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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a resolution \(\pi:(\tilde X,E) \longrightarrow (X,o)\) of a normal surface singularity and an element \(h\in {\mathfrak m}_{X,o} \subset {\mathcal O}_{X,o}\), we consider a cycle \(\sum\limits_{j=1}^rv_{E_j}(h\circ\pi)E_j\) on \(E\), where \(E=\bigcup\limits_{j=1}^rE_j\) is the irreducible decomposition of \(E\) and \(v_{E_j}(h\circ\pi)\) is the vanishing order of \(h\circ\pi\) on \(E_j\). Among all such cycles, the minimal one \(M_E\) is called the maximal ideal cycle on \(E\) (S.S.T.Yau). In generally, we have \(Z_E\leqq M_E\) for the fundamental cycle \(Z_E\). It is a natural problem to compare \(M_E\) and \(Z_E\). There are several results on it (see references of this paper).
If \((X,o)\) is a hypersurface singularity defined by \(z_1^{a_1}+\cdots +z_n^{a_n}=0\) (\(a_j\geqq 2\) for any \(j\)), then it is called a hypersurface singularity of Brieskorn type. For 2-dimensional hypersurface singularities of Brieskorn type (i.e., \(n=3\)), \textit{K. Konno} and \textit{D. Nagashima} [Osaka J. Math. 49, No. 1, 225--245 (2012; Zbl 1246.14051)] determined \(M_E\) for the minimal good resolution of \((X,o)\) considering a cyclic covering of \({\mathbb C}^2\). Namely, they determined the \(M_E\) from \(a_1,a_2,a_3\), and compared \(M_E\) and \(Z_E\). They obtain a necessary and sufficient condition to coincide them.
A complete intersection singularity of Brieskorn type is defined by \((X,o)=\{(x_1,\cdots,x_m)\mid q_{j1}x_1^{a_1}+\cdots +q_{jm}x_m^{a_m}=0, j=3,\cdots,m\}\), where \(a_i\geq 2\) are integers. The condition to be complete intersection is written as a condition on conficients \(\{q_{i,j}\}_{i,j}\). The class of these singularities is important in normal surface singularity theory. For example, \textit{W. D. Neumann} [in: Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 233--243 (1983; Zbl 0519.32010)] proved that the universal abelian cover of a weighted homogeneous normal surface singularity with rational homology sphere link is a complete intersection singularity of Brieskorn type.
In this paper, Men-Okuma generalized Konno-Nagashima's results to the case of complete intersection singularities of Brieskorn type according to the similar way. They determined the \(M_E\) from \(a_1,\cdots,a_m\), and compared \(M_E\) and \(Z_E\), and obtained a necessary and sufficient condition to coincide them. surface singularities; weighted homogeneous singularities; Brieskorn complete intersections; maximal ideal cycles Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Singularities in algebraic geometry The maximal ideal cycles over complete intersection surface singularities of Brieskorn type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(F\) be an algebraically closed field of characteristic \(p>0\), and let \(B=F[x,y,z]/h\) be the homogeneous coordinate ring of a nodal cubic \(C\). Suppose \(J\) is a homogeneous ideal of \(B\). For \(n>0\), let \(q=p^n\) and let \(J^{[q]}\) be the ideal generated by \(u^q\) for \(u \in J\). The function \(e_n\) given by the dimension of \(B/J^{[q]}\) over \(F\) is called the Hilbert-Kunz function of \(C\). The author expresses \(e_n\) as a quadratic polynomial of \(q\) with the coefficients explicitly computed. The idea is to consider the kernel bundle \(W\) of \(J\) on \(C\), as was done earlier in [\textit{H. Brenner}, Math. Ann. 334, No. 1, 91--110 (2006; Zbl 1098.13017)] and [\textit{V. Trivedi}, J. Algebra 284, No. 2, 627--644 (2005; Zbl 1094.14024)]. Applying the results of \textit{I. Burban} [``Frobenius morphism and vector bundles on cycles of projective lines'', \url{arXiv:1010.0399}] on the classification of indecomposable locally free sheaves on \(C\) to \(W\), the author obtains the result after some calculations. Hilbert-Kunz vector bundles; nodal cubics \beginbarticle \bauthor\binitsP. \bsnmMonsky, \batitleHilbert-Kunz theory for nodal cubics, via sheaves, \bjtitleJ. Algebra \bvolume346 (\byear2011), page 180-\blpage188. \endbarticle \endbibitem Plane and space curves Hilbert-Kunz theory for nodal cubics, via 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 This survey investigates the geometry of singularities from the viewpoint of conformal and topological quantum field theory and string theory.
First, some classical results concerning simple surface singularities are collected, paying special attention to the ubiquitous ADE theme. For conformal field theory, recent progress both on axiomatic and on constructive issues is discussed, as well as a well established classification result, which is also related to the ADE theme, but not complete. Special focus concerning constructive results is owed to superconformal field theories associated to \(K3\) surfaces and some of their higher dimensional cousins. Finally, for topological quantum field theories, their role between conformal field theory and singularity theory is reviewed, along with the origin of \(tt^*\) geometry, and some of its applications. conformal field theory; topological field theory; singularity theory Global theory and resolution of singularities (algebro-geometric aspects), McKay correspondence, \(K3\) surfaces and Enriques surfaces, Virasoro and related algebras, Deformations of complex singularities; vanishing cycles, Modifications; resolution of singularities (complex-analytic aspects), Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Topological field theories in quantum mechanics On the geometry of singularities in quantum field theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((X,L)\) be a complex polarized threefold, and \(K_X\) its canonical bundle. In the complex affine space spanned by the divisors of \(X\) (modulo numerical equivalence) consider the affine space \(\Pi\) spanned by \(L\) and \(K_X\), generically a plane. The Hilbert curve \(\Gamma_{(X,L)}\) is defined in \(\Pi\) by the equation \(\chi((u+1/2)K_X+vL)=0\), where \(\chi(D)\) stands for the Euler characteristic of \(D\). The explicit expression of the equation turns to be, by Riemann-Roch, of degree three with only homogeneous parts of degree \(3\) and \(1\) (see Section 2 of the paper under review and references therein). This Hilbert curve has special properties when \(X\) admits some special fibrations. In particular, it is known that if there exists a fibration \(\phi:X \to C\) over a smooth curve such that \(K_X+2L=\phi^*A\) for \(A\) an ample \({\mathbb Q}\)-line bundle on \(C\), then \(\Gamma_{(X,L)}\) is reducible. The paper under review deals with the question of the reducibility of \(\Gamma_{(X,L)}\) when \(X\) admits a geometric conic fibration, that is, \(\phi:X \to S\) over a smooth surface \(S\) such that every fiber is isomorphic to a conic and \(K_X+L=\phi^*A\) for \(A\) ample on \(S\). As a first result, when \(S\) is bielliptic or abelian, and under some numerical assumption (see Prop. 5.3 for details), the reducibility of \(\Gamma_{(X,L)}\) is shown to be equivalent to the fact that \(\phi:X \to S\) does not have singular fibers. More generally, for any surface \(S\), it is not only the reducibility but a particular line as a component which plays the game, to be concrete (see Theorem 1.1): the line defined by the equation \(u-v=0\) is a component of \(\Gamma_{(X,L)}\) if and only if \(\phi:X \to S\) does not have singular fibers; moreover, the Hilbert curve has a triple point if and only if \(\phi\) does not have singular fibers and \(K_X^3+48\chi({\mathcal O}_X)=0\). Examples constructed as quadric sections of cones with a different line as a component of \(\Gamma_{(X,L)}\) are presented. conic fibration; scroll; Hilbert curve Divisors, linear systems, invertible sheaves, Adjunction problems, \(4\)-folds, Special varieties Hilbert curves of conic fibrations over smooth 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 these notes, we study properties of the multiplicity at points of a variety \(X\) over a perfect field. We focus on properties that can be studied using the ramification method, such as discriminants and some generalized discriminants that we introduce. We also show how these methods lead to an alternative proof of resolution of singularities for varieties over fields of characteristic zero. Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry An introduction to resolution of singularities via 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 A classical result of geometric invariant theory is the Hilbert-Mumford semistability criterion. In one form, it deals with a linear action of a reductive algebraic group \(G\) on a vector space over any field \(k\). Here, a transparent elementary proof is given that is valid over \(\mathbb{C}\). An elementary positivity lemma in linear algebra is proved and used to deduce the proof of the criterion over \(\mathbb{C}\). The following lemma may be of independent interest:
