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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type \(\mathbb{A}\), we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type \(\mathbb{A}\) quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation. Syzygies, resolutions, complexes and commutative rings, Determinantal varieties, Representations of quivers and partially ordered sets, Singularities in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Grassmannians, Schubert varieties, flag manifolds Free resolutions of orbit closures of Dynkin quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper gives a detailed analysis of the resolution of an isolated double point of a surface, i.e. an isolated singularity of multiplicity two of an algebraic surface. These double points are locally given by an equation \( z^2=f(x,y),\) where \(f\) is a square-free polynomial such that the curve \(f=0\) has a singular point at \((0,0)\). The paper explains and complements the papers by \textit{D. J. Dixon} [Pac. J. Math. 80, 105--117 (1979; Zbl 0422.14003)] and \textit{H. B. Laufer} [Isr. J. Math. 31, 315--334 (1978; Zbl 0415.14003)] on the same subject.
The results of the paper are: 1. relations between the canonical and the minimal resolution of a double point, 2. relations and formulas for cycles connected to the resolution process: fundamental cycle and fiber cycle, 3. computations of conditions that a double point imposes to canonical and pluricanonical systems of a surface. surface singularity; double point; resolution of singularity; fundamental cycle; fiber cycle A. Calabri - R. Ferraro, Explicit resolutions of double point singularities of surfaces, Collect. Math. 53 (2002), 99--131. Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Explicit resolutions of double point singularities of surfaces. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the minimal model program, the main classes of singularities of pairs \((X,B)\) are: terminal, canonical, log terminal, and log canonical. The first three classes are rational and hence Cohen-Macaulay (CM). However, log canonical singularities are neither, in general. (Note however that log canonical singularities are Du Bois [\textit{J. Kollár} and \textit{S. J. Kovács}, J. Am. Math. Soc. 23, No. 3, 791--813 (2010; Zbl 1202.14003)]). In the paper under review, the authors study rationality and depth conditions of non-klt centers of a log canonical pair \((X,B)\). Similar results were obtained by S. J. Kovács using different methods in [\textit{S. J. Kovács}, Pure Appl. Math. Q. 7, No. 4, 1495--1515 (2011; Zbl 1316.14031)].
Let \(X\) be an algebraic variety over an algebraically closed field of characteristic zero and let \(f: Y\rightarrow X\) be a resolution. The sheaves \(R^if_*\mathcal{O}_X\), \(i\geq0\), are coherent on \(X\) and do not depend on the choice of the resolution. A normal variety \(X\) has rationality singularities if \(R^if_*\mathcal{O}_X=0\) for all \(i>0\). In the paper under discussion, the authors study the non-rational centers, which are the subvarieties defined by the associated primes of \(R^if_*\mathcal{O}_X\), \(i>0\). (See Definition 1.1.) The first main result of the paper, Theorem 1.2, asserts that: For a log canonical pair \((X,B)\), every non-rational center of \(X\) is a non-klt center of \((X,B)\). Note that the closed set of non-rational singularities is a subset of the closed set of non-klt singularities, but the theorem is not obvious.
For the depth condition, instead of Serre's condition \(S_n\), the authors study the condition \(C_n\), where \(C\) stands for a ``closed point.'' A coherent sheaf \(\mathcal{F}\) on \(X\) satisfies condition \(C_n\) if for every closed point \(x\in\) \text{Supp}\((\mathcal{F})\), one has \text{depth}\( (\mathcal{F}_x)\geq n\). Note that if \(n\leq\dim X\), then \(S_n\) is stronger than \(C_n\). However, there is an example where \(X\) is \(C_n\) but not \(S_n\) (Example 2.2). The condition \(C_n\) is convenient to work with for projective varieties due to the following simple criterion: For \(\mathcal{F}\) a coherent sheaf on a projective scheme \(X\) with an ample invertible sheaf \(L\), \(\mathcal{F}\) is \(C_n\) if and only if \(\mathrm{H}^i(X,\mathcal{F}(-sL))=0\) for any \(i<\dim X\) and \(s\gg0\) (Lemma 2.3, or \textit{R. Hartshorne} [Algebraic geometry. Graduate Texts in Mathematics. 52. New York-Heidelberg-Berlin: Springer-Verlag. XVI. (1977; Zbl 0367.14001) III, 7.6]). The second main result of the paper under review, Theorem 1.5, asserts: For \(X\) a normal variety of \(\dim X\geq d+2\). Assume that the pair \((X,B)\) is log canonical and that every non-klt centers of \((X,B)\) has dimension \(\geq d\). Then for each \(i>0\), the sheaf \(R^if_*\mathcal{O}_X\) is \(C_{d+1-i}\) and \(X\) is \(C_{d+2}\).
The proof uses a special resolution \(f:Y\rightarrow X\) that factors through a ``dlt-blowup'' \(f':Y'\rightarrow X\) of \((X,B)\), which is constructed by running a minimal model program over \(X\). See Theorem 3.1 and Proposition 3.7. To get the first result, a vanishing theorem for global embedded simple normal crossing pairs of \textit{F. Ambro} [Proc. Steklov Inst. Math. 240, 214--233 (2003); translation from Tr. Mat. Inst. Im. V. A. Steklova 240, 220--239 (2003; Zbl 1081.14021)] and \textit{O. Fujino} [``Introduction to the log minimal model program for log canonical pairs'', \url{arXiv:0907.1506}] is applied. For the second result, the method of \textit{J. Kollár} [Ann. Math. (2) 124, 171--202 (1986; Zbl 0605.14014)] is applied to the extent of variation of mixed Hodge structures in [\textit{Kawamata, Y.}, ``On algebraic fiber spaces'', \url{arXiv:math/0107160}, ``Semipositivity theorem for reducible algebraic fiber spaces'', \url{arXiv:0911.1670}, ``Hodge theory on generalized normal crossing varieties'', \url{arXiv:1104.0524}]. singularities; non-klt centers; log canonical centers V. Alexeev and C. D. Hacon, ''Non-rational centers of log canonical singularities,'' J. Algebra, vol. 369, pp. 1-15, 2012. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays) Non-rational centers of log canonical singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders [the authors, Invent. Math. 161, No. 2, 427--452 (2005; Zbl 1078.14005)]. These were classified independently by \textit{M. Artin} [Manuscr. Math. 58, 445--471 (1987; Zbl 0625.16005)] (in terms of ramification data) and \textit{I. Reiten} and \textit{M. Van den Bergh} [Two-dimensional tame and maximal orders of finite representation type. Providence, RI: American Mathematical Society (AMS) (1989; Zbl 0677.16002)] (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups \(G \subset \operatorname{GL}_2\), explicitly computing \(H^2 (G, k^*)\), and then matching these up with Artin's list of ramification data and Reiten-Van den Bergh's AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in the first author et al. [Proc. Lond. Math. Soc. (3) 98, No. 1, 83--115 (2009; Zbl 1226.14006)] to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let \(B = k_{\zeta} [[x, y]]\) be the skew power series ring where \(\zeta\) is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form \(A = B/(f)\) where \(f \in Z(B)\) which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers. maximal Cohen-Macaulay modules; matrix factorizations Noncommutative algebraic geometry, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), McKay correspondence, Representations of quivers and partially ordered sets, Cohen-Macaulay modules, Group rings of finite groups and their modules (group-theoretic aspects), Singularities in algebraic geometry, Ordinary representations and characters Low dimensional orders of finite representation type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities M. Auslander initiated the use of noncommutative algebras of finite dimension by studying the representation theory of Cohen-Macaulay rings, that is, more generally \textit{orders}. Two classes of noncommutative algebras were singled out, the Auslander algebras and the non-singular orders. The representation theory of finite-dimensional representations is encoded in the structure of their Auslander algebras. All Cohen-Macaulay modules are projective, so that the study of the non-singular orders are more basic.
The study of such algebras gives applications in algebraic geometry. A first example is Van den Bergh's definition of \textit{Noncommutative Crepant Resolution}, NCCR: Let \(R\) be a commutative noetherian normal domain. Then a reflexive \(R\)-module \(M\) is said to give a NCCR of \(\text{Spec}R\) if \(\Lambda=\text{End}_R(M)\) is a nonsingular \(R\)-order, which means that \(\Lambda_{\mathfrak p}=\text{End}_{R_{\mathfrak p}}(M_{\mathfrak p},M_{\mathfrak p})\) is a maximal Cohen-Macaulay \(R_{\mathfrak p}\)-module for each \(\mathfrak p\in\text{Spec}R\).
This article considers the slightly simpler concept of NCR, that is \textit{Noncommutative resolution}: A finitely generated module \(M\) over a commutative noetherian ring \(R\) is called a NCR of \(\text{Spec}(R)\) if \(M\) is faithful and \(\text{End}_R(M)\) has finite global dimension. NCRs exist when \(R\) is artinian, or reduced and one-dimensional. This article treats the question on which rings \(R\) that have an NCR. The conditions are given in the terms of the Grothendieck group of the category of finitely generated \(R\)-modules, and its subcategories. The existence of an NCR forces strong constraints on the singularities of \(R\). The formulations of the conditions on \(R\) in terms of the Grothendieck group leads to influence and use of results from algebraic K-theory. This leads to one of the main results stating that for surface singularities over an algebraically closed field, the existence of a NCR characterise rational singularities. Thus the rationality of a surface singularity can be tested on the existence of a NCR.
The main results of the article, more or less verbatim, is as follows:
{Theorem 2.5.} Let \(R\) be a semilocal ring and assume that \(M\) gives a NCR of \(R\). Let \(\mathcal C_M\) be the full subcategory of mod\(R\) consisting of \(X\) satisfying \(\text{supp}X\subset\text{NG}(M).\) Then \(K_0(R)/\langle\mathcal C_M\rangle\) is a finitely generated abelian group. (\(K_0(R)\) denotes the Grothendieck group of \(R\), \(\text{NG}(M)\) is the nongenerating locus of \(M\).)
{Theorem 3.11.} Let \(R\) be a normal, Cohen-Macaulay standard graded algebra over a subfield \(k\) of \(\mathbb C\). Let \(\mathfrak m\) be the irrelevant ideal of \(R\). Suppose that \(\text{Spec}R\setminus\{\mathfrak m\}\) has only rational singularities. Suppose moreover that there exists an \(R\)-module \(M\) giving a NCR. Then \(\text{Spec}R\) has only rational singularities.
A few relevant, explicit examples are given, and all in all this article gives nice results from the noncommutative algebraic geometry to the commutative. Also, a really nice historical survey is given in the introduction, and a good list of references ends the article. non-commutative resolution; NCR; non-commutative crepant resolution; NCCR; non-generating locus; semilocal ring Dao, H., Iyama, O., Takahashi, R., Vial, C.: Non-commutative resolutions and Grothendieck groups. J. Noncommut. Geom. \textbf{9}(1), 21-34 (2015) Grothendieck groups, \(K\)-theory and commutative rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Representations of orders, lattices, algebras over commutative rings, Homological dimension (category-theoretic aspects) Non-commutative resolutions and Grothendieck groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove a localization theorem for the type \( \mathbf{A}_{n-1}\) rational Cherednik algebra \( \text{H}_c=\text{H}_{1,c}(\mathbf{A}_{n-1})\) over \( \overline{\mathbb{F}}_p\), an algebraic closure of the finite field. In the most interesting special case where \( c\in \mathbb{F}_p\), we construct an Azumaya algebra \( \text{H}_c\) on \( \text{Hilb}^n{\mathbb{A}}^2\), the Hilbert scheme of \( n\) points in the plane, such that \( \Gamma(\text{Hilb}^n{\mathbb{A}}^2, \,\text{H}_c)=\text{H}_c\). Our localization theorem provides an equivalence between the bounded derived categories of \( \text{H}_c\)-modules and sheaves of coherent \( \text{H}_c\)-modules on \( \text{Hilb}^n{\mathbb{A}}^2\), respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] and \textit{M. Haiman} [Invent. Math. 149, 371--407 (2002; Zbl 1053.14005)]. Bezrukavnikov, R.; Finkelberg, M.; Ginzburg, V., Cherednik algebras and Hilbert schemes in characteristic \textit{p}, Represent. Theory, 10, 254-298, (2006), with an appendix by Pavel Etingof Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Associative rings and algebras arising under various constructions, Derived categories, triangulated categories Cherednik algebras and Hilbert schemes in characteristic \(p\). With an appendix by Pavel Etingof | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 Nash problem asks for the bijectivity of the Nash map that is an injective map from the set of irreducible components of the space of arcs passing through the singular locus to the set of the essential divisors. This problem was negatively solved by the reviewer and \textit{J. Kollár} in genera [Duke Math. J. 120, No. 3, 601--620 (2003; Zbl 1052.14011)]; i.e., the Nash map is not surjective in general. Therefore the next problem is to determine the image of the Nash map. In this paper, the authors prove that a non-uniruled exceptional divisor belongs to the image of the Nash map. arcs; wedges; resolution of singularities; Nash map; essential divisors; uniruled variety M. Lejeune-Jalabert and A. J. Reguera-López, Exceptional divisors which are not uniruled belong to the image of the Nash map, 2008. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Modifications; resolution of singularities (complex-analytic aspects) Exceptional divisors that are not uniruled belong to the image of the Nash map | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A closed subscheme \(C\) of a regular algebraic variety \(X\) (or its defining sheaf of ideals \(\mathcal I\)) is tame if \(X'\), the blowing-up of \(X\) with centre \(C\) (or \({\mathcal I}\)), is again regular. If \(C\) is regular then it is tame, but this is not always true if \(C\) is not regular (in particular, when \(C\) is non-reduced). The authors address the following question: when is \(C\) (or its ideal \({\mathcal I}\)) tame? They work over a base field \(K\) of characteristic zero, with \(X={\text{Spec}}(K[x_1, \dots, x_n])={\mathbf A}^n\) (affine \(n\)-space) and \(C\) defined by an ideal generated by monomials \(x_1 ^{a_1}, \dots, {x_n}^{a_n}\), \(a_i \geq 0\) an integer, for all \(i\). When the centre is a monomial ideal of this form, the blow-up variety \(X'\) can be covered by open affine sets isomorphic to toric varieties. This fact allows them to use special techniques, e.g., combinatorial ones.
The main general result of the paper is a theorem giving (among other things) a criterion for smoothness of the blow-up of \({\mathbf A}^n\) with centre a monomial ideal as above, in terms of ideal tangent cones. The ideal tangent cone to \(\mathcal I\) at \({\mathbf a}=(a_1,\dots,a_n)\), where \(x_1^{a_1} \cdots x_n^{a_n} \in {\mathcal I}\) is a certain subset of \({\mathbb Z}^n\) that the authors had introduced earlier in the paper.
This theorem is applied to obtain examples and tameness results involving specific types of monomial ideals. These include:
(a) The case of products of coordinate ideals (i.e., defined by products \(x_{i_1} \cdots x_{i_q}\) of the variables, without repeated factors).
(b) Building sets. These are certain finite collections of linear subspaces of \({\mathbf A}^n\), related to work on arrangements by De Concini, Procesi, Fulton, Mac Pherson, etc. The authors prove that the product of coordinate ideals for which the corresponding linear subspaces form a building set is tame.
(c) The so-called permutohedral ideals are tame. These are monomial ideals which are not related to building sets. smoothness; blowing-up; toric variety; arrangements; permutohedra Faber, E.; Westra, D. B., Blowups in tame monomial ideals, J. Pure Appl. Algebra, 215, 8, 1805-1821, (2011) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Rational and birational maps, Configurations and arrangements of linear subspaces Blowups in tame monomial ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a singular algebraic variety defined over a perfect field \(k\), with quotient field \(K(X)\). Let \(s\geq 2\) be the highest multiplicity of \(X\) and let \(F_s(X)\) be the set of points of multiplicity \(s\). If \(Y\subset F_s(X)\) is a regular center and \(X\leftarrow X_1\) is the blow up at \(Y\), then the highest multiplicity of \(X_1\) is less than or equal to \(s\). A sequence of blow ups at regular centers \(Y_i\subset F_s (X_i)\), say \(X\leftarrow X_1\leftarrow\cdots\leftarrow X_n\), is said to be a \textit{simplification} of the multiplicity if the maximum multiplicity of \(X_n\) is strictly lower than that of \(X\), that is, if \(F_s(X_n)\) is empty. In characteristic zero there is an algorithm which assigns to each \(X\) a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive. In this paper we will study finite dominant morphisms between singular varieties \(\beta : X^\prime \to X\) of generic rank \(r\geq 1\) (i.e., \([K( X^\prime):K(X)]=r)\). We will see that, when imposing suitable conditions on \(\beta\), there is a strong link between the strata of maximum multiplicity of \(X\) and \(X^\prime\), say \(F_s(X)\) and \(F_{rs}(X^\prime)\) respectively. In such case, we will say that the morphism is strongly transversal. When \(\beta :X^\prime\to X\) is strongly transversal one can obtain information about the simplification of the multiplicity of \(X\) from that of \(X^\prime\) and vice versa. Finally, we will see that given a singular variety \(X\) and a finite field extension \(L\) of \(K(X)\) of rank \(r\geq 1\), one can construct (at least locally, in étale topology) a strongly transversal morphism \(\beta :X^\prime\to X\), where \(X^\prime\) has quotient field \(L\). finite morphisms; multiplicity; Rees algebras; singularities Integral closure of commutative rings and ideals, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Finite morphisms and simultaneous reduction of the 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 From the authors' introduction: ``This article is part of the authors' program whose purpose is to prove the following conjecture on Resolution of Singularities of threefolds in mixed characteristic. The conjecture is a special case of Grothendieck's Resolution conjecture for quasi-excellent schemes.
Conjecture 1.1 Let \(C\) be an integral regular excellent curve with function field \(F\). Let \(S/F\) be a reduced algebraic projective surface and \(\mathcal X\) be a flat projective \(C\)-scheme with generic fiber \({\mathcal X}_F =S\). There exists a birational projective \(C\)-morphism \(\pi :{\mathcal Y} \to \mathcal X\) such that {\parindent=6mm \begin{itemize}\item[(i)] \({\mathcal Y}\) is everywhere regular. \item[(ii)] \(\pi^{-1}(\text{Reg} {\mathcal X}) \to \text{Reg} {\mathcal X} \) is an isomorphism.''
\end{itemize}} In a previous paper [J. Algebra 320, No. 3, 1051--1082 (2008; Zbl 1159.14009)], the authors have developed equicharacteristic techniques which extend to that situation. Using classical invariants introduced by Hironaka, they present a proof of the following:
Main Theorem 1.3. Let \((R, {\mathcal M}, k=k(x):= R/{\mathcal M} )\) be an excellent regular local ring of dimension four, \((Z,x):= (\text{Spec} R,{\mathcal M})\) and \((X,x):= (\text{Spec} R/(h),x)\) be a reduced hypersurface. Assume that the multiplicity \(m(x)\) of \((X,x)\) satisfies \(m(x) < p:=\text{char} k(x)\). Let \(v\) be a valuation of \(K(X)\) centered at \(x\). Then there exists a finite sequence of local blowing ups
\[
(X,x)=:(X_0,x_0) \leftarrow (X_1,x_1) \leftarrow \dots \leftarrow (X_n,x_n) ,
\]
where \(x_i\in X_i\), \(0\leq i \leq n\) is the center of \(v\), each blowing up center \(Y_i\subset X_i\) is permissible at \(x_i\) (in Hironaka's sense), such that \(x_n\) is regular.
The authors point out that the methods applied are global in nature and thus an extension of the main theorem to a global version should be possible. arithmetic varieties; Hironaka; resolution of singularities; blowing up; local uniformization V. Cossart, O. Piltant, Resolution of Singularities of Arithmetical Threefolds II. ArXiv e-prints, Dec. 2014. Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Resolution of singularities of threefolds in mixed characteristic: case of small 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 resolution of singularities of algebraic varieties defined over a field of characteristic zero by a sequence of blow-ups is a famous result proved by \textit{H. Hironaka} [Ann. Math. (2) 79, 109-203, 205-326 (1964; Zbl 0122.38603)], which has a lot of applications. Probably, relatively few mathematicians read the 200 page proof. This can change now. The article is arranged in a similar way to a talk in a colloquium (25\% should be understood by everyone, 25\% is for people who are interested, the next 25\% is for the specialists and the rest only the speaker will understand) with one difference: that also the rest is understandable. It starts with an overview explaining the result and giving rough ideas of the proof. The author suggests that very busy people should only read this.
