a set of about thirty related, carefully selected working problems, the solutions of which are to provide some important enrichment, refinement, and completion of the respective main text, apart from serving as a highly valuable testing ground for the reader's understanding of the core material.
Finally, the reader finds a rich bibliography of related textbooks, a detailed list of contents of the single chapters, an extensive index of symbols used in the course of the text, and a sweeping, virtually complete subject index at the end of the book.
To sum up, the book under review furnishes an excellent introduction to the basic theory of algebraic schemes and their morphisms. While presenting the highly abstract and technically utmost demanding material in full generality, and therefore in its widest applicability, the authors have consistently striven to keep it accessible even for beginners in the field. In regard to its comprehensiveness, lucidity, depth, versatility and didactical conception, this book is rather unique within the relevant textbook literature, and therefore an utmost valuable replenishment of the latter. Moreover, the authors have produced the so far most profound introduction to the original EGA volumes by Grothendieck and Dieudonné, which certainly must be seen as another rewarding result of their overall work.
As the title of the current book suggests, there will be a forthcoming second volume, with focus on the cohomology of schemes and its applications. To doubt, these two volumes together will represent another significant standard text in contemporary algebraic geometry and its allied areas within mathematics as a whole. textbook (algebraic geometry); schemes and morphisms; prevarieties; quasi-coherent sheaves; vector bundles; divisors; algebraic curves; determinantal varieties; singularities Görtz, U., Wedhorn, T.: Algebraic Geometry I. Vieweg+Teubner, Wiesbaden (2010) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Schemes and morphisms, Divisors, linear systems, invertible sheaves, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Group schemes, Determinantal varieties, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry Algebraic geometry I. Schemes. With examples and exercises | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 a codimension-1 algebraic cycle obtained from the Poincaré line bundle, Beauville defined the Fourier transform on the Chow groups of an abelian variety \(A\) and showed that the Fourier transform induces a decomposition of the Chow ring \(\mathrm{CH}^*(A)\). By using a codimension-2 algebraic cycle representing the Beauville-Bogomolov class, in book gived evidence for the existence of a similar decomposition for the Chow ring of hyperKähler varieties deformation equivalent to the Hilbert scheme of length-2 subschemes on a \(K3\) surface. Further was establish the existence of such a decomposition for the Hilbert scheme of length-2 subschemes on a \(K3\) surface and for the variety of lines on a very general cubic fourfold.
The manuscript is divided into three parts. In Chapter 1 the Beauville-Bogomolov class \(\mathfrak{B}\) is introduced and establish in Proposition 1.3 the quadratic equation. The core of Part 1 consists in Theorems 2.2 \& 2.4 and Theorem 3.3, where was considered a hyperKähler variety \(F\) of \(K3\) -type endowed with a cycle \(L \in \mathrm{CH}^2 (F \times F)\) representing the Beauville-Bogomolov class \(\mathfrak{B}\), and show that the conclusion of Theorem 2 holds for \(F\) and that the resulting Fourier decomposition on the Chow groups of \(F\) is in fact induced by a Chow-Künneth decomposition. In Chapter 6 was shown that the multiplicativity property of the Fourier decomposition boils down to intersection-theoretic properties of the cycle \(L\) and deduced that the Fourier decomposition is a birational invariant for hyperKähler varieties of \(K3^{[2]}\)-type. This approach is used in Chapter 7 to give new insight on the theory of algebraic cycles on abelian varieties by showing how Beauville's Fourier decomposition theorem is a direct consequence of a recent theorem of \textit{P. O'Sullivan} [J. Reine Angew. Math. 654, 1--81 (2011; Zbl 1258.14006)]. In Section 8 introduces the notion of multiplicative Chow-Künneth decomposition and its relevance is discussed. Part 1 ends with Chapter 9 where a proof of the algebraicity of the Beauville-Bogomolov class \(\mathfrak{B}\) is given for hyperKähler varieties of \(K3^{[n]}\)-type.
Part 2 and Part 3 are devoted to proving Theorem 1, Theorem 2 and Theorem 3 for the Hilbert scheme of length-2 subschemes on a \(K3\) surface and for the variety of lines on a cubic fourfold, respectively. In both cases, the strategy for proving Theorem 1 and Theorem 2 consists in first studying the incidence correspondence \(I\) and its intersection-theoretic properties, and then in constructing a cycle \(L \in \mathrm{CH}^2 (F \times F)\) very close to \(I\) representing the Beauville-Bogomolov class satisfying some known hypotheses. Chow ring; Beauville-Bogomolov class; Beauville's Fourier decomposition; cohomological Fourier transform M. Shen and C. Vial, The Fourier transform for certain hyperKähler fourfolds, Mem. Amer. Math. Soc. 240 (2016). Parametrization (Chow and Hilbert schemes), \(4\)-folds, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry The Fourier transform for certain hyperkähler fourfolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the extended version of the authoors in [Sel. Math., New Ser. 23, No. 3, 1909--1930 (2017; Zbl 1397.17015)]. stable and canonical bases; Leclerc-Thibon involution; Hilbert schemes Quantum groups (quantized enveloping algebras) and related deformations, Parametrization (Chow and Hilbert schemes), Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of forms Infinitesimal change of stable basis | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the behaviour of analytic branches of a projective variety with respect to hypersurface sections and give conditions under which their number and their orders are preserved. Singularities of curves, local rings, Singularities in algebraic geometry Analytic branches and hypersurface sections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 algebraic methods to study systems of linear partial differential equations with constant coefficients. equations and systems with constant coefficients; Hilbert schemes General theory of PDEs and systems of PDEs with constant coefficients, Overdetermined systems of PDEs with constant coefficients, Parametrization (Chow and Hilbert schemes) Linear differential operators with constant coefficients and Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For part I see Invent. Math. 94, No. 1, 163-169 (1988; Zbl 0701.14002).]
Let \(M\) be a smooth projective \(n\)-fold, let \({\mathcal O}_ M(1)\) be an ample line bundle on \(M\), and let \(X\subseteq M\) be a generic complete intersection of type \((m_ 1,\ldots,m_ k)\). The main result of the paper shows that if \(m_ 1+\cdots+m_ k\geq\dim(X)+n+1\) (respectively \(\dim(X)+n)\), then the desingularization of every subvariety of \(X\) is of general type (respectively, every subvariety of \(X\) has non-zero geometric genus). It is also noted that for surfaces the result is best possible. subvariety of general type; generic complete intersection; desingularization; geometric genus Ein L.: Subvarieties of generic complete intersections. II. Math. Ann. 289, 465--471 (1991) Complete intersections, Parametrization (Chow and Hilbert schemes), Surfaces of general type Subvarieties of generic complete intersections. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Teissier-Plücker formula for the class of an algebraic hypersurface with isolated singularities is generalized to arbitrary projective varieties with isolated singularities. The generalization uses the Buchsbaum-Rim multiplicity of the jacobian ideal instead of the ordinary one. Two different proofs are given. The first one, closer to the classical spirit, works only for complete intersections. Teissier-Plücker formula; projective varieties with isolated singularities; Buchsbaum-Rim multiplicity Steven L. Kleiman, A generalized Teissier-Plücker formula, Classification of algebraic varieties, Contemporary Mathematics 162, American Mathematical Society, 1992, p. 249-260 Enumerative problems (combinatorial problems) in algebraic geometry, Hypersurfaces and algebraic geometry, Singularities in algebraic geometry, Multiplicity theory and related topics A generalized Teissier-Plücker formula | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author proves that a homeomorphism \(\phi\), that gives a right equivalence \(g=f\circ \phi\), preserves the multiplicity, if \(\phi(z)/z\) is bounded on a line segment intersecting the zero locus of \(f\) and its tangent cone only at the origin. isolated complex hypersurface singularities; multiplicity; topological equivalence Complex surface and hypersurface singularities, Singularities in algebraic geometry A note on the Zariski multiplicity 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 [For the entire collection see Zbl 0607.00005.]
In this paper the authors give a very exhaustive survey of the known properties in the homotopy theory of arrangements of hyperplanes in \({\mathbb{C}}^ N \)(examples: K(\(\pi\),1)-arrangements, simplicial arrangements, free arrangements, etc.). A table with the various properties and the implications among them is constructed, in which it is also indicated which implications are not known and which are of significant interest. Sketch of proofs or counterexamples and references are given for all the implications. In the final section the authors construct a commutative diagram connecting the cohomology of the complement to an arrangement with those of its 1-minimal model and of the associated formal arrangement and K(\(\pi\),1)-space. At last several conjectures are formulated. isolated singularities; homotopy theory of arrangements of hyperplanes; K(\(\pi \) ,1)-space Michael Falk and Richard Randell, On the homotopy theory of arrangements, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam, 1987, pp. 101 -- 124. Homotopy theory and fundamental groups in algebraic geometry, Projective techniques in algebraic geometry, Singularities in algebraic geometry On the homotopy theory of 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 article under review concerns quotients of affine varieties by torus actions. For an affine variety \(X\) endowed with an action of a torus \(T\) the authors study its toric Chow quotient, toric Hilbert scheme and the Chow morphism between them restricted to the main components. They introduce the main component \(H_0\) of the toric Hilbert scheme, which parametrizes general \(T\)-orbit closures in \(X\) and their flat limits. They also give a geometric description of the Altmann-Hausen family of \(T\)-varieties over the normalization \((X/_CT)_0^{\mathrm{norm}}\) of the main component of the toric Chow quotient of \(X\) by \(T\), introduced by \textit{K. Altmann} and \textit{J. Hausen} [Math. Ann. 334, No. 3, 557--607 (2006; Zbl 1193.14060)]. Define
\[
W_x = \overline{\{(x,q(x)) : x \in X^{\mathrm{ss}}\}} \subseteq X\times (X/_CT)_0,
\]
where \(q\) is the quotient map and \(X^{\mathrm{ss}}\) is the set of points which are semistable with respect to all characters of \(T\). Then the Altmann-Hausen family is shown to be the normalization of \(W_X\) together with the projection to \((X/_CT)_0^{\mathrm{norm}}\).