Let \(m_{ij}\), \(1\leq i\leq r\), \(1\leq j\leq n\), be integers satisfying the following property: If \(b_1,\dots, b_r\) are real numbers (not all zero) such that \(b_1m_{1j}+\cdots+b_rm_{rj}=0\) for all \(j=1, \dots, n\), then at least two of the \(b_i\) must have opposite signs. Then there are real numbers (and, therefore, also integers) \(c_i\) such that \(m_{i1}c_1+ \cdots +m_{in} c_n>0\) for all \(i\leq r\). semistability criterion; algebraic one-parameter groups; geometric invariant theory; Hilbert-Mumford semistability Geometric invariant theory, Linear programming, Linear and multilinear algebra; matrix theory An elementary proof of the Hilbert-Mumford criterion | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 normal complex projective threefold with only quotient singularities in codimension two. Such a threefold admits two different approaches to define its second Chern class: the class \(\widetilde c_ 2 (X)\) defined in the terms of the \(Q\)-tangent bundle of the locus of orbifold points on \(X\), and the class \(c_ 2 (X)\), defined in terms of a crepant in codimension two resolution of the singularities of \(X\). In this paper the authors prove the following main theorem: Let \(X\) be a 3- fold with canonical singularities with numerically trivial canonical class \(K_ X\), and with numerically trivial \(\widetilde c_ 2 (X)\). Then \(X\) is isomorphic to a quotient of an abelian threefold by a finite group acting freely in codimension one. This result is a generalization of a similar result for smooth varieties, but is perhaps unexpected because \(X\) is not assumed a priori to have quotient singularities. The key step of the proof is to establish that such \(X\) has only quotient singularities (this way proving the correctness of the notion \(\widetilde c_ 2 (X))\). The main corollary of the theorem shows that the triviality of \(c_ 2 (X)\) is stronger than that of \(\widetilde c_ 2 (X)\):
Let \(X\) be as above, but \(c_ 2 (X) = 0\). Then \(X\) is a quotient of an abelian threefold by a finite group acting freely in codimension two. In particular, these results are a generalization of the structure theorem for smooth Calabi-Yau threefolds with trivial \(c_ 2\). threefold with only quotient singularities; second Chern class; quotient of an abelian threefold; smooth Calabi-Yau threefolds Shepherd-Barron N.I., Wilson P.M.H.: Singular threefolds with numerically trivial first and second Chern classes. J. Alg. Geom. 3, 265--281 (1994) \(3\)-folds, Homogeneous spaces and generalizations, Characteristic classes and numbers in differential topology, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Singular threefolds with numerically trivial first and second Chern classes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities To each variety \(X\) and a nonnegative integer \(m\), there is a space \textit{\(X_m\)} over \(X\), called the jet scheme of \(X\) of order \(m\), parametrizing \(m\)-th jets on \(X\). Its fiber over a singular point of \(X\) is called a singular fiber. For a surface with a rational double point, Mourtada gave a one-to-one correspondence between the irreducible components of the singular fiber of \textit{\(X_m\)} and the exceptional curves of the minimal resolution of \(X\) for \(m \gg 0\). In this article, for a surface \(X\) over \(\mathbb{C}\) with a singularity of \textit{\(A_n\)} or \(D_4\)-type, we study the intersections of irreducible components of the singular fiber and construct a graph using this information. The vertices of the graph correspond to irreducible components of the singular fiber and two vertices are connected when the intersection of the corresponding components is maximal for the inclusion relation. In the case of \textit{\(A_n\)} or \(D_4\)-type singularity, we show that this graph is isomorphic to the resolution graph for \(m \gg 0\). jet scheme; rational double point singularities Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry On the configuration of the singular fibers of jet schemes of 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 The author proves a formula for cyclotomic polynomials which was required in the study of the characteristic polynomial of the monodromy of an isolated hypersurface singularity. cyclotomic polynomials; monodromy; isolated hypersurface singularity Polynomials in number theory, Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties Cyclotomic polynomials and singularities of complex 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 Let \(k\) be a commutative ring and let \(R\) be a commutative \(k\)-algebra. Let \(A\) be a \(R\)-algebra. We discuss the connections between the coarse moduli space of the \(n\)-dimensional representations of \(A\), the non-commutative Hilbert scheme on \(A\) and the affine scheme which represents multiplicative homogeneous polynomial laws of degree \(n\) on \(A\).
We build a norm map which specializes to the Hilbert-Chow morphism on the geometric points when \(A\) is commutative and \(k\) is an algebraically closed field. This generalizes the construction done by Grothendieck, Deligne and others. When \(k\) is an infinite field and \(A= k\{x_1,\dots, x_m\}\) is the free \(k\)-associative algebra on \(m\) letters, we give a simple description of this norm map. Hilbert-Chow morphism; divided Schemes and morphisms, Parametrization (Chow and Hilbert schemes), Representation theory of associative rings and algebras Moduli of linear representations, symmetric products and the non commutative Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article studies seminormal schemes which were discussed earlier by \textit{C.~Traverso} [Ann. Sc. Norm. Sup. Pisa 24 585--595 (1970; Zbl 0205.50501), \textit{S. Greco} and \textit{C.~Traverso} [Compos. Math. 40, 325--365 (1980; Zbl 0412.14024)].
Following Greco-Traverso [loc. cit.], a ring \(A\) is said to be Mori if it is reduced and its integral closure \(A^\nu\) in \(Q(A)\) is finite over \(A\). In this case, the largest subring \(A^{sn}\) such that
(1) \(A\subseteq A^{sn} \subseteq A^\nu\),
(2) \({\text{ Spec}} A^{sn} \to {\text{ Spec}} A\) is bijective,
(3) all maps on the residue fields are isomorphisms
is said to be the semi-normalization of \(A\).
The following is shown:
Given a Noetherian and Mori ring \(A\), the semi-normalization of \(A\) is exactly the subring of \(\prod_{ {\mathbf p} \in {\text{ Spec}} A} \kappa({ p})\) consisting of pointwise functions which vary algebraically along DVRs.
As a corollary, a characterization of morphisms of a seminormal scheme is deduced which relates them to certain compatible set maps of points over DVRs.
Another application is a theorem on the \(a^2, a^3 \Rightarrow a\in A\) characterization.
Furthermore, the article gives a direct proof of the equivalence of the pointwise property and the simplicial characterization of the semi-normalization, as described by \textit{M. Saito} [Math. Ann. 316, No. 2, 283--331 (2000; Zbl 0976.14011)]. The final section gives an application to characterize the Chow variety using the pointwise characterization of semi-normalization. seminormal scheme; Mori ring; semi-normalization A. Röllin and N. Ross. A probabilistic approach to local limit theorems with applications to random graphs. , 2010. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Local theory in algebraic geometry A characterization of seminormal schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers the affine real variety \(V\) in \(\mathbb R^n\) defined as a zero level set of the mapping \(F\) into \(\mathbb{R}^2\), build of two quadratic forms. A complete topological description of \(V\) in all generic cases and the topology of the intersection of \(V\) with a half-space and the topology of various deformations of \(V_t\) are found. For the intersection of \(V\) and the half-space \(Z\), the author provides a complete description of its topological type in all but three isolated cases in dimension 4, and also in the diagonal case for generic \(V_t\). singularities; quadratic mappings; affine varieties, smoothing López de Medrano, S, Singularities of homogeneous quadratic mappings, RACSAM, 108, 95-112, (2014) Real algebraic sets, Critical points of functions and mappings on manifolds, Singularities of surfaces or higher-dimensional varieties Singularities of homogeneous quadratic mappings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The articles of this volume will be reviewed individually.
This volume comprises three essentially independent, but related, papers treating the foundations of Grothendieck duality on noetherian formal schemes and on not-necessarily noetherian ordinary schemes. In the preface it is explained briefly, what is done and what is left undone. Duality; Noetherian formal schemes; Non-noetherian ordinary schemes Tarrío, L. A.; López, A. J.; Lipman, J., Studies in duality on noetherian formal schemes and non-Noetherian ordinary schemes, Contemp. Math., vol. 244, (1999), American Mathematical Society Providence, RI Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, (Co)homology theory in algebraic geometry Studies in duality on noetherian formal schemes and non-noetherian ordinary schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is well known that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but solvable and nilpotent Lie algebras are not. In the paper under review, the authors give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. In particular, they get the correspondence between the nilpotent Lie algebras of dimension less than or equal to \(7\) and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to \(1\). derivation; nilpotent Lie algebra; isolated singularity; \(k\)-th Yau algebras Singularities in algebraic geometry, Local complex singularities, Solvable, nilpotent (super)algebras, Lie algebras of vector fields and related (super) algebras Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the preview in Zbl 0527.14030. double planes; surface singularity; Milnor number Singularities of surfaces or higher-dimensional varieties, Local complex singularities, Singularities in algebraic geometry, Complex singularities, Differentiable maps on manifolds, Theory of singularities and catastrophe theory On singularities of double planes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities All algebras considered here are standard graded k-algebras, that is, of form \(k[X_ 0,...,X_ n]/I\) with k a field, \(\deg (X_ i)=1\) for all i, and I a homogeneous ideal. It is well known that a function \(H:\quad {\mathbb{Z}}\to {\mathbb{Z}}\) is the Hilbert function of some algebra A if and only if H is a 0-sequence.