The next chapter gives an introduction to the main problems (choice of the centre of the blow-up, equiconstant points, improvement of singularities under blow-up including many examples). This is for the next 25.
The next chapter (constructions and proofs) gives the technical details. It contains also several examples and a section on problems in positive characteristic. In an appendix, necessary basic facts from commutative algebra and the theory of blow-ups used in the previous chapters are collected.
It is advisable for everybody interested in resolution of singularities to read this article. Hironaka resolution of singularities; blowing up H.HAUSER,\textit{The Hironaka theorem on resolution of singularities (or: A proof we always wanted to} \textit{understand)}, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 323--403.http://dx.doi.org/ 10.1090/S0273-0979-03-00982-0.MR1978567 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) The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 of the edition published by Springer (1987) in Zbl 0686.14008. Cano, F.: Desingularization strategies for a three-dimensional vector field. Lect. notes in math. 1259 (1987) Global theory and resolution of singularities (algebro-geometric aspects), Ordinary differential equations in the complex domain, Research exposition (monographs, survey articles) pertaining to algebraic geometry, 2-person games, Singularities in algebraic geometry, \(3\)-folds Desingularization strategies for three-dimensional vector fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper we characterize the Blowing-up maps of ordinary singularities for which there exists a natural Gysin morphism, i.e. a bivariant class \(\theta \in Hom_{D(Y)}(R\pi_*\mathbb {Q}_X, \mathbb {Q}_Y)\), compatible with pullback and with restriction to the complement of the singularity. bivariant theory; Gysin morphism; blowing-up; derived category; Borel-Moore homology; isolated singularities; projective contractions Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Topological properties in algebraic geometry On the existence of a Gysin morphism for the blow-up of an ordinary singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the second part of the authors, ibid., 109-160 (1997; Zbl 0879.58007), see the review above. It presents numerous results about mappings of cusp type or, in other words, mappings \(F\), which in a suitable nonlinear system of ``coordinates'' \(\mathbb{R}^2\times E\) can be represented in the form \(F(s,t,v) =(s^3-ts,t,v)\); some other singularities are also presented in this part. The account is also based on the abstract global characterization of the cusp maps which was obtained by the authors in 1993.
The contents of this part are: (13) Introduction; (14) Critical values of Fredholm mappings; (15) Applications of critical values to nonlinear differential equations; (16) Factorization of differentiable maps; (17) Local structure of cusps; (18) Some local cusp results (Lazzeri-Micheletti cusp study, Cafagna-Tarantello multiplicity results, Lupo-Micheletti cusp, other local cusp results); (19) von Kármán equation; (20) Abstract global characterization of the cusp map; (21) Mandhyan integral operator cusp map; (22) Pseudo-cusp; (23) Cafagna and Donati theorems on ordinary differential equations (Cafagna-Donati global cusp map, Donati pendulum cusp, Cafasgna-Donati generalized Riccati equation); (24) Micheletti cusp-like map; (25) Cafagna Dirichlet example; (26) \(u^3\) Dirichlet map -- initial results; (27) \(u^3\) Dirichlet map -- the singular set and its image; (28) \(u^3\) Dirichlet map -- the global results; (29) Ruf \(u^3\) Neumann cusp map; (30) Ruf's higher order singularities; (31) Damon's work in differential equations.
The second part of this survey is written with the same accuracy and fullness as the first one; the acquaintence with both parts of this survey is undoubtedly useful to all specialists in the field and all who study Nonlinear Analysis. survey; mappings of cusp type; singularities Church, P. T.; Timourian, J. G.: Global structure for nonlinear operators in differential and integral equations. I. folds; II. Cusps: topological nonlinear analysis. (1997) Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Equations involving nonlinear operators (general), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Global structure for nonlinear operators in differential and integral equations. II: Cusps | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If \((X,p)\) is an isolated Gorenstein threefold singularity admitting a small resolution \(\pi:Y\to X\), i.e. a resolution such that the exceptional set is the union of smooth rational curves, then we know that \((X,p)\) is locally given by the equation \(f(x,y,z)+tg(x,y,z,t)\) where \(f\) defines a rational double point \((S,p)\), and we say that \((X,p)\) is a \(cDV\).
In this article the author studies the converse question of which \(cDV\) singularities admit small resolutions, when \(f\) defines an \(A_ n\) or a \(D_ n\) surface singularity. --- To obtain his results, the author studies the versal deformation of the surfaces \(S\), \(\overline S=\pi^{- 1}(S)\) which is also a surface with only \(RDP\) singularities, and \(S\tilde S\) where \(\lambda:\tilde S\to\overline S\) is the minimal resolution of \(\overline S\). We denote respectively by \({\mathcal B}\), \(\overline{{\mathcal B}}\) and \(\tilde {\mathcal B}\) and \({\mathcal S}\), \(\overline{{\mathcal S}}\) and \(\tilde{\mathcal S}\) the versal deformation spaces and the versal families of \(S\), \(\overline S\) and \(\tilde S\), and we get:
\[
\begin{matrix} \widetilde{\mathcal S} & \longrightarrow & \overline{{\mathcal S}} & \longrightarrow & \strut{\mathcal S} \\ @VVV @VVV @VVV \\ \widetilde{\mathcal B} & \longrightarrow & \overline{{\mathcal B}} & \longrightarrow & \strut{\mathcal B}. \end{matrix}
\]
The \(cDV\) singularity \((X,p)\) is represented by a map \(\nu:D\to{\mathcal B}\), where \(D\) is a smooth curve and the small resolution is represented by lifting \(\nu\) to \(\overline\nu:D\to\overline{{\mathcal B}}\), such that the space \(Y\) given by \(\overline\nu\) is smooth. The author analyses the lifting criterion explicitly in the \(cA_ n\) and \(cD_ n\) cases in terms of the equation of \((X,p)\) and he obtains:
Theorem: If \((X,p)\) is an isolated \(cA_ n\) singularity admitting a small resolution \(Y\to X\), then the exceptional curve in \(Y\) is a chain of \(n\) rational curves, i.e. the surface \(\overline S\) is smooth, and \(X\) has the form \(xy+g(z,t)\), where \(g(z,t)\) has \(n+1\) distinct branches at the origin. --- Conversely, any \(X\) as above admits a small resolution.
Let \((X,p)\) be an isolated \(cD_ n\) singularity given by \(\nu:D\to{\mathcal B}\), where \({\mathcal B}\) is \(k^ n({\mathbf t})=k^ n(t_ 1,\dots,t_ n)\) and \({\mathcal S}\to{\mathcal B}\) is given by: \({\mathcal S}=\{(x,y,z,{\mathbf t})\mid f(x,y,z,{\mathbf t})=x^ 2+y^ 2z-t_ 1z^{n-2}-\cdots-t_{n-1}+2t_ ny=0\}\). If \(t\) is a local parameter on the smooth curve \(D\), we view the \(t_ i\) as analytic functions of \(t\), and we associate to \(f\) the function \(F(z,t)=z^ n+t_ 1z^{n-1}+\cdots+t_{n-1}z+t^ 2_ n\).
Theorem: The \(cD_ n\) singularity admits a small resolution \(Y\to X\) such that the surface \(\overline S\) is smooth if and only if \(F(z,t)\) factors into \(n\) distinct factors, each tangent to \(z=0\) with even multiplicity.
The author studies more precisely the case of a \(cD_ 4\) singularity, he analyses the function \(F(z,t)\) and finds all the singularities that the surface \(\overline S\) could admit. \(cDV\) singularity; Gorenstein threefold singularity; small resolution; versal deformation spaces Katz, S.: Small resolutions of Gorenstein threefold singularities. Algebraic geometry: Sundance 1988, pp. 61-70, Contemp. Math., vol. 116, Am. Math. Soc., Providence, RI (1991) Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Formal methods and deformations in algebraic geometry, Deformations of singularities, Singularities in algebraic geometry Small resolutions of Gorenstein threefold singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0504.00008.]
This is a continuation of a previous paper [Journées de géométrie algebrique, Angers/France 1979, 273-310 (1980; Zbl 0451.14014)] here quoted as C.3. These two papers together give new, very interesting, insight in the difficult problems regarding the classification of 3-folds of ''general type''. In the case of surfaces (of general type) it is well known that the canonical model X may have only very simple singularities, the minimal model S is non singular, \(K_ S\) is ''numerically effective and free'' \((=:nef)\) and the morphism \(f: S\to X\) is such that \(f^*\omega_ X=\omega_ S\) (f is ''crepant''). In this paper a definition of minimal model S for a 3-fold X is proposed (X of f. g. general type (C.3.) and \(\kappa_{num}\geq 0)\) which preserves the maximum of the previous properties for surfaces. S may have singularities of a specified simple type called ''quick'' and is obtained by blowing up the canonical models X. Before stating the main result we need some more definitions. \(f: Y\to X\) is a partial resolution if it is a proper birational morphism in which Y is always assumed normal. If f is a partial resolution an exceptional prime divisor of f is any prime divisor \(\Gamma\) \(\subset Y\) such that \(co\dim f(\Gamma)\geq 2.\) Let X be a variety of dimension 3 with canonical singularities (C.3.), \(P\in X\) is a terminal singularity if it has a resolution \(f: Y\to X\) such that (i) f has at least one exceptional prime divisor and (ii) if \(K_ Y=f^*K_ X+\Delta\) every exceptional prime divisor of f appears in \(\Delta\) with strictly positive coefficient.
The main theorem is the following: 1. Let \(P\in X\) a 3-fold point then P is terminal if and only if it is quick. - 2. Let X be a 3-fold with canonical singularities. Then there exists a partial resolution \(f: S\to X\) such that (a) f is crepant, and (b) S has quick singularities. Furthermore this f can be chosen as the composite of certain elementary steps (blow-ups) which are intrinsic to X and is then uniquely determined and projective. - The paper contains many other results of interest in themselves and many appealing conjectures and open problems. classification of 3-folds of general type; numerically effective canonical divisor; crepant resolution; canonical models; partial resolution; exceptional prime divisor; terminal singularity; quick singularities M. Reid, \textit{Minimal models of canonical} 3\textit{-folds}, in \textit{Algebraic varieties and analytic varieties (Tokyo, 1981)}, \textit{Adv. Stud. Pure Math.}\textbf{1} (1983) 131, North-Holland, Amsterdam, The Netherlands. \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry Minimal models of canonical 3-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 Let \(V\) be an irreducible affine algebraic variety over a field \(k\) of characteristic zero, and let \((f_0,\dots ,f_m)\) be a sequence of elements of the coordinate ring. There is probably no elementary condition on the \(f_i\) and their derivatives which determines whether the blow-up of \(V\) along \((f_0,\dots ,f_m)\) is non-singular. The result of the current paper is that there indeed is such an elementary condition, involving the first and second derivatives of the \(f_i,\) provided we admit certain singular blow-ups, all of which can be resolved by an additional Nash blow-up.
This paper is the promised sequel of an earlier paper [\textit{J. A. Moody}, J. Algebra 189, No. 1, 90-100 (1997; Zbl 0891.14004)], in which the same program was carried out for individual vector fields. Indeed, the current paper generalizes the result of the mentioned paper to algebraic foliations of arbitrary codimension, and the case of codimension zero foliations corresponds to the problem of resolving the singularities of \(V.\)
The results have a close connection with a question of Nash concerning resolutions. We now describe this briefly following a paper by \textit{J. Milnor} [Math. Intell. 17, No. 3, 11-17 (1995; Zbl 0846.01016)] where further references may be found. Let \(r=\dim(V).\) Suppose \(V=V_0 \subset W_0\) is an embedding in a non-singular variety over \(k.\) Then \(V_0\) lifts to a subvariety \(V_1\subset W_1=\text{Grass}_r(W_0)\) of the variety of \(r\)-planes in the tangent bundle of \(W_0\). The natural map \(\pi:V_1\to V_0\) is called the Nash blow-up of \(V_0\). It is the lowest blow-up where \(\pi^*(\Omega_{V_0/k})/\)torsion is locally free. Now we can repeat the process, giving a variety \(V_2\subset W_2=\text{Grass}_r(W_1)\) and so-on, and the question is whether eventually \(V_i\) is non-singular.
There is a particular explicit sequence of ideals \(R=J_0, J_1, J_2,\dots \subset R\) so that \(V_0= \text{Bl}_{J_0}V\), \(V_1=\text{Bl}_{J_1}V\), \(V_2=\text{Bl}_{J_2}V,\dots .\) with \(J_i|J_{i+1}\) for all \(i.\) Applying our earlier paper [\textit{J. A. Moody}, Ill. J. Math. 45, No. 1, 163-165 (2001; Zbl 0989.13001)], \(V_i\) is non-singular if and only if the ideal class of \(J_{i+1}\) divides some power of the ideal class of \(J_i.\)
The present paper brings things down to earth considerably: Such a divisibility of ideal classes implies that for this value of \(i\) and for some \(N\geq r+2\)
\[
J_i^{N-r-2}J_{i+1}^{r+3}=J_i^NJ_{i+2}.
\]
Yet note that this identity in turn implies \(J_{i+2}\) is a divisor of some power of \(J_{i+1}.\) Thus although \(V_i\) may fail to be non-singular, when the identity holds the next variety \(V_{i+1}\) must be non-singular. Thus the Nash question is equivalent to the assertion that the identity above holds for some sufficiently large \(i\) and \(N.\) Nash question; Jacobian condition; non-singular blow-up; Nash blow-up J.A. Moody, On resolving singularities, Maths Institute, Warwick University, 2001, preprint. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) On resolving 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 the article under review, the author introduces a notion of local volume for Cartier divisors.
Let \(X\) be a normal quasiprojective variety of dimension \(n\geq 2\) over \(\mathbb{C}\), and let \(x\) be a point on \(X\). Fix a projective birational morphism \(\pi: X'\rightarrow X\), and let \(D\) be a Cartier divisor on \(X'\). The local volume of \(D\) at \(x\) is defined as
\[
\mathrm{vol}_x (D) := \overline{\lim}_{m\rightarrow +\infty} \frac{n ! \cdot h_x ^1 (m D)}{m^n} ,
\]
where \(h_x ^1 (m D) := \dim H_{{x}} ^1 (X, \pi_* \mathcal{O}_{X'} (D))\).
The author proved several important properties of the local volume. As a main theorem, the author proved that \(\mathrm{vol}_x\) is a well-defined \(n\)-homogeneous, and locally Lipschitz continuous on \(N^1 (X' /X)_{\mathbb{R}}\). The author also studies the vanishing and convexity for \(\mathrm{vol}_x ^{\frac{1}{n}}\).
In the last part of the article, the author compare the local volume with another notion of volume defined in [\textit{S. Boucksom, T. De Fernex} and \textit{C. Favre}, Duke Math. J. 161, No. 8, 1455--1520 (2012; Zbl 1251.14026)]. Let \((\widetilde{X}, E)\rightarrow (X, x)\) be a log-resolution of a normal isolated singularity. We set \(\mathrm{vol} (X, x) := \mathrm{vol}_x (K_{\widetilde{X} +E})\) and set \(\mathrm{vol}_{BdFF} (X,x)\) be the local volume defined in [loc. cit.]. In general, we have \(\mathrm{vol} (X, x) \leq \mathrm{vol}_{BdFF} (X,x)\). The author proved that these two definitions coincide in the numerically Gorenstein case. local volumes; Hilbert-Samuel multiplicity; plurigenera; asymptotic invariants; Okounkov body M. Fulger, ''Local volumes on normal algebraic varieties,'' arXiv: 1105.2981 [math.AG]. Rational and birational maps, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local cohomology and algebraic geometry, Local complex singularities Local volumes of cartier divisors over normal algebraic varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In 1956, Abhyankar stated his local weak simultaneous resolution theorem for algebraic surfaces. By using this theorem, he was able to eliminate the use of Zariski's factorization theorem for proving the local uniformization theorem for algebraic surfaces (a step in Zariski's plan for proving resolution of singularities). The author thinks that generalizing the local weak simultaneous resolution theorem could be useful to show local uniformization for higher-dimensional varieties since Zariski's factorization theorem is not true for these varieties. In this sense, he gives a partial generalization which does not account for all possible rational ranks. The concrete result (section 3 of the paper) which is proved from a monomialization theorem for ordered semigroups (section 2) is the following:
Let \(K\) be an \(n\)-dimensional function field over an algebraically closed field \(k\). Assume that either the characteristic of \(k\) is zero or that it is positive and \(n\leq 3\). Let \(L/K\) be a finite algebraic extension. Let \(v\) be a valuation of \(L/k\) with \(k\)-dimension zero, rational rank \(r\), and valuation ring \(V\). Let \((S,N)\subset L\) be an \(n\)-dimensional local ring over \(k\) which is birationally dominated by \(V\). Let \(R=S\cap K\), \(M=N\cap K\), \(U=V\cap K\) and \(Q=MS\).
(1) If \(r=n\), then \(S\) can be replaced by an iterated monoidal transform along \(v\) so that \(Q\) is \(N\)-primary;
(2) If \(r<n\), then \(S\) can be replaced by an iterated monoidal transform along \(v\) so that \(\text{ht} Q\geq r+1\). In particular, if \(r=n-1\), then for such a transform \(Q\) is \(N\)-primary. simultaneous resolution; valuations; monomialization; local uniformization; function field DOI: 10.1006/jabr.1996.7014 Valuations and their generalizations for commutative rings, Arithmetic theory of algebraic function fields, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Algebraic functions and function fields in algebraic geometry, Regular local rings Local weak simultaneous resolution for high rational ranks | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(n\) be a positive integer. The Hilbert scheme of \(n\) points in the complex plane has a natural \((\mathbb C^*)\)-action induced by the action of the torus \(T:=(\mathbb C^*)^2\) on \(\mathbb C^2\). Let now \(T:=\left\{(t^a,t^b)\in T\,\,|\,\, t\in \mathbb C^* \right\}\), where \(\gcd(a,b)=1\) and \(a,b\geq 1\) be a one dimensional subtorus of \(T\). The set of fixed points under the action of such a subtorus has the structure of a smooth variety, denoted by \((\mathbb C^2)_{a,b}^{[n]}\).\newline In the paper under review, the author generalizes \textit{A. Iarrobino} and \textit{J. Yaméogo} [Commun. Algebra 31, No. 8, 3863--3916 (2003; Zbl 1048.14003)] by providing a formula for the class of the irreducible components of \((\mathbb C^2)_{1,k}^{[n]}\) in terms of polynomials in \(\mathbb L\), the class of \(\mathbb A_{\mathbb C}^1\) in the Grothendieck ring of complex quasi-projective varieties. Based on computer calculations, the author also makes a conjecture on a possible formula for the Grothendieck ring classes of the more general varieties \((\mathbb C^2)_{a,b}^{[n]}\).\newline Another result in the paper is an interesting relation between the classes of certain open strata of \((\mathbb C^2)^{[n]}\) and the \((q,t)\)-Catalan numbers.\newline Finally, using a well known quiver description of \((\mathbb C^2)^{[n]}\), the author provides sufficient conditions under which a \((1,k)\)-quasi-homogeneous Hilbert scheme of points is isomorphic to a homogeneous nested Hilbert scheme of points. The latter result generalizes \textit{J. Cheah} [Pac. J. Math. 183, No. 1, 39--90 (1998; Zbl 0904.14001)]. Hilbert schemes of points; torus action; \((q,t)\)-Catalan numbers; quiver varieties A. Buryak, ''The classes of the quasihomogeneous Hilbert schemes of points on the plane,'' Mosc. Math. J., 12:1, 21--36; http://arxiv.org/abs/1011.4459 . Parametrization (Chow and Hilbert schemes), Combinatorial aspects of partitions of integers, Representations of quivers and partially ordered sets The classes of the quasihomogeneous Hilbert schemes of points on the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let S be a reduced surface embedded in \({\mathbb{C}}^ N\), and \(o\in S\) an isolated singular point of S. Define the topological invariant of (S,o), \(Eu_ o(S)\), as the Euler-Poincaré characteristic of \(S\cap H\cap B_{\delta}\), where \(B_{\delta}\) is the ball in \({\mathbb{C}}^ N\) of center o and radius \(\delta>0\) sufficiently small, and H a general hyperplane passing sufficiently near the point o. Let \(\pi: X\to S\) be a resolution of the singularity (S,o) such that m\({\mathcal O}_ X\) is invertible, where m is the maximal ideal of the local ring \({\mathcal O}_{S,o}\). - Then the first result of this paper consists in the equality \(Eu_ o(S)=(Z_ o\cdot(Z_ o-| Z_ o| -K)),\) where \(Z_ o\) is the effective divisor with support in \(\pi^{-1}(o)\) such that \(m{\mathcal O}_ X={\mathcal O}_ X(-Z_ o), | Z_ o|\) is the reduced associated cycle of \(Z_ o\), and K is a canonical divisor of X. - The second result asserts that if (S,o) is a normal point of S, then \(Eu_ o(S)\leq 1\), and \(Eu_ o(S)=1\) iff the point \(o\in S\) is smooth. Finally, the author illustrates these results by considering certain examples of singularities such as: rational, cyclic, or conical singularities. isolated surface singularities; resolution; maximal cycles; effective divisor González-Sprinberg, G., Cycle maximal et invariant d'Euler local des singularités isolées de surfaces, Topology, 21, 4, 401-408, (1982) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Topological properties in algebraic geometry, Singularities in algebraic geometry, Local complex singularities Cycle maximal et invariant d'Euler local des singularités isolées de 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 first part of the authors' article [ibid. 231, No. 5, 3022--3053 (2012; Zbl 1257.14002)] should have been published before the second part [ibid. 231, No. 5, 3003--3021 (2012; Zbl 1262.14003)]. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) Erratum to ``Resolution except for minimal singularities. I'' [Adv. Math. 231 (2012) 3022-3053] | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 first-order Thom-Boardman singularity sets (and some higher-order ones) of the dual mapping of an arbitrary smooth hypersurface into the complex projective space \({\mathbb{P}}_ n\) are investigated.