The main result of the article describes the relation between the universal family \(U_0 \rightarrow H_0\) and the family \(W_X \rightarrow (X/_CT)_0\). The authors prove that the restriction of the toric Chow morphism \(H_0 \rightarrow (X/_CT)_0\), coming from a generalization of the construction given by \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)], lifts to a birational projective morphism \(U_0 \rightarrow W_0\). The last section concerns the case when \(X\) is a toric variety and \(T\) is a subtorus of its big torus. The results from previous sections are rephrased in terms of fans. In particular, an explicit description of the fan of the Altmann-Hausen family is given. torus action; toric variety; toric Chow quotient; toric Hilbert scheme O.V. Chuvashova, N.A. Pechenkin, Quotients of an affine variety by an action of a torus. February 2012. ArXiv e-prints arXiv:1202.5760. Geometric invariant theory, Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies Quotients of an affine variety by an action of a torus | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Enriques surfaces can be characterised as those compact complex surfaces that are not simply connected with a \(K3\) surface as universal cover. In this article, the authors introduce Enriques manifolds as those compact complex manifolds that are not simply connected with a hyperkähler manifold as universal cover. A hyperkähler (or holomorphic symplectic) manifold is a simply connected compact Kähler manifold \(X\) such that \(H^{0}(X, \Omega_{X}^{2})\) is generated by a closed non-degenerate \(2\)-form. Their theory largely runs parallel to the theory of \(K3\) surfaces [\textit{D. Huybrechts}, Invent. Math. 135, No. 1, 63--113 (1999; Zbl 0953.53031)]. A two-dimensional Enriques manifold is an Enriques surface in the usual sense.
Among others, the following basic properties of Enriques manifolds are shown: if \(Y\) is an Enriques manifold with universal cover \(X\rightarrow Y\), then \(\dim(Y)=2n\), \(\pi_1(Y)\) is a finite cyclic group of order \(d\mid n+1\) and \(\varphi(d)<b_2(X)\) (\(\varphi\) is Euler's phi-function). The order of \(\pi_1(Y)\) is called the index of \(Y\).
The main part of this paper consists of the construction of examples of Enriques manifolds. They are obtained as quotients of moduli spaces of sheaves on certain \(K3\) surfaces and as quotients of Hilbert schemes of points on bielliptic surfaces. All the examples constructed here have index \(2\), \(3\), or \(4\). It remains open which indices can actually occur. This seems mainly due to the small number of known examples of hyperkähler manifolds.
The authors continue to study Enriques manifolds in [Pure Appl. Math. Q. 7, No. 4, 1631--1656 (2011; Zbl 1316.32011)].
Enriques manifolds were introduced and studied independently by \textit{S. Boissière} et al. [J. Math. Pures Appl. (9) 95, No. 5, 553--563 (2011; Zbl 1215.14046)]. Enriques surfaces; hyperkähler manifold; Hilbert scheme; bielliptic surface Oguiso, K.; Schröer, S., Enriques manifolds, J. Reine Angew. Math., 661, 215-235, (2011) Moduli, classification: analytic theory; relations with modular forms, Special Riemannian manifolds (Einstein, Sasakian, etc.), Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Calabi-Yau theory (complex-analytic aspects), Parametrization (Chow and Hilbert schemes), \(K3\) surfaces and Enriques surfaces Enriques manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let (X,0) be an isolated singularity of the complete intersection variety defined by \(X={\mathfrak f}^{-1}(0)\) where \({\mathfrak f}=(f_ 1,...,f_ k): ({\mathbb{C}}^{n+k},0)\to ({\mathbb{C}}^ k,0)\). For sufficiently small \(\epsilon\) and generic \({\mathfrak t}=(t_ 1,...,t_ k)\) \((0<t_ j\ll \epsilon)\), \(X_ t={\mathfrak f}^{-1}({\mathfrak t})\cap {\mathfrak B}_{\epsilon}\) is a smooth \((n-1)-\)connected n-dimensional complex manifold which is called the Milnor fiber of X. The homology group \(H=H_ n(X_ t,{\mathfrak Z})\) with the intersection pairing is called the Milnor lattice. Let \(\mu_+\) and \(\mu_-\) be the dimension of maximal positive, respectively negative, definite subspace of \(H_{{\mathfrak R}}=H\otimes {\mathfrak R}\) and let \(\mu_ 0\) be the rank of ker H. (X,0) is called to be parabolic if \(\mu_{-\epsilon}=0\) and \(\mu_ 0>0\) and hyperbolic if \(\mu_{-\epsilon}=1\) where \(\epsilon =(-1)^{n/2}\). The author proves that
Theorem (3.1). Assume that (X,0) is not a hypersurface singularity.
(i) Then (X,0) is parabolic if and only if (X,0) is of type \(\tilde D_{n+3}\) and
(ii) The singularity (X,0) is hyperbolic if and only if (X,0) is of type \(T^ 2_{p,q,r,s}\) (2\(\leq p\leq r\), \(2\leq q\leq s\), \(3\leq s)\) or \(T^ n_{2,q,2,s}\) (2\(\leq q\leq s\), \(3\leq s).\)
(iii) If (X,0) is neither parabolic nor hyperbolic, then \(\mu_{- \epsilon}\geq 2\). polar curve; isolated singularity; complete intersection; Milnor lattice; parabolic; hyperbolic W. Ebeling, ''The monodromy groups of isolated singularities of complete intersection,''Lect. Notes Math.,1923 (1987). Local complex singularities, Singularities in algebraic geometry, Complete intersections Vanishing lattices and monodromy groups of isolated complete intersection singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We compute the Brauer groups of several moduli spaces of stable quiver representations. quiver representations; moduli spaces; Brauer groups; tautological bundles; quadrics Fine and coarse moduli spaces, Brauer groups of schemes, Representations of quivers and partially ordered sets Brauer groups for 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 The content of this volume consists of new results and survey articles concerning real and complex algebraic geometry, singularities of curves and hypersurfaces, invariants of singularities, algebraic theory of derivaations and other topics.
This volume is dedicated to three mathematicians celebrating in 2017 the jubilees of 70th and 60th birth anniversaries: Arkadiusz Płoski, Kamil Rusek and Krzysztof Kurdyka. Festschriften, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Semialgebraic sets and related spaces, Singularities in algebraic geometry, Proceedings of conferences of miscellaneous specific interest Preface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under reviewed is concerned with the geometry of (irreducible) hyperkähler manifolds (i.e. compact simply-connected Kähler manifolds whose space of global holomorphic 2-forms is generated by a symplectic form) and in particular with the following question raised by \textit{A. Beauville} [Springer Proc. Math. 8, 49--63 (2011; Zbl 1231.32012)].
\textit{Question:} If \(X\) is a hyperkähler manifold and \(L\subset X\) a Lagrangian torus, is \(L\) a fibre of an (almost holomorphic) Lagrangian fibration \(f:X\rightarrow B\)?
A submanifold \(L\) of \(X\) is said to be Lagrangian if \(\mathrm{dim}(X)=2\,\mathrm{dim}(L)\) and if the restriction of the symplectic form to \(L\) vanishes identically.\newline The main result of this paper is a positive answer to the latter question in non-projective case (building on the earlier work [\textit{F. Campana, K. Oguiso} and \textit{T. Peternell}, J. Differ. Geom. 85, No. 3, 397--424 (2010; Zbl 1232.53042)]).
Theorem. (Th. 4.1) If \(X\) is a non-projective hyperkähler manifold and \(L\) a Lagrangian torus, then \(X\) has algebraic dimension \(n\) and \(L\) is a fibre of some algebraic reduction \(f:X\rightarrow B\).
The main tools used to prove this result are infinitesial computations in the cycle space (Barlet space) and a criterion of non-simplicity (in addition to the results of Campana et al. [loc. cit.]. In the projective case, the natural idea is to try to deform the pair \((X,L)\) to a non-projective one. This is easily achieved when the absolute situation is considered but the presence of the submanifold \(L\) makes the analysis harder. However the authors are able to derive the following criterion.
Theorem. (Th. 5.3) Let \(X\) be a hyperkähler manifold and \(L\) a Lagrangian subtorus. Then \(L\) is a fiber of a Lagrangian fibration if and only if there exists an effective divisor \(D\) in \(X\) such that \(c_1(\mathcal{O}_X(D))\) belongs to
\[
\mathrm{Ker}\big(\mathrm{H}^{1,1}(X,\mathbb{R})\longrightarrow\mathrm{H}^{1,1}(L,\mathbb{R})\big).
\]
It is worth noting here that this last assumption has been recently proved in [\textit{J.-M. Hwang} and \textit{R. Weiss}, Invent. Math. 192, No. 1, 83--109 (2013; Zbl 1276.14059)], then providing us with a complete and positive answer to the question of Beauville.\newline The final section is devoted to proving the existence of a nice model of the Lagrangian fibration (when it exists). The abutment of a carefully chosen MMP with scaling is actually a smooth hyperkähler manifold endowed with a holomorphic model of the Lagrangian fibration. hyperkähler manifolds; Lagrangian fibration; cycle spaces; deformation of pairs Greb, D.; Lehn, C.; Rollenske, S., \textit{Lagrangian fibrations of hyperkähler manifolds, on a question of Beauville}, Ann. Sci. Éc. Norm. Supér. (4), 46, 375-403, (2013) Compact Kähler manifolds: generalizations, classification, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Deformations of submanifolds and subspaces, Parametrization (Chow and Hilbert schemes), Minimal model program (Mori theory, extremal rays) Lagrangian fibrations on hyperkähler manifolds -- question of Beauville | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 analytic families of germs of isolated complete-intersection singularieties (ICIS), or ICIS germs. Their main goal is to develop some new algebraic tools and a geometric point of view that enables them to describe some standard equisingularity conditions (Whitney's condition A and Thom's Condition \(A_f)\) in terms of suitable numerical invariants of isolated singularities.
The basic numerical invariants used in this context are certain homological Buchsbaum-Rim multiplicities for modules, the general theory of which is recalled and considerably advanced in Sections 1 and 2 of this paper, culminating in a generalization (from ideals to modules) of B. Teissier's so-called ``principle of specialization of integral dependence'' [cf. \textit{B. Teissier}, Astérisque 7-8, 285-362 (1974; Zbl 0295.14003)]. Originally, B. Teissier had established this principle as an equivalent condition for Whitney equisingularity, and formulated it in terms of the Jacobian ideal of an analytic family of hypersurface germs with isolated singularities.
In the sequel, the authors apply their generalization of this principle of specialization of integral dependence to the study of ICIS germs, i.e, to higher codimension. For this purpose, they introduce the more general concept of the Jacobian module of an ICIS germ, the integral closure of which is then used to describe equisingularity conditions based on the behavior of the limit tangent hyperplanes to the general member of the family.
After a thorough treatment of strict dependence (à la M. Lejeune-Jalabert and B. Teissier) in the more general context, which is carried out in Section 3 of the paper, the authors study Whitney's Equisingularity Condition A and establish a generalization of it to families of ICIS germs in Section 4.
The concluding Section 5 concerns Thom's Condition \(A_f\) for function germs \(f\) and culminates in a generalization of the Lê-Saito theorem to families of ICIS germs. This generalization is based upon a more recent construction by \textit{A. J. Parameswaran} [Compos. Math. 80, No. 3, 323-336 (1991; Zbl 0751.14005)] and yields, among other important results, a refinement of an earlier theorem of \textit{J. Briançon}, \textit{P. Maisonobe} and \textit{M. Merle} [Invent. Math. 117, 531-550 (1994; Zbl 0920.32010)].