The authors are interested in additional properties that guarantee the existence of a reduced or integral algebra with a given Hilbert function. Not all O-sequences will do; the authors show that if k is finite then there are even differentiable O-sequences that are not Hilbert functions of any reduced algebra. If, however, H is the Hilbert function of N points in projective space then under certain conditions on k there is a domain with Hilbert function H. The technique of first choice for producing domains with given Hilbert function is to find any algebra with Hilbert function \(\Delta\) H, then to lift this algebra to one that is reduced or integral, as desired. Unfortunately it can happen that there is no such lifting; even when a Hilbert function \(G=H_ A\) is liftable (that is, \(\int G \) is the Hilbert function of a reduced or integral algebra, the algebra A itself may not be liftable to such an algebra). This leads the authors to investigate liftability of certain algebras (for example monomial complete intersections) over various fields. reduced algebra; integral algebra with a given Hilbert function; Hilbert function of N points in projective space Roberts L, Roitman M. On Hilbert Function of Reduced and of Integral Algebra, J Pure Appl Algebra, 1989, 56: 85--104 Integral domains, Special varieties, Polynomial rings and ideals; rings of integer-valued polynomials, Multiplicity theory and related topics, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) On Hilbert functions of reduced and of integral 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 This survey reports on recent developments regarding the global structure of complex varieties which occur in the minimal model program. log-canonical space; klt singularity; reflexive differential; flat bundle; étale fundamental group Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Vanishing theorems in algebraic geometry, Topological properties in algebraic geometry, Global theory of complex singularities; cohomological properties, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants Aspects of the geometry of varieties with 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 We prove homological mirror symmetry for Milnor fibers of simple singularities in dimensions greater than 1, which are among the log Fano cases of Conjecture 1.5 of the authors' manuscript [``Homological mirror symmetry for Milnor fibers via moduli of \(A_\infty\)-structures'', \url{arXiv:1806.04345}]. The proof is based on a relation between matrix factorizations and Calabi-Yau completions. As an application, we give an explicit computation of the Hochschild cohomology group of the derived \(n\)-preprojective algebra of a Dynkin quiver for any \(n\geqslant 1\), and the symplectic cohomology group of the Milnor fiber of any simple singularity in any dimension greater than 1. homologial mirror symmetry; simple singularity; symplectic cohomology; Dynkin quiver Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Mirror symmetry (algebro-geometric aspects) Homological mirror symmetry for Milnor fibers of simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth variety and \(G\) be an effective \(\mathbb{Q}\)-divisor on \(X\). We construct an embedded resolution for \(G\), \(f:Y\to X\). We consider the following \(\mathbb{Q}\)-divisor on \(Y\): \(R=K_Y-f^* (K_X+G) =\sum a_jF_j\), where we assume that the \(F_j\)'s are distinct irreducible smooth divisors and their supports are in normal crossing. Then \(\lceil R\rceil =\sum \lceil a_j\rceil F_j\). We write \(\lceil R\rceil= P-N\) or \(R=P-N-\Delta\) where \(P\) and \(N\) are effective integral divisors with no common components, and \(\Delta\) is an effective \(\mathbb{Q}\)-divisor with all its coefficients between 0 and 1 and whose support is a divisor in normal crossing. We observe that
\[
K_Y+ \Delta \equiv f^*(K_X+G) +P-N.
\]
We note that \(P\) is \(f\)-exceptional. By a well known lemma of Fujita, we know that \(f_*({\mathcal O}_P (P))=0\). Then
\[
f_*\bigl( {\mathcal O}_Y(P- N)\bigr) =f_*\bigl({\mathcal O}_Y(-N)\bigr)\subset f_*({\mathcal O}_Y)= {\mathcal O}_X.
\]
Hence, \(f_*({\mathcal O}_Y(P-N))\) is an ideal sheaf. We call this the multiplier ideal of \(G\). Suppose that \(Z(G)\) is the scheme defined by this ideal. We will denote the multiplier ideal by \(I_{Z(G)}\). The purpose of this note is to give a survey of the algebraic properties of multiplier ideals, and illustrate some of their applications to classical projective geometry.
We start in \S 1 and \S 2 by recalling the basic constructions of multiplier ideals. Then we study the behavior of these ideals under various standard geometric operations. In \S 3, we apply these results to study the singularities of theta divisors. In particular, we show that an irreducible theta divisor has only rational singularities. All the results in this section are joint work with R. Lazarsfeld. In \S 4, we study adjoint linear systems. These results generalize the classical theorems of Kodaira, Bombieri and Reider on linear systems on surfaces. In the last section, we give a simple proof of a theorem of Levine on the invariance of plurigenera under deformations. vanishing theorems; multiplier ideals; theta divisors Lawrence Ein, Multiplier ideals, vanishing theorems and applications, Algebraic geometry --- Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 203 -- 219. Divisors, linear systems, invertible sheaves, Theta functions and abelian varieties, Minimal model program (Mori theory, extremal rays), Vanishing theorems in algebraic geometry Multiplier ideals, vanishing theorems and 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 If \(X\subset {\mathbb P}^N\) is a smooth irreducible nondegenerate projective variety and \(X^*\subset {\mathbb P}^N\) is its dual variety, then the locus of bitangent hyperplanes \(h\in X^*\) (i.e., such that \(h\) is tangent to at least two points of \(X\)) is a component of the singular locus of \(X^*\). The author provides a sufficient condition for this component to be of maximal dimension and shows how it can be used to determine which dual varieties of Grassmannians are normal. The main result of this paper is the following
Theorem. Let \(X=G(k, n)\subset {\mathbb P}^N\), \(N={\binom{n}{k}-1}\), be the Grassmannian of \(k\)-planes in \({\mathbb C}^n\), \(k\leqslant n-k\), \(k\geqslant 3\). Then \(X^*\) is normal if and only if \((k, n)=(3, 6), (3, 7)\), or \((3, 8)\). projective space; Grassmannian; dual variety; singularity Frédéric Holweck, ``Singularities of duals of Grassmannians'', J. Algebra337 (2011) no. 1, p. 369-384 Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Classical problems, Schubert calculus Singularities of duals of Grassmannians | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((X,0)\) be the germ of an isolated singularity and \(\pi:Y\longrightarrow X\) be a resolution with exceptional divisors \(E_1,\ldots,E_n\). The singularity is called terminal if \(mK_X\) is a Cartier divisor for some integer \(m>0\) and \(K_Y \equiv f^*K_X + \sum a_iE_i, a_i>0\) for all i. The finite subgroups \(G \subset \text{Gl}_4(\mathbb C)\) are classified such that \(\mathbb C^4/G\) has only terminal Gorenstein singularities. action of a finite subgroup; Klein group; irreducible representation Anno, R., Four-dimensional terminal Gorenstein quotient singularities, Math. Notes, 73, 769, (2003) Singularities in algebraic geometry, Minimal model program (Mori theory, extremal rays), Singularities of surfaces or higher-dimensional varieties Four-dimensional terminal Gorenstein quotient singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives an example of a real Hilbert ring \(A\) such that the polynomial ring \(A[X]\) is not a real Hilbert ring. He also constructs a real maximal ideal in \(A[X]\), whose contraction to \(A\) is not a real maximal ideal. The above are counterexamples to claims contained in a paper by \textit{J. V. Leahy} and \textit{J. V. Leahy} IV, Kobe J. Math. 5, No. 2, 185--192 (1988; Zbl 0678.12008). real Hilbert ring; polynomial ring; real maximal ideal Other special types of modules and ideals in commutative rings, Real algebraic and real-analytic geometry, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) Remarks on real Hilbert rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the author studies the relationship between the automorphism group of a rational surface and that of its Hilbert scheme of points. For a divisor \(D\) on a smooth projective variety \(X\), let \(\kappa(X, D)\) be the Iitaka dimension of \(D\) and \(K_X\) be the canonical divisor of \(X\). The Hilbert scheme \(S^{[n]}\) of \(n\) points on a smooth projective surface \(S\) is known to be smooth projective of dimension \(2n\). An automorphism of \(S^{[n]}\) is defined to be natural if it commutes with the Hilbert-Chow morphism \(S^{[n]} \to S^{(n)}\) where \(S^{(n)}\) is the \(n\)-th symmetric product of \(S\). Fix a rational surface \(S\) with \(\kappa(S, -K_S) \ge 1\). The author proves that if \(T\) is a smooth projective surface and \(S^{[n]} \cong T^{[n]}\) for some \(n \ge 2\), then \(S \cong T\). Moreover, if \(S \not \cong \mathbb P^1 \times \mathbb P^1\) and \(n \ge 2\), or if \(S = \mathbb P^1 \times \mathbb P^1\) and \(n \ge 3\), then all the automorphisms of \(S^{[n]}\) are natural (hence, \(\mathrm{Aut}(S^{[n]}) \cong\mathrm{Aut}(S)\)). The remaining case \((\mathbb P^1 \times \mathbb P^1)^{[2]}\) does have a non-natural automorphism which leaves the exceptional divisor of the Hilbert-Chow morphism globally invariant. The main idea in the proofs is to use the anti-canonical divisors and the properties of the Hilbert-Chow morphism. Hilbert schemes of points; rational surfaces; automorphism Parametrization (Chow and Hilbert schemes), Automorphisms of surfaces and higher-dimensional varieties, Rational and ruled surfaces Automorphisms of the Hilbert schemes of n points of a rational surface and the anticanonical Iitaka dimension | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(F\) denote a binary form of order \(d\) over the complex numbers. If \(r\) is a divisor of \(d\) , then the Hilbert covariant \(H_{r,d} (F) \) vanishes exactly when \(F\) is the perfect power of an order \(r\) form. In geometric terms, the coefficients of \(H\) give defining equations for the image variety \(X\) of an embedding \(P^r \hookrightarrow P^d\). In this paper the authors describe a new construction of the Hilbert covariant; and simultaneously situate it into a wider class of covariants called the Göttingen covariants, all of which vanish on \(X\). They prove that the ideal generated by the coefficients of \(H\) defines \(X\) as a scheme. Finally, they exhibit a generalisation of the Gottingen covariants to \(n\)-ary forms using the classical Clebsch transfer principle. binary forms; covariants; \(SL_2\)-representations A. Abdesselam and J. Chipalkatti. On Hilbert covariants. Canad. J. Math., 66(1):3-- 30, 2014. Actions of groups on commutative rings; invariant theory, Geometric invariant theory On Hilbert covariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective variety of dimension \(n\) over an algebraically closed field \(k\). Let \({\mathcal C}_d(X)\) be the \textit{Chow variety} parametrizing \(d\)-dimensional cycles on \(X\) (see, for example, the book of [\textit{J. Kollár}, Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 32. Berlin: Springer-Verlag (1995; Zbl 0877.14012)] for its construction). For \(a+b = n-1\), let \({\mathcal I}\subset {\mathcal C}_a(X)\times {\mathcal C}_b(X)\) be the incidence variety parametrizing the pairs \((A,B)\) with \(A\cap B \neq \emptyset\). When \(k = {\mathbb C}\), B. Mazur constructed, in 1993, using intersection theory operations on the universal cycles, a Weil divisor on \({\mathcal C}_a(X)\times {\mathcal C}_b(X)\) supported on \(\mathcal I\) and posed the problem whether \(\mathcal I\) is the support of a Cartier divisor, satisfying some additional properties. \textit{B. Wang} [Compos. Math. 115, No. 3, 303--327 (1999; Zbl 0982.14017)] showed that the Weil divisor \((n-1)!\, {\mathcal I}\) is Cartier.