The main idea is to use the Hessian mapping and the relationships between dual and Hessian mappings. - Results are obtained regarding connectedness, stratifications, numerical invariants and so on. Thom-Boardman singularity; Hessian mapping A. D. R. Choudary and A. Dimca, On the dual and Hessian mappings of projective hypersurfaces , Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 3, 461-468. Singularities in algebraic geometry, Projective techniques in algebraic geometry, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Global theory and resolution of singularities (algebro-geometric aspects), Complex singularities, Singularities of surfaces or higher-dimensional varieties On the dual and hessian mappings of projective hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review is intended to describe the explicit constructions of crepant resolutions of higher-dimensional orbifolds with Gorenstein quotient singularities, that is, the algebraic (or analytic) varieties such that the analytic type of each singular point is described as \({\mathbb C}^n / G\), where \(G\) is a (non-trivial) finite subgroup of the special linear group \(\text{SL}_n ({\mathbb C})\). For the 2-dimensional case, finite subgroups \(G\) of \(\text{SL}_2 ({\mathbb C})\) were classically classified into ADE series. It is well known that \({\mathbb C}^2 / G\) are always hypersurface singularities, and the minimal resolutions of them give the desired crepant resolutions. For the 3-dimensional case, the required crepant resolutions were found for all finite subgroups \(G \subset \text{SL}_3 ({\mathbb C})\) by virtue of \textit{Y. Ito} [Proc. Japan Acad., Ser. A 70, 131--136 (1994; Zbl 0831.14006)], \textit{D. Markushevich} [Math. Ann. 308, 279--289 (1997; Zbl 0899.14016)], \textit{S. S. Roan} [Int. J. Math. 5, 523--536 (1994; Zbl 0856.14005)] and \textit{S. S. Roan} [Topology 35, 489--508 (1996; Zbl 0872.14034)], which are depending on the classical result on the classification of finite subgroups of \(\text{SL}_3 ({\mathbb C})\) due to Miller-Blichfeldt-Dickson [\textit{Y. A. Miller, H. F. Blichfeldt} and \textit{L. E. Dickson}, ``Theory and application of finite groups''. New York, Wiley (1915; JFM 45.0255.12)]. But, for such a higher-dimensional case, the non-uniqueness of crepant resolutions happens due to the existence of certain kinds of codimension 2 birational operations known as flops. In order to understand higher-dimensional crepant resolutions qualitatively, the development has resulted in the theory of \(G\)-Hilbert schemes \(\text{Hilb}^G ({\mathbb C}^n)\) associated to the quotient singularities \({\mathbb C}^n / G\) as in \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad.,Ser. A 72, 135--138 (1996; Zbl 0881.14002)]: the crepant resolutions of \({\mathbb C}^n / G\) would be related to \(G\)-Hilbert schemes \(\text{Hilb}^G ({\mathbb C}^n)\) of \(G\)-stable 0-dimensional subschemes of \({\mathbb C}^n\) of length equal to the order \(| G | \) of \(G\). As a result, the structure of \(\text{Hilb}^G ({\mathbb C}^n)\) now yields that \(\text{Hilb}^G ({\mathbb C}^3)\) is a toric crepant resolution of \({\mathbb C}^3 / G\) for every finite abelian subgroup \(G \subset \text{SL}_3 ({\mathbb C})\) by virtue of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, 535--554 (2001; Zbl 0966.14028)], \textit{Y. Ito} and \textit{H. Nakajima} [Topology 39, 1155--1191 (2000; Zbl 0995.14001)] and \textit{I. Nakamura} [J. Alg. Geom. 10, 757--779 (2001; Zbl 1104.14003)]. Whereas, for the cases \(n \geq 4\), there are very few results concerning the crepant resolutions of \({\mathbb C}^n / G\) and the structure of \(\text{Hilb}^G ({\mathbb C})\) for finite subgroups \(G \subset \text{SL}_n ({\mathbb C})\). The authors restricted themselves to the case where \(G\) is the subgroup \(A_r (n)\) of \(\text{SL}_n({\mathbb C})\) consisting of all diagonal matrices of order \(r+1\). In an earlier paper [Int.. J. Math. Math. Sci. 26, 649--669 (2001; Zbl 1065.14018)], the authors studied the case of \(n=4\) and \(G=A_r (4)\), and obtained crepant resolutions of \({\mathbb C}^4 / A_r (4)\) through the detailed investigations of the structure of \(\text{Hilb}^{A_r (4)} ({\mathbb C}^4)\). In the present paper, the authors study the case of \(n=4, 5\) and \(r=1\). More precisely, for \(n=4\) and \(r=1\), they describe the toric variety structure of \(\text{Hilb}^{A_1 (4)} ({\mathbb C}^4)\) which is NOT crepant. Then, blowing-down the divisor \({\mathbb P}^1 \times {\mathbb P}^1 \times {\mathbb P}^1\) on \(\text{Hilb}^{A_1 (4)} ({\mathbb C}^4)\) onto \({\mathbb P}^1 \times {\mathbb P}^1\) in different ways, they obtain three different toric crepant resolutions of \({\mathbb C}^4 / A_1(4)\). These three crepant resolutions are related to each other by 4-fold flops. For the case \(n=5\) and \(r=1\) also, as in the 4-dimensional case just above, they describe the toric variety structure of \(\text{Hilb}^{A_1(5)} ({\mathbb C}^5)\) which is NOT crepant, and obtain twelve mutually different crepant resolutions of \({\mathbb C}^5 / A_1(5)\), all of which are dominated by \(\text{Hilb}^{A_1(5)}({\mathbb C}^5)\). These twelve crepant resolutions are related to each other by 5-fold flops. \(G\)-Hilbert schemes; \(G\)-clusters; toric crepant resolution Chiang, L.; Roan, S. -S.: Crepant resolutions of \(Cn/A1(n)\) and flops of n-folds for n=4,5, Fields inst. Commun. 38, 27-41 (2003) Homogeneous spaces and generalizations, Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Parametrization (Chow and Hilbert schemes) Crepant resolutions of \(\mathbb{C}^n/A_1(n)\) and flops of \(n\)-folds for \(n=4,5\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Every three-dimensional complex algebraic variety with isolated singular point has a resolution factoring through the Nash blowup and the blowup of the maximal ideal over which the second Fitting ideal sheaf is locally principal. In such resolutions one can construct Hsiang-Pati coordinates and thus obtain generators for the Nash sheaf that are the differentials of monomial functions. Taalman, L., The Nash sheaf of a complete resolution, Manuscr. Math., 106, 249-270, (2001) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) The Nash sheaf of a complete resolution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author builds ``Tsuchihashi cusps'' [\textit{H. Tsuchihashi}, Tôhoku Math. J., II. Ser. 35, 607-639 (1983; Zbl 0585.14004)] (this is a generalization of Hilbert modular cusp singularities). Such a singularity is defined by a pair \((C,\Gamma)\) of an open convex cone \(C\subset\mathbb{R}^ n\) and a discrete group \(\Gamma\subset GL(n,\mathbb{Z})\) with good conditions. The author defines and studies the notion of ``semi-integral stellable polyhedral cones'' \(C\), the group \(\Gamma\) generated by the reflections with respect to the facets of such a \(C\) gives rise to a good pair \((C,\Gamma)\). There is a duality among stellable cones, the corresponding singularities are dual in the sense of Tsuchihashi [loc. cit.].
At the end, the author gives effective examples of his singularities and computes the arithmetic genus default \(\chi_ \infty\) and the Ogata zeta zero \(Z(0)\) and verifies on his examples the Ogata-Satake conjecture: the \(\chi_ \infty\) of a cusp is equal to the \(Z(0)\) of its dual. A proof of this conjecture is announced as forthcoming.
[See also: \textit{E. B. Vinberg}, Math. USSR, Izv. 5(1971), 1083-1119 (1972); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1072-1112 (1971; Zbl 0247.20054) and \textit{T. Satake} and \textit{S. Ogata}, in Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math. 15, 1-27 (1989; Zbl 0712.14009)]. Tsuchihashi cusps; arithmetic genus default; zeta zero Ishida, M.-N.: Cusp singularities given by stellable cones. Int. J. Math.2, 635--657 (1991) Singularities in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Global theory and resolution of singularities (algebro-geometric aspects) Cusp singularities given by reflections of stellable cones | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In [\textit{R. Bédard} and \textit{R. Schiffler}, Represent. Theory 7, 481--548 (2003; Zbl 1060.17001)] one gives the characterization of the orbits of representations of quivers of type \(A\) with Zariski closures which are rationally smooth. In the paper under review the author continues these investigations and studies the local rational smoothness of these closures. He obtains a description of the orbits with the property that the projectivization of their Zariski closures are rationally smooth.
The main idea of the method is to use that the change of basis between canonical, and PBW-basis of the positive part of the quantized enveloping algebra of type \(A_n\) has a geometric interpretation in terms of local intersection cohomology of some affine algebraic varieties, namely the Zariski closures of orbits of representations of a quiver of type \(A_n\). Then the author applies some methods already used in his paper with Bédard, and some results from there. Comparing the results of both papers, it turns out that the only non-smooth, projectively rationally smooth orbit closures are of type \(A_2\) and \(A_3\). projective representations of quivers; varieties of representations; rational smoothness; quantum groups; quantized enveloping algebra Quantum groups (quantized enveloping algebras) and related deformations, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Singularities in algebraic geometry, Representations of quivers and partially ordered sets, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Projective rational smoothness of varieties of representations for quivers of type \(A\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the notion of the Gröbner cell for the Hilbert scheme of points in the plane, as well as that of the punctual Hilbert scheme is comprehensively defined. An explicit parametrization of the Gröbner cells in terms of minors of a matrix is recalled. The main core of this paper shows that the decomposition of the Punctual Hilbert scheme into Grönber cells induces that of the compactified Jacobians of plane curve singularities. As an important application of this decomposition, the topological invariance of an analog of the compactified Jacobian and the corresponding motivic superpolynomial for families of singularities is concluded. Hilbert schemes; affine plane; Grothendieck-Deligne map; Gröbner cells; zeta functions; plane curve singularities Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Plane and space curves, Exact enumeration problems, generating functions, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Jacobians, Prym varieties, Hecke algebras and their representations, Combinatorial aspects of representation theory, Braid groups; Artin groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Gröbner cells of punctual Hilbert schemes in dimension two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We present a concise proof for the existence and construction of a strong resolution of excellent schemes of finite type over a field of characteristic zero. Our proof is based on earlier work of \textit{O. Villamayor} [Ann. Sci. Éc. Norm. Supér., IV. Ser. 22, 1--32 (1989; Zbl 0675.14003)], \textit{S. Encinas} and \textit{O. Villamayor} [Acta Math. 181, 109--158 (1998; Zbl 0930.14038)] and \textit{E. Bierstone} and \textit{P. Milman} [Invent. Math. 128, 207--302 (1997; Zbl 0896.14006)]. It proposes some substantial simplifications which may be helpful for a better understanding of how to prove Hironaka's famous theorem on embedded resolution of singularities. algorithmic resolution; centers of blowup; resolution datum; embedded resolution Encinas, S. and Hauser, H.: Strong resolution of singularities in characteristic zero. Comment. Math. Helv. 77 (2002), no. 4, 821-845. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Strong resolution of singularities in characteristic zero | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article gives a short introduction to the problems of local uniformization and local monomialization in their interaction with the general problem of resolution of singularities. Some details are sketched, with references to recent and forthcoming papers of the author. It starts with classical results of Zariski (his fundamental theorem on local uniformization, the theorem on resolution of surface-singularities), illustrates first difficulties, mentions Abhyankar's results on resolution in dimension \(\leq 3\) and Hironaka's famous theorem on the existence of a resolution over a field of characteristic 0. The second part of the paper explains the problem of monomialization (which in general does not have a solution in positive characteristics). Theorem 3.1 [proved in the author's book ``Local monomialization and factorization of morphisms'', Astérisque No. 260 (Paris: Société Mathématique de France) (1999; Zbl 0941.14001)] gives an affirmative answer to a question of Abhyankar on simultaneous resolution along a valuation in characteristic 0. The final section is devoted to several results on monomialization of morphisms. Together with an overview of the proof, the author announces the existence of a global monomialization of a proper dominant morphism from a 3-fold to a surface (in characteristic 0). resolution of singularities; uniformization Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Singularities in algebraic geometry, Morphisms of commutative rings Local monomialization. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C \subseteq \mathbb{A}^ 3_ k\) be an irreducible curve defined by the prime ideal \({\mathfrak p} \subseteq k[x,y,z] = : R\), \(k\) algebraically closed. Assume \(C\) is an almost complete intersection. Let \({\mathfrak m} \subseteq R\) be a maximal ideal and consider the function \(\chi (r,s) = \dim_ k {\mathfrak m}^ r {\mathfrak p}^ s/{\mathfrak m}^{r + 1} {\mathfrak p}^ s\). For large \(r\) and \(s\) it holds: \(\chi (r,s) = H(r,s)\), a polynomial of degree 2 with rational coefficients.
Let \(H(r,s) = e_ 0 {r \choose 2} + e_ 1 rs + e_ 2 {s \choose 2} +\) lower degree terms. The coefficients \(e_ 0, e_ 1, e_ 2\) (the so- called mixed multiplicities of \({\mathfrak m}\) and \({\mathfrak p})\) determine the multiplicities of the local rings associated to the blow ups. The purpose of the paper is to calculate the function \(\chi (r,s)\). blowing up of curve singularities; almost complete intersection curve; multiplicities of blow-ups Katz D., Comm. Algebra 22 pp 721-- (1994) Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Plane and space curves, Singularities in algebraic geometry, Complete intersections On the multiplicity of blow-ups associated to almost complete intersection space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this note, a simple construction is described that yields a family of affine toric \(d\)-folds with terminal singularities for each dimension \(d \geq 4\). This answers a question on terminal singularities of toric 4- folds raised by \textit{K. Matsuki} [Am. J. Math. 113, No. 5, 835-859 (1991; Zbl 0746.14017)] and contrasts with earlier results of Reid, Kawamata and Kollár in dimension 3. It implies that a method different from the one described by Matsuki is needed to establish the flop conjectures I, II for 4-folds with canonical singularities. toric \(d\)-folds with terminal singularities; flop conjectures Sturmfels, B.: Toric ideals. Lecture notes (1994) Toric varieties, Newton polyhedra, Okounkov bodies, \(4\)-folds, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry A note on lattice simplices and toric varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a closed subvariety of a smooth variety over a perfect field \(k\), \(m\) the maximum value of the multiplicity \(m(x)\) of \(X\) at point \(x\), and \(M(X)\) the (closed) subset of \(X\) of points \(y\) where \(m(y)=m\). A possible problem is to find a sequence of monoidal transforms \(V=V_0, \ldots, V_r\), such that if \(X_i\) is the strict transform of \(X\) to \(V_i\), the centers satisfy \(C_i \subset M(X_i)\) and the maximum multiplicity of \(X_r\) is \(<m\). If this can be done, iterating the process we resolve the singularities of \(X\).
The problem has affirmative solution in characteristic zero and remains open, for \(\dim(X) > 3\), in positive characteristic. The present paper is a contribution to this open problem when \(X\) is a hypersurface in a smooth variety \(V\). As in other articles of the authors, the question is rephrased in terms of \textit{Rees algebras} (certain graded subagebras of the polynonial algebra \({\mathcal O}_V[T]\)).
In previous work the authors (and Ana Bravo) have shown that, in any characteristic, it is possible to ``improve'' the singularities of a Rees algebra \(\mathcal G\) (over a regular variety \(V\)). More precisely, let \(e(\mathcal G)\) denote the minimum value of \({\tau}_{\mathcal G,x}\), for \(x\) a singular point of \( \mathcal G\), (\(\tau\) is a version of a numerical invariant introduced by Hironaka). By descending induction on \(e(\mathcal G)\) it is possible to associate to \(\mathcal G\), locally in the etale topology: (a) sequences
\[
(1) \qquad V=V_0 \leftarrow \cdots \leftarrow V_s \, ,
\]
\[
(2) \qquad V'_0 \leftarrow \cdots \leftarrow V'_s \, ,
\]
of monoidal transforms with regular centers, (c) transversal smooth projections \(\beta _i:V_i \to V'_i\), \(\dim (V'_i)=\dim(V_i)-e\), for all \(i\), and (d) Rees algebras \({\mathcal R}_{i}\), such that \({\mathcal R}_s\) is monomial. The algebra \(\mathcal{R}_i\) is the \textit{elimination algebra of} \({\mathcal G}_i\) (the transform of \({\mathcal G}\) to \(V_i\)) with respect to \(\beta _i\), a concept thoroughly studied by Villamayor. Monomial algebras correspond to \textit{monomial ideals}, a particularly simple type of invertible sheaf, which can be resolved pretty easily essentially in a combinatorial way. In this case, one says that \(\mathcal G _s\) is in the \textit{monomial situation}. If the characteristic of the base field \(k\) is zero, the mentioned result leads to an extension of the sequence (1) which resolves \(\mathcal G\), but in positive characteristic there are difficulties. Trying to overcome these, in the present paper the authors introduce a concept of ``strong monomial situation''. If in (1) \(\mathcal G _s\) is in the strongly monomial situation, even in positive characteristic it is possible to extend the sequence (1) so that a resolution of \(\mathcal G\) is achieved. The authors give a numerical criterion to decide whether the strong monomial situation has been reached.
Both the notion of strong monomial situation and the mentioned criterion require some new auxiliary concepts, introduced and developed in this paper. For instance: (i) the notion of \textit{\(p\)-presentation} of a Rees algebra \(\mathcal G\) over \(V\) of dimension \(d\), relative to a transversal projection \(\beta : V \to V'\), \(\dim V' = d-1\), which is a way to describe \(\mathcal G\) (locally, in the etale topology) in terms of the elimination algebra \( \mathcal R\) of \(\mathcal G\) and a monic polynomial, whose degree is a power of \(p\), with coefficients defined on \(V'\); (ii) the notion of \textit{slope} of \(\mathcal G\) relative to a p-presentation; (iii) the notion of H-ord at a point of \(V\), defined in terms of slopes of suitable general presentations. The authors sometimes work under the assumption that \(e(\mathcal G)=1\), and announce that the general situation will be discussed in future papers.