Altogether, the methods and results developed in the paper under review must be seen as a major step toward the general study of equisingularity. hypersurface singularities; invariants of singularities; analytic families; Buchsbaum-Rim multiplicities; equisingularity; Jacobian module Terence Gaffney & Steven L. Kleiman, ``Specialization of integral dependence for modules'', Invent. Math.137 (1999) no. 3, p. 541-574 Equisingularity (topological and analytic), Singularities in algebraic geometry, Multiplicity theory and related topics Specialization of integral dependence for modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 singular locus of the variety of degenerate hypermatrices of an arbitrary format. Our main result is a classification of irreducible components of the singular locus. Equivalently, we classify irreducible components of the singular locus for the projectively dual variety of a product of several projective spaces taken in the Segre embedding. hyperdeterminant; singular locus; cusp type singularities; node type singularities; projectively dual variety; Segre embedding DOI: 10.5802/aif.1526 Singularities in algebraic geometry, Determinantal varieties Singularities of hyperdeterminants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite group, \(K^0(G)\) the Grothendieck group of real algebraic varieties with \(G\)-action. A homomorphism \(\beta^G :K^0(G) \longrightarrow \mathbb Z[[u^{-1}]][u]\) , the \(G\)-equivariant virtual Poincaré series, is constructed. For compact non-singular \(X\) the coefficients of \(\beta^G(X)\) coincide with the \(G\)-equivariant Betti numbers with coefficients in \(\mathbb Z /2\). The \(G\)-equivariant virtual Poincaré series is also defined for the Grothendieck group of arc-symmetric sets endored with a \(G\)-action. As an application the equivariant zeta-function with signs (in analogy to the motivic zeta-function of Denef and Loeser) is introduced to study singularities of Nash function germs. equivariant homology; arc symmetric sets; motivic integration; blow-Nash equivalence Fichou, G.: Equivariant virtual Betti numbers. Ann. de l'Inst. Fourier \textbf{58}(1), 1-27 (2008) Singularities in algebraic geometry, Nash functions and manifolds, Topology of real algebraic varieties, Global theory of complex singularities; cohomological properties Equivariant virtual Betti numbers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A=\mathbb{C} [[x_ 1,\dots,x_ n]]\) be the formal power series ring in \(n\) variables over the complex numbers, and \(G\) a finite abelian group acting linearly and faithfully on \(A\). The aim of this paper is to study the Grothendieck group \(K_ 0\pmod R\) of the category of finitely generated modules over the invariant ring \(R=A^ G\). \textit{M. Auslander} and \textit{I. Reiten} [J. Pure Appl. Algebra 39, 1-51 (1986; Zbl 0576.18008)] showed that \(K_ 0\pmod R\) is finitely generated by at most \(c(G)\) elements, where \(c(G)\) denotes the class number of \(G\). In particular, \(K_ 0\pmod R\) is a factor group of \(\mathbb{Z} [G^*]\) where \(G^*\) denotes the character group of \(G\). The authors prove that \(K_ 0\pmod R \simeq \mathbb{Z} [G^*]/K\), where \(K\) is defined as in the paper of \textit{J. Herzog} and \textit{H. Sanders} [Singularities, representation of algebras, and vector bundles, Proc. Symp., Lambrecht 1985, Lect. Notes Math. 1273, 134-149 (1987; Zbl 0652.14016)]. Grothendieck group; quotient singularity J. Herzog, E. Marcos and R. Waldi, On the Grothendieck group of a quotient singularity defined by a finite abelian group,J. Algebra 149 (1992), 122--138 \(K_0\) of group rings and orders, Grothendieck groups (category-theoretic aspects), Actions of groups on commutative rings; invariant theory, Singularities in algebraic geometry, Formal power series rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry On the Grothendieck group of a quotient singularity defined by a finite abelian group | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A new class of non-isolated hyperplane singularities which have the transversal type \(A_ k\) is introduced. Special deformations are constructed which preserve the topology of the Milnor fibre, and their analytic properties are investigated. This enables one to compute the number of nondegenerate singular points and to describe the homotopy type of the Milnor fibre. non-isolated hyperplane singularities; topology of the Milnor fibre; homotopy type of the Milnor fibre Topology of vector bundles and fiber bundles, Algebraic topology on manifolds and differential topology, Deformations of singularities, Singularities of differentiable mappings in differential topology, Singularities in algebraic geometry, Local complex singularities On hyperplanar singularities of the transversal type \(A_ k\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review provides an elegant ``reasonably self-contained'' survey to the classification problem of analytic function germs. In 1985 \textit{T.-C. Kuo} [Invent. Math. 82, 257--262 (1985; Zbl 0587.32018)] already introduced the idea of blow-analytic map and blow-analytic equivalence: a \textit{blow-analytic map} is a map which becomes real analytic when it is composed with a locally finite number of blowing-ups. A homeomorphism \(h: ({\mathbb R}^n,0)\to ({\mathbb R}^n,0)\) is a \textit{blow-analytic equivalence} if \(h\) and \(h^{-1}\) are blow-analytic maps.
The paper under review begins with some historical motivatation to the classification problem, and discuss recent progress including motivic integration. The authors give triviality theorem for some analytic families of germs and provide some blow-analytic invariants for analytic function germs, useful for the classification problem. Then, they discuss properties of blow-analytic maps and their relation with Lipschitz maps, obtaining in particular that a blow-analytic homeomorphism need not be bilischitz, a fact already discovered by S. Koike. In the end of the paper, \textit{blow-analytic isomorphisms} are introduced and a version of the inverse mapping theorem for them is suggested.
The paper is very readable and clear, and it provides a number of exotic examples. In the last section, some interesting open problems are suggested. real analytic germ; blow-analytic; arc lifting property; equisingularity Toshizumi Fukui and Laurentiu Paunescu, On blow-analytic equivalence, Arc spaces and additive invariants in real algebraic and analytic geometry, Panor. Synthèses, vol. 24, Soc. Math. France, Paris, 2007, pp. 87 -- 125 (English, with English and French summaries). Nash functions and manifolds, Semi-analytic sets, subanalytic sets, and generalizations, Singularities in algebraic geometry On blow-analytic 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 Some years ago, Florian Pop showed that a field which is finitely generated over its prime field is determined up to isomorphism by its absolute Galois group (allowing a finite purely inseparable extension in positive characteristic). This theorem, whose pedigree can be traced back to investigations by Neukirch concerning Galois groups of number fields in the early 1970's, gives a positive answer to the so-called ``birational anabelian conjecture'' of A. Grothendieck formulated in 1983. In work in progress [Prog. Math. 181, 519--532 (2000; Zbl 1022.14006); ``Pro-1 birational anabelian geometry over algebraically closed fields. I'', preprint, \\url{arxiv:math.AG/0307076}] \textit{F. Pop} extends the above result to fields of finite type and of dimension at least 2 over the algebraic closure of the prime field; the case of dimension 2 was also considered recently by Bogomolov et Tschinkel.
The lecture will survey the known results in the area and then present the main ideas entering Pop's proofs. absolute Galois group; function field; anabelian geometry Szamuely, T., Groupes de Galois de corps de type fini (d'après pop), Astérisque, 294, 403-431, (2004) Separable extensions, Galois theory, Global theory and resolution of singularities (algebro-geometric aspects), Arithmetic ground fields for curves, Arithmetic ground fields for surfaces or higher-dimensional varieties Galois groups of fields of finite type (following Pop). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities One constructs an explicit set of independent infinitesimal deformations of a two-dimensional cusp singularity and one formulates the conjecture that this spans the whole space \(T^ 1\) of infinitesimal deformations. As a corollary one gets Karras' result about non-rigidity of such singularities (due to Freitag and Kiehl, cusp singularities of dimension greater than two are rigid). Later, the author has been able to prove the conjecture and thus an explicit description of \(T^ 1\) is now available. rigidity; infinitesimal deformations of a two-dimensional cusp singularity [Be 1] K. Behnke. Infinitesimal deformations of cusp singularities. Math. Ann. 265, 407--422, 1983 Deformations of singularities, Infinitesimal methods in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local deformation theory, Artin approximation, etc., Singularities in algebraic geometry, Local complex singularities Infinitesimal deformations of cusp singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove a general method to compactify certain open varieties by adding normal crossing divisors. This is done by showing that blowing up along an arrangement of subvarieties can be carried out. Important examples such as Ulyanov's configuration spaces and complements of arrangements of linear subspaces in projective spaces, etc., are covered. Intersection ring and (nonrecursive) Hodge polynomials are computed. Furthermore, some general structures arising from the blowup process are also described and studied. Y. Hu, A compactification of open varieties, Trans. Amer. Math. Soc. 355 (2003), 4737--4753. JSTOR: Parametrization (Chow and Hilbert schemes), Enumeration in graph theory, Configurations and arrangements of linear subspaces A compactification of open 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 Dimer models are introduced by string theorists to study four-dimensional \(N=1\) superconformal field theories. See, e.g., a review by \textit{K. D. Kennaway} [Int. J. Mod. Phys. A 22, No. 18, 2977-3038 (2007; Zbl 1141.81328)] and references therein for a physical background. A dimer model is a bipartite graph on a real two-torus which encodes the information of a quiver with relations. A typical example of such a quiver is the McKay quiver determined by a finite Abelian subgroup \(G\) of \(\text{SL}(3,\mathbb{C})\) [see \textit{M. Reid}, ``McKay correspondence'', \url{arXiv:alg-geom/9702016v3}, \textit{K. Ueda} and \textit{M. Yamazaki}, Commun. Math. Phys. 301, No. 3, 723-747 (2011; Zbl 1211.81090)]. In this case, the moduli space of representations of the McKay quiver (for the dimension vector \((1,1,\dots,1)\)) coincides with the moduli space of \(G\)-constellations considered by \textit{A. Craw} and \textit{A. Ishii} [Duke Math. J. 124, No. 2, 259-307 (2004; Zbl 1082.14009)]. For a generic choice of a stability parameter \(\theta\), the moduli space of \(G\)-constellations is a crepant resolution of the quotient singularity \(\mathbb{C}^3/G\) and the derived category of coherent sheaves on the moduli space is equivalent to the derived category of finitely generated modules over the path algebra of the McKay quiver. It is expected that these kinds of statements can be generalized to the case of dimer models that are `consistent' in the physics context, which should be called `brane tilings'. In this note, we discuss a slightly weaker notion of non-degenerate dimer models, which is strong enough to ensure that the moduli space is a crepant resolution of the three-dimensional toric singularity determined by the Newton polygon of the characteristic polynomial (see Theorem 6.4). We expect that one has to impose further conditions to prove the derived equivalence.