In the paper under review, the author proposes a new approach to Mazur's question. Let \({\mathcal H}_d(X)\) be the \textit{Hilbert scheme} parametrizing \(d\)-dimensional subschemes of \(X\). Let \({\mathcal U}_a\), \({\mathcal U}_b\) be the closed subschemes of \(X\times {\mathcal H}_a(X)\times {\mathcal H}_b(X)\) obtained by pulling back the universal families over \({\mathcal H}_a(X)\) and \({\mathcal H}_b(X)\). Using the \textit{determinant functor} constructed by \textit{F. Knudsen} and \textit{D. Mumford} [Math. Scand. 39, No. 1, 19--55 (1976; Zbl 0343.14008)], one gets a line bundle \({\mathcal L} := \text{det}\, \text{R}\, pr_{23\ast}({\mathcal O}_{{\mathcal U}_a}\otimes^{\text{L}} {\mathcal O}_{{\mathcal U}_b})\) on \({\mathcal H}_a(X)\times {\mathcal H}_b(X)\). Let \(U\subset {\mathcal H}_a(X)\times {\mathcal H}_b(X)\) be the open subset over which the fibers of \({\mathcal U}_a\) and \({\mathcal U}_b\) are disjoint. One expects that, for \(a+b = n-1\), \(U\) is dense. Since \(\text{R}\, pr_{23\ast}({\mathcal O}_{{\mathcal U}_a}\otimes^{\text{L}} {\mathcal O}_{{\mathcal U}_b})\) is acyclic on \(U\), the ``Div'' construction of Knudsen and Mumford shows that \(\mathcal L\) is the invertible sheaf associated to a canonically defined Cartier divisor on \({\mathcal H}_a(X)\times {\mathcal H}_b(X)\).
One faces, now, the problem of showing that \(\mathcal L\) descends to a line bundle on \({\mathcal C}_a(X)\times {\mathcal C}_b(X)\) (via the product of the Hilbert-Chow morphisms). In order to solve this problem, the author studies the morphism of Picard groups induced by a seminormal proper hypercovering of a seminormal scheme, using some results of \textit{L. Barbieri-Viale} and \textit{V. Srinivas} [``Albanese and Picard 1-motives'', Mém. Soc. Math. Fr., Nouv. Sér. 87 (2001; Zbl 1085.14011)].
Then, by analysing the Hilbert-Chow morphism for 0-cycles, using the results of [\textit{B. Iversen}, Linear determinants with applications to the Picard scheme of a family of algebraic curves.Lecture Notes in Mathematics. 174. Berlin-Heidelberg-New York: Springer-Verlag. (1970; Zbl 0205.50802)], and for codimension 1 cycles, using the results of Knudsen and Mumford, the author shows that, in the case \(a=0\), \(b = n-1\), \(\mathcal L\) descends from \({\mathcal H}_0(X)\times {\mathcal H}_{n-1}(X)\) to \({\mathcal C}_0(X)\times {\mathcal C}_{n-1}(X)\). Chow variety; incidence divisor; Hilbert-Chow morphism; determinant functor; seminormal variety; simplicial Picard functor Parametrization (Chow and Hilbert schemes), Picard groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The Hilbert-Chow morphism and the incidence divisor: zero-cycles and 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 papers will not be reviewed individually.]
Contents: \textit{S. Tsuyumine}, Rings of automorphic forms which are not Cohen-Macaulay (pp. 1-12); \textit{H. Terao}, The characteristic polynomial of arrangements and logarithmic multi-vector fields (pp. 13-20); \textit{H. Sato}, A topological property of analytic spaces (pp. 21-26); \textit{S. Izumi}, On the rank condition for homomorphisms of analytic algebras (pp. 27-29); \textit{S. Ogata}, Infinitesimal deformations of Tsuchihashi cusp singularities (pp. 30-38); \textit{G. Ishikawa}, Topologically extremal real surfaces in \({\mathbb{P}}^ 2\times {\mathbb{P}}^ 1\) and \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) (English) (pp. 39-58); \textit{M. Ishida}, Invariants of toric divisors (pp. 59-73); \textit{S. Kitagawa}, On the classification of complex reflection groups of parabolic type (pp. 74- 98); \textit{M. Oka}, Examples of algebraic surfaces with \(q=0\) and \(p_ g\leq 1\) which are locally hypersurfaces (English) (pp. 99-111); \textit{M. Tomari} and \textit{K. Watanabe}, On 2-dimensional normal singularities with a ``star-shaped'' resolution (pp. 112-142); \textit{K. Saito}, Algebraic surfaces for regular systems of weights (English) (pp. 143-192). Complex analytic singularities; Singularities; Commutative rings; Proceedings; Symposium; Kyoto/Japan Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to commutative algebra Complex analytic singularities and commutative rings. Proceedings of a symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto (Japan), December 2-4, 1985 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f:\mathbb{C}^n \to\mathbb{C}\) be a polynomial of degree \(d\). One replaces \(f\) by a proper mapping \(t:X\to \mathbb{C}\) (which depends on the chosen system of coordinates on \(\mathbb{C}^n)\), where \(X\) is the closure in \(\mathbb{P}^n \times\mathbb{C}\) of the graph of \(f\). Let \(H_\infty\) denote the hyperplane at infinity and let \(X_\infty: =X\cap (H_\infty \times\mathbb{C})\). The least fine Whitney stratification \({\mathcal W}\) of \(X\) that contains the stratum \(X\backslash X_\infty\) is called the canonical Whitney stratification at infinity of \(X\). The authors say that the polynomial \(f\) has isolated \({\mathcal W}\)-singularities at infinity if \(t\) has isolated singularities with respect to the stratification \({\mathcal W}\).
The main result of the paper is the following: Let \(f:\mathbb{C}^n \to \mathbb{C}\) be a polynomial with isolated \({\mathcal W}\)-singularities at infinity. Then the general fibre of \(f\) is homotopy equivalent to a bouquet of spheres of real dimension \(n-1\).
The existence of an isolated \({\mathcal W}\)-singularity at infinity is related to the existence of a certain polar curve as germ at that point. One proves that, if the germ of such polar curve is nonvoid, then the point is a singularity where a certain positive number of cycles of a general fibre vanish. singularities at infinity; vanishing cycles; polar curves; canonical Whitney stratification Siersma, D; Tibăr, M, Singularities at infinity and their vanishing cycles, Duke Math. Journal, 80, 771-783, (1995) Deformations of complex singularities; vanishing cycles, Milnor fibration; relations with knot theory, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Singularities in algebraic geometry Singularities at infinity and their vanishing cycles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(F\) be a real quadratic field of class number 1 and discriminant \(q\), and let \(X/{\mathbb{Q}}\) be a smooth complete variety obtained from the coarse moduli space parameterizing polarized abelian surfaces with real multiplication by the ring of integers \(O_F\), obtained by resolving the cusps and finite quotient singularities. One may extend \(X\) to a smooth scheme over \({\mathbb{Z}}_p\) for \(p\in U\) - a set containing all but finitely many primes \(p\). The author makes the very restrictive hypothesis that the Doi-Naganuma lift \(S_2(\Gamma_0(q), \varepsilon_q) \rightarrow S_2(\text{SL}_2(O_F))\) is surjective. Under the hypotheses above the author proves some very strong theorems concerning zero-cycles on these varieties, the Tate conjecture for divisors on the reduction modulo \(p\) and more. In particular:
Theorem A: Let \(p\) be a prime that splits in \(F\) and such that \(X\) has good reduction at \(p\) with closed fiber \(X_p\). Then the Tate conjecture holds for divisors on \(X_p\). That is: \(NS(X_p)\otimes {\mathbb{Q}}_\ell \cong H^2_{et}(\overline{X_p}, {\mathbb{Q}}_\ell(1))^{Fr_p}\).