If \(\dim X =2\), in other articles they showed that, even in positive characteristic, with the methods just reviewed a resolution of singularities is obtained. resolution of singularities; positive characteristic; differential operators; Rees algebras; monomial ideals. Benito, Angélica; Villamayor U., Orlando E., Monoidal transforms and invariants of singularities in positive characteristic, Compos. Math., 149, 8, 1267-1311, (2013) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Graded rings, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes Monoidal transforms and invariants of 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 The book under review consists of six chapters, in addition to foreword, preface, prologue, bibliography including 132 items (among them 11 original works of H.Hironaka), and index. In short, ``a complete and self-contained proof of the theorem of desingularization for complex-analytic varieties'' is precisely the main purpose that the authors have to achieve. \par The book begins with the foreword written by B. Teissier, where he gives a brief historical background and discusses fundamental ideas in or around the general desingularization problem. In the preface H.Hironaka explains the crucial points of his original approach (see [\textit{H. Hironaka}, Actes Congr. internat. Math. 1970, No. 2, 627--631 (1971; Zbl 0231.32007); H. Hironaka, Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)]) and compares his results with several later ones (see, e.g., [\textit{E. Bierstone} and \textit{P. D. Milman}, Invent. Math. 128, No. 2, 207--302 (1997; Zbl 0896.14006); \textit{A. J. de Jong}, Publ. Math., Inst. Hautes Étud. Sci. 83, 51--93 (1996; Zbl 0916.14005)]). He also describes how the idea of creating this book appeared and how it was implemented. \par In the detailed (of 14 pages) prologue, the authors explain very carefully the main ideas presented in the book and illustrate the corresponding material with a series of interesting and highly non-trivial pictures, non-formal commments, remarks, etc. \par The first chapter is an introduction to the theory of complex-analytic spaces adapted to the subject. Among other things, the authors discuss the notions of horizontal and vertical morphisms, cone-fibered spaces, blow-ups, blowing, normal and tangent cones, and other related constructions. \par The chapter 2 is devoted to a generalization of the classical Weierstrass preparation theorem which is an important technical tool for desingularization of general complex-analytic singularities. The authors' goal here is to show that ``the desingularization problem (or rather its crucial inductive step) for an arbitrary singular variety $X$ can be interpreted equivalently as the same problem'' for a system of hypersurfaces $\{H_j\}_{1\leq j\leq m}$ satisfied certain conditions and such that $X = \bigcap_{1\leq j\leq m} H_j$. \par The third chapter develops another important ingredient of the proof, the theory of maximal contact in dimensions two and higher. In particular, the existence of maximal contact for hypersurfaces is proved; it plays a key role in the inductive proof of desingularization theorems. The authors study the characteristic cones, continuity for the maximal contact, contact stability theorems, etc. They also emphasize that their version is stronger than a similar notion of maximal contact proposed in [\textit{J. Giraud}, Math. Z. 137, 285--310 (1974; Zbl 0275.32003)] and that there is no good analogy of this notion in the case of base fields of positive characteristics. \par The chapter 4, called ``Groves and Polygroves'', introduces the basic concepts of the theory of infinitely near singularities and presents properties of trees, forests, groves, polygroves, soil, gardens and other structures associated with complex spaces, modifications, ideals, etc. (see [\textit{H. Hironaka}, in: Algebraic Geom., Oslo 1970, Proc. 5th Nordic Summer School Math. 315--332 (1972; Zbl 0247.32008)]). \par The fifth chapter is a detail description of the induction process, which gives a complete proof of the desingularization theorem. \par The chapter 6, entitled ``Epilogue: Singularities of Differential Equations'', begins with a humorous epigraph where Hironaka formulates the general problem ``Solve all the differential equations''. This chapter contains a detailed description of a number of ideas and results relating to the theory of complex analytical foliations, which are based on the theory of resolution of singularities of complex analytical foliations. Among them, the general theory of foliations, vector fields and valuations, the concept of Newton polygon associated with a differential equation, the reduction of singularities of foliations in dimension two and codimension-one in three-dimensional spaces towards Thom's conjecture (see [\textit{C. Camacho} and \textit{P. Sad}, Ann. Math. (2) 115, 579--595 (1982; Zbl 0503.32007)]), and others. \par It is appropriate to complete the review by quoting the following conclusion from Teissier's foreword: ``This text will be particularly useful and interesting for readers of the younger generation who wish to understand one of the most fundamental results in algebraic and analytic geometry and invent possible extensions and applications of the methods created to prove it''. resolution of singularities, hypersurfaces; complex analytic spaces; infinitely near singularities; modifications; horizontal morphisms, vertical morphisms, cone-fibered spaces, blow-ups, blowing cones, normal cones; tangent cones; Weierstrass preparation theorem; normal flatness; maximal contact; idealistic exponents; characteristic cones; continuity of maximal contact; contact stability theorems; trees; forests; groves; polygroves; soil; gardens; allées; normal crossings; Samuel stratification; complex analytic foliations; valuations; Newton polygon; Thom's conjecture Research exposition (monographs, survey articles) pertaining to algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects), Complex spaces, Global theory and resolution of singularities (algebro-geometric aspects), Dynamical aspects of holomorphic foliations and vector fields, Singularities of holomorphic vector fields and foliations, Foliations (differential geometric aspects), Contact manifolds (general theory), Foliations in differential topology; geometric theory, Singularities in algebraic geometry Complex analytic 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 In the paper basic elementary properties of blowing-ups of two-dimensional complex manifolds and an analytic proof of the Zariski theorem on equisingularity for curves are given. blowing-ups of two-dimensional complex manifolds; equisingularity Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Blowing-ups and bifurcation points. I: The Zariski theorem on equisingularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{D. Mumford} [Publ. Math., Inst. Hautes Étud. Sci. 9, 5--22 (1961; Zbl 0108.16801)] defined a rational pullback for Weil divisors on normal surfaces, which is linear, respects effectivity, and satisfies the projection formula. In higher dimensions, the existence of small resolutions of singularities precludes such general results. We single out a higher-dimensional situation that resembles the surface case and show for it that a rational pullback for Weil divisors exists, which is also linear, respects effectivity, and satisfies the projection formula. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Divisors, linear systems, invertible sheaves, Schemes and morphisms A higher-dimensional generalization of Mumford's rational pullback for Weil divisors | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives a definition of regular topological triviality for a germ of functions and proves this property for a wide class of germs. For example, all unimodal and bimodal families of singularities in V. I. Arnold's classification are regular topologically trivial. topological classification; germ of functions; singularities Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On topological equivalence of germs of functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author extends Viro's method of glueing polynomials in order to keep singular or critical points in the process. The input for the glueing method is a subdivision \(\{\Delta_i\}\) of a nondegenerate Newton polyhedron and a compatible system of polynomials \(F_i\) with support on the \(\Delta_i\). A sufficient condition for the existence of a polynomial with the same singularities as the \(F_i\) is roughly that for each \(i\) the equisingular locus in the space of all polynomials is smooth and transversal to the space of polynomials with the given Newton diagram and coinciding with \(F_i\) for all monomials in \(\Delta_i\).
As example plane curves with the maximal number of cusps are constructed for degree eight (\(\kappa=15\)) and nine (\(\kappa=20\)). Another application is the asymptotically complete solution to the problem of possible collections of critical points of real polynomials in two variables without critical points at infinity. glueing polynomials; Newton polyhedron; equisingular locus; plane curves; maximal number of cusps; critical points E. Shustin, Gluing of singular and critical points. \textit{Topology}\textbf{37} (1998), 195-217. MR1480886 Zbl 0905.14008 Global theory and resolution of singularities (algebro-geometric aspects), Equisingularity (topological and analytic), Topological properties in algebraic geometry, Singularities in algebraic geometry, Singularities of curves, local rings Gluing of singular and critical points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0644.00012.]
The author presents certain results and examples concerning the fundamental group \(\pi_ 1(X-D)\), where X is a smooth complex space and \(D\subset X\) an analytic hypersurface.
The central example in the global case is the discriminant D(P) of an irreducible polynomial P in \({\mathbb{C}}^ n\), and in the local case the result of \textit{Lê Dũng Tráng} and \textit{K. Saito} [Ark. Math. 22, 1- 24 (1984; Zbl 0553.14006)]. local singularities; fundamental group Local complex singularities, Complex singularities, Fundamental group, presentations, free differential calculus, Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry On the fundamental group of the complement of an analytic hypersurface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give a method to resolve a quotient surface singularity which arises as the quotient of a product action of a finite group on two curves. In the characteristic zero case, the singularity is resolved by means of a continued fraction, which is known as the Hirzebruch-Jung desingularization. We develop the method in the positive characteristic case where the square of the characteristic does not divide the order of the group. surface singularity; positive characteristic; wild quotient; desingularization; partial desingularization; toric geometry Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Positive characteristic ground fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Quotient singularities of products of two curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A well known theorem of K. Saito states that an isolated hypersurface singularity admits a \({\mathbb{C}}^*\) action if and only if the defining function belongs to its own Jacobian ideal. This is equivalent to equality of the Milnor and Tyurina numbers, \(\mu\) and \(\tau\) respectively. Here the result (in this form) is generalized to isolated complete intersection surface singularities. More generally, the difference \(\mu\)-\(\tau\) is expressed in terms of other (nonnegative) analytic invariants. The theory is applied to obtain results about the irregularity and about equisingular deformations. The proof depends on choosing a component C of the exceptional divisor which either has positive genus or meets at least three other components, and extending a derivation from C to the whole divisor. quasi-homogeneous Gorenstein surface singularities; equality of the Milnor and Tyurina numbers; isolated complete intersection surface singularities; irregularity; equisingular deformations J. M. Wahl, A characterization of quasihomogeneous Gorenstein surface singularities, Compos. Math., 55 (1985), 269--288. MR799816 (87e:32013) Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects), Complete intersections, Deformations of singularities, Global theory and resolution of singularities (algebro-geometric aspects), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities in algebraic geometry A characterization of quasi-homogeneous Gorenstein surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In their earlier paper [\textit{A.Némethi} and \textit{L.Nicolaescu}, Geom. Topol. 6, 269--328 (2002; Zbl 1031.32023)] the authors analyzed a hypothetical upper bound for the geometric genus of a normal surface singularity whose link \(M\) is a rational homology sphere; the bound is expressed in terms of the resolution of the singularity and the Seiberg-Witten invariant associated to the canonical spin\(^c\) structure of \(M.\) They conjectured that if the link of a \(\mathbb Q\)-Gorenstein singularity is a rational homology sphere then the geometric genus is in fact equal to the bound.
In the paper under review this conjecture is verified for normal surface singularities with good \(\mathbb C^*\)-action. Among other things a topological interpretation of the generalized Batyrev's stringy invariant of such a singularity introduced by \textit{W. Veys} [J. Algebr. Geom. 13, No. 1, 115--141 (2004; Zbl 1060.14021)] is obtained. links of normal surface singularities; Q-Gorenstein singularities; geometric genus; plumbing graph; spin structure; Seiberg-Witten invariants of Q-homology spheres; Reidemeister-Turaev torsion; Seifert invariants; complete intersections singularities; log canonical singularities Némethi, András; Nicolaescu, Liviu I., Seiberg--Witten invariants and surface singularities. II. Singularities with good \(\mathbb{C}^*\)-action, J. London Math. Soc. (2), 69, 3, 593-607, (2004) Complex surface and hypersurface singularities, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Knots and links in the 3-sphere, Singularities of surfaces or higher-dimensional varieties, Invariants of knots and \(3\)-manifolds Seiberg-Witten invariants and surface singularities. II: Singularities with good \(\mathbb{C}^*\)-action | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \((R,M)\) be a regular local ring of arbitrary dimension and \(I\) an \(M\)-primary ideal. If \(R \to B\) is a quadratic transform of \(R\), \(I_ 1\) is the proper transform of \(I\) in \(B\) with \(I_ 1\) primary to the maximal ideal of \(B\), then it is known that multiplicity \(e(I_ 1R) < e(IR)\). In this paper the author associates to each ideal \(I\) in a regular local noetherian ring \(R\) an invariant \(w (I,R)\) \((\max_ i V_ i (I))\), when the integral closure \(\overline I\) has the Rees decomposition in terms of valuation ideals \(V_ i\). In this \(v_ i\) stands for the valuation corresponding to \(V_ i\).
The following theorem is the main result of this paper. Let \((R,M,K)\) be a regular local ring of positive dimension \(d\), \(I\) an ideal of \(R\) such that \(\dim (R) = s(I)\), the analytic spread of \(I\). If \((B,N)\) is a quadratic transform of \(R\) along 1, and \(I_ 1\) is the proper transform of \(I\) in \(B\), then either \(s(I_ 1) < s(I)\) or \(w (I_ 1,B) < w (I,R)\). There are some problems proposed in the concluding section. regular local ring; proper transform; multiplicity; valuation ideals; quadratic transform DOI: 10.1080/00927879408825128 Regular local rings, Multiplicity theory and related topics, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Improving ideals by means of quadratic transformations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The existence of a certain type of equivalence between triangulated categories of singularities for varieties of different dimensions is proved. This class of equivalences generalizes the so-called Knörrer periodicity. As a consequence, equivalences between the categories of \(D\)-branes of type \(B\) on Landau-Ginzburg models of different dimensions are obtained. D. O. Orlov, Triangulated categories of singularities, and equivalences between Landau-Ginzburg models, Mat. Sb. 197 (2006), no. 12, 117 -- 132 (Russian, with Russian summary); English transl., Sb. Math. 197 (2006), no. 11-12, 1827 -- 1840. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Triangulated categories of singularities and equivalences between Landau-Ginzburg 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 paper is divided into three parts. In Section 2, the authors discuss a way to measure the dimension of singularities of polynomials at infinity and a method of constructing such polynomials, motivated by a paper of the second author [Compos. Math. 111, No. 1, 89-109 (1998; Zbl 0901.58003)]. In Section 3, the authors consider the homotopy groups of the complements to hypersurfaces which may have singularities at infinity, expanding the results of the first author [Ann. Math. (2) 139, No. 1, 117-144 (1994; Zbl 0815.57017)]. Some results on the homology of smoothings and homotopy groups of the complements are presented in Section 4. singularities at infinity Libgober, A.; Tibăr, M.: Homotopy groups of complements and nonisolated singularities. Internat. math. Res. notices 17, 871-888 (2002) Singularities of differentiable mappings in differential topology, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Homotopy groups of complements and nonisolated singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) an algebraically closed field, \(char\;k = 0\). Let \(C\) be an irreducible nonsingular curve such that \(rC = S\cap F\), \(r\in \mathbb {N}\), where \(S\) and \(F\) are two surfaces in \(\mathbb {P}^3\) and all the singularities of \(F\) are of the form \(z^p =x^{ps}-y^{ps}\), \(p\) prime, \(s\in \mathbb {N}\). We prove that \(C\) can never pass through such kind of singularities of a surface, unless \(r = pa\), \(a\in \mathbb {N}\). These singularities are Kodaira singularities. Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Complex surface and hypersurface singularities, \(3\)-folds, \(4\)-folds, \(n\)-folds (\(n>4\)), Hypersurfaces and algebraic geometry Multiple structures on smooth 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 Given a singular algebraic surface \(S=\{f=0\}\) in affine 3-space, one constructs an embedded resolution \(h:X \to\mathbb A^3\) of \(S\) as a composition of blowing-ups \(\pi\) with centre \(C\) a point or a smooth curve, and exceptional locus \(E\) the projective plane or a ruled surface (over C), respectively. Denote (in the resolution stage just after the creation of \(E\)) by \(D\) the total intersection of E with the strict transform of \(S\) and of all previously created exceptional surfaces. When \(C\) is a point, there is no restriction on the possible curve configurations \(D \subset E=\mathbb P^2\). But when \(C\) is a projective curve, not every configuration \(D \subset E\) on every isomorphism class of \(E\) can occur in this context of embedded resolution. In this paper the author investigates the case when \(D\) does not contain any fibre of the ruled surface \(E\). He proves in particular that then the \(e\)-invariant of \(E\) [see e.g. \textit{R. Hartshorne}, ``Algebraic geometry'' (1977; Zbl 0367.14001), V.2] must satisfy \(e>0\), and he relates this \(e\)-invariant to the genus of \(C\).
This (quite special) case is very interesting from the point of view of applications to certain singularity invariants of \(f\). The motivic zeta function of \(f\) (or its specializations Hodge and topological zeta function) is a rational function that can be described in terms of the embedded resolution \(h\). We just mention that \(\{-\nu_i/N_i\}\) is a (complete) list of its candidate poles. Here \(E_i, i\in T\) are the irreducible components of \(h^{-1}\{f=0\}\), and \(N_i\) and \(\nu_i -1\) the multiplicities of \(E_i\) in the divisor of \(f\circ h\) and \(h^*(dx\wedge dy\wedge dz)\), respectively. Typically however, many of these candidates are no actual poles; this phenomenon is related to the famous monodromy conjecture. In particular, when the topological Euler characteristic \(\chi(E\setminus D) =0\), one generally expects that \(-\nu/N\) is not a pole. For several years a \` problem configuration\'\ contradicting this philosophy was \(D\) being three nonintersecting sections of a ruled surface \(E\) over an elliptic curve \(C\). But now, as a consequence of the positivity of the \(e\)-invariant, the author shows that \(D\) can never consist of three nonintersecting curves.
Moreover \` generic\'\ non-connected configurations \(D\) on a ruled surface \(E\) with \(\chi(E\setminus D) =0\) (that can occur in \(h\)) are classified, and it is shown that then \(-\nu/N\) does not contribute to the poles of the zeta functions above. ruled surface; embedded resolution; motivic zeta function Global theory and resolution of singularities (algebro-geometric aspects), Rational and ruled surfaces, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects) Ruled exceptional surfaces and the poles of motivic zeta functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0619.00007.]
For a cyclic two dimensional quotient singularity (X,0) the author computes the Chern-classes of the vector bundles on the resolution \(\pi: X\to X\) that are of the form \(\pi\) *(M)/torsion, where M is an indecomposable reflexive module over \({\mathcal O}_{X,0}\). In this way [and in his paper in Math. Ann. 279, No.4, 583-598 (1988; Zbl 0616.14001)] he generalizes a result of \textit{G. Gonzales-Sprinberg} and \textit{J. L. Verdier} [Ann. Sci. Éc. Norm. Supér., II. Sér. 16, 409-449 (1983; Zbl 0538.14033)] who computed these Chern-classes for rational double points and in this way gave a geometric interpretation of the correspondance of \textit{J. McKay} [Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 183-186 (1980; Zbl 0451.05026)]. McKay correspondance; cyclic two dimensional quotient singularity; Chern- classes of the vector bundles; resolution; reflexive module J. Wunram, Reflexive modules on cyclic quotient surface singularities, in \(Singularities, Representation of Algebras, and Vector Bundles\) (Springer, Berlin, 1987), pp. 221-231 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Other special types of modules and ideals in commutative rings, Singularities of surfaces or higher-dimensional varieties, Group actions on varieties or schemes (quotients) Reflexive modules on 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 This paper continuous our study on the blowing ups of 3-dimensional terminal singularities \(X\) of indices \(m\geq 2\). In the previous part I [\textit{T. Hayakawa}, Publ. Res. Inst. Math. Sci 35, 515-570 (1999; Zbl 0969.14008)], we introduced the notion of pseudo weighted valuation, which consists of an embedding \(j:X\hookrightarrow \mathbb{C}^4/ \mathbb{Z}_m\) and a weight \(\sigma\). By using these data, we blow up \(X\) and get divisorial blow ups of \(X\) with small discrepancies. We also showed that, in most cases, there is a one-to-one correspondence between these divisorial blow ups and a certain set of pseudo weighted valuations, and remarked that this correspondence does not necessarily hold in the case \(X\) is of type \((cD/2)\). Our purpose here is to study the blowing ups of terminal singularities of type \((cD/2)\) and to determine all divisorial blow ups of \(X\) with discrepancies \(1/2\). The paper under review combined with the results in part I, covers all 3-dimensional terminal singularities of indices \(m\geq 2\). Indeed, if \(X\) is a 3-dimensional terminal singularity of index \(m\geq 2\), we can determine all prime divisors with discrepancies \(1/m\) and in particular we obtain the following:
Theorem 1. Let \(X\) be a germ of a 3-dimensional terminal singularity of index \(m\geq 2\). Then there exists at least one divisorial blow up \(\pi:\overline X\to X\) with discrepancy \(1/m\). Furthermore \(\pi\) does not increase axial weights.