For the proof, we use a generalization of the description of a torus-fixed point on the moduli space in terms of a choice of a covering by hexagons of the fundamental region of a real 2-torus due to \textit{I. Nakamura} [J. Algebr. Geom. 10, No. 4, 757-779 (2001; Zbl 1104.14003)]. Many of the arguments are similar to those of \textit{A. Ishii} [in Clay Mathematics Proceedings 3, 227-237 (2005; Zbl 1156.14308)]. There is also a physics paper by \textit{S. Franco} and \textit{D. Vegh} [Moduli spaces of gauge theories from dimer models: proof of the correspondence, \url{arXiv:hep-th/0601063v2}] which deals with the relation between brane tilings and moduli spaces. dimer models; superconformal field theories; bipartite graphs; quivers with relations; McKay quivers; moduli spaces; representations of quivers; crepant resolutions; quotient singularities A. Ishii and K. Ueda, \textit{On moduli spaces of quiver representations associated with dimer models}, arXiv:0710.1898. Representations of quivers and partially ordered sets, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, String and superstring theories; other extended objects (e.g., branes) in quantum field theory On moduli spaces of quiver representations associated with dimer models. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The articles of this volume will not be indexed individually.
Contents:
\textit{Iku Nakamura}, ''Duality of cusp singularities'' (pp. 1-18); \textit{Peter Slodowy}, ''Chevalley groups over \({\mathbb{C}}((t))\) and deformations of simply elliptic singularities'' (pp. 19-38); \textit{Etsuo Yoshinaga} and \textit{Masahiko Suzuki}, ''Normal forms of nondegenerate quasihomogeneous functions with inner modality \(\leq 4''\) (pp. 39-47); \textit{Alan H. Durfee}, ''A naive guide to mixed Hodge theory'' (pp. 48-63); \textit{Kei-ichi Watanabe}, ''Rational singularities with \({\mathbb{C}}^*\)-action'' (pp. 64- 76); \textit{Kimio Watanabe}, ''On plurigenera of normal isolated singularities''. I (pp. 78-138); \textit{Shigeki Ohyanagi}, ''On two- dimensional normal singularities of type \({}_*A_ n\), \({}_*D_ n\) and \({}_*E_ n''\) (pp. 139-158); \textit{Toshitake Kohno}, ''On the rational K(\(\pi\),1)-properties of open algebraic varieties'' (pp. 159- 176); \textit{Hiroaki Terao}, ''New exponents and Betti numbers of complement of hyperplanes'' (pp. 177-195); \textit{Norbert A'Campo}, ''Limits of intrinsic metrics on the vanishing variety of curve singularities'' (p. 196); \textit{Lê Dũng Tráng}, ''Geometry of tangents, local polar varieties and Chern classes'' (pp. 197-212); \textit{F. Lazzeri}, ''Conjugation schemes and second homotopy group'' (p. 213-225). cusp singularities; Rational singularities; plurigenera; normal singularities Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Singularities in algebraic geometry, Complex singularities, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Local complex singularities, Singularities of curves, local rings, Singularities of surfaces or higher-dimensional varieties Complex analysis of singularities. Proceedings of a Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, September 10-12, 1980 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Was ist ``undergraduate algebraic geometry''? Im Falle des vorliegenden Buches der gelungene Versuch, auf gut 100 Seiten eine elementare Einführung in das fundamentale, technisch komplizierte, hochabstrakte und daher anspruchsvolle Gebiet der Algebraischen Geometrie zu geben, das der Autor selbst ``a monolithic block'' nennt, which is ``colonising adjacent areas of mathematics''. In der Tat, ``algebraic geometry was able to absorb practically all the advances made in topology, homological algebra, number theory, etc.'', ganz zu schweigen von Kategorientheorie, Komplexer Analysis, Differentialgeometrie usw. Der Versuch, einen elementaren Zugang zu diesem Gebiet zu finden, scheint daher von vornherein zum Scheitern verurteilt zu sein. Daß er vom Autor dennoch erfolgreich unternommen worden ist, verdient Beachtung. Schon die existierenden, zumeist recht umfangreichen Lehrbücher über ``Graduate Algebraic Geometrie'' weisen Kompromisse auf, indem z.B. die Stoffauswahl eingeschränkt und/oder die Darstellung teilweise skizzenhaft ist, und ``étudier les EGAs'' \((=\acute Elements\) de géométrie algébrique, Publ. Math., Inst. Hautes Étud. Sci.) nach Grothendieck erfordert das Durcharbeiten eines großen Stapels von paperbacks, ``many of which still remain to be written up in an approachable way''. Wie hat nun der Autor dieses Problem gelöst? Durch konsequente, exemplarische, dabei klassiche Stoffauswahl und ausführliche Erörterung geschickt ausgewählter Beispiele. Behandelt werden u.a. ebene Kegelschnitte, elliptische Kurven, das Geschlecht von Kurven (recht kurz), affine Varietäten nebst Nullstellensatz, Funktionen auf Varietäten, projektive Varietäten, birationale Äquivalenz, Nicht-Singularität und Dimension. Dabei sind etwa die Ausführungen über Kegelschnitte exemplarisch für beliebige rationale Kurven, werden die kubischen Flächen als Musterbeispiele von rationalen del Pezzo Flächen untersucht und dienen elliptische Kurven als einfachste Beispiele abelscher Varietäten.
Im Hinblick auf den Gegensatz ``Computation versus Theory'' hat sich der Autor für die Betonung von ``Computation'' entschieden: ``When general theory proves the existence of some construction, then doing it in terms of explicit coordinate expressions is a useful exercise that helps one to keep a grip on reality''. In einem lesenswerten Anhang wird die neueste Entwicklung der Algebraischen Geometrie aufgezeigt und kommentiert. Hier geht der Autor auf die modernen Begriffe ein, indem er z.B. erläutert, inwieweit affine Schemata allgemeiner sind als affine Varietäten. Wünschenswert wäre eventuell noch die Aufnahme der Garbentheorie im Zusammenhang mit rationalen Funktionen auf Varietäten sowie die Behandlung quasiprojektiver Varietäten im Anschluß an die projektiven Varietäten gewesen, aber das hätte wohl den Umfang des Bändchens zu sehr vergrößert. Der Stil des nicht immer leicht lesbaren Buches ist locker und witzig (der Leser findet u.a. psychologische Betreuung, und selbst Expräsident Reagan kommt im Text vor). Selbst wer einen weniger flapsigen Tonfall bevorzugt, wird Gefallen an dieser kleinen, doch inhaltsreichen Monographie finden. curve; variety; genus; singularity; dimension; rational function; tangent space; surface; birational equivalence Reid, M.: Undergraduate algebraic geometry. London mathematical society student texts (1988) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Curves in algebraic geometry, Varieties and morphisms, Singularities in algebraic geometry, Surfaces and higher-dimensional varieties Undergraduate 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 Der Grundkörper ist \(k=\mathbb{C}\), und \({\mathcal H}:=H_{d,g}= \text{Hilb}^P (\mathbb{P}^3_k)\), \(P(T)= dT-g+1\), ist das (volle) Hilbertschema der Raumkurven vom Grad \(d\) und Geschlecht \(g\). Für \(3\leq d\leq 11\) definiert man \(g(d)\) durch die Tabelle
\[
\begin{matrix} d & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11\\ g(d) & -2 & 0 & 1 & 2 & 4 & 6 & 9 & 11 & 15\end{matrix}
\]
und für \(d\geq 12\) durch die Formel \(g(d)={1\over 6}d(d-3)\). Die vorliegende Arbeit bringt zunächst eine Ergänzung und verschiedene Verbesserungen zu früheren Arbeiten des Autors [\textit{G. Gotzmann}, ``Der kombinatorische Teil der ersten Chowgruppe eines Hilbertschemas von Raumkurven'' und ``Der algebraische Teil der ersten Chowgruppe eines Hilbertschemas von Raumkurven'' (Münster 1994; Zbl 0834.14004 und Münster 1997; Zbl 0954.14002)], und man erhält als Zusammenfassung:
Satz I. Wenn \(d\geq 3\) und \(g\leq g(d)\) ist, dann ist
(i) \(\dim_\mathbb{Q} A_1({\mathcal H})\otimes_\mathbb{Z} \mathbb{Q}=3\);
(ii) \(\text{Pic} ({\mathcal H}) \simeq \mathbb{Z}^3\oplus \mathbb{C}^r\), mit \(r=\dim_kH^1 ({\mathcal H},{\mathcal O}_{\mathcal H})\);
(iii) \(\text{NS}({\mathcal H})\simeq\mathbb{Z}^3\).
Der Satz ist eine Folgerung aus etwas genaueren Ergebnissen in Abschnitt 6, wo explizite Basen von \(A_1({\mathcal H})\otimes\mathbb{Q}\) und \(\text{NS}({\mathcal H})\) bestimmt werden. -- Man kann die in Satz I genannten Ergebnisse für \({\mathcal H}\) auf die zugehörige universelle \({\mathcal C}\subset{\mathcal H}\times\mathbb{P}^3\) übertragen, und man erhält:
Satz II. Wenn \(d\geq 3\) und \(g\leq g(d)\) ist, dann ist
(i) \(\dim_\mathbb{Q} A_1({\mathcal C})\otimes_\mathbb{Z} \mathbb{Q}=4\);
(ii) \(\text{Pic}({\mathcal C})\simeq \mathbb{Z}^4\oplus \mathbb{C}^r\), mit \(r=\dim_kH^1 ({\mathcal C},{\mathcal O}_{\mathcal C})\);
(iii) \(\text{NS}({\mathcal C})\simeq\mathbb{Z}^4\). Néron-Severi group; Hilbert scheme; universal curve; Picard group Parametrization (Chow and Hilbert schemes), Plane and space curves The Néron-Severi group of a Hilbert scheme of space curves and the universal curve | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review studies the category of vector bundles \(vect-\mathbb{X}\) over \(\mathbb{X}:=\mathbb{X}(a,b,c)\), the weighted projective line of weight type \((a,b,c)\), over an algebraically closed field. The category of coherent sheaves \(coh-\mathbb{X}\) can be obtained by applying Serre's construction to the triangle singularity \(x_{1}^{a}+x_{2}^{b}+x_{3}^{c}\).
The article shows that \(vect-\mathbb{X}\) is a Frobenius category, where the line bundles are the indecomposable projective-injective objects. Also, the category \(\underline{vect}-\mathbb{X}\) obtained by taking the quotient of \(vect-\mathbb{X}\) by the line bundles is triangulated and has a tilting object, and it is fractional Calabi Yau, of CY-dimension a function of the Euler characteristic. The main point to prove this is the characterization of certain exceptional objects in the category. It is proven that each indecomposable rank \(2\) bundle \(F\) is exceptional in \(coh-\mathbb{X}\) and in \(vect-\mathbb{X}\) and, up to line bundle twist, \(F\) is isomorphic to an extension bundle. It is also shown explicitly that certain class of bundles form a strong exceptional sequence hence a tilting object.