Theorem B: The boundary map \(\partial ': H^1(X, K_2)\otimes \mathbb{Q} \rightarrow \oplus_\ell \text{Pic}(X_\ell) \otimes \mathbb{Q}\) is surjective, where the summation on the r.h.s. is over all primes \(\ell \in U\) that are split in \(F\).
Theorem C: For all split primes \(p\in U\) the prime to \(p\) torsion in \(\text{Ch}_0(X\times_{{\mathbb{Z}}_p} {\mathbb{Q}}_p)\) is finite.
Under additional hypothesis involving Beilinson's conjectures, Tate's conjecture at inert primes and more, the author obtains the following
Theorem D: For all \(p\in U-\{2, 3\}\) the \(p\)-torsion part of \(\text{Ch}_0(X)\) is finite.
The proof of the Tate conjecture is based on reduction to an analogous statement for the intersection cohomology, which, in turn, is studied via modular forms. In this study the assumption on the Doi-Naganuma lift is used to reduce the study of Galois modules, a priori associated to Hilbert modular forms, to Galois modules associated to elliptic modular forms. zero-cycle; Hilbert-Blumenthal surface; Doi-Naganuma lift; Tate conjecture A. Langer, Zero-cycles on Hilbert--Blumenthal surfaces, Duke Math. J. 103 (2000), no. 1, 131--163. Arithmetic aspects of modular and Shimura varieties, Algebraic cycles, Modular and Shimura varieties Zero-cycles on Hilbert-Blumenthal 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 We give a survey on the known results about the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer \(d\) and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree \(d\) having singular points of the given type as its only singularities. The set of all such curves is a quasiprojective variety, which we call an equisingular family, denoted by ESF. We describe, in terms of numerical invariants of the curves and their singularities, the state of the art concerning necessary and sufficient conditions for the non-emptiness and \(T\)-smoothness (i.e., being smooth of expected dimension) of the corresponding ESF. The considered singularities can be arbitrary, but we pay special attention to plane curves with nodes and cusps, the most studied case, where still no complete answer is known in general. An important result is, however, that the necessary and the sufficient conditions show the same asymptotics for \(T\)-smooth equisingular families if the degree goes to infinity. plane algebraic curves; many singularities; equisingular families; existence problem; \(T\)-smoothness problem; irreducibility problem; deformation problem Singularities of curves, local rings, Plane and space curves Plane algebraic curves with prescribed 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 smooth Kähler threefold with trivial canonical bundle. In the paper under review, the author uses Kuranishi theory to produce a construction of the local analytic Hilbert scheme of curves on \(X_0\), at a fixed smooth irreducible curve \(Y_0\), as a gradient scheme. More precisely, a versal deformation \(s:X\to X'\) of \(X_0\) over an analytic polydisc \(X'\) is considered, then an analytic neighborhood \(Y'\) of \(Y_0\) in the relative Hilbert scheme of proper curves in \(X/X'\). The universal curve \(p:Y\to Y'\) induces a map \(\pi:Y'\to X'\), which can be extended to a smooth morphism \(U'\to X'\), where \(U'\) is an analytic polydisc, containing \(Y'\) as a closed analytic subscheme. The dimension of \(U'\) is \(\dim X'+h^0(N_{Y_0/X_0})+1\). Finally \(\tilde X'\) denotes the manifold obtained by removing the zero-section from the total space of the line bundle on \(X'\) \(s_*\Omega^3_{X/X'}\), and \(\tilde U', \tilde Y'\) are obtained from \(U',Y'\) via base change by \(\tilde X'/X'\).
The author constructs a potential function \(\Phi\) on \(\tilde U'\) such that the relative Hilbert scheme \(\tilde Y'\) is given by the gradient ideal of \(\Phi\), that is, its germ is the zero-scheme of the section \(d_{\tilde U'/\tilde X'}\Phi\) of \(\Omega^1_{\tilde U'/\tilde X}\). Moreover, using a result of Donagi--Markman, a new formulation is given of the Abel--Jacobi map into the intermediate Jacobian.
The last two sections of the paper, written respectively by Richard Thomas and Claire Voisin, contain analogous results in two related settings, i.e. the deformation problem of the pair \((E_0,X_0)\), where \(E_0\) is a holomorphic bundle on \(X_0\), and that of the triple \((X_0, S_0,L_0)\), where \(S_0\) is a smooth very ample divisor on \(X_0\) and \(L_0\) is a line bundle on \(S_0\) satisfying some suitable conditions. Calabi-Yau threefold; local analytic deformation theory; gradient scheme; Abel-Jacobi map Clemens H.: Moduli schemes associated to K-trivial threefolds as gradient schemes. J. Algebraic Geom. 14(4), 705--739 (2005) Families, moduli, classification: algebraic theory, Parametrization (Chow and Hilbert schemes), Calabi-Yau manifolds (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, \(3\)-folds Moduli schemes associated to \(K\)-trivial threefolds as gradient schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of this paper is to study certain non-reduced subschemes of projective space supported on very simple objects: certain sets of points (lying on rational normal curves, elliptic linearly normal curves or rational normal scrolls) and disjoint unions of rational normal curves or linear spaces. Namely, the non-reduced subschemes are given by the ``fattening'' of these objects, i.e.\ the \(m_i\)-th infinitesimal neighborhoods (an \(m_i\) for each component). The goal is to give bounds on the Hilbert function, the degrees of the generators of the homogeneous ideal, and the higher syzygies. In the introduction there is a very nice motivational paragraph for studying non-reduced schemes, mentioning the interpolation of polynomials, the construction of vector bundles, limits of flat families, etc. The methods are a refinement of a paper by \textit{M. V. Catalisano, Ngô Viêt Trung} and \textit{G. Valla} [Proc. Am. Math. Soc. 118, No. 3, 717-724 (1993; Zbl 0787.14030) and by \textit{Ngô Viêt Trung} and \textit{G. Valla} ``Upper bounds for the regularity index of fat points with uniform position'' (preprint), see also J. Algebra 176, No. 1, 182-209 (1995; Zbl 0835.14020)]. non-reduced subschemes; Hilbert function; generators of the homogeneous ideal; higher syzygies; fat point; infinitesimal neighbourhood Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves, Infinitesimal methods in algebraic geometry On the homogeneous ideal of unreduced projective 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 \(k\) be a field, \(X=(X_{ij})\) an \((n,n)\)-matrix of indeterminates over \(k\), and \(r\) an integer with \(1\leq r<\min(m,n)\). Let \(R\) be the factor ring of \(k[X_{11},\ldots,X_{mn}]\) with respect to the ideal \(I_{r+1}\) generated by all \((r+1)\)-rowed minors of \(X\), and let \(D\) be the module of Kähler differentials of \(R\) over \(k\). \textit{U. Vetter} [Commun. Algebra 11, 1701-1724 (1983; Zbl 0513.13013)] showed that \(\text{depth}(D)=(m+n-r+1)(r-1)+2\).