As a consequence of this result, we obtain the following:
Theorem 2. Let \(X\) be a germ of a 3-dimensional terminal singularity of index \(m\geq 2\). Then there is a projective birational morphism \(\psi:Y\to X\) such that
(i) \(Y\) has only Gorenstein terminal singularities, and
(ii) \(\psi\) is a composition of divisorial blow ups of points of indices \(\geq 2\) such that their discrepancies are minimal. discrepancy; blowing ups; 3-dimensional terminal singularities; pseudo weighted valuation T. Hayakawa, Blowing ups of \(3\)-dimensional terminal singularities, II, Publ. Res. Inst. Math. Sci. 36 (2000), 423--456. Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds Blowing ups of 3-dimensional terminal singularities. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author give an explicit geometric description to some of H. Nakajima's quiver varieties. More precisely, let \(X = {\mathbb{C}}^2\), \(\Gamma \subset \text{SL}({\mathbb{C}}^2)\) be a finite subgroup, and \(X_\Gamma\) be a minimal resolution of \(X/\Gamma\). The main result states that \(X^{\Gamma [n]}\) (the \(\Gamma\)-equivariant Hilbert scheme of \(X\)) and \(X_\Gamma^{[n]}\) (the Hilbert scheme of \(X_\Gamma\)) are quiver varieties for the affine Dynkin graph corresponding to \(\Gamma\) via the McKay correspondence with the same dimension vectors but different parameters.
In section two, basic concepts such as the definition of quivers, quiver varieties, representation of quivers and the construction of Crawley-Boevey were reviewed. In section three, the author reproduced in a short form a geometric version of the McKay correspondence based on investigation of \(X_\Gamma\), and proved a generalization of certain result of \textit{M. Kapranov} and \textit{E. Vasserot} [Math. Ann. 316, No. 3, 565--576 (2000; Zbl 0997.14001)]. The main result mentioned above was verified in section four. In particular, it follows that the varieties \(X^{\Gamma [n]}\) and \(X_\Gamma^{[n]}\) are diffeomorphic. In section five, \(({\mathbb{C}}^* \times {\mathbb{C}}^*)\)-actions on \(X^{\Gamma [n]}\) and \(X_\Gamma^{[n]}\) for cyclic \(\Gamma \cong {\mathbb{Z}}/d {\mathbb{Z}}\) were considered. The author proved the combinatorial identity \(UCY(n, d) = CY(n, d)\) where \(UCY\) and \(CY\) denote the number of uniformly colored diagrams and the number of collections of diagrams respectively. quiver varieties; Hilbert schemes; McKay correspondence; moduli space Kuznetsov, A.: Quiver varieties and Hilbert schemes. Moscow Math. J. \textbf{7}, 673-697 (2007). arXiv:math.AG/0111092 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Quiver varieties and Hilbert schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite subgroup of \(\text{SL}(n,\mathbb{C})\). The generalized McKay correspondence aims to relate the geometry of crepant (i.e. with trivial canonical divisor) resolutions of singularities of the quotient \(\mathbb{C}^n/G\) to the representations of the group \(G\). This paper deals with the natural candidate given by the Hilbert scheme of \(G\)-regular orbits introduced by \textit{I. Nakamura} [J. Algebr. Geom. 10, No.~4, 757--779 (2001; Zbl 1104.14003)], parametrizing generalized \(G\)-orbits on \(\mathbb{C}^n\), denoted by \(G\text{-Hilb}(\mathbb{C}^n)=:Y\). By a theorem of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], for \(n=3\) this provides the required resolution of singularities. The McKay correspondence is realized as follows: there exists a natural integral basis of the Grothendieck group \(K(Y)\) given by natural bundles \(\mathcal{R}_k\) indexed by the irreducible representations of \(G\). This provides, through Chern character, a rational basis of the cohomology \(H^*(Y,\mathbb{Q})\) in one-to-one correspondence with the irreducible representations of \(G\). It is still an open problem to get a similar correspondence for the integral cohomology \(H^*(Y,\mathbb{Z})\). Reid conjectured that some ``cookery'' with the Chern classes of these bundles \(\mathcal{R}_k\) should provide an integral basis. This paper establishes explicitly this integral McKay correspondence for all abelian subgroups \(A\) in \(\text{SL}(3,\mathbb{C})\) (Theorem 1.1). The method follows the recipe introduced by Reid, uses previous work of \textit{Y. Ito, H. Nakajima} [Topology 39, No.~6, 1155--1191 (2000; Zbl 0995.14001)] and an explicit algorithm of computation of \(A\text{-Hilb}(\mathbb{C}^3)\) already described by \textit{A. Craw} and \textit{M. Reid} [in: Geometry of toric varieties. Lect. summer school. Grenoble. 2000, Sémin. Congr. 6, 129--154 (2002; Zbl 1080.14502)] and extending the initial work of Nakamura [loc.cit.], based upon a decoration of the toric fan of \(A\text{-Hilb}(\mathbb{C}^3)\) with the characters of the group \(A\). The integral basis of \(H^2(Y,\mathbb{Z})\) is then given by the first Chern classes of some \(\mathcal{R}_k\)'s indexed by specific non-trivial characters (Proposition 7.1). In order to base \(H^4(Y,\mathbb{Z})\), the author computes all relations between the line bundles (since the group is abelian) \(\mathcal{R}_k\) in \(\text{Pic}(Y)\), and introduces a family of virtual bundles \(\mathcal{V}_m\) indexed by the remaining non-trivial irreducible representations, whose second Chern classes will give the expected integral basis (Proposition 7.3). Hilbert scheme of orbits; toric geometry Craw, A., An explicit construction of the McKay correspondence for \(A\)-Hilb \({\mathbb{C}^3}\), J. Algebra, 285, 682-705, (2005) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(3\)-folds, Toric varieties, Newton polyhedra, Okounkov bodies, Classical real and complex (co)homology in algebraic geometry, Ordinary representations and characters An explicit construction of the McKay correspondence for \(A\)-Hilb \(\mathbb C^3\) | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\pi\) :\(X\to Y\) be a proper birational morphism between surfaces with rational double points. We first show that \(\pi\) can be factored into a sequence of ''elementary'' morphisms, each of which contracts a smooth rational curve to a point. (This is a special case of a theorem of F. Sakai.) We then study conormal sheaves of smooth curves on surfaces with rational double points, and give necessary and sufficient conditions for an ''elementary'' contraction to exist in terms of conormal sheaves. This generalizes the classical case of smooth surfaces, in which the ''elementary'' morphism is the blow-up of a smooth point, and a smooth rational curve C can be contrasted to a smooth point if and only if its conormal sheaf is \({\mathcal O}_ C(1)\). The paper also contains an application of these results to the study of Gorenstein threefold singularities with small resolutions. decomposition of birational morphism; smooth curves on surfaces with rational double points; blow-up of a smooth point; Gorenstein threefold singularities D. Morrison: ''The birational geometry of surfaces with rational double points'', Math. Ann., Vol. 271, (1985), pp. 415--438. Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry The birational geometry of surfaces with rational double points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Translation from Mat. Zametki 35, No.4, 579-588 (Russian) (1984; Zbl 0543.58011). topological classification; germs of functions; singularities Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Topological equivalence of germs of functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0607.00005.]
In Hironaka's resolution of singularities process, as the author explains, there are two important parts: the first one consist to find good numerical invariants and the second one to prove that these numerical invariants are improved by a finite number of permissible blow ups. The author makes a good review of the question in \(characteristic p\neq 0\), collects together results on infinitely very near points due to Bennett, Giraud, Hironaka and others.
Let \(\pi\) : Z'\(\to Z\), Z regular, be the blowing up along a center \(Y\subset X\subset Z\) permissible for X, i.e. Y is regular and X is normally flat along Y, X' is the proper transform and x'\(\in X'\), \(\pi (x')=x\), then the main result consists in choosing a good standard base for \(in_ x(X,Z)\) (equations for the tangent cone of X in Z) and for \(in_{x'}(X',Z')\) related to each other in a nice manner if x' is an infinitely very near point of x. resolution of singularities; characteristic p; infinitely very near point Tadao Oda, Infinitely very near singular points, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam, 1987, pp. 363 -- 404. Global theory and resolution of singularities (algebro-geometric aspects), Infinitesimal methods in algebraic geometry, Singularities in algebraic geometry Infinitely very near singular points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article gives a classification (within an isotopy in the class of algebraic surfaces) for surfaces of degree 4 with a nonsimple singularity. All connected components of the space of surfaces (according to their sets of singularities) are listed (4184 components realizing 2523 configurations of singularities).
Essential tool is the projection from a fixed singular point of the surface. Closely related are questions on curves of degree 6 (many of them can be obtained as sets of critical values of such projections. The author obtains the following theorem: The rigid isotopy type of a curve of degree 6 having at least one nonsimple singular point is determined by the set of singularities. curve of degree 6; quartic surface; nonsimple singularity; rigid isotopy type Degtyarev, AI, Classification of surfaces of degree four having a nonsimple singular point, Math. USSR-Izv., 35, 607-627, (1990) Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Special surfaces, Singularities in algebraic geometry, Complex surface and hypersurface singularities Classification of surfaces of degree four having a nonsimple singular 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 The article under review contributes to the structure theory of minimal compact complex surfaces \(S\) in class VII\(_0^+\) (\(b_1(S)=1, \kappa(S)=-\infty, b_2(S)=:n>0\)) with global spherical shells. The latter means that there exists a biholomorphic embedding \(\varphi:U\rightarrow S\) of an open neighborhood \(U\subset\mathbb C^2\backslash\{0\}\) of the 3-sphere \(S^3\) such that \(S\backslash\varphi(S^3)\) is connected. The author studies the (normal) singularities obtained by blowing down the maximal divisor \(D=\sum_{i=0}^{n-1}D_i\), the sum of the \(n\) irreducible rational curves \(D_i\) on \(S\). These singularities are of genus \(1\) or \(2\) and are \(\mathbb Q\)-Gorenstein if and only if \(H^0(S,K^{-m}_S)\not=0\) for some \(m\geq 1\). They are numerically Gorenstein if and only if \(H^0(S,K_S^{-1}\otimes L)\not=0\) for some topologically trivial line bundle \(L\) on \(S\). A central part of the paper is devoted to the description of the discriminant of the quadratic form associated to a singularity. Let \(M(S)=(D_i.D_j)\) be the self-intersection matrix of \(D\). The singularities can be parametrized by the configuration of their dual graphs, represented by finite sequences \(\sigma\) of integers.
The author introduces a family of polynomials \(P_\sigma\) in \(N=N(\sigma)\) variables, \(P_\sigma(\mathbb Z^N)\subset\mathbb Z\), and numbers \(k_i=k_i(\sigma)\in\mathbb N\), \(0\leq i\leq N-1\). He obtains the existence of a sequence \(\sigma\) with the property that \(\det M(S)=(-1)^n(P_\sigma(k_0,\dots,k_{N-1}))^2\) and \([H_2(S,\mathbb Z):\sum_{i=0}^{n-1}\mathbb ZD_i]=P_\sigma(k_0,\dots,k_{N-1})\). The number \(\triangle_\sigma:=P_\sigma(k_0,\dots,k_{N-1})+1\), the so-called twisting coefficient of the singularity, is a multiplicative invariant, i.e. \(\triangle_{\sigma\sigma'}=\triangle_\sigma\triangle_{\sigma'}\), and equals the product of the determinants of the intersection matrices of the branches of the divisor \(D\). The author points out the close connection of these invariants to global topological and analytical properties of surfaces with global spherical shells and to the classification of singular contracting germs of mappings and dynamical systems, see e.g. [\textit{G. Dloussky} and \textit{K. Oeljeklaus}, Ann. Inst. Fourier 49, No. 5, 1503--1545 (1999; Zbl 0978.32021)] and [\textit{Ch. Favre}, J. Math. Pures Appl., IX. Sér. 79, No. 5, 475--514 (2000; Zbl 0983.32023)]. minimal compact complex surfaces in class \(\text{VII}_0^+\); global spherical shells; singularities; \(\mathbb Q\)-Gorenstein; numerically Gorenstein; twisting coefficient [7] Dloussky G., Quadratic forms and singularities of genus one or two. Annales de la faculté des sciences de Toulouse vol 20 (2011), p15-69. Minimal model program (Mori theory, extremal rays), Compact complex surfaces, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Rational and birational maps Quadratic forms and singularities of genus one or two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a regular irreducible scheme of finite type over a field \(k\).
Studying the resolution of singularities Hironaka introduced idealistic exponents \(\mathbb E=(J,b)\) on \(X\) as a pair consisting of a quasi-coherent ideal sheaf \(J\) on \(X\) and a positive integer \(b\in \mathbb Z\).
Based on Hironaka's work idealistic exponents are studied over Spec\((\mathbb Z)\). An idealistic interpretation of the tangent cone, the directrix and the ridge
is given. The notion of characteristic polyhedra od idealistic exponents is given and studied. singularities; idealistic exponents; characteristic polyhedra; Newton polyhedra; resolution of singularities Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Idealistic exponents: tangent cone, ridge, characteristic polyhedra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article is an introductory survey on symplectic reflection algebras written mostly from the algebraic point of view. Here symplectic reflection algebras are introduced as deformations of orbit space singularities and this is used as motivation for most of the constructions and results of the paper. The latter include classification of symplectic singularities admitting symplectic resolutions; category \(\mathcal O\), the KZ-functor, highest weight covers and finite Hecke algebras; derived equivalences for some quiver varieties; the construction of quantizations of Hilbert schemes of points on the plane and associated geometric interpretations of representations of symplectic reflection algebras. Special cases of deformed preprojective algebras, rational Cherednik algebras and the symmetric group are discussed. symplectic reflection algebras; orbit space singularities; category \(\mathcal O\); KZ-functors; Hecke algebras; derived equivalences; Hilbert schemes; rational Cherednik algebras; deformations Gordon, I.G., Symplectic reflection algebras, (Trends in representation theory of algebras and related topics, EMS ser. congr. rep., (2008), Eur. Math. Soc. Zürich), 285-347 Associative rings and algebras arising under various constructions, Representations of quivers and partially ordered sets, Quantum groups (quantized enveloping algebras) and related deformations, Hecke algebras and their representations, Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Deformations of associative rings Symplectic reflection algebras. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The existence problem for unimodular triangulations of lattice polytopes is investigated. These triangulations correspond to the crepant resolutions of toric singularities -- resolutions that preserve the canonical divisor.
After the introduction of the basic concepts and tools, the thesis is divided into three chapters. Chapter 2. The so-called empty lattice simplices are the obstacles to a unimodular triangulation. It is known that their lattice width is bounded by a constant \(w(d)\) that only depends on the dimension. Another constant, \(W(d)\) -- the maximal width of almost empty simplices, is introduced. The construction of an infinite family of \(d\)-dimensional empty simplices out of an almost empty \((d-1)\)-dimensional simplex shows the monotonicity of both constants and disproves a conjecture of Bárány.
A computer search in dimension 4 yields exactly one empty simplex of width 4 and suggests that the determinant of empty width 3 simplices does not exceed 179. Together with a proof of \(W(3)=2\) this supports a modified conjecture.
Chapter 3. A unimodular triangulation is constructed for the polytopes that are associated with toric local complete intersections, thus generalizing a result of \textit{D. I. Dais, M. Henk} and \textit{G.-M. Ziegler}
[Adv. Math. 139, No.2, 194--239 (1998; Zbl 0930.14006)]. Furthermore, these polytopes are shown to have the Koszul property.
Chapter 4. The string theoretic Hodge numbers of \textit{V. V. Batyrev} and \textit{D. I. Dais} [Topology 35, No. 4, 901--929 (1996; Zbl 0864.14022)]
are computed for two series of (hypersurfaces in the projective toric varieties corresponding to) reflexive polytopes. The first series is given by the pseudo symmetric Fano polytopes. Their polar duals give rise to smooth hypersurfaces, so by mirror-symmetry, formulae of
\textit{V. I. Danilov} and \textit{A. G. Khovanskij} [Math. USSR, Izv. 29, 279-298 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 5, 925--945 (1986; Zbl 0669.14012)] can be used. These dual polytopes admit unimodular triangulations and they have the Koszul property. The second series consists of pyramids over reflexive polytopes. In this case one really has to use the stringy version. Haase, C.: Lattice polytopes and unimodular triangulations, Ph.D. thesis, Technical University of Berlin (2000) Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Global theory and resolution of singularities (algebro-geometric aspects), Complete intersections, Singularities in algebraic geometry Lattice polytopes and triangulations. With applications to toric geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be a field and \(X=(x_{ij})\) a \((m\times n)\)-matrix of indeterminates over \(K\), \(n\geq m\). With \(S=K[x_{ij}]\), \(X\) determines the generic \(S\)-linear map \(\phi:S^n\rightarrow S^m.\) Let \(\text{Spec}R\) be the locus in \(\text{Spec}S\) where \(\phi\) has non-maximal rank: \(R\) is the quotient of \(S\) given by the maximal minors of \(X\), and is the generic determinantal variety.
The classical \(R\)-modules \(M_a=\text{cok}\bigwedge^a_S\phi\) are maximal Cohen-Macaulay and are resolved by the Buchsbaum-Rim complex. In this article the authors prove that the \((M_a)_a\) yields a kind of non-commutative desingularization of the singular variety \(\text{Spec} R\): For \(1\leq a\leq m\) put \(M_a=\text{cok}\bigwedge^a_S\phi\) and \(M=\bigoplus_a M_a\). Then \(E=\text{End}_R(M)\) is maximal Cohen-Macaulay as an \(R\)-module with finite global dimension. That is, \(E\) is a non-commutative desingularization of \(\text{Spec} R\).
If \(m=n\) then \(R\) is the hypersurface \(R=S/(\det\phi)\) and so \(R\) is Gorenstein and the non-commutative desingularization is an example of a non-commutative crepant resolution.
The authors give a description by generators and relations of the non-commutative resolution \(E\) by stating that \(E\) as a \(K\)-algebra is isomorphic to the path algebra \(K\tilde Q\) of some quiver \(\tilde Q\).
The results above are purely algebraic, but are proved by relating them to algebraic geometry. The classical fact that \(\text{Spec} R\) has a Springer type resolution of singularities is frequently used: Define the incidence variety
\[
\mathcal Z=\{([\lambda],\theta)\in\mathbb P^{m-1}(K)\times M_{m\times n}(K)|\lambda\theta=0\}
\]
with projections \(p^\prime:\mathcal Z\rightarrow\mathbb P^{m-1}\) and \(q^\prime:\mathbb Z\rightarrow\text{Spec} R\). The key geometric facts then include: The scheme \(\mathcal Z\) is projective over \(\text{Spec} R\), which is of finite type over \(K\). The \(\mathcal O_{\mathcal Z}\)-module
\[
\mathcal T := p^{\prime\ast}\left(\bigoplus^m_{a=1}\left(\bigwedge^{a-1}\Omega_{\mathbb P^{m-1}}\right)(a)\right)
\]
is a classical tilting bundle on \(\mathcal Z\) , i.e.
(1) \(\mathcal T\) is a locally free sheaf, in particular, a perfect complex on \(\mathcal Z\),
(2) \(\mathcal T\) generates the derived category \(\mathcal D(\text{Qch}(\mathcal Z))\); \(\text{Ext}^\bullet_{\mathcal O_{\mathcal Z}}(\mathcal T, C)=0\) for a complex \(C\) in \(\mathcal D(\text{Qch}(\mathcal Z))\) implies \(C\cong 0\),
(3) \(\text{Hom}_{\mathcal O_{\mathcal Z}}(\mathcal T,\mathcal T[i])=0\) for \(i\neq 0\),
(4) \(M\cong \mathbf{R}q^\prime_\ast\mathcal T\) and
(5) \(E\cong\text{End}_{\mathcal Z}(\mathcal T)\).
These geometric considerations leads to an interesting and important result stating that the variety \(\mathcal Z\) is the fine moduli space for the \(\tilde Q\)-representations \(W\) of dimension vector \((1,m-1,\left(\begin{smallmatrix} m-1\\2\end{smallmatrix}\right),\dots,1)\) that are generated by the last component \(W_m\).
The proofs of the results depends mostly on the explicit computation of the cohomology of certain homogeneous bundles on \(\mathcal P^{m-1}\), determination of higher direct images of twisted bundles of homomorphisms between the modules of differential forms and other technical results. The article is more or less self contained, containing e.g. the construction of the projective tautological Koszul complex. In addition, of interest in itself is a construction of projective resolutions from sparse spectral sequences. This is then used in to construct the non-commutative desingularization \(E\) above, with algebra structure given by the quiverized Clifford algebra and its presentation.