The applications of this are, a factorization property showing that many morphisms in the category of vector bundles factor through a direct sum of line bundles, a very simple explanation of Happel-Seidel symmetry, and the setup of a framework for the ADE-chain problem. triangle singularity; Kleinian singularity; Fuchsian singularity; weighted projective line; vector bundle; singularity category; Cohen-Macaulay module; stable category; ADE-chain; Nakayama algebra; Happel-Seidel symmetry D. Kussin, H. Lenzing, and H. Meltzer, \emph{Triangle singularities, {ADE}-chains, and weighted projective lines}, Adv. Math. \textbf{237} (2013), 194--251. \MR{3028577} Singularities of surfaces or higher-dimensional varieties, Representation type (finite, tame, wild, etc.) of associative algebras, Derived categories, triangulated categories, Representations of quivers and partially ordered sets, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Triangle singularities, ADE-chains, and weighted projective lines | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(n\) and let \(L\) be an ample line bundle on \(X\). The global generation of adjoint bundles of the form \(K_X+mL\) has been largely investigated in connection with the Fujita freeness conjecture. A stronger version of this conjecture concerns \(K_X+L\). For \(n=2\) the problem is settled by Reider's theorem. For \(n=3\) and \(4\), \textit{Y. Kawamata} [Math. Ann. 308, No. 3, 491--505 (1997; Zbl 0909.14001)] proved the global generation of \(K_X+L\) at a point \(x \in X\) under the assumption that \(L^d \cdot Z\) satisfies an appropriate bound from below for any subvariety \(Z \subset X\) of dimension \(d\) containing \(x\). In general, the work of \textit{U. Angehrn} and \textit{Y.-T. Siu} [Invent. Math. 122, No. 2, 291--308 (1995; Zbl 0847.32035)] shows that if \(L^d \cdot Z > \binom{n+1}{2}^d\) for every subvariety \(Z \subset X\) of dimension \(d\), \(1 \leq d \leq n\), then \(K_X+L\) is globally generated. In the paper under review the authors prove the following Kawamata-type result for \(5\)-folds. Let \(n=5\), and \(x\in X\). If \(L^5 > 7^5\) and \(L^d \cdot Z \geq 7^d\) for any subscheme \(Z \subset X\) of dimension \(d, 1 \leq d \leq 4\), containing \(x\), then \(K_X+L\) is globally generated at \(x\). The proof relies on techniques of multiplier ideals; in particular, a crucial role is played by the notion of critical variety for an effective \(\mathbb Q\)-divisor at a point \(x\) and the related concept of deficit, introduced by \textit{L. Ein} [Proc. Symp. Pure Math. 62, 203--219 (1997; Zbl 0978.14004)] and by \textit{S. Helmke} [Duke Math. J. 88, No. 2, 201--216 (1997; Zbl 0876.14004)], independently. adjoint line bundle; global generation; 5-fold; multiplier ideal; critical variety Divisors, linear systems, invertible sheaves, Multiplier ideals, Singularities in algebraic geometry, Adjunction problems Global generation of adjoint line bundles on projective 5-folds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(f:(\mathbb{C}^n,0)\rightarrow (\mathbb{C},0)\) a germ of an analytic function with an isolated singularity at the origin, \(S:=\mathbb{C}\{ x_1,\ldots ,x_n\}\) and \(J_f:=(\partial f/\partial x_1,\ldots ,\partial f/\partial x_n)\), one defines the Milnor and Tjurina numbers as \(\mu :=\dim (S/J_f)\) and \(\tau :=\dim (S/(J_f,f))\). The author shows that \(\mu /\tau \leq n\), with equality if and only if Ker\((f)=(f^{n-1})\). As an application a lower bound of \(\tau\) in terms of \(n\) and the multiplicity of \(f\) at the origin is given. isolated singularity; Milnor number; Tjurina number; multiplicity Germs of analytic sets, local parametrization, Local complex singularities, Singularities in algebraic geometry Milnor and Tjurina numbers for a hypersurface germ with isolated singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We add a supplementary argument to [\textit{O. Fujino}, J. Math. Sci., Tokyo 18, No. 3, 299--323 (2011; Zbl 1260.14006)]. Singularities in algebraic geometry, Minimal model program (Mori theory, extremal rays) Addendum to ``On isolated log canonical singularities with index one'' | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this survey article we present connections between Picard-Lefschetz invariants of isolated hypersurface singularities and Blanchfield forms for links. We emphasize the unifying role of Hermitian Variation Structures introduced by Némethi. Seifert forms; Hodge numbers; Milnor fibration; linking pairings; Blanchfield pairings Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Singularities in algebraic geometry, Knot theory, Higher-dimensional knots and links, Singularities of surfaces or higher-dimensional varieties Real Seifert forms, Hodge numbers and Blanchfield pairings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 Zbl 0743.00050.]
This survey paper consists of three parts and an extensive bibliography. The first part is a general introduction to the theory of singularities of complex analytic spaces, which centers on the many characterisations of the \(ADE\)-singularities. The by now classical highlights are treated: the connection with Lie groups, quotient singularities and the platonic solids, the resolution graph and the Dynkin diagram, simpleness in the sense of Arnol'd. The author also covers the characterisation of the \(ADE\)-singularities by finite representation type. In many places references to newer and further developments are given.
The last two chapters introduce more advanced topics of current research in a careful discussion of problems and questions. The first one is the deformation space of rational surface singularities. The last chapter concerns moduli spaces for singularities and modules over local rings, a subject to which the author has contributed substantially. Whereas in the global case moduli spaces are constructed with Geometric Invariant Theory, and the groups appearing are reductive, one deals here with unipotent groups. simple singularities; deformation theory; bibliography; platonic solids; Dynkin diagram; moduli spaces Greuel, G.-M., Deformation und klassifikation von singularitäten und moduln, 177-238, (1992), Stuttgart Local complex singularities, Complex surface and hypersurface singularities, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients) Deformation and classification of singularities and modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\subset \mathbb{P}^n\) be a general complex hypersurface of degree \(d\). It is known that if \(d\) is sufficiently large, \(d\geq 2n - 2\), then \(X\) does not contain rational curves [see \textit{H. Clemens}, Ann. Sci. Éc. Norm. Supér., IV Sér. 19, No.~4, 629--636 (1986; Zbl 0611.14024); \textit{C. Voisin}, J. Differ. Geom. 44, 200--213 (1996; Zbl 0883.14022) and 49, No. 3, 601--611 (1998; Zbl 0994.14026)]. For \(d = 2n - 3\), \(n\geq 6\), \(X\) contains a finite number of lines but does not contain rational curves of degree \(e \geq 2\) [\textit{G. Pacienza}, J. Algebr. Geom. 12, 245--267 (2003; Zbl 1054.14057)]. In this paper the authors prove the following relevant result for low degree: For \(n\geq 2\) and \(d < (n + 1)/2\), a general complex hypersurface \(X\) in \(\mathbb P^n\) of degree \(d\) has the property that for each integer \(e\geq 1\) the scheme \(R_e(X)\) parametrizing degree \(e\) smooth rational curves on \(X\) is an integral local complete intersection scheme of ``expected'' dimension \((n+1-d)e +(n-4)\). The scheme \(R_e(X)\) is embedded as an open subscheme in the Kontsevich moduli space \(\overline{\mathcal M}_{0,0}(X,e)\) parametrizing stable maps to \(X\) and a partition of \(\overline{\mathcal M}_{0,0}(X,e)\) into locally closed subsets is used as in \textit{K. Behrend} and \textit{Yu. Manin} [Duke Math. J. 85, 1--60 (1996; Zbl 0872.14019)]. The authors use classical results about lines on hypersurfaces and include a new result about flatness of the projection map from the space of pointed lines. Moreover they use the deformation theory of stable maps, properness of the stack \(\overline{\mathcal M}_{0,r}(X,e)\) and a version of Mori's bend-and-break lemma. The authors use dual graphs associated to pointed curves and stable \(A\)-graphs as in the above article of K. Behrend and Yu. Manin (loc. cit.). dual graphs; Hilbert schemes; Kontsevich moduli spaces of stable maps; stacks Harris, J; Roth, M; Starr, J, Rational curves on hypersurfaces of low degree, J. Reine Angew. Math., 571, 73-106, (2004) Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Hypersurfaces and algebraic geometry, Parametrization (Chow and Hilbert schemes) Rational curves on hypersurfaces of low 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 The explicit computation of the intersection cohomology IH(X) à la Goresky-MacPherson of a complex space X is usually difficult. Nevertheless, according to Goresky-MacPherson, if a small resolution of the singularities \(\tilde X\to X\) of X exists, then IH(X) is roughly speaking the same as the cohomology \(H(\tilde X)\) of \(\tilde X.\) The author proves by an explicit construction the existence of a small resolution for any Schubert cell and therefore obtains a combinatorial description of the intersection cohomology. intersection cohomology; small resolution for any Schubert cell Zelevinskiĭ, A. V.: Small resolutions of singularities of Schubert varieties. Funct. anal. Appl. 17, No. 2, 142-144 (1983) Global theory and resolution of singularities (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, (Co)homology theory in algebraic geometry Small resolutions of singularities of Schubert varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A geometric description is given for the tangent cone to a Pfaffian quadratic singularity of the theta-divisor of the Prym variety for a given two-sheeted non-ramified covering of non-singular curves. A class of singularities with tangent cone of rank 5 is also considered which are close to Mumford singularities. Pfaffian quadratic singularity; theta-divisor of the Prym variety; singularities with tangent cone; Mumford singularities V. Kanev,Quadratic singularities of the Pfaffian theta divisor of a Prym variety, Math. Notes of the Ac. of Sc. of the USSR,31 (1982), 301--305. Jacobians, Prym varieties, Singularities in algebraic geometry, Singularities of curves, local rings, Picard schemes, higher Jacobians Quadratic singularities of the Pfaffian theta divisor of a Prym variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 semialgebraic map \(f:X\rightarrow Y\) between two real algebraic sets is called blow-Nash if it can be made Nash (i.e., semialgebraic and real analytic) by composing with finitely many blowings-up with nonsingular centers.
We prove that if a blow-Nash self-homeomorphism \(f:X\rightarrow X\) satisfies a lower bound of the Jacobian determinant condition then \(f^{-1}\) is also blow-Nash and satisfies the same condition.