In the paper under review the authors consider the case of a generic symmetric matrix \(X\). In the case treated by \textit{U. Vetter} (loc. cit.) there is an obvious filtration which can be used to give a lower bound for the depth; it is the right-hand side of the equality above. In the case of a generic symmetric matrix the authors use a filtration which comes from a combinatorial structure of the module of differentials given by the Jacobians of \(I_{r+1}\) to get a lower bound for the depth. This lower bound cannot be expected to be sharp. determinantal singularity; Kähler differentials; algebras with straightening law; depth Vetter, U.; Warneke, K.: Differentials of a symmetric generic determinantal singularity. Comm. algebra 25, 2193-2209 (1997) Linkage, complete intersections and determinantal ideals, Dimension theory, depth, related commutative rings (catenary, etc.), Determinantal varieties Differentials of a symmetric generic determinantal 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 Let \(S=\text{spec} R\) where \(R\) is a Noetherian normal complete local ring containing an algebraically closed field isomorphic to the residue field of \(R\). Such an \(S\) is called a normal surface singularity, and it is called a rational surface singularity if furthermore there is a desingularization \(\pi:X\to S\) such that the stalk of \(R^1\pi_*{\mathcal O}_X\) at the closed point is zero. Let \(\pi:X\to S\) be any desingularization of a rational surface singularity \(S\). In the present paper, normal \(S\)-schemes \(Y\) factoring \(\pi\) are related to complete ideals on \(S\) and the semifactorization theory for the latter is used to get a characterization of \(S\)-isomorphisms between \(S\)-schemes \(Y\). isomorphisms between schemes; normal surface singularity; rational surface singularity; desingularization; complete ideals; semifactorization Cossart, V.; Piltant, O.; Reguera-López, A. J.: On isomorphisms of blowing-ups of complete ideals of a rational surface singularity. Manuscripta math. 98, No. 1, 65-73 (1999) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Rational and birational maps On isomorphisms of blowing-ups of complete ideals of a rational surface 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 In the paper under review, the author proves several results concerning the minimal model program for varieties with bad singularities, i.e. log canonical singularities, following the analysis in [\textit{O. Fujino}, Publ. Res. Inst. Math. Sci. 47, No. 3, 727--789 (2011; Zbl 1234.14013)]. In particular, in the main theorems the authour proves that the class of rational log canonical singularities is stable under the main operations of the minimal models program, i.e. divisorial and Fano contractions in Theorem 1.1 and flips in Theorem 1.2. We should note that the singularities of \(X\) are not automatically rational when \((X, \Delta)\) is a log canonical pair. Moreover, even if we start from a log canonical pair, its log canonical model might not have rational singularities, see Example 5.1. In Theorem 4.1, the author gives a supplementary result in the case of log canonical surfaces. In particular he shows that given a log canonical surface \((X, \Delta)\) and \(f : X \to Y\) a projective birational morphism such that \(Y\) is normal and \(-(K_X+\Delta)\) is \(f\)-ample, then the exceptional locus of \(f\) passes through no nonrational singular points of \(X\). In particular, the number of nonrational log canonical singularities never decreases under a minimal model program, see Corollary 4.2. rational singularities; log canonical singularities; minimal model program; log canonical surfaces Minimal model program (Mori theory, extremal rays), Singularities of surfaces or higher-dimensional varieties On log canonical rational 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 completes the classification of isolated canonical cyclic quotient singularities in dimension three which was begun earlier by the author and \textit{G. Stevens} [Proc. Am. Math. Soc. 90, 15-20 (1984)]. The main theorem states that every such singularity is either Gorenstein or terminal, or is isomorphic to one of two ''exceptional'' singularities: the quotient by the cyclic group of order 9 acting via \(diag(e^{2\pi i/9},e^{8\pi i/9},e^{14\pi i/9}),\) and the quotient by the cyclic group of order 14 acting via \(diag(e^{\pi i/7},e^{9\pi i/7},e^{11\pi i/7}).\) The first two types were previously known to be characterized among isolated cyclic quotient singularities by a property of the determinant of the action: for Gorenstein singularities, the determinant is identically 1, while for terminal singularities, the determinant gives a faithful representation of the group. The proof uses a characterization of canonical quotient singularities due to \textit{M. Reid} [Journées de Géometrie algébrique, Angers/Fr. 1979, 273-310 (1980; Zbl 0451.14014)], \textit{N. I. Shepherd-Barron}, and \textit{Y.-S. Tai} [Invent. Math. 68, 425-439 (1982; Zbl 0508.14038)] together with some combinatorics which arose in a study of algebraic cycles on Fermat varieties made by \textit{T. Shioda} [J. Fac. Sci., Univ. Tokyo, Sect. IA 28, 725-734 (1981)] and \textit{N. Aoki} [Math. Ann. 266, 23-54 (1983; Zbl 0506.14030)]. canonical singularity; Fermat variety; Gorenstein ring; terminal singularity; classification of isolated canonical cyclic quotient singularities; Gorenstein singularities D. R. Morrison, Canonical quotient singularities in dimension three, Proc. Amer. Math. Soc. 93 (1985), 393--396. Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Geometric invariant theory Canonical quotient singularities in dimension three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This monograph presents a local theory of planar and space curve singularities both from an algebraic and geometric point of view, being motivated by possible extensions of the Zariski equisingularity theory to the case of space curves and by making links to the theory of arc spaces, resolution of singularities, valuations. The book covers the following topics. The first chapter is devoted to branches at points, defined in several different ways and characterized by parameterizations, notably the Hamburger-Noether and Puiseux expansions. The second chapter treats the geometric features and numerical invariants extracted from local rings, and among them semigroup of values, Arf closures (multiplicity sequences by successive blow-ups), saturation (invariants of plane projection), including their computation through parameterizations. The third chapter introduces sequences of infinitely near points and their representation via Hamburger-Noether matrices. In chapters four and five, infinitely near points are studied geometrically by means of the divisorial theory of singularities. Many results hold over arbitrary perfect fields, which makes sense from the arithmetic and computational point of view. In general, the presentation is clear and self-contained, and satisfies most of readers' requirements. planar and space curve singularities; branches and parameterizations; Hamburger-Noether matrices; resolution of singularities; infinitely near points Campillo, A., Castellanos, J.: Curve Singularities. An Algebraic and Geometric Approach. Actualités Mathématiques, 126pp. Hermann, Paris (2005) Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Computational aspects of algebraic curves, Modifications; resolution of singularities (complex-analytic aspects) Curve singularities. An algebraic and geometric 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 The article of \textit{S. M. Gusein-Zade, F. Delgado} and \textit{A. Campillo} [Funct. Anal. Appl. 33, No. 1, 56--57 (1999); translation from Funkts. Anal. Prilozh. 33, No. 1, 66--68 (1999; Zbl 0967.14017] initiated an intense activity regarding different multi-variable filtrations associated with divisorial filtrations of exceptional divisors of different resolutions.
In the present article, the author defines a Poincaré series associated with a germ of a toric or analytically irreducible quasi-ordinary hypersurface singularity via a finite sequence of monomial valuations, provided that at least one of them is centered at the origin. This involves the definition of a multi-graded ring associated with the analytic algebra of the singularity by the sequence of the valuations. The author proves that the corresponding Poincaré series is a rational function with integer coefficients. Moreover, it can also recovered as an integral with respect to the Euler characteristic of a function defined (by the valuations) on the projectivization of the analytic algebra of the germ. This shows that the Poincaré series associated with the set of divisorial valuations of the essential divisors is an analytic invariant of the singular germ.
In the quasi-ordinary hypersurface case, the author shows that the Poincaré series is equivalent with the normalized sequence of characteristic monomials (in the analytic case, this set is a complete invariant of the embedded topological type). quasi-ordinary singularities; Poincaré series; multi-graded rings; valuations; divisorial valuations; characteristic monomials; hypersurface singularities; Nash map; toric singularities Singularities of surfaces or higher-dimensional varieties, Invariants of analytic local rings, Toric varieties, Newton polyhedra, Okounkov bodies Quasi-ordinary singularities, essential divisors and Poincaré series | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the paper, the authors construct a Mathematica program to classify 3-dimensional simple \(K3\) singularities. In Tohoku Math. J., II. Ser. 42, 351-380 (1990; Zbl 0733.14017), \textit{T. Yonemura} calculated manually the weights of such hypersurface singularities. three-dimensional simple \(K3\) singularity; programming; hypersurface singularities \(K3\) surfaces and Enriques surfaces, Singularities of surfaces or higher-dimensional varieties, Software, source code, etc. for problems pertaining to algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Effectivity, complexity and computational aspects of algebraic geometry Programming of classification theorems for simple \(K3\) singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors generalise a result of \textit{T. Tomaru} for Brieskorn-Pham singularities [Pac. J. Math. 170, No. 1, 271--295 (1995; Zbl 0848.14017)] to the case of Brieskorn complete intersections. The minimal cycle on a resolution of a normal surface singularity is the smallest cycle with the same genus as the fundamental cycle. It coincides with the fundamental cycle for Brieskorn complete intersections of type \((a_1,\dots,a_m)\) if \(\text{lcm}((a_1,\dots,a_{m-1})\leq a_m < 2 \text{lcm}((a_1,\dots,a_{m-1})\). The proof is based on the explicit description of the resolution graph and the divisor of the function \(z_m\). normal surface singularities; cyclic quotient singularities; Brieskorn complete intersections; fundamental cycle; minimal cycle Fanning Meng, Wenjun Yuan and Zhigang Wang The Minimal Cycles over Brieskorn Complete Intersection Surface Singularities Taiwanese J. Math. 20 (2016), 277--286.DOI: 10.11650/tjm.20.2016.6434 Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities The minimal cycles over Brieskorn complete intersection 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 What kind of singularities can a plane curve of degree \(d\) admit? In this note, we focus on the sum of Milnor numbers of all singularities. We work over the field of complex numbers \(\mathbb{C}\). Let \(C\) be a plane curve of degree \(d\). Let \(\mu_p\) denote the Milnor number of a singularity \((C,p)\). We define \(\mu(C)=\sum_{p\in\text{Sing}(C)}\mu_p\) and call it the total Milnor number of \(C\). Let \(\overline\kappa\) denote the log-Kodaira dimension of \(\mathbb{P}^2-C\).
Theorem A. Let \(\nu\) be the maximum of the multiplicities of all singularities on \(C\). Then we have \(\mu(C)<{\nu\over 2\nu+1} (2d^2-3d)\) unless \(\overline\kappa=-\infty\). In particular, if \(C\) is irreducible and singular, then the inequality holds without exception.