Particularly nice is the treatment of the noncommutative desingularization as a moduli space for representations. It is really interesting to notice that the points in \(\mathcal Z\) corresponding to the simple representations in \(W\) as those lying over the non-singular locus of \(\text{Spec} R\).
The article is strongly recommended to anyone who will understand this level of representation theory in the algebraic geometric view. Be prepared to use a lot of effort to go through all proofs in detail. non-commutative desingularization; sparse spectral sequence; simple representations; tautological Koszul complex; maximal Cohen-Macaulay; quivers; Clifford algebra Abhyankar, S.: Uniformization in a \( p\)-cyclic extension of a two dimensional regular local domain of residue field characteristic \( p\) . Festschrift zur Gedächtnisfeier für Karl Weierstrass 1815 - 1965, Wissenschaftliche Abhandlungen des Landes Nordrhein-Westfalen \textbf{33} (1966), 243-317, Westdeutscher Verlag, Köln und Opladen Noncommutative algebraic geometry, Cohen-Macaulay modules, Global theory and resolution of singularities (algebro-geometric aspects), Riemann-Roch theorems, Rings arising from noncommutative algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Representations of quivers and partially ordered sets Non-commutative desingularization of determinantal varieties. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smoothing of a rational surface singularity and \(K_X\) the canonical divisor. J. Kollár conjectured that the canonical algebra \(\bigoplus_k \mathcal O_X(kK_X)\) is finitely generated. Here the author studies Kollár's conjecture for sandwiched singularities. It is shown that the canonical algebra of sandwiched singularities is finitely generated when the symbolic algebra of a curve in three-space is finitely generated. This could be useful to prove Kollár's conjecture for sandwiched singularities. rational surface singularity; Kollár's conjecture; symbolic algebra; modifications Deformations of singularities, Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties On the canonical algebra of smoothings of sandwiched 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 resolution of singularities in characteristic \(p>0\) or mixed characteristic is still an open problem. There are only results in small dimension and special cases.The author discusses to what extent local techniques of resolution of singularities in characteristic zero can be applied to positive characteristic. One application is the embedded resolution of determinantal varieties. The article is expository and also addressed to non-experts. resolution of singularities; idealistic exponents; positive characteristic; determinantal varieties Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Determinantal varieties Partial local resolution by characteristic zero methods | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0742.00082.]
The author describes the deformation theory of a simple singularity of type \(E_ 8\) from his point of view. The concept of the universal polynomial of type \(E_ 8\) plays the central role. It is a polynomial of degree 240 (= the number of root vectors in the root system of type \(E_ 8\)) with coefficients in the field of rational functions with 8 variables, and has the Galois group isomorphic to the Weyl group of type \(E_ 8\). Fix a point in the parameter space of the semi-universal deformation family of the simple singularity of type \(E_ 8\) and consider the fiber lying over it. Every fiber has a natural compactification \(S\). \(S\) has a structure of a rational elliptic surface with a section and with a singular fiber \(F\) of type II. \(F\) is isomorphic to the plane nodal cubic curve. Obviously the class \([F]\) of the singular fiber \(F\) in the Picard group \(\text{Pic}(S)\) of \(S\) satisfies \([F]^ 2=0\). The orthogonal complement \(L^*\) of \(L=\mathbb{Z}[F]\) contains \(L\) and the quotient module \(L^*/L\) with the induced bilinear form is isomorphic to the root lattice \(Q(E_ 8)\) of type \(E_ 8\) up to the signature of the bilinear forms. Thus the root system of \(L^*/L\), i.e., the collection of elements \(x\) in \(L^*/L\) with \(x^ 2=-2\), has 240 elements, which are called root vectors. The restriction morphism \(\text{Pic}(S)\to\text{Pic}(F)\) induces a morphism \(L^*/L\to\text{Pic}^ 0(F)\cong\mathbb{C}\) and we have 240 numbers in \(\mathbb{C}\) as the images of 240 root vectors. By definition the universal polynomial is the polynomial with these 240 numbers as roots. Its coefficients depend on the values of the parameters. The author shows that the every coefficient of this polynomial is a polynomial with rational coefficients of the parameters of the deformation family.
He told me that he would like to apply this concept to the non-abelian arithmetic theory of the field of rational numbers and the field of rational functions. However, this application is not developed in this article. universal polynomial of type \(E_ 8\); semi-universal deformation family of the simple singularity of type \(E_ 8\); universal polynoial of type \(E_ 8\) T. Shioda, Mordell-Weil lattices of type E 8 and deformation of singularities, in: Lecture Notes in Math. 1468 (1991), 177-202. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Mordell-Weil lattices of type \(E_ 8\) and deformation 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 We give a complete system of invariants for the topological conjugacy of polynomials of \(\mathbb{C}^2\) outside a big enough compact set in the two possible versions: as foliations (forgetting the values of the fibers) and as functions. These invariants are described as a weighted and colored tree, that is obtained after reduction of singularities of the polynomial in the line of infinity. We give regularity criterion for the values of a polynomial and a description of the topology of its fibers used in the construction of the topological conjugacy from the tree. singularities; resolution of singularities; algebraic plane curves; knots and links L. Fourrier, \(Topologie d'un polynome de deux variables complexes au voisinage de l'infini\), Annales de l'institut Fourier, 46 (1996), 645-687 Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Knots and links in the 3-sphere Topology of a polynomial of two complex variables at the neighbourhood of infinity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities From the introduction: ''Let \(X\) be a variety over a field \(k\). An alteration of \(X\) is a dominant proper morphism \(X'\to X\) of varieties over \(k\), with \(\dim X=\dim X'\). We prove that any variety has an alteration which is regular. This is weaker than resolution of singularities in that we allow finite extensions of the function field \(k(X)\). In fact, we can choose \(X'\) to be a complement of a divisor with strict normal crossings in some regular projective variety \(\overline X'\) (see theorem 4.1 and remark 4.2.). If the field \(k\) is local, we can find \(X' \subset \overline X'\) such that \(\overline X'\) is actually defined over a finite extension \(k\subset k'\) and has semi-stable reduction over \({\mathcal O}_{k'}\) in the strongest possible sense (see theorem 6.5).
As an application, we note that theorem 4.1 implies that for any variety \(X\) over a perfect field \(k\), there exist \((\alpha)\) a simplicial scheme \(X_\bullet\) projective and smooth over \(k\), \((\beta)\) a strict normal crossings divisor \(D_\bullet\) in \(X_\bullet\); we put \(U_\bullet=X_\bullet\setminus D_\bullet\), and \((\gamma)\) an augmentation \(a:U_\bullet\to X\) which is a proper hypercovering of \(X\). In case \(k\) is local, we may assume that the pairs \((X_n,D_n)\) are defined over finite extensions \(k_n\) of \(k\) and extend to strict semi-stable pairs over \({\mathcal O}_{k_n}\) (see 6.3). This should be interpreted as saying that in a suitable category \({\mathcal M}{\mathcal M}_k\) of mixed motives over \(k\), any variety \(X\) may be replaced by a complex of varieties which are complements of strict normal crossing divisors in smooth projective varieties.
We give a short sketch of the argument that proves our results in case \(X\) is a proper variety. The idea is to fibre \(X\) over a variety \(Y\) such that all fibres are curves and work by induction on the dimension of \(X\). After modifying \(X\), we may assume \(X\) is projective and normal and we can choose the fibration to be a kind of Lefschetz pencil, where the morphism is smooth generically along any component of any fibre. Next one chooses a sufficiently general and sufficiently ample relative divisor \(H\) on \(X\) over \(Y\). After altering \(Y\), i.e. we take a base change with an alteration \(Y'\to Y\), we may assume that \(H\) is a union of sections \(\sigma_i:Y\to X\). The choice of \(H\) above gives that for any component of any fibre of \(X\to Y\), there are at least three sections \(\sigma_i\) intersecting it in distinct points of the smooth locus of \(X\to Y\). The generic fibre of \(X\to Y\), together with the points determined by the \(\sigma_i\) is a stable pointed curve. By the existence of proper moduli spaces of stable pointed curves, we can replace \(Y\) by an alteration such that this extends to a family \({\mathcal C}\) with sections \(\tau_i\) of stable pointed curves over \(Y\). An important step is to show that the rational morphism \({\mathcal C}\cdots\to X\) extends to a morphism, possibly after replacing \(Y\) by a modification; this follows from the condition on sections hitting components of fibres above. Thus we see that we may replace \(X\) by \({\mathcal C}\). We apply the induction hypothesis to \(Y\) and we get \(Y\) regular. However, our induction hypothesis is actually stronger and we may assume that the locus of degeneracy of \({\mathcal C}\to Y\) is a divisor with strict normal crossings. At this point it is clear that the only singularities of \({\mathcal C}\) are given by equations of the type \(xy=t_1^{n_1} \cdot\dots \cdot t_d^{n_d}\). These we resolve explicitly.
Section 2 contains definitions and results, which we assume known in the rest of the paper. In section 3 we resolve singularities for a family of semi-stable curves over a regular scheme, which is degenerate over a divisor, with normal crossings. This we use in section 4, where we prove the theorem on varieties. Section 5 deals with the problem of altering a family of curves into a family of semi-stable curves. This we use in section 6, where we do the relative case, i.e. the case of schemes over a complete discrete valuation ring.
In the final two sections we indicate how to refine the method of proof of theorem 4.1 and theorem 6.5 to get results where one has additional restraints or works over other base schemes. In section 7 we prove that our method works (over algebraically closed fields) to get resolution of singularities up to quotient singularities and purely inseparable function field extensions. In fact we deal with the situation where there is a finite group acting. In section 8 we do the arithmetic case. In particular, we show that any integral scheme \(X\), flat and projective over \(\text{Spec} \mathbb{Z}\) can be altered into a scheme \(Y\) which is semi-stable over the ring of integers in a number field (theorem 8.2).
In a follow-up of this article the author proves that one can alter any family of curves into a semi-stable family of curves [see \textit{A. J. de Jong}, Ann. Inst. Fourier 47, No. 2, 599-621 (1977; Zbl 0868.14012)]. This is stronger than the result of section 5. In this cited paper the author deals with group actions as well. Thus the reader can find therein a number of results that extend the results of this article to (slightly) more general situations. For example it is shown that regular alterations exist of schemes of finite type over two-dimensional excellent base schemes.
A.Nobile:
This very important article describes a solution to a weak version of the desingularization problem for algebraic varieties over a field, as well as generalizations. The key concept to express the main results is that of alteration. An alteration of an integral scheme \(S\) is an integral scheme \(S'\) together with a morphism \(\varphi:S'\to S\) which is surjective, proper and such that, for a suitable open dense set \(U\subseteq X\), the induced morphism \(\varphi_U:\varphi^{-1}(U)\to U\) is finite.
The basic main result of this paper says: Given a variety \(X\) over an arbitrary field \(k\) and a proper closed set \(Z\subset X\), then there is an alteration \(\varphi:X_1\to X\) such that \(X_1\) is an open set of a regular scheme \(X_1'\), projective over \(k\), such that \(\varphi^{-1}(Z) \cup(X_1'-X_1)\) is a strict normal crossings divisor \(D\) of \(X_1'\) (i.e., the irreducible components of \(D\) are regular and meet transversally). There are also: (a) a \(G\)-equivariant version, where \(G\) is a finite group acting on \(X\), (b) a ``relative'' version, where \(X\) is an irreducible, separated scheme, flat and of finite type over \(S=\text{Spec}\,R\), with \(R\) a complete discrete valuation ring; (c) an arithmetic version, where the basic situation is as in part (b), but now \(R\) is a Dedekind domain whose field of fractions is a global field. -- The result of (b) may be interpreted as ``weak'' semistable reduction theorem (weak because certain alterations are allowed), without restrictions on the characteristics involved. As a tool to obtain (b) or (c), some interesting results about improving a family of curves via alterations are discussed.
This is the main technique to show the relevant theorems, sketched in the case of varieties over a field.
Moreover, to simplify \(X\) is assumed projective. One tries to find an alteration \(f:X'\to X\) such that there is flat morphism \(g:X'\to T\), with \(T\) regular, \(\dim(T)= \dim (X')-1\), such that the general fiber of \(g\) is regular and any fiber is a curve with, at worst, ordinary double points as singularities, moreover the set of points of \(T\) where \(g^{-1}(t)\) is singular is contained in a strict normal crossings divisor. Then, to desingularize such a \(X'\) by means of monoidal transformations is easy. An alteration as above is obtained by using some classical projective techniques, the theory of moduli for pointed semi-stable curves and an induction hypothesis (applied to \(T\), whose dimension is one less than that of \(X)\).
These results, although in a sense weak (because they involve alterations and not birational morphisms) are strong enough to solve a number of cohomological problems that ``require'' resolutin) of singularities. Some are described in the introduction. See also: \textit{P. Berthelot}, ``Altérations de varietés algébriques, Séminaire Bourbaki, Volume 1995/96, Exposé No. 815, Astérisque 24l, 273--311 (1997; Zbl 0924.14087)]. alteration; resolution of singularities; semi-stable family of curves de Jong, A. J., Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci., 83, 51-93, (1996) Global theory and resolution of singularities (algebro-geometric aspects), Families, moduli of curves (algebraic), Singularities in algebraic geometry Smoothness, semi-stability and alterations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Authors' abstract: A tame ideal is an ideal \(I\subseteq k[x_1,\ldots,x_n]\) such that the blowup of the affine space \(\mathbb{A}^n_k\) along \(I\) is regular. In this paper, we give a combinatorial characterization of tame square free monomial ideals. More precisely, we show that a square free monomial ideal is tame if and only if the corresponding clutter is a complete \(d\)-partite \(d\)-uniform clutter. Equivalently, a square free monomial ideal is tame, if and only if the facets of its Stanley-Reisner complex have mutually disjoint complements. Also, we characterize all monomial ideals generated in degree at most \(2\) which are tame. Finally, we prove that tame square free ideals are of fiber type. blowup algebra; resolution of singularity; tame ideal; clutter Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Symbolic computation and algebraic computation, Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Singularities in algebraic geometry On the blowup of affine spaces along monomial ideals: tameness | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a report about some results on uniformization of singularities in characteristic \(p\). Given any singularity defined by the equation: \(T^p-f (x,y,z)=0\) there is a number \(n\) such that along any valuation, the multiplicity will be smaller in \(n\) steps of blow-ups. The main tool is the Newton polygon. The same equation was studied by \textit{V. Cossart} in his Ph. D. thesis (Orsay 1987) and probably the proof of canonical uniformization in characteristic \(p\) announced by Spivakovsky works in this case, too. characteristic \(p\); uniformization of singularities; Newton polygon Moh, T.T.: On a Newton polygon approach to the uniformization of singularities of characteristic \(p\). In: Algebraic geometry and singularities (La Rábida, 1991), 49-93, Progr. Math.\textbf{134}, Birkhäuser (1996) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Valuation rings, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure On a Newton polygon approach to the uniformization of singularities of characteristic \(p\) | 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 Although the problem of the existence of a resolution of singularities in characteristic zero was already proved by Hironaka in the 1960s and although algorithmic proofs of it have been given independently by the groups of Bierstone and Milman and of Encinas and Villamayor in the early 1990s, the explicit construction of a resolution of singularities of a given variety is still a very complicated computational task. In this article, we would like to outline the algorithmic approach of Encinas and Villamayor and simultaneously discuss the practical problems connected to the task of implementing the algorithm. Frühbis-Krüger, A., Pfister, G., October 2004. Practical aspects of algorithmic resolution of singularities. Tech. Rep. 33, Centre for Computer Algebra, University of Kaiserslautern, online available at http://www.mathematik.uni-kl.de/~zca Global theory and resolution of singularities (algebro-geometric aspects), Computational aspects of higher-dimensional varieties, Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Algorithmic resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The global and local topological zeta functions are singularity invariants associated to a polynomial \(f\) and its germ at 0, respectively. By definition, these zeta functions are rational functions in one variable and their poles are negative rational numbers.
In this paper we study their poles of maximal possible order. When \(f\) is non degenerate with respect to its Newton polyhedron we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; concerning the global zeta function we give a similar result. Moreover for any \(f\) we show that poles of maximal possible order are always of the form \(-1/N\) with \(N\) a positive integer. topological zeta functions; singularity invariants Grothendieck, A.: \textit{Revêtements étales et groupe fondamental}. Séminaire de géométrie algébrique du BoisMarie 1960-61 (SGA 1). Updated and annotated reprint of the 1971 original. Vol. 3 of Documents Mathématiques (Paris). Société Mathématique de France, Paris (2003) Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On the poles of maximal order of the topological zeta function | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For classifying singularities, suitable sets of invariants are required. In the case of curve singularities, the invariants one often uses have a valuative nature, since one has a canonically associated finite set of valuations. Thus, the semigroup of values of a plane curve singularity characterizes Zariski's equisingularity type, and the semigroup of the Arf closure (resp. the saturation) characterizes the multiplicity sequence of the resolution process (resp. the equisingularity type of a generic plane projection) for space curve singularities. Saturation has been introduced by Zariski and by Pham-Teissier in a different way. One of the authors gave another definition of saturation which is very appropriate to handle the value semigroups arithmetically [\textit{A. Campillo}, in Singularities, Summer Inst. Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 1, 211-220 (1983; Zbl 0553.14013 and in Singularities, Banach Cent. Publ. 20, 121-137 (1988; see the preceding review)]. As a consequence the semigroups of one dimensional C. M. local rings have a nice structure which is very much simpler than that of the semigroups of plane curves and, even, they have a more natural geometrical interpretation. This nice structure follows from Arf property which is satisfied by the saturated rings. In this communication, we show how the definitions of Arf and saturation can be extended to the relative situation of an arbitrary (local) ring and finitely many discrete valuations of it, in such a way that the actual semigroups also have a nice structure providing reasonable invariants. As an application, one obtains a classification method for singularities having a canonical resolution. classifying singularities; space curve singularities; saturation; value semigroups; semigroups of one dimensional C. M. local rings; canonical resolution Campillo, A.; Delgado, F.; Núñez, C. A.: The arithmetic of arf and saturated semigroups, Appl. rev. Real acad. Cienc. exact. Fís. natur. Madrid 82, No. 1, 161-163 (1988) Singularities of curves, local rings, Singularities in algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Global theory and resolution of singularities (algebro-geometric aspects) Arithmetic of the Arf semigroups and saturations. Applications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of the paper is to prove that in dimension 3 singularities of reflexive sheaves can be resolved by blowing up points and staying in each step within the class of reflexive sheaves: Let \(X\) be a 3-dimensional algebraic variety over an algebraically closed field of characteristic \(0\), \(\mathcal F\) a reflexive sheaf of rank 2 on \(X\) and \(S=\text{Sing} (\mathcal F)=\{x\in X\mid \mathcal F_x\) is not free\}. Define \(X_i\), \(S_i\) and \(\mathcal F_i\) inductively by \(X_0=X\), \(S_0=S\), and \(\pi_i: X_i\to X_{i-1}\) the blowing up of \(X_{i-1}\) with center \(S_{i-1}\), \(\mathcal F_i\) the reflexive transformed of \(\mathcal F_{i-1}\), i.e. \(\mathcal F_i=\pi_i^\ast(F_{i-1})^{\vee\vee}\). There is an \(r\) such that \(S_r=\emptyset\), i.e. \(\mathcal F_r\) is locally free. Reflexive sheaf; reflexive module; locally free sheaf; resolution of singularity; quadratic transform Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Other special types of modules and ideals in commutative rings, Multiplicity theory and related topics Singularities of reflexive sheaves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Sei \((Z,o)\) eine dreidimensionale terminale Singularität vom Typ \(cA/r\), \(r \geq 1,\) und \(\phi:\widetilde{Z} \to Z\) eine Auflösung. Weiters sei \(S \subset Z\) ein exzeptioneller Divisor mit Zentrum\((S)= o\).