The proof relies on motivic integration arguments and on the virtual Poincaré polynomial of \textit{C. McCrory} and \textit{A. Parusiński} [C. R., Math., Acad. Sci. Paris 336, No. 9, 763--768 (2003; Zbl 1073.14071)] and \textit{G. Fichou} [Compos. Math. 141, No. 3, 655--688 (2005; Zbl 1080.14070)]. In particular, we need to generalize Denef-Loeser change of variables key lemma to maps that are generically one-to-one and not merely birational. Campesato, JB, An inverse mapping theorem for blow-Nash maps on singular spaces, Nagoya Math. J., 223, 162-194, (2016) Semialgebraic sets and related spaces, Singularities in algebraic geometry, Arcs and motivic integration, Real algebraic sets, Nash functions and manifolds An inverse mapping theorem for blow-Nash maps on singular spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Poincaré series of multi-index filtration on the ring of germs defined by Gusein-Zade and Ebeling for the germ of a function in terms of its Newton diagram is considered. Examples of functions of two variables are described in the paper. These examples show that Poincaré series for the germ of a function depends not only on the type of the diagram, but also on the germ of the function. germs of holomorphic functions; Poincaré series Local complex singularities, Singularities in algebraic geometry Poincaré series of filtration associated with Newton diagram and topological types of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this short note (2 pages) the authors sketch the proof of the Bloch's formula: \(A^ p(X)=H^ p(X_ 1,\underline K_ p)\) where X is a regular noetherian scheme, \(A^ p(X)\) is the Chow group generated by sous varieties of codimension n of X modular linear equivalence and Ḵ\({}_ p\) defined by \textit{M. Karoubi} and \textit{O. Villamayor} [Math. Scand. 28, 265-307 (1971; Zbl 0231.18018)]. Karoubi-Villamayor K-functor; Bloch's formula; Chow group Parametrization (Chow and Hilbert schemes), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) Equivalence of the p-th component of the Chow ring and the cohomology groups in their \(K_ p\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a connected complex manifold of dimension \(n\). A singular meromorphic foliation on \(X\) of codimension \(q\) is given by the inclusion of a coherent sheaf \(A\) of rank \(q\) of \(\Omega^ 1_ X\) together with an integrability condition; such a foliation is said to be of type \(A\). The author generalises a construction of \textit{X. Gomez-Mont} and \textit{J. Mucino} [Lect. Notes Math. 1345, 129-162 (1988; Zbl 0681.58032)] to show that the set of foliations of type \(A\) has a natural structure of algebraic variety. In the case where \(X\) is projective with \(H\) an ample line bundle on \(X\), he then obtains a bound for the number of irreducible components of this variety, depending only on the numerical invariants of \(X\), \(H\) and \(A\). complex manifold; singular meromorphic foliation Singularities of holomorphic vector fields and foliations, Parametrization (Chow and Hilbert schemes), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Families of singular meromorphic 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 We construct a combinatorial algorithm of resolution of singularities for binomial ideals, over a field of arbitrary characteristic. This algorithm can be applied to any binomial ideal. This means ideals generated by binomial equations without any restriction, including monomials and \(p\)th powers, where \(p\) is the characteristic of the base field. In particular, this algorithm works for toric ideals. However, toric geometry tools are not needed, and the algorithm is constructed following the same point of view as the Villamayor algorithm of resolution of singularities in characteristic zero.
For part I, cf. [Math. Nachr. 285, No. 11-12, 1316--1342 (2012; Zbl 1274.14014)]. Blanco, R, Desingularization of binomial varieties in arbitrary characteristic. part II: combinatorial desingularization algorithm, Q. J. Math., 63, 771-794, (2012) Global theory and resolution of singularities (algebro-geometric aspects) Desingularization of binomial varieties in arbitrary characteristic. II: Combinatorial desingularization algorithm | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f:{\mathbb C}^2 \to \mathbb C\) be a nonconstant polynomial in two complex variables with a finite set of critical points; let \(C^{ t}\) be the projective closure of the fiber \(f^{-1}(t)\), let \(L_{ \infty}\) in \({\mathbb P}^2 ({\mathbb C})\) be the line at infinity and let \(C_{ \infty}= C^{ t} \cap L_{ \infty}\). The set \(\Lambda(f) = \{ t \in {\mathbb C}:\mu_{p}^t > \mu_{p}^{min} \;\text{ for a } p \in C_{ \infty} \}\), where \(\mu_{p}^t = \mu_{p}(C^t)\) is the Milnor number and \(\mu_{p}^{min}= \inf_{t \in {\mathbb C}} \mu_{p}^t\), is called the set of irregular values of \(f\) at infinity.
In this note the authors characterize polynomials \(f\) with no critical points and one irregular value at infinity: improving a recent result by \textit{A. Assi} [see Math. Z. 230, No. 1, 165--183 (1999; Zbl 0934.32017)], they give a description of the irregular fiber of such a polynomial. This result is applied to the estimation of the number of points at infinity of a polynomial with no critical points and at most one irregular value at infinity. The authors give also a discriminant criterion for polynomials to have one irregular value and present a list of open questions. affine curves; irregular value J. Gwozdziewicz and A. Ploski, On the singularities at infinity of plane algebraic curves, Rocky Mountain J. Math. 32 (2002), 139--148. Singularities of curves, local rings, Milnor fibration; relations with knot theory, Singularities in algebraic geometry On the singularities at infinity of plane algebraic 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 book is a selfcontained introduction to some of the combinatorical techniques for dealing with multigraded polynomial rings, semi group rings, and determinantal rings. An important role play combinatorically defined ideals and their quotients with the aim to compute numerical invariants and resolutions using gradings more refined than the standard grading.
The book is subdivided in three parts: Monomial Ideals, Toric Algebras, and Determinants. It has altogether 18 chapters containing homological invariants of monomial ideals and their polyhedral resolutions, toric varieties, local cohomology, Hilbert schemes among other subjects, to show how the tools developed can be used for studying algebraic varieties with group actions. Each chapter begins with an overview and ends with notes and references. The book assumes the knowledge of commutative algebra (graded rings, free resolutions, Gröbner bases) and a little simplicial topology and polyhedral geometry. It is interesting for a wide audience of students and researchers.
The book may serve as a basis for a full year course on this topic. It contains a lot of exercises and hints for further studies. monomial ideal; toric algebra; Hilbert scheme; local cohomology; multigraded polynomial rings E. Miller and B. Sturmfels, \textit{Combinatorial commutative algebra}, Graduate Texts in Mathematics volume 227, Springer, Germany (2005). Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Polynomial rings and ideals; rings of integer-valued polynomials, Computational aspects and applications of commutative rings, Toric varieties, Newton polyhedra, Okounkov bodies, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Parametrization (Chow and Hilbert schemes), Linkage, complete intersections and determinantal ideals Combinatorial commutative algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A nonnormal surface singularity occurs as (general) hyperplane section of a normal three-dimensional isolated singularity if this singularity is not Cohen-Macaulay. Such singularities can occur as result of small contractions. In higher dimensions a resolution (with normal crossings exceptional divisor) is in general not the correct tool for understanding the singularity. But it may happen that a small resolution exists, meaning (in dimension three) that the exceptional set is only a curve. The simplest case is when the curve is a smooth rational curve.
In this interesting paper, the author studies nonnormal surface isolated singularities. In Section~2, the author discusses invariants for nonnormal singularities. In Section~3, the author describes equations and deformations for surface singularities with the \(\delta\)-invariant being one, whose normalisation is a double point. In Section~4, a relation between deformations of a (partial) resolution and of the singularity itself is given. Finally, surface singularities with exceptional set being a smooth rational curve are studied in Section~5. nonnormal singularities; simultaneous normalisation; small modifications Local complex singularities, Complex surface and hypersurface singularities, Singularities in algebraic geometry, Deformations of singularities, Deformations of complex singularities; vanishing cycles Deforming nonnormal isolated surface singularities and constructing threefolds with \(\mathbb{P}^{1}\) as exceptional set | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 splitting (or toric) non-commutative crepant resolutions (= NCCRs) of some toric rings. In particular, we consider Hibi rings, which are toric rings arising from partially ordered sets, and show that Gorenstein Hibi rings with class group \(\mathbb{Z}^2\) have a splitting NCCR.\par In the appendix, we also discuss Gorenstein toric rings with class group \(\mathbb{Z}\), in which case the existence of splitting NCCRs is already known. We especially observe the mutations of modules giving splitting NCCRs for the three dimensional case, and show the connectedness of the exchange graph. non-commutative crepant resolutions; Hibi rings; class groups Cohen-Macaulay modules, Rings arising from noncommutative algebraic geometry, Algebraic aspects of posets, Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects) Non-commutative crepant resolutions of Hibi rings with small class group | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((V, 0)\) be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras \(L_k(V)\) associated to \((V, 0)\). Its dimension is denoted as \(\lambda_k(V)\). The \(L_k(V)\) is a generalization of the Yau algebra \(L(V)\) and \(L_0(V)=L(V)\). These numbers \(\lambda_k(V)\) are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute \(L_1(V)\) for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of \(\lambda_1(V)\). We also formulate a sharp upper estimate conjecture for the \(L_k(V)\) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture and prove it for binomial and trinomial singularities. isolated hypersurface singularity; Lie algebra; moduli algebra Local complex singularities, Singularities in algebraic geometry, Complex surface and hypersurface singularities, Solvable, nilpotent (super)algebras, Deformations of singularities Inequality conjectures on derivations of local \(k\)-th Hessain algebras associated to isolated hypersurface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The moduli spaces of stable sheaves on elliptic surfaces have been investigated by \textit{R. Friedman} [Invent. Math. 96, No. 2, 283--332 (1989; Zbl 0671.14006)]. In this paper, the author further studies the singularities and Kodaira dimensions of these moduli spaces.
Let \(X\) be a minimal surface over \(\mathbb C\) whose Kodaira dimension \(\kappa(X)\) equals \(1\) and irregularity \(q(X)\) equals \(0\), so there is an elliptic fibration \(\pi: X \to \mathbb P^1\).
Let \(\Lambda(X)\) be the number of multiple fibers of \(\pi\).
Up to deformation equivalence, one can always assume that every singular fiber of \(X\) is either a rational integral curve with one node or a multiple fiber with smooth reduction.
Fix an integer \(c_2 > 0\), and let \(H\) be an ample line bundle which is \(c_2\)-suitable (that is, there is no wall of type \((0, c_2)\) separating \(c_1(H)\) and the class of a fiber).
Let \(\mathfrak M(c_2)\) be the moduli space of of rank-\(2\) \(H\)-stable sheaves \(E\) with Chern classes \(c_1(E) = 0\) and \(c_2(E) = c_2\).
It is known that the restriction \(E_\eta\) of a sheaf \(E \in \mathfrak M(c_2)\) to the generic fiber can be classified into one of the three cases:
\begin{itemize}
\item[(1)] \(E_\eta\) has no sub-line bundle with fiber degree \(0\);
\item[(2)] \(E_\eta\) has a sub-line bundle with fiber degree \(0\), but \(E_\eta\) is not decomposable;
\item[(3)] \(E_\eta\) is decomposable into line bundles with fiber degree \(0\).