Theorem B. Let \(C\) be a plane curve of degree \(d\geq 6\) having only simple singularities. Then
\[
\mu(C)\leq {5\over 6}d^2-d+\sum_p{2\over \bigl|\Gamma (p) \bigr|} -\#\bigl(\text{Sing} (C)\bigr),
\]
where \(p\) runs over the ADE singularities and \(\Gamma\) denotes the local uniformization group.
Theorem C. Given a positive integer \(k\) dividing \(d\), we can construct a \(k\)-fold covering \(W\) of \(\mathbb{P}^2\) branched along \(C\). Under the assumption that the irregularity of \(W\) vanishes, we have
\[
\mu(C)<{4k+ 1\over 6k}d^2-{3\over 2}d+ {k \over k-1}+ {1\over k-1} \sum_{q\in\text{Sing}(W)}c_q,
\]
where \(c_q=2\cdot \dim(R^1 \pi_*{\mathcal O}_{\widetilde W})_q-\dim H^1 (\pi^{-1}(q), \mathbb{R})\) for \(\pi:\widetilde W\to W\) a resolution of singularities.
Corollary. Let \(C\) be a plane curve of degree \(d\) having only ADE singularities. Then
\[
\mu(C) <\begin{cases} {3\over 4}d^2- {3\over 2}d+2 \quad &\text{if }d\text{ is even} \\ {3\over 4}d^2-d+{1\over 4}\quad &\text{if }d\text{ is odd}.\end{cases}.
\]
singularities of plane curve; total Milnor number; ADE singularities Sakai F., Singularities of plane curves (1990) Singularities of curves, local rings, Plane and space curves, Global theory and resolution of singularities (algebro-geometric aspects) Singularities of plane curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this book the author studies properties of germs of plane curves, i.e., power series in \({\mathbb C}[[x, y]]\), the ring of formal power series in two indeterminates over the complex numbers, or power series in \({\mathbb C}\{x, y\}\), the ring of convergent power series. After some introductory remarks, the first chapter deals with Puiseux's theorem and the Newton-Puiseux algorithm, and the second chapter introduces branches of a germ and the notion of intersection multiplicity of two curves. Blowing up and infinitely near points are the subject of the third chapter; Enrique's definition of infinitely near points is also mentioned. This chapter deals also with the notion of proximate points, free and satellite points, and the resolution of singularities for germs of curves. There are also algebraic descriptions of the rings in the successive neighborhoods of the ring of a plane germ (cf. section 3.10 and 3.11). The notion of virtual multiplicity of clusters is defined in chapter 4; this chapter contains also a proof of Noether's \(Af + B\varphi\) theorem. Chapter 5 deals with characteristic exponents, the position of infinitely near points and the semigroup of values of an irreducible germ. In particular, it is shown that two irreducible germs are equisingular (i.e., have the same characteristic exponents) iff they have the same semigroup. (The well-known necessary and sufficient condition for a numerical semigroup to be the semigroup of a plane irreducible germ is given as exercises 5.10 and 5.11.) Polar germs are dealt with in chapter 6. Here the author gives Pham's example (showing that equisingular curves may have non-equisingular polars), proves Merle's result on the polar quotients, introduces the Milnor number, and shows [using result of his two papers: Math. Ann. 287, 429-454 (1990; Zbl 0675.14009); Comp. Math. 89, 339-359 (1993; Zbl 0806.14021)] that in the case of a single characteristic exponent, the polars of a germ may be used for unveiling properties of a germ that depend on its isomorphism class and not only on its equisingularity class. In section 6.10 and 6.11 the author treats the polar invariants of a reduced germ \(\xi\) and shows how to compute them from an Enrique diagram of \(\xi\).
Linear families of germs are the subject of chapter 7. In section 7.2 the author introduces the weighted clusters of base points of pencils and proves their main properties. Section 7.5 and 7.6 deals with the notion of \(E\)-sufficiency (or \(C^0\)-sufficiency): A positive integer \(n\) is said to \(E\)-sufficient for a reduced \(f \in {\mathbb C} [[x,y]]\) iff all \(h \in f + m^n\) are non-zero, reduced and equisingular to \(f\) (\(m\) is the maximal ideal of \({\mathbb C} [[x, y]]\)). ({}The letter \(E\) stands for equisingularity: there is a similar notion with regard to being analytically isomorphic which is dealt with in section 7.7.) The last chapter presents -- in the context of this book -- a classification of valuations of \({\mathbb C}\{x,y\}\) and a proof of Zariski's factorization theorem for complete ideals in \({\mathbb C}\{x, y\}\).
This book covers a lot of classical material in a modern fashion; in particular, it clarifies some obscure definitions, remarks and statements in older papers from the end of the 19th century and the beginning of the 20th century. As prerequisites for reading this book a course in algebra should be enough. The clear presentation of the proofs makes life for a reader very easy; exercises at the end of each chapter are either examples to give the reader the possibility of proving for himself that he has mastered the text or provide new theorems not dealt with in the main text.
Taken all together: A book making nice reading; it belongs to the shelf of every mathematician interested in curves and their singularities. Puiseux's theorem; Newton-Puiseux algorithm; infinitely near points; proximate points; satellite and free points; resolution of singularities; equisingularity; semigroup of a curve; characteristic exponents; linear families of germs; clusters; polar germ; formal power series; convergent power series E. Casas-Alvero, Singularities of plane curves, London Math. Soc. Lecture Note Ser. 276, Cambridge University Press, Cambridge 2000. Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces, Equisingularity (topological and analytic), Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Plane and space curves, Germs of analytic sets, local parametrization Singularities of plane curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Using combinatorial properties of complex reflection groups we show that, if the group \(W\) is different from the wreath product \(\mathfrak S_n\wr\mathbb Z/m\mathbb Z \) and the binary tetrahedral group (labelled \(G(m, 1, n)\) and \(G_{4}\), respectively, in the Shephard-Todd classification), then the generalised Calogero-Moser space \(X_{\mathbf c}\) associated to the centre of the rational Cherednik algebra \(H_{0, \mathbf c}(W)\) is singular for all values of the parameter \(\mathbf c\). This result and a theorem of Ginzburg and Kaledin imply that there does not exist a symplectic resolution of the singular symplectic variety \(\mathfrak h \times \mathfrak h ^*/W\) when \(W\) is a complex reflection group different from \(\mathfrak S_n\wr\mathbb Z/m\mathbb Z \) and the binary tetrahedral group (where \(\mathfrak h\) is the reflection representation associated to \(W\)). Conversely, it has been shown by Etingof and Ginzburg that \(X_{\mathbf c}\) is smooth for generic values of \(\mathbf c\) when \(W\cong \mathfrak S_n\wr\mathbb Z/m\mathbb Z \). We show that this is also the case when \(W\) is the binary tetrahedral group. A theorem of Namikawa then implies the existence of a symplectic resolution in this case, completing the classification. Finally, we note that the above results, together with the work of Chlouveraki, are consistent with a conjecture of Gordon and Martino on block partitions in the restricted rational Cherednik algebra. Bellamy, G., \textit{on singular Calogero-Moser spaces}, Bull. Lond. Math. Soc., 41, 315-326, (2009) Noncommutative algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Rings arising from noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Reflection and Coxeter groups (group-theoretic aspects) On singular Calogero-Moser spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author proves that the last Hilbert-Samuel coefficient of an \(\mathbf m\)-primary ideal of a \(d\)-dimensional Cohen-Macaulay local ring \(R\) of characteristic zero is bounded by the geometric genus of \(X=\text{Spec}(R)\) (assumed to be an isolated singularity), provided that the associated graded ring of \(R\) with respect to \(I^n\) is Cohen-Macaulay for \(n\gg 0\). The ideals attaining the maximal value, are those for which the blow-up of the singularity centered at the ideal has only rational singularities. The one-dimensional case is studied separately in the last section. Hilbert-Samuel function; resolution of singularities; rational singularity; Cohen-Macaulay ring Singularities in algebraic geometry, Multiplicity theory and related topics, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) On the last Hilbert-Samuel coefficient of isolated singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(F_1,F_2,\dots,F_s\) be general forms in \(R = k[x_0,x_1,\dots,x_n]\), where \(k\) is an algebraically closed field of characteristic zero. (Notice that for ``general'' to make sense, the degrees of the forms have to be specified in advance.) In this paper, a \textit{star configuration} is defined as the union of the complete intersection, codimension two subvarieties defined by the ideals \((F_i,F_j), i \neq j\). It is not hard to see that the ideal of such a configuration is of the form \((\tilde F_1,\dots,\tilde F_s)\), where \(\tilde F_i = \frac{\prod_{j=1}^s F_j}{F_i}\), for \(i = 1,\dots,s\), and the minimal free resolution has a simple form. Most of this paper generalizes previous work of the author with \textit{J. Ahn} [J. Korean Math. Soc. 49, No. 2, 405--417 (2012; Zbl 1242.13020)], where it was assumed that all the forms have the same degree and that \(n=2\) (so the configurations are sets of points in \(\mathbb P^2\)). In this paper, the author still assumes that \(n=2\), but the assumption about the degrees is dropped, and the forms are assumed to have degrees \(d_1,\dots,d_s\) resp. The first main result is that the configuration has generic Hilbert function if and only if \(d_i \leq 2\) for all \(i\). The other two main results of this paper give conditions for when it holds that if \(\mathbb X\) and \(\mathbb Y\) are both star configurations defined by general forms of degree \(\leq 2\) then \(R/(I_{\mathbb X} + I_{\mathbb Y})\) has the Weak Lefschetz property, and finding specific Lefschetz elements for this algebra. Hilbert functions; minimal free resolutions; Lefschetz elements; star-configurations; Weak Lefschetz Property , Star-configurations in P2having generic Hilbert function and the weak Lefschetz property, Comm. Algebra 40 (2012), no. 6, 2226--2242. Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Low codimension problems in algebraic geometry, Complete intersections Star-configurations in 2 having generic Hilbert function and the weak Lefschetz property | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article contains a survey of results on ``generic singularities''. Many of the results discussed were obtained by the author. No proofs are presented.