In der vorliegenden kurzen Note wird gezeigt, dass dann die zugehörige diskrete Bewertung des Funktionenkörpers \(K(Z)\) rational ist, d.h. \(S\) eine rationale Fläche. Yu. G. Prokhorov, ''A remark on the resolution of three-dimensional terminal singularities,'' Uspekhi Mat. Nauk [Russian Math. Surveys], 57 (2002), no. 4, 815--816. Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry A remark on the resolution of three-dimensional terminal singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We construct a relative Chow group \(CH_ 0(X,Y)\) associated to a singular variety X with a closed subset Y containing the singular locus of X. This comes equipped with a cycle map \(\gamma: X-Y\to CH_ 0(X,Y),\) and our main geometric result is that the relation of rational equivalence is \(\sigma\)-closed. This is, \(\gamma^{-1}(0)\) is a countable union of closed subsets of X-Y. We apply this to the case in which \(X=Spec(A)\) is a singular affine surface and obtain our main algebraic result: the set of maximal ideals of A which are complete intersections is \(\sigma\)-closed in the set of all regular maximal ideals of A. \(K_ 0\); \(G_ 0\); relative Chow group; singular variety; rational equivalence; complete intersections Levine, M.; Weibel, C., \textit{zero cycles and complete intersections on singular varieties}, J. Reine Angew. Math., 359, 106-120, (1985) Algebraic cycles, Complete intersections, Parametrization (Chow and Hilbert schemes), (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Singularities in algebraic geometry Zero cycles and complete intersections 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 The problem of resolution of singularities in characteristic 0 was solved by Hironata at the beginning of the sixties of the last century. Much later algorithmic methods have been developed for solving this problem. The study of algorithmic equiresolution started about 15 years ago. Some basic results in this direction can be found in the article of \textit{S. Encinas, A. Nobile} and \textit{O. E. Villamayor} [Proc. Lond. Math. Soc., III. Ser. 86, No. 3, 607--648 (2003; Zbl 1076.14020)]. Here families of ideals or of embedded schemes, parametrized by smooth varieties are studied. The equiresolution proposed required that the centers for the transformations are smooth over the parameter variety (condition AE) or required the local constancy of a certain invariant associated to each fibre (condition \(\tau\)).
In this paper a definition of equiresolution is proposed for families parametrized by not necessarily reduced schemes, called condition E. Other approaches are proposed, condition A, C and F. Condition A corresponds to AE mentioned above. The main objective of the paper is to prove that when the parameter space is regular all these conditions are quivalent. Assuming the properness of certain projections it is proved that they are also equivalent to \(\tau\) mentioned above. resolution algorithm; embedded variety; coherent ideal; basic object Nobile, A.: Simultaneous algorithmic resolution of singularities, Geom. dedic. 163, 61-103 (2013) Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, fibrations in algebraic geometry Simultaneous algorithmic resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is concerned with normal two dimensional singularities (V,p), in particular with elliptic singularities in the sense of Wagreich \((p_ a=1)\). The goal is, to compare Zariski's canonical resolution (successive blowing up points and normalization) with the minimal resolution of (V,p). One main result is the following:
Let (V,p) be elliptic and Gorenstein and let E be the minimal elliptic cycle on the minimal resolution of (V,p). - \((i):\quad (V,p)\) is absolutely isolated (i.e. no normalization occurs in Zariski's canonical resolution) if and only if \(E^ 2\leq -3\). - \((ii):\quad Zariski's\) canonical resolution gives the minimal resolution if and only if \(E^ 2\leq -2\). - This extends known result of H. B. Laufer and S. S.-T. Yau. Moreover the author gives a precise description of the relation between Zariski's canonical resolution and the minimal resolution without any assumption on \(E^ 2.\)
In the proofs the author uses a formula for the geometric genus \(p_ g\) of a normal 2-dimensional hypersurface singularity in terms of resolution data obtained by successive blowing up smooth centers. This formula is proved in the first part of the paper. normal two dimensional singularities; elliptic singularities; Zariski's canonical resolution; minimal resolution M. Tomari, A \(p_g\)-formula and elliptic singularities, Publ. Res. Inst. Math. Sci. 21 (1985), no. 2, 297--354. Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry A \(p_ g\)-formula and elliptic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is devoted to prove that a rational triple point of dimension two is isomorphic to the normalization of the local surface defined by a cubic equation. To do it, the authors provide concrete equations for the nine classes of dual graphs given by Artin for those points. Graphs of the final steps of the canonical resolution of the singularity corresponding with the provided equations are showed. rational triple points; cubic equations Z. Chen, R. Du, S.-L. Tan and F. Yu, Cubic equations of rational triple points of dimension two , in American Mathematical Society, Providence, RI, 2007, 63-76. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities Cubic equations of rational triple points of dimension two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Our main aim is to study the singularities of space curves, i.e. curves which cannot be embedded into the plane. We consider algebroid curves over an algebraically closed field \(k\) of any characteristic. The most important fact in relation with the singularities of spaces curves is that they can be solved by quadratic transformations (blowing up in closed points) and, consequently, we find of interest to describe several invariants associated with the resolution process.
The descriptions of the singularities are also very important, because they can be defined by a large number of equations, as they are not necessarily complete intersection. In this paper we will work with a kind of parametrization of the Hamburger-Noether matrix which describes the resolution process. space curves; resolution; singularities; Hamburger-Noether matrix Singularities of curves, local rings, Plane and space curves, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On the singularities of space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The problem of resolving the singularities of an algebraic variety by a sequence of birational transformations has a long history. A ground- breaking step forward was made in 1940, when O. Zariski developed an ingenious method for uniformizing hypersurface singularities (in characteristic zero) by a process of successive substitutions of variables in polynomial or power series [cf. \textit{O. Zariski}, Ann. Math., II. Ser. 41, 852-896 (1940; Zbl 0025.21601)]. Then, in 1964, H. Hironaka gave an affirmative answer to the whole problem of resolving singularities in characteristic zero, essentially by generalizing Zariski's approach to a general process of successive ``permissible'' blow-up transformations, expressible in the full scheme-theoretic framework [cf. \textit{H. Hironaka}, Ann. Math., II. Ser. 79, 109-326 (1964; Zbl 0122.386)]. After that celebrated paper of Hironaka's, many attempts have been undertaken to analyse the constructiveness of his process of desingularization in various concrete situations. Among them are the papers of \textit{S. S. Abhyankar} [cf. ``Weighted expansions for canonical desingularization'', Lect. Notes Math. 910 (1982; Zbl 0479.14009)], \textit{O. E. Villamayor} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 1-32 (1989; Zbl 0675.14003)], \textit{E. Bierstone} and \textit{P. Milman} [J. Am. Math. Soc. 2, No. 4, 801-836 (1989; Zbl 0685.32007)] and others.
In the present paper, the author provides another approach to the presentation and uniformization of hypersurface singularities. Generalizing Zariski's method and systematizing Hironaka's ``quasi- canonical resolution procedure'' for hypersurface singularities with a normal crossing factor, he constructs a numerical sequence for any hypersurface singularity, which classifies the singularity completely and, moreover, describes a permissible resolution procedure in a very concrete and effective way. As the author points out, his systematized approach has the advantage of being applicable to the study of hypersurface singularities in positive characteristic, too [cf. the author, Publ. Res. Inst. Math. Sci. 23, No. 6, 965-973 (1987; Zbl 0657.14002)]. resolving the singularities; uniformization of hypersurface singularities [M]Moh, T. T., Canonical uniformization of hypersurface singularities of characteristic zero.Camm. Algebra 20 (1992), 3207--3251. Global theory and resolution of singularities (algebro-geometric aspects), Hypersurfaces and algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry Quasi-canonical uniformization of hypersurface singularities of characteristic zero | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The motivic zeta function \(Z_{\text{mot}}(f;s)\) of \textit{J. Denef} and \textit{F. Loeser} [J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)] is essentially a singularity invariant, associated to a non-constant polynomial \(f \in \mathbb C[x_1,\dots,x_n]\). There is an explicit formula in terms of an embedded resolution \(h\) of \(\{f=0\}\) in affine \(n\)--space; we have in particular that each irreducible component of \(h^{-1}\{f=0\}\) induces a candidate pole (of order at most \(n\)) of \(Z_{\text{mot}}(f;s)\) and that each pole is obtained in this way. However, usually most of these candidate poles are superfluous. When \(n=2\) there is a geometric criterion to decide whether such a candidate pole is really a pole [\textit{W. Veys}, Manuscr. Math. 87, No. 4, 435--448 (1995; Zbl 0851.14012)]. Finding nice geometric conditions in higher dimensions, assuring that a candidate pole is really a pole, is a difficult problem.
In this paper the author proves in arbitrary dimension a necessary and sufficient (geometric) condition for a candidate pole to be a pole of order \(n-1\) or \(n\). In fact he shows this first for the more manageable and concrete Hodge zeta function of \(f\). His criterion turns out to be a generalization of the result for \(n=2\), mentioned above.
One can associate similar zeta functions to a non-constant regular function \(f\) on a normal surface (germ), with again all candidate poles induced by an embedded resolution. In this context a geometric determination of all poles was obtained in [\textit{B. Rodrigues} and \textit{W. Veys}, Proc. London Math. Soc. 87, No. 1, 164--196 (2003; Zbl 1048.14002)]. We want to mention the remarkable fact that this criterion in dimension 2 on a singular ambient surface is reasonably similar to the author's criterion in dimension \(n\) for poles `of high order\'\ on a smooth ambient variety. Hodge and motivic zeta functions; poles Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects) Geometric determination of the poles of highest and second highest order of Hodge and motivic zeta functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review studies some properties of the notion of multiplicity of a point of an algebraic variety. First, it reviews some basic results, including proofs of some facts that often are mentioned without verification. For instance, if \(X\) is an algebraic variety over a field \(k\), or more generally an equidimensional algebraic scheme over \(k\) (i.e., a scheme of finite type over \(k\)), then the function \({\text{mult}}_X:X \to \mathbb N\), where for \(x \in X\) \({\text{mult}}_X(x)\) denotes the multiplicity of the local ring \({\mathcal {O}}_{X,x}\), is upper-semicontinuous. Hence \(X\) (or its underlying topological space) can be stratified, each (locally closed) stratum consisting of points where \({\text{mult}}_X\) is constant. This function is nonincreasing if we blow-up \(X\) with a regular center along which \({\text{mult}}_X\) is constant. He also proves that \({\text{mult}}_X\) and \({\text{mult}}_{X_{\mathrm{red}}}\) yield the same stratification of the common underlying topological space. But the main results have to do with the notion of \textit{representation}.
Namely, if \(X\) is an equidimensional algebraic scheme over a field \(k\) we say that \(X\) is \textit{globally representable} if there is a closed immersion \(X \subset V\), where \(V\) is a regular algebraic \(k\)-scheme, and a Rees algebra \({\mathcal G}_X\) on \(V\) such that \(\mathrm{Sing} ({\mathcal G}_X)=\mathrm{Max}({\text{mult}}_X)\) (where \(\mathrm{Sing} ({\mathcal G}_X)\) and \(\mathrm{Max}({\text{mult}}_X)\) denote the singular locus of \({\mathcal G}_X\) and the set of points of maximum multiplicity of \(X\) respectively); moreover it is required that similar equalities remain valid when we take permissible transformations so that the maximum value of the multiplicity does not drop. If the base field is perfect and \(X\) does not have embedded components a result of Villamayor says that each point of \(X\) has an étale neighborhood \(U\) such that \(U\) is globally representable.
After reviewing these results, Abad proves his main result. It says that if \(X\) is globally representable, say via a closed immersion \(X \subset V\) and a Rees algebra \({\mathcal G}_V\), then the Rees algebra (over \(X\)) \({\mathcal G}_X := {\mathcal G}_V | X\) is intrinsic to \(X\). That is, if \(X\) is also represented via \(X\subset V'\) and \({\mathcal G}_{V'}\), then \({\mathcal G}_V | X = {\mathcal G}_{V'} | X\) have the same integral closure.
In this result, \(X\) may be singular. There is much interesting work on Rees algebras, but most of it is done under the assumption that they are defined over nonsingular schemes. To better evaluate Abad's theorem, it would be interesting to study how, and if, these results extend to the singular case.
Finally, the author shows that if \(\phi:X' \to X\) is a finite morphism of algebraic varieties over a perfect field \(k\) of generic degree \(n\), with \(m\), \(m'\) denoting the maximum value of \({\text{mult}}_X\) and \({\text{mult}} _{X'}\) respectively, we always have \(m' \leq nm\). Moreover, if \(m'=nm\) then:
(i) \(\phi(\mathrm{Max}({\text{mult}}_{X'})) \subseteq \mathrm{Max}({\text{mult}} _X)\),
(ii) if \(F\) is a closed set in \(X'\) then the induced mapping \(F \to \phi(F)\) is a homeomorphism and \(F\) is regular if and only if \(\phi(F)\) is regular,
(iii) if the Rees algebras \({\mathcal G}_X\) and \({\mathcal G}_{X'}\) (discussed above) are defined, then \({\phi}^{*}({\mathcal G}_{X}) \subseteq {\mathcal G}_{X'}\).
There are some illustrative examples. multiplicity; Rees algebra; integral closure; singularity; blow-up; stratification C.ABAD,\textit{On the highest multiplicity locus of algebraic varieties and Rees algebras}, J. Algebra 441 (2015), 294--313.http://dx.doi.org/10.1016/j.jalgebra.2015.07.010.MR3391929 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Multiplicity theory and related topics, Integral closure of commutative rings and ideals, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics On the highest multiplicity locus of algebraic varieties and Rees algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a complex quasi-projective surface and let \(n\) be a positive integer. The Hilbert scheme \(\hbox{Hilb}^n(X)\) parametrizes the zero-dimensional subschemes of \(X\) of length \(n\). The author studies the topological Euler characteristics of these Hilbert schemes, which he collects in a generating series
\[
Z_X(q) = \sum_{n \geq 0} q^n \chi(\hbox{Hilb}^n(X)).
\]
When \(X\) is a smooth surface, these have been carefully studied by Fogarty, and Göttsche described the generating series of the Poincaré polynomial of these Hilbert schemes in terms of the Betti numbers of \(X\). To extend this, the author considers a particular action of \(\mathbb Z_p\) (\(p\) a positive integer) on \(\mathbb C^2\) involving another integer \(q\) coprime to \(p\), and denotes by \(X(p,q)\) the quotient variety. The main result of this paper is a representation of \(Z_{X(p,1)}(q)\) as a coefficient of a two-variable generating function, obtained by studying a torus action on \(X(p,1)\). The author links the combinatorics of this problem to \(p\)-fountains, a generalization of the notion of a fountain of coins. Hilbert scheme; cyclic quotient singularity; \(p\)-fountain Gyenge, Á, Hilbert scheme of points on cyclic quotient singularities of type \((p, 1)\), Period. Math. Hungar., 73, 93-99, (2016) Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Hilbert scheme of points on cyclic quotient singularities of type {\((p,1)\)} | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme \(\text{Hilb}^4 \mathbb{P}^3\) of zero-dimensional subschemes of length 4 in \(\mathbb{P}^3\) is singular along the smooth locus parametrising fat points of maximal embedding dimension, and the transverse singularity is the cone over the Grassmannian \(G (2,6)\) in its Plücker embedding. The author gives a less computational proof of this fact using the \(PGL (3)\)-action on the Grassmannian and on the versal deformation of the fat point in question.
In the second section the ranks of the Chow groups of \(\text{Hilb}^4 \mathbb{P}^3\) and the Betti numbers of its desingularisation are determined. Betti numbers of desingularisation; fat points of maximal embedding dimension; Chow groups Katz, S.: The desingularization of hilb4p3 and its Betti numbers. Zero-dimensional schemes, 231-242 (1994) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Topological properties in algebraic geometry The desingularization of \(\text{Hilb}^ 4 \mathbb{P}^ 3\) and its 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 See the review of the 2014 English edition in [Zbl 1308.14001]. S. Ishii, Introduction to singularities (in Japanese), Springer-Verlag Tokyo, Tokyo, 1997. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Deformations and infinitesimal methods in commutative ring theory, Minimal model program (Mori theory, extremal rays) Introduction to 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 gives some overview on SINGULAR, a computer algebra system for polynomial computations with special emphasis on the needs of commutative algebra, algebraic geometry and singularity theory, which has been developed under the guidance of \textit{G.-M. Greuel}, \textit{G. Pfister} and the second author [\texttt{http://www.singular.uni-kl.de}]. We draw the bow from SINGULAR's early years to its latest features. Moreover, we present some explicit calculations, focusing on applications in singularity theory. Computational aspects in algebraic geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Invariants of analytic local rings 21 years of SINGULAR experiments in mathematics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(V\) a quasi-projective variety let \(Z(V, \cdot)\) be the simplicial abelian group defined, in degree \(n\) by algebraic cycles on \(V \times \Delta^n\) whose support is meeting all faces properly: the homotopy groups are, by definition, the higher Chow groups of \(V\). For a given \(X\) let \(U\) be a Zariski open subset of \(X\) and \(Y = X - U\); we then have a canonical map \(Z(X, \cdot)/Z (Y, \cdot) \to Z (U, \cdot)\). The ``moving lemma'' claims that the map above is a homotopy equivalence, yielding the expected long exact sequence of higher Chow groups of \(Y\), \(X\) and \(U\). The basic idea in the proof is to move by blowing up faces but this is achieved only after a combination of extremely delicate simplicial arguments. moving lemma; higher Chow groups Spencer Bloch, ``The moving lemma for higher Chow groups'', J. Algebr. Geom.3 (1994) no. 3, p. 537-568 Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Cycles and subschemes The moving lemma for higher Chow groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For \(X\) quasi-projective over a separably closed field \(k\) and \(Y\) a closed subset containing the singular locus which is contained in an open affine set, let us consider the relative Chow group \(\text{CH}^ n(X,Y)\) of codimension-\(n\) cycles on \(X\) which miss \(Y\), \(n=\dim X>2\). The following results are obtained:
(i) finiteness of torsion zero-cycles (prime-to-char\((k))\),
(ii) if \(X\) is affine then \(\text{CH}^ n(X,Y)\) is torsion free prime- to-char\((k)\),
(iii) (Bloch's formula) if \(Y\) is a finite set of closed points then there is an isogeny (i.e. a surjective map with finite kernel) \(\text{CH}^ n (X,Y) \to H^ n (X,{\mathcal K}_ n)\;(\text{char} (k)=0\) or \(\text{char} (k) \geq n)\) and if moreover \(X\) is affine then the group \(H^ n(X,{\mathcal K}_ n)\) is uniquely divisible prime-to-char\((k)\).
For \(X\) a surface with ordinary multiple curves over \(k\) algebraically closed of zero characteristic the torsion group \(_ mH^ 2(X,{\mathcal K}_ 2)\) is finite for all \(m\). zero-cycles on singular varieties; Bloch's formula; relative Chow group; torsion group DOI: 10.1016/0022-4049(92)90015-8 Algebraic cycles, Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry Zero-cycles on singular varieties: Torsion and Bloch's 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 We relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed. equianalytic deformation; equisingular deformation; equisingular ideal; free curve; Jacobian ideal; nearly free curve; rational cuspidal cirve Plane and space curves, Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings, Deformations and infinitesimal methods in commutative ring theory Deformations of plane curves and Jacobian syzygies | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(f\in \mathbb{C} [x_1, x_2, \dots, x_n]\) and fix an embedded resolution \(h: X\to \mathbb{C}^n\) of \(f^{-1} (0)\) such that \(h^{-1} (f^{-1} (0))\) is a normal crossing divisor. We denote by \(E_i\), \(i\in T\), the reduced irreducible components of \(h^{-1} (f^{-1} (0))\), and by \(N_i\) and \(\nu_i -1\) the multiplicities of \(E_i\) in the divisor of \(f\circ h\) and \(h* (dx_1 \wedge dx_2 \wedge \cdots \wedge dx_n)\) on \(X\), respectively. For \(I\subset T\) denote also \(E_I= \bigcap_{i\in I} E_i\) and \(\overset \circ E_I= E_I\setminus (\bigcup_{j\not\in I} E_j)\). The rational function
\[
Z_{\text{top}, 0} (s)= \sum_{I\subset T} \chi(E_I \cap h^{-1} (0)) \prod_{i\in I} {1\over {\nu_i+ sN_i}}
\]
is called the topological zeta function of the germ of \(f\) at 0, where \(\chi(\;)\) denotes the Euler-Poincaré characteristic. It does not depend on the chosen resolution.