\end{itemize}
The author proves that if \(\Lambda(X) \le \max(2, 7(\chi(\mathcal O_X)+2)/4)\) and if \(E \in \mathfrak M(c_2)\) is a singular point such that \(E_\eta\) satisfies Case (1), then \(E\) is a canonical singularity of \(\mathfrak M(c_2)\).
Next, assume that \(\chi(\mathcal O_X) = 1\) and \(X\) contains exactly two multiple fibers with multiplicities \(2\) and \(m \ge 3\).
Then it is verified that \(E_\eta\) satisfies Case (1) whenever \(E \in \mathfrak M(c_2)\) is a singular point. Moreover, if \(c_2 \ge 3\) and if \(\mathfrak M(c_2)\) is projective (for example, when \(c_2\) is odd), then the Kodaira dimension of \(\mathfrak M(c_2)\) is equal to \((\dim \mathfrak M(c_2) +1)/2\).
Section 2 recalls background materials, including some facts from birational geometry. Section 3 is devoted to the study of the pluricanonical maps on the moduli space \(\mathfrak M(c_2)\).
Section 4 verifies a sufficient condition for a singular point of \(\mathfrak M(c_2)\) to be canonical.
Section 5 estimates the rank of \(H^1(\mathrm{ad}(f))\) for a singular point \(E \in \mathfrak M(c_2)\) of Case (1). Here \(f: E \to E(K_X)\) is a nonzero traceless homomorphism, and \(H^1(\mathrm{ad}(f)): \mathrm{Ext}^1(E, E) \to \mathrm{Ext}^1(E, E(K_X))\) is the induced map. The main results of the paper are confirmed in Section 6 and Section 7. Existence of singular points \(E \in \mathfrak M(c_2)\) satisfying Case (1) is dealt with in Section 8.
In Section 9, the author considers singular points \(E \in \mathfrak M(c_2)\) satisfying Case (2). moduli of stable vector bundles; elliptic surface; singularities; obstruction; Kodaira dimension Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Algebraic moduli problems, moduli of vector bundles, Complex-analytic moduli problems, Singularities in algebraic geometry, Birational geometry The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we study the following problem: ``Find the projective manifolds V/G with trivial canonical bundle and zero first Betti number by the construction of resolving singularities of the quotient of a complex torus V by a finite abelian group G. Computer their Euler numbers.''
By the toroidal desingularization, we find a criterion for the existence of such V/G for those (V,G) with trivial dualizing sheaf \(\omega_{V/G}\). Because the dimension of V is 2 or 3, such V/G can always be constructed. We also derive a formula of the Euler number of V/G in terms of the Euler numbers of fixed points of elements of G, which was suggested by string theorists. Kummer surfaces; resolving singularities of the quotient of a complex torus by a finite abelian group; Euler numbers; toroidal desingularization; string J. Halverson, C. Long and B. Sung, \textit{Algorithmic universality in F-theory compactifications}, \textit{Phys. Rev.}\textbf{D 96} (2017) 126006 [arXiv:1706.02299] [INSPIRE]. Special surfaces, Global theory and resolution of singularities (algebro-geometric aspects), Topological properties in algebraic geometry On the generalization of Kummer surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(F\in \mathbb{Q}[X]\) be a polynomial of degree \(d\) such that \(S=V (F=0)\) admits one regular real solution (i.e. the gradient of \(F\) does not vanish). The problem is treated of finding real solutions of \(F=0\) in \(\mathbb{R}^n\). In case that \(F\) is squarefree and the real variety defined by \(F\) is smooth there exist already algorithms of intrinsic complexity to find real solutions of \(F=0\) (cf. [\textit{B. Bank, M. Guisti, J. Heintz} and \textit{G. M. Mbakop}, ``Polar varieties, real equation solving and data structures: the hypersurface case'', J. Complexity 13, No. 1, 5--27 (1997; Zbl 0872.68066)].
In this case an intrinsic invariant \(\delta(F)\) is used which essentially determines the complexity of the algorithm and is a combination of the degree \(d\) with the maximal degree of the generic polar varieties of suitable type. The introduction of the dual polar variety was necessary for the case when \(S_{\mathbb{R}}\) is unbounded. In this situation some of the generic polar varieties of \(S\) may have an empty intersection with \(S_{\mathbb{R}}\). The dual polar varieties can be considered as the complex counterpart of Lagrange-multipliers.
The aim of the paper is to solve the problem in case of the existence of singular solutions.
Two discrete families of algorithms are presented which solve the problem in the particular case of a complex hypersurface containing smooth real points and possibly also real singularities, one in the case that \(S\) is compact and another without this assumption. real polynomial solving; intrinsic complexity; singularities; polar and bipolar varieties; degree of varieties B. Bank, M. Giusti, J. Heintz, L. Lehmann, and L. M. Pardo, \textit{Algorithms of intrinsic complexity for point searching in compact real singular hypersurfaces}, Found. Comput. Math. \textbf{12} (2012), no. 1, 75-122. Real algebraic sets, Singularities in algebraic geometry, Deformations of singularities, Symbolic computation and algebraic computation Algorithms of intrinsic complexity for point searching in compact real singular hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities An example is given to show that not every derivation in the nilradical of the Lie algebra of derivations of moduli algebras can be liftable and the dimension of the nilradical of the Lie algebra of derivations of moduli algebras is not a topological invariant for an isolated hypersurface singularity. liftable derivation; moduli algebras; isolated hypersurface singularity Chen Hao. A remark on liftable derivation of moduli algebras of isolated hypersurface singularities. to appear in Proc. AMS Complex surface and hypersurface singularities, Local complex singularities, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry A remark on the liftable derivation of moduli algebras of isolated hypersurface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities By the result of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], there exists an equivalence between the derived category of \(G-\)equivariant coherent sheaves on a quasiprojective variety \(M\) with the derived category of coherent sheaves on the irreducible component \(Y\) of the \(G-\)Hilbert scheme of \(M\) that contains free orbits. The equivalence holds under the assumption that \(G\) is a finite group acting on \(M\) such that the canonical bundle on \(M\) is locally trivial as a \(G-\)sheaf and \(\text{dim} Y\times_{(M/G)} Y \leq \dim M +1.\) The article under review generalizes this result to the case of any smooth Deligne-Mumford stack \(\mathcal X\) with coarse moduli space \(X\) which is a quasiprojective Gorenstein variety. Denote by \(\text{Hilb}({\mathcal X})\) a scheme representing the Hilbert functor studied by \textit{M. Olsson} and \textit{J. Starr} [Commun. Algebra 31, No. 8, 4069--4096 (2003; Zbl 1071.14002)]. Then, the role of scheme \(Y\) is played by the component \(\text{Hilb}'({\mathcal X})\subset \text{Hilb}({\mathcal X})\) containing non-stacky points in \(\mathcal X.\)
In the above setting, the main theorem asserts that if \(\text{dim} \text{Hilb}'({\mathcal X})\times_X \text{Hilb}'({\mathcal X}) \leq \dim X +1,\) then \(\text{Hilb}'({\mathcal X})\) is smooth and there is an equivalence between the categories \(D^b({\mathcal X})\) and \(D^b(\text{Hilb}'({\mathcal X}))\) given by the integral functor with universal object over \(\text{Hilb}'({\mathcal X})\) as a kernel.
Moreover, in the setting of Bridgeland, King and Reid, the authors prove the twisted version of equivalence in the sense of \textit{V. Baranovsky} and \textit{T. Petrov} [Adv. Math. 209, No. 2, 547--560 (2007; Zbl 1113.14033)]. McKay correspondence; derived categories; stacks; Hilbert scheme; Brauer group Chen, J-C; Tseng, H-H, A note on derived mckay correspondence, Math. Res. Lett., 15, 435-445, (2008) McKay correspondence, Stacks and moduli problems, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects), Generalizations (algebraic spaces, stacks), Brauer groups of schemes A note on derived McKay correspondence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \( S\) be a \( K3\) surface and Aut \(D(S)\) the group of auto-equivalences of the derived category of \( S\). We construct a natural representation of Aut \(D(S)\) on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on \( S\). The main result of this paper is the precise relation of this action with the monodromy of the Hilbert schemes \( S^{[n]}\) of points on the surface. A formula is provided for the monodromy representation in terms of the Chern character of the universal sheaf. Isometries of the second cohomology of \( S^{[n]}\) are lifted, via this formula, to monodromy operators of the whole cohomology ring of \( S^{[n]}\). Eyal Markman, On the monodromy of moduli spaces of sheaves on \?3 surfaces, J. Algebraic Geom. 17 (2008), no. 1, 29 -- 99. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, \(K3\) surfaces and Enriques surfaces, Parametrization (Chow and Hilbert schemes), Structure of families (Picard-Lefschetz, monodromy, etc.) On the monodromy of moduli spaces of sheaves on \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors describe methods for the computation of the quasiadjunction polytopes for plane curves singularities. These polytopes give refinements of the zero sets of multivariable Alexander polynomials.
As a consequence, some hyperplanes are determined on which all the polynomials in the multivariable Bernstein ideal vanish. plane curve singularities; Alexander invariants; local systems; mixed Hodge structure [12] Pierrette Cassou-Noguès &aAnatoly Libgober, &Multivariable Hodge theoretical invariants of germs of plane curves&#xJ. Knot Theory Ramifications20 (2011) no. 6, p. 787Article | &MR 28 | &Zbl 1226. Mixed Hodge theory of singular varieties (complex-analytic aspects), Singularities in algebraic geometry Multivariable Hodge theoretical invariants of germs 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 \(Q_0\) and \(Q_1\) be, respectively, the set of non negative integers not bigger than \(n\) and the set of the first \(n\) positive integers. Identify the pair \(Q=(Q_0,Q_1)\) with an oriented graph, where \(Q_0\) is the set of vertices and \(Q_1\) is the set of arrows. Let \(t:Q_1\rightarrow Q_0\) and \(h:Q_1\rightarrow Q_0\) be, respectively, the tail and the head functions on the arrows. Assume that, \(\{t(a),h(a)\}=\{a-1,a\}\), so that one may have leftwards arrows or rightward arrows, depending on \(\delta(a)=h(a)-t(a)\) being negative or positive. Any such a graph \(Q\) is said to be a quiver of type \(A\) (a chain of vertices with arrows between them). To each such a quiver one can associate a suitable set of quiver representations: They form an affine space (direct sum of certain spaces of homomorphisms of \(\mathbb{C}\)-vector spaces) which is naturally acted on by a group \(G\) which is a product of linear groups. As the authors remark, the \(G\)-orbits of such representations are classified by the lace diagrams of \textit{S. Abeasis} and \textit{A. Del Fra} [J. Algebra 93, 376--412 (1985; Zbl 0598.16030)].