More precisely, generic singularities are those that appear when we consider a sufficiently general linear projection \(\pi\) of a smooth \(r\)-dimensional projective variety \(X \subset {\mathbb P}^n\) into a linear subspace \( V \) (\( \approx {\mathbb P}^{m}\)) of \( {\mathbb P}^n\), where \( r+1 \leq m \leq 2r\), and (letting \(Y=\pi (X)\)) we exclude points of \(Y=\pi (X)\) in a suitable lower dimensional subvariety. Expanding results of M. Noether, E. Lluis, etc, J. Roberts developed the basic theory of these singularities in the 1970's, see \textit{J. Roberts} [Trans. Am. Math. Soc. 212, 229--268 (1975; Zbl 0314.14003)]. After briefly reviewing this theory, Zaare-Nahandi discusses some of his contributions. For instance, using the notation above, if \(y=\pi(x), x \in X\), is an analytically irreducible generic singularity, he has obtained a very explicit description of the induced homomorphism \(\pi ^* : {\hat {\mathcal O}}_{V,y} \to {\hat {\mathcal O}}_{X,x}\) and of the \textit{local defining ideal} of the singularity \(y\), namely Ker(\(\pi ^*\)). The local defining ideal is expressed in terms of minors of an associated matrix \({\mathcal M}\) with coefficients in \({\hat {\mathcal O}}_{V,y}\). The ring \({\hat {\mathcal O}}_{V,y}\) is isomorphic to a power series ring, and specializing some of the variables the matrix \(\mathcal M\) induces a matrix \({\mathcal M}_0\) with interesting properties. For instance, the defining ideal becomes a square-free monomial ideal. A simplicial complex may be associated to it, some of its properties are studied. As an application, a formula for the depth of \({\mathcal O}_{Y,y}\) is obtained and a partial answer to a conjecture of Andreotti, Bombieri and Holme weak normality of certain points of \(Y\) is gotten.
The author works over an algebraically closed field but if the characterisitc is positive the embedding \(X \subset {\mathbb P}^n\) must satisfy some extra conditions. generic projections; generic singularities; local defining ideal Singularities in algebraic geometry, Local theory in algebraic geometry, Projective techniques in algebraic geometry, Determinantal varieties, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes Algebraic properties of generic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Theorem: Let \(i: X\to X^*\) be an open dense immersion of schemes over a field \(k\), such that \(X\) is of finite type over \(k\) and \(X^*\) is separated and noetherian. Then the following assertions are equivalent:
(i) \(X^*\) is proper over \(k\).
(ii) Every integral closed subscheme \(C\subseteq X^*\) of dimension one and of finite type over \(k\) is proper over \(k\).
(iii) For every coherent \({\mathcal O}_{X^*}\)-module \(\mathbb{F}\), \(\dim_ k H^ 0(X^*, \mathbb{F})<\infty\).
(iv) If \(\mathcal I\) is the ideal of an integral closed subscheme \(C\) of dimension one and of finite type over \(k\), then \(\dim_ kH^ 0(X^*, {\mathcal O}_{X^*}/{\mathcal I})<\infty\).
(v) The canonical morphism \(X^*\to \text{Spec }k\) is universally closed. properness for schemes Schemes and morphisms A criterion for properness for schemes generically of finite type over a field | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Editorial remark: Due to personal communication of M. Roczen, this article is almost identical to his joint paper with the second author and \textit{B. Dgheim} [in: Topics in algebra. Part 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw/Poland 1988, Banach Cent. Publ. 26, Part 2, 27--30 (1990; Zbl 0741.14019)], with the difference that the assumption \(\mathrm{char} (k)\neq 2\) is dropped but necessary since the considered equations do not define isolated singularities for \(\mathrm{char} (k)=2\). Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Singularities in algebraic geometry, Picard groups, Finite ground fields in algebraic geometry Divisors of the canonical resolutions of some 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 generalize the results of \textit{C. P. Kahn} [Math. Ann. 285, No. 1, 141--160 (1989; Zbl 0662.14022)] about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame. Cohen-Macaulay modules; vector bundles; non-commutative surface singularities Noncommutative algebraic geometry, Cohen-Macaulay modules in associative algebras, Representation type (finite, tame, wild, etc.) of associative algebras, Rings arising from noncommutative algebraic geometry On Cohen-Macaulay modules over non-commutative surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article is an overview of the vanishing cycles method in number theory over function fields. We first explain how this works in detail in a toy example and then give three examples which are relevant to current research. The focus will be a general explanation of which sorts of problems this method can be applied to. vanishing cycles; function fields Arithmetic theory of algebraic function fields, Zeta and \(L\)-functions in characteristic \(p\), Structure of families (Picard-Lefschetz, monodromy, etc.) Singularities and vanishing cycles in number theory over function 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 note, the authors tackle a problem of determining singularities of a plane rational curve \(C\) from its parameterization \(\mathbf{f}=(f_0:f_1:f_2)\). In order to study the singularities of \(C\) of degree \(n\), they use the fact that the parameterization of \(C\) defines a projection \(\pi:\mathbb{P}^n\dashrightarrow\mathbb{P}^2\), which is generically one-to-one from the rational normal curve \(C_n\subset\mathbb{P}^n\) onto its image \(\pi(C_n)=C\subset\mathbb{P}^2\). Then they explore the secant varieties to \(C_n\). In particular, they define via \(\mathbf{f}\) certain \(0\)-dimensional schemes \(X_k\subset\mathbb{P}^k\), \(2\leq k\leq (n-1)\), which encode all information on the singularities of multiplicity \(\geq k\) of \(C\).
The authors give also a series of algorithms which allow one to obtain information about the singularities of \(C\) from such schemes \(X_k\). The algorithms presented in the paper in pseudo code can be easily implemented in symbolic algebra programs such as CoCoA, Macaulay2 or Bertini.
Editorial remark: in [\textit{A. Gimigliano} and \textit{M. Idà}, Geom. Dedicata 217, No. 1, Paper No. 5, 14 p. (2023; Zbl 07612177)], the last two authors give a counterexample to Lemma 4.2. rational curves; parameterizations; singularities; algorithms; projections Bernardi, A.; Gimigliano, A.; Idà, M., Singularities of plane rational curves via projections, J. Symb. Comput., 86, 189-214, (2018) Computational aspects of algebraic curves, Symbolic computation and algebraic computation Singularities of plane rational curves via projections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Um die Natur eines singulären Punktes \(x^{0} y^{0}\) der Function \(F (x,y) = 0\) zu erkennen, benutzt der Verfasser die Differentialgleichungen, welche durch wiederholte Differentiation von \(0 = F(x,y)\) entstehen, und für die einen allgemeinen Ausdruck aufstellt. Ist nun der Punkt \(x^0 y^0\) ein \(i\)-facher Punkt der Curve, so liefert die erste der Differentialgleichungen, die nicht identisch verschwindet, die Werthe des ersten Differentialquotienten \(y_1^0\), welche für jenen Punkt stattfinden; mit einem dieser Werthe ergiebt dann die nächste nicht identische Gleichung die zugehörigen Werthe von \(y_{2}^{0}\), u. s. w. Es wird bewiesen, dass schliesslich einmal die \(i\) Zweige sich von einander trennen müssen, d. h. dass ein Differentialquotient existiren muss, der für jeden der \(i\) Zweige einen anderen Werth hat. Um nun die Entwickelungen zu finden, setzt man, wenn für \(g\) Zweige die Differentialquotienten \(y_1^0 y_2^0 \ldots y_{h-1}^{0}\) denselben Werth haben,
\[
x = x^{0} + \xi, \quad y = y^{0} + y_{1}^{0} \xi + \cdots + \frac{y_{h}^{0} + z}{h!} \; \xi^{h},
\]
und erhält dann eine Gleichung zwischen \(\xi\) und \(z\), die im Punkte \(r_{\xi}=0, z=0\) höchstens einen \((i-1)\)-fachen Punkt hat, so dass eine Reduction auf einen einfacheren Fall eingetreten