In this article, the author determines all poles of \(Z_{\text{top}, 0} (s)\) for \(n=2\) and for any \(f\in \mathbb{C} [x_1, x_2]\). Assume that \((X, h)\) is the canonical embedded resolution of \(f\), i.e., the minimum one in the set of all birational morphisms \(h: X\to \mathbb{C}^2\) such that \(h^{-1} (f^{-1} (0))\) is a normal crossing divisor with smooth components. The main results are the following two theorems:
Theorem 1. \(Z_{\text{top}, 0} (s)\) has at most one pole of order 2. Moreover \(s_0\) is a pole of order 2 if and only if there exist two intersection components \(E_i\) and \(E_j\) with \(s_0= -{\nu_i \over N_i}=- {\nu_j \over N_j}\), and in that case \(s_0\) is the pole closest to the origin.
Theorem 2. A complex number \(s_0\) is a pole of \(Z_{\text{top}, 0} (s)\) if and only if \(s_0=- {\nu_i \over N_i}\) for some exceptional curve \(E_i\) intersecting at least three times other components or \(s_0=- {1\over N_i}\) for some irreducible component \(E_i\) of the strict transform of \(f^{-1} (0)\). The verification relies on consideration of the resolution graph and the numbers \({\nu_i \over N_i}\). curve singularity; embedded resolution; topological zeta function; resolution graph [21] Veys W., Determination of the poles of the topological zeta function for curves, Manuscripta Math. \textbf{87} (1995), no. 4, 435-448. Singularities of curves, local rings, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Determination of the poles of the topological zeta function for curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A weak wild arithmetic quotient singularity arises from the quotient of a smooth arithmetic surface by a finite group action, where the inertia group of a point on a closed characteristic \(p\) fiber is a \(p\)-group acting with smallest possible ramification jump. In this paper, we give complete explicit resolutions of these singularities using deformation theory and valuation theory, taking a more local perspective than previous work has taken. Our descriptions answer several questions of \textit{D. Lorenzini} [Algebra Number Theory 8, No. 2, 331--367 (2014; Zbl 1332.14029)]. Along the way, we give a valuation-theoretic criterion for a normal snc-model of \(\mathbb{P}^1\) over a discretely valued field to be regular. Singularities of surfaces or higher-dimensional varieties, Positive characteristic ground fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Explicit resolution of weak wild quotient singularities on arithmetic 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 author sketches his construction for the Chern classes of a projective subvariety \(V_d\) \((d= \) dimension) of the complex projective space \(\mathbb{P}_n\subset\mathbb{C})\) [Shuxue Jinzhan 8, 395--409 (1965)]. One uses the cohomology ring (Ehresmann symbols) of the flag manifold \{point \(\subset\) linear subspace of dimension \(d\}\) and the variety determined by the pairs \(\{x,T_x\}\), where \(x\) is a generic point of \(V_d\) and \(T_x\) the corresponding tangent space.
[For the entire collection see Zbl 0534.00009.] singular projective variety; Chern classes; Ehresmann symbols; flag manifold; tangent space Singularities in algebraic geometry, Characteristic classes and numbers in differential topology, Classical real and complex (co)homology in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Chern classes on algebraic varieties with arbitrary 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 is a report on recent work on resolution of singularities of algebraic varieties defined over fields of positive characteristic. It mainly discusses the work of the author on the subject. He is an expert in the field, who for a number of years has been investigating the obstructions to extend the arguments leading to a resolution process in characteristic zero to the general case, and ways to overcome them. Probably the main problem is the lack, in the situation where the base field \(k\) has characteristic \(\mathrm{ch}(k)=p >0\), of \textit{hypersurfaces of maximal contact} (HMC).
Given a closed subvariety \(X\) of a variety \(W\), smooth over \(k\), a HMC is a hypersurface \(H\) of \(W\) containing the singular locus of \(X\), so that this property is preserved when we blow-up \(W\), using a suitable center, and take strict transforms of \(X\) and \(H\) respectively. They are available (locally, near a singular point \(x\) of \( X\)) if \(\mathrm{ch}(k)=0\) and they allow us, by looking at an induced situation where \(W\) is replaced by \(H\), and using induction on the dimension of the ambient variety, to introduce a useful numerical invariant that ``controls'' a desingularization process. Indeed, after a blowing-up with an appropriate centre, this invariant decreases. But an HMC may not exist if \(\mathrm{ch}(k)>0\).
The author proposes a ``weaker'' substitute for these HMC, available in any characteristic, which allows him to introduce a natural analogue of the mentioned invariant in the characterisitc zero case. Unfortunately, this new invariant does not work so well if \(\mathrm{ch}(k) =p >0\). Sometimes, when we blow-up what seems to be a perfectly acceptable centre, there are points \(z\) lying over \(x\), a point of \(X\) of maximum multiplicity, such that the invariant at \(z\) is strictly bigger that that at \(x\). Hauser calls them \textit{kangaroo} and \textit{antelope} points respectively (and the singularity \(x \in X\) is called \textit{wild}).
In the more technical part of the article, he studies necessary conditions for the presence of wild singularities (the \textit{Kangaroo Theorem}), and ways to modify the basic invariants, trying to control these jumping phenomena. He is able to complete this project if \(X\) is two-dimensional, obtaining a new proof of desingularization for surfaces.
The article is partially expository. The first sections review basic known results, mainly without proofs, and is a good introduction to the subject. Later (more technical) sections include precise definitions and proofs of several results, e.g., the mentioned Kangaroo Theorem. There are numerous interesting examples, and a brief but clear discussion of the ongoing efforts of other geometers in a similar direction (e.g., among others, Hironaka, Kawanoue, Villamayor, who in general use other methods). The concluding comments and the given bibliography should be useful to readers interested in learning more on these topics. algebraic variety; positive characteristic; resolution of singularities; wild singularity; kangaroo point ,\textit{On the problem of resolution of singularities in positive characteristic (or: a proof we are still} \textit{waiting for)}, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 1, 1--30.http://dx.doi.org/10. 1090/S0273-0979-09-01274-9.MR2566444 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Polynomials in real and complex fields: location of zeros (algebraic theorems), Singularities of surfaces or higher-dimensional varieties On the problem of resolution of singularities in positive characteristic (Or: a proof we are still waiting for) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 approaches the classification problem for invariant subspaces of nilpotent linear operators (studied by C. M. Ringel and M. Schmidmeier) through categorical methods, obtaining a wide range of results based on the study of vector bundles on a weighted projective line.
Let \(\mathbb X=\mathbb X(2,3,p)\), \(p\geq 2\), be the weighted projective line of the corresponding weights and denote the category coh-\(\mathbb X\) of coherent sheaves on \(\mathbb X\), obtained by applying Serre's construction to the triangle singularity \(x_1^2+x_2^3+x_3^p\). The Picard group of \(\mathbb X\) is the rank one Abelian group \(\mathbb L=\mathbb L(2,3,p)\) on three generators and the corresponding relations and, up to isomorphism, the line bundles are given by the system of twisted structure sheaves \(\mathcal Ox\), \(x\in\mathbb L\), which is denoted by \(\mathcal L\). Then, the stable triangulated category \(\underline{\text{vect}}\text{-}\mathbb X/[\mathcal L]\) has a tilting object \(T\) which is constructed in the appendix of the article, and has Serre duality.
The goal of the article is to show that the functor
\[
\Phi\colon\mathrm{vect-}\mathbb X\to\mathrm{mod-}\underline{\mathcal P},\quad E\mapsto\underline{\mathcal P}(-,E)
\]
induces an equivalence of the bounded derived categories \(D^b(\mathrm{coh-}\mathbb X)\) and \(D^b(\mathrm{mod-}A)\), where \(\mathcal P\) is the class of persistent line bundles in \(\mathcal L\) and \(A\) is the finite dimensional endomorphism algebra of \(T\), which is the representation-finite Nakayama algebra \(A(2(p-1),3)\) of certain quiver with nilpotency relations, which completes a relating picture between triangle singularities, the invariant subspace problem and representation theory of quivers.
The result allows to prove properties of the corresponding categories such as the action of the Picard group, the construction of tilting objects, the calculation of the fractional Calabi-Yau dimension, the Euler characteristic and finally to show that the categories form an ADE-chain, for \(p\geq 2\). A special role plays the case \(p=6\), having Euler characteristic zero, and whose classification of indecomposable bundles over the weighted projective line is very similar to the classical one by Atiyah of bundles over elliptic curves. weighted projective lines; nilpotent operators; invariant subspace problem; triangle singularities; tilting objects; ADE-chains; Calabi-Yau fractional categories; stable categories of vector bundles; vector bundles on smooth elliptic curves Kussin, D.; Lenzing, H.; Meltzer, H., Nilpotent operators and weighted projective lines, J. Reine Angew. Math., 685, 33-71, (2013) Representations of associative Artinian rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Module categories in associative algebras, Derived categories and associative algebras, Representations of quivers and partially ordered sets, Singularities in algebraic geometry, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Representation type (finite, tame, wild, etc.) of associative algebras Nilpotent operators 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 Da in den letzten zwei Dekaden konstruktive Beweise des Hironakaschen Theorems über die Auflösung von Singularitäten algebraischer Varietäten gefunden wurden, war es naheliegend, Computerimplementierungen dieser Algorithmen zu realisieren. Den Auflösungsprozess (das ``Aufblasen'') charakterisieren verschiedene ``Kontroll-Invarianten''. Bodnar und Schicho haben im Jahre 2000 die Invariante von Villamayor implementiert, die Autoren der vorliegenden Arbeit 2004 eine Variante davon.
Mit diesem Desingularisierungsalgorithmus ist es nun den Autoren möglich, weitere Invarianten direkt zu berechnen: Sie bestimmen die Schnittmatrix des Ausnahmedivisors, die Spektralzahlen einer Hyperflächensingularität und die Denef-Loesersche Zetafunktion. Für die beiden letzteren Anwendungen gibt es allerdings auch alternative Zugänge. resolution of singularities; practical point of view Frühbis-Krüger, A.; Pfister, G., Resolve.lib - a {\scsingular} 4-0-2 library for the resolution of singularities, (2005), Singular distribution Computational aspects of algebraic surfaces, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Some applications of resolution of singularities from the practical point of view | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a projective surface over an algebraically closed field and \(p\in S\) a non-singular point. Denote by \(\text{H}(n)\) the reduced punctual Hilbert scheme parameterizing length \(n\) zero-dimensional subschemes of \(S\) supported at \(p\). By a result of \textit{J. Briançon} [Invent. Math. 41, 45--89 (1977; Zbl 0353.14004)], \(\text{H}(n)\) is irreducible of dimension \(n-1\). In fact, \(\text{H}(1)\) is a point, \(\text{H}(2)\cong {\mathbb P}_1\) but \(\text{H}(n)\) is singular for \(n\geq 3\) and contains as a dense open smooth subset the curvilinear schemes.
This paper is concerned with the description of the birational model for \(\text{H}(n)\) introduced by \textit{A. S. Tikhomirov} [Proc. Steklov Inst. Math. 208, 280--295 (1995; Zbl 0884.14001)]: the reduced moduli space \(\text{HF}(n)\) of length \(n\) complete flags \(\xi_1\subset \cdots\subset\xi_n\) of subschemes of \(S\) supported at \(p\) projects onto \(\text{H}(n)\), so that the closure \(\text{HF}'(n)\) in \(\text{HF}(n)\) of the inverse image of the curvilinear points is the unique component of \(\text{HF}(n)\) mapping birationally onto \(\text{H}(n)\).
In order to get a better understanding of the variety \(\text{HF}'(n)\), the author introduces the reduced moduli space \(\text{HMF}(n)\) of multiplicative complete flags \(\xi_1\subset\cdots\subset\xi_n\), that is with the property that \(I_i I_j\subset I_{i+j}\) where \(I_i\) denotes the ideal sheaf of \(\xi_i\) (\S 4) and identifies the model \(\text{HF}'(n)\) as a component of \(\text{HMF}(n)\). He shows for \(n\leq 7\) that \(\text{HMF}(n)\) is irreducible, so that \(\text{HF}'(n)=\text{HMF}(n)\) (Question 5.5). He proves further for \(n\leq 4\) that \(\text{HMF}(n)\) is smooth (Theorem 6.1), so that \(\text{HF}'(n)\) is a resolution of singularities of \(\text{H}(n)\), but that \(\text{HMF}(5)\) is singular along a curve (Theorem 6.2).
In the first sections, the author gives a detailed construction of the moduli space of (multiplicative) complete flags. punctual Hilbert scheme; complete flags; curvilinear schemes Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects) A partial resolution of the punctual Hilbert scheme of a nonsingular surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities There are many generalizations of the McKay correspondence for higher dimensional Gorenstein quotient singularities and there are some applications to compute the topological invariants today. But some of the invariants are completely different from the classical invariants, in particular for non-Gorenstein cases. In this paper, the author discusses the McKay correspondence for 2-dimensional quotient singularities via ``special'' representations which gives the classical topological invariants, and gives a new characterization of the special representations for cyclic quotient singularities in terms of combinatorics. \(G\)-clusters; \(G\)-Hilbert schemes; quotient singularities; special representations Ito, Yukari, Special {M}c{K}ay correspondence, Geometry of Toric Varieties, Sémin. Congr., 6, 213-225, (2002), Soc. Math. France, Paris Classical groups (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Special 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 [For the entire collection see Zbl 0607.00004.]
An expanded version of this note is to be found in the author's paper in Compos. Math. 64, 311-327 (1987; Zbl 0648.14005). index of regularity; Hilbert function; plurigenera; number of; integral points; Newton polyhedra; resolution; complete; intersection singularity Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Complete intersections A note about plurigenera | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives a reader-friendly introduction to the McKay correspondence, based on his earlier article [Hokkaido Math. J. 32, No. 2, 317--333 (2003; Zbl 1046.14002)]. Most proofs are omitted, but several examples and explicit calculations are included, thus guiding the reader along the historical development through the various levels of this beautiful subject: From the original construction of McKay (including comprehensive calculations in the tetrahedral, octahedral and icosahedral case) to more geometric approaches, culminating in the results of \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)] and the interpretation in terms of derived categories [\textit{A. Ishii}, J. Reine Angew. Math. 549, 221--233 (2002; Zbl 1057.14057); \textit{T. Bridgeland, A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)]. Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Derived categories, triangulated categories McKay correspondence for 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 Let \((X,0)\) be a normal complex surface singularity and \((C,0)\subset (X,0)\) a reduced curve. \textit{M. Morales} [in: Noeuds, tresses et singularités, C. R. Sémin., Plans-sur-Bex 1982, Monogr. Enseign. Math. 31, 191--203 (1983; Zbl 0542.14001)] proved a formula for the \(\delta\)-invariant \(\delta(C,0)\) applying the Riemann-Roch formula on the resolution space of the singularity \((X,0)\).
The article under review considers the case that \((X,0)\) is a rational singularity. The authors provide a formula for \(\delta(C,0)\) similar to the formula above and prove that the equality \(\kappa_X(C,0)=\delta(C,0)\), where \(\kappa_X\) is a generalization of the \(\kappa\)-invariant: see [\textit{J. I. Cogolludo-Agustín}, Topological invariants of the complement to arrangements of rational plane curves. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1038.32025); \textit{J. I. Cogolludo-Agustín} and \textit{J. Martín-Morales}, Rev. Mat. Complut. 32, No. 2, 419--450 (2019; Zbl 1432.32036)]. These results clarify that \(\delta(C,0)\) is determined by the embedded topological type of the pair \(C\subset X\). The last section of the article provides exemples and some interesting facts about formulas for the \(\delta\)-invariant. normal surface singularities; rational surface singularities; delta invariant of curves; kappa invariant of curves; Riemann-Roch theorem Singularities in algebraic geometry, Invariants of analytic local rings, Complex surface and hypersurface singularities, Global theory and resolution of singularities (algebro-geometric aspects) Delta invariant of curves on rational surfaces. I: An analytic approach | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a perfect field having positive characteristic \(p \). Let \(X\) be a \(k\)-variety, i.e., a reduced separated \(k\)-scheme of finite type; let \(X_1,\ldots,X_c\) be the irreducible components of \(X\). Let \(X_{\infty}\) be the space of arcs of \(X\); it represents the functor on \(k\)-algebras \(A\mapsto X(A[[t]])\). The \(k\)-scheme \(X_\infty\) is not of finite type if \(\dim(X)>0\), but satisfies some finiteness properties [\textit{A. J. Reguera}, Compos. Math. 142, No. 1, 119--130 (2006; Zbl 1118.14004); Am. J. Math. 131, No. 2, 313--350 (2009; Zbl 1188.14010)]. There is a natural map \(j:X_\infty \to X\). Let \(Z\) be a subscheme of \(X\); then \(X^Z_\infty\) denotes the subscheme \(j^{-1}(Z) \) of \(X_\infty \). As a subset, \(X^Z_\infty\) consists of all arcs with center inside \(Z\). There is an inclusion \(Z_\infty\subset X_\infty\) which is strict if \(Z\neq X\).
We cite the authors: ``\((\mathrm{Sing}(X))_\infty\) may contain some of the irreducible components of \(X_{\infty} ^{\mathrm{Sing}}(X)\). Understanding these ``\#small'' components is the main purpose of this article.'' Any small component is the Zariski closure \(Z^0_\infty\) in \(X_\infty\) of a subset of the form \((Z\setminus \mathrm{Sing} (Z))_\infty\).
Now let \(Z\subset X\) be a nowhere dense subvariety of \(X\) such that
\[
\bigcup_{i=1}^c \mathrm{Sing}(X_i) \subset Z.
\]
Then the natural map \(X_\infty\setminus Z_\infty\to X\) induces a bijection on irreducible components. In particular, \(X_\infty\) has finitely many irreducible components.
In section 3 the concept of arc-sharpness of a \(k\)-variety \(X\) at a point \(\zeta\in X\) is introduced. Theorem 3.10 and Theorem 3.11 provide two criteria which imply that a point \(\zeta\in X\) is not arc-sharp.
Section 4 deals with the existence of small irreducible components of \(X_{\infty}\). For \(X\) with \(\dim(X)=2\) there is a decomposition of \(X_\infty\) into irreducible components. Lastly, in section 5 some open problems are mentioned; there are also some examples. resolution of singularities; arc sharp spaces; valuations; small irreducible components Valuations and their generalizations for commutative rings, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Small irreducible components of arc spaces in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The McKay correspondence establishes a bijection between non-trivial irreducible representations of a finite subgroup \(\Gamma\) of \(\text{SL}(2,\mathbb{C})\) and irreducible components of the exceptional divisor of the minimal resolution of the quotient surface singularity \(X_\Gamma:= \mathbb{C}^2/\Gamma\). This was discovered by J. McKay around 1980, using a rather formal approach. There was much subsequent work trying to give a more geometric description of the procedure. Among others, there were relevant contributions of Gonzalez-Springerg, Verdier, Esnault, Artin, working with suitable auxiliary reflexive sheaves. Later Itô and Nakamura used certain Hilbert schemes of zero-dimensional subschemes of the plane and Kapranov and Vasserot introduced suitable derived categories to describe the correspondence. Also attempts were made to generalize the theory. For instance, Riemenschneider conjectured that a similar correspondence could be obtained by using certain finite subgroups of \(\text{GL}(2,\mathbb{C})\) (namely, the small ones, i.e. those freely acting on \(\mathbb{C}^2-(0, 0)\)) and certain representations thereof (the so-called special ones). The conjecture was proved by \textit{A. Ishii} [J. Reine Angew. Math. 549, 221--233 (2002; Zbl 1057.14057)], using Hilbert schemes and derived categories techniques.
In the present article, the author reviews these and other developments and shows that for a cyclic quotient singularity Ishii's theorem can be proved relatively easily [by using results of \textit{R. Kidoh}, Hokkaido Math. J. 30, No. 1, 91--103 (2001; Zbl 1015.14004)]. The author also presents a theorem giving several equivalent characterizations of the notion of special representation. These results had been obtained by the author several years ago, but had not been presented in a mathematical journal before.
The author also proposes a problem, namely to describe the deformation theory of \(X_\Gamma\) in terms of representations of \(\Gamma\). He announces some partial results in this direction. quotient surface singularity; reflexive sheaves; special representation; McKay correspondence; Hilbert scheme O. Riemenschneider, Special representations and the two-dimensional McKay correspondence, Hokkaido Math. J. 32 (2003), 317--333. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects) Special representations and the two-dimensional McKay correspondence | 0 |
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