The main result of this paper is the proof (Section 2) of a very explicit formula for the \(G\)-equivariant cohomology class of the closure of any orbit of such an action. Nicely, the authors also show how their formula can be interpreted in terms of classes of degeneracy loci defined by a quiver of vector bundles. The proof of the main result is completed in Section 3, while section 4 is devoted to conjectural speculations, regarding the \(T\)-equivariant Grothendieck class of orbits closure with respect to the action referred to above. In spite of being addressed to people already knowing a bit about the subject, the paper is, on the other hand, written in an especially friendly way and, therefore, it may also serve as a stimulus to walk some initial steps into the beautiful theory of quivers and their representations. quiver orbits; equivariant cohomology Buch, Anders Skovsted; Rimányi, Richárd, A formula for non-equioriented quiver orbits of type \(A\), J. Algebraic Geom., 16, 3, 531-546, (2007) Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Singularities of differentiable mappings in differential topology, Grassmannians, Schubert varieties, flag manifolds A formula for non-equioriented quiver orbits of type \(A\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The earlier work of the first and the third name authors [J. Am. Math. Soc. 31, No. 3, 661--697 (2018; Zbl 1387.05265)] introduced the algebra \(\mathbb{A}_{q,t}\) and its polynomial representation. In this paper we construct an action of this algebra on the equivariant \(K\)-theory of certain smooth strata in the flag Hilbert scheme of points on the plane. In this presentation, the fixed points of the torus action correspond to generalized Macdonald polynomials, and the matrix elements of the operators have an explicit presentation. Graded rings and modules (associative rings and algebras), Combinatorial aspects of groups and algebras, Parametrization (Chow and Hilbert schemes), Derived categories and associative algebras, Hermitian \(K\)-theory, relations with \(K\)-theory of rings, Orthogonal polynomials and functions associated with root systems, Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) The \(\mathbb{A}_{q,t}\) algebra and parabolic flag Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((A,\mathfrak{m})\) be a normal two-dimensional local ring and \(I\) an \(\mathfrak{m}\)-primary integrally closed ideal with a minimal reduction \(Q\). Then we calculate the numbers: \(\mathrm{nr}I)=\min\{n\mid\overline{I^{n+1}}=Q\overline{I^n}\}\), \(\bar{r}(I)=\min\{n\mid\overline{I^{N+1}}=Q\overline{I^N},\forall N\geq n\},\mathrm{nr}(A)\), and \(\bar{r}(A)\), where \(\mathrm{nr}(A)\) (resp. \(\bar{r}(A))\) is the maximum of \(\mathrm{nr}(I)\) (resp. \(\bar{r}(I))\) for all \(\mathfrak{m}\)-primary integrally closed ideals \(I\subset A\). Then we have that \(\bar{r}(A)\leq p_g(A)+1\), where \(p_g(A)\) is the geometric genus of \(A\). In this paper, we give an upper bound of \(\bar{r}(A)\) when \(A\) is a cone-like singularity (which has a minimal resolution whose exceptional set is a single smooth curve) and show, in particular, if \(A\) is a hypersurface singularity defined by a homogeneous polynomial of degree \(d\), then \(\bar{r}(A)= \mathrm{nr}(\mathfrak{m})=d-1\). Also we give an example of \(A\) and \(I\) so that \(\mathrm{nr}(I)=1\) but \(\bar{r}(I)=\bar{r}(A)=p_g(A) +1=g+1\) for every integer \(g\geq 2\). normal reduction number; two-dimensional singularity; homogeneous hypersurface singularity Integral closure of commutative rings and ideals, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties The normal reduction number of two-dimensional cone-like 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 surface ruled by conics \(S\subset \mathbb{P}^n\) is substantially the data of a map (ruling) \(\sigma:S\to \mathbb{P}^1\) such that each fibre of the ruling is a conic in \(\mathbb{P}^n\) and the general fibre is smooth. These notions will be made precise in section 1. The first natural question is to describe such surfaces, their possible singularities, and their special fibres.
Theorem. Let \(S\subset \mathbb{P}^n\) \((n\geq 6)\) be a rational projective surface ruled by conics. If \(S\) is linearly normal and the scroll generated by the fibres of \(S\) is not a cone, then the general fibre of \(S\) is a smooth conic and the special fibres of \(S\) are:
\(\delta\) simply degenerate conics and \(S\) is smooth at every point of these fibres;
\(s\) simply degenerate conics and \(S\) is singular at the singular point of each conic;
\(d\) doubly degenerate conics and \(S\) is singular at exactly two points of each conic.
Moreorer, \(S\) is smooth away from the special fibres and its singularities are ordinary double points.
It is well known that \(S\) can be obtained from a (geometrically) ruled surface by a finite number of blowing-up's and contractions, or blowing-down's Nevertheless, an explicit method allowing one to obtain a birational ruled model of \(S\) seems to be unknown up to now. Therefore, we were looking for a somewhat ``canonical'' method, conceived to ``arrange'' the sequence of blowing-up's and contraction, in order to obtain a (hopefully unique) minimal model, i.e., a geometrically ruled surface not containing exceptional curves of the first kind. It is clear that to obtain such model it is necessary to blow up the singular points of \(S\) and then to contract the right number of exceptional curves. We will say that a geometrically ruled surface \(T\) is a primitive model of \(S\) if it is obtained in this way and has minimum invariant \(t\), i.e., \(T\cong R_{1,t+1}\). We introduce another invariant say \(r\), where \(-r\) is the minimum self-intersection of a directrix of \(S\). We prove:
Theorem. If \(S\) is smooth, then the surface \(T\) is unique if and only if \(r\geq\delta\); in this case \(t=r-\delta \).
The construction used to obtain \(T\) in the case \(r\geq\delta\) can be performed also in the range \(r<\delta\). In this situation, there arises naturally the problem of the uniqueness of the directrix of minimum self-intersection.
Theorem. If \(S\) is smooth and \(r<\delta<2r\), then there exists a unique directrix of self-intersection \(-r\).
Finally in some particular cases we characterize the very ample divisors on projective surfaces ruled by conics. surface ruled by conics; singularities; special fibres; primitive model DOI: 10.1081/AGB-120022436 Rational and ruled surfaces, Projective techniques in algebraic geometry, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties On rational surfaces ruled by conics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 the first half of the author's work devoted to the systematic treatment of canonical spin polynomial invariants of algebraic surfaces. The notion of a spin polynomial of a smooth simple connected compact complex algebraic surface \(S\) was introduced in the author's recent works [see, e.g., Russ. Acad. Sci., Izv., Math. 42, No. 2, 333-369 (1994); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 2, 125-164 (1993; Zbl 0823.14031)] in the differential-geometric setting. In the present paper the spin polynomials are defined in the algebro-geometric setting. The paper consists of two chapters.
In chapter 1 a treatment of Jacobians \(J_k^i(S)\) and theta-loci \(\Theta_k^i(S)\) is given. Here \(i\) indexes the spin chambers of the Kähler cone of \(S\), \(J_k^i(S)\) is the Gieseker-Maruyama moduli space \(M^i(2, c_1=K_S, c_2=c_2(S)+k)\) with respect to the almost-canonical polarization lying in the \(i\)-th chamber, and \(\Theta_k^i(S)= \{[F]\in J_k^i(s)\mid h^0(F)\geq 1\}\). Then the spin polynomials \(s\gamma(k,n,i)\in S^dH^2(S,\mathbb{Z})\), \(d=3c_2(S)- K_S^2+3k-p_a(S)- 2n+1\), \(n\geq 0\), are defined by the cohomological correspondence using the discriminant of the universal quasifamily of sheaves on \(S\times \Theta_k^i(S)\). -- In chapter 2 the differential-geometric construction of spin polynomials is discussed and its coincidence with the algebro-geometric one is proved; the spin polynomials \(s\gamma(k,n,i)\) are then interpreted as ``fundamental'' algebraic classes of intermediate dimension \(2d\) in the cohomology ring of the Hilbert scheme \(\text{Hilb}^dS\). Jacobian; canonical spin polynomial invariants; theta-loci; Gieseker-Maruyama moduli space; almost-canonical polarization; Hilbert scheme Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Differentiable structures in differential topology, Moduli, classification: analytic theory; relations with modular forms, Parametrization (Chow and Hilbert schemes), Theta functions and abelian varieties Canonical spin polynomials of an algebraic surface. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors calculate the Betti numbers of the Hilbert scheme of points in the plane. Observe that the maximal torus of SL(3) acts on \(Hilb^ d({\mathbb{P}}^ 2)\) with isolated fixed points. It follows from a result of Birula-Białynicki that \(Hilb^ d({\mathbb{P}}^ 2)\) has a cellular decomposition. Then the calculation of the Betti numbers reduces to a careful study of the representation of the torus at the tangent spaces of the fixed points. As a by-product to their method, the authors also obtain similar results about the punctual Hilbert scheme and the Hilbert scheme of points in the affine plane. Betti numbers of the Hilbert scheme of points in the plane; punctual Hilbert scheme Ellingsrud, Geir; Strømme, Stein Arild, On the homology of the Hilbert scheme of points in the plane, Invent. Math., 87, 343-352, (1987) Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry On the homology of the Hilbert scheme of points in the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper treats a generalization of the Briançon-Skoda theorem about integral closures of ideals for graded systems of ideals satisfying a certain geometric condition. This work divides in two sections, the first contains some definitions and basic properties of stable graded systems of ideals. The main result of section 2 is the following:
Let \(a_{\bullet}=\{a_n\}_{n\in \mathbb N}\) be a stable graded system of ideals. Then there exists an integer \(C\) so that for all \(n\gg 0\), \(\mathcal{J}(Cn.{a}_{\bullet})\subseteq {a}_{n},\) where \(\mathcal{J}(Cn.a_{\bullet})\) denotes the level \(n\) asymptotic multiplier ideal attached to graded system of ideal \(a_{\bullet}\).
As asymptotic multiplier ideals are integrally closed, the fact that \(\overline{a}_m \subseteq \mathcal{J}(m.a_{\bullet})\) then implies the following generalization of the aforementioned result of Briançon and Skoda.
Let \(a_{\bullet}=\{a_n\}_{n\in \mathbb N}\) be a stable graded system of ideals. Then there exists a positive integer \(C\), such that for all \(n\gg 0\), \({\overline{a}}_{Cn}\subseteq a_n\). asymptotic order of vanishing; ideal sheaf; level-\(n\) asympotic multiplier ideal; graded system of ideals; smooth complex variety; symbolic power of an ideal Ideals and multiplicative ideal theory in commutative rings, Integral closure of commutative rings and ideals, Structure, classification theorems for modules and ideals in commutative rings, Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves, Vanishing theorems in algebraic geometry A Briançon-Skoda type theorem for graded systems of ideals | 0 |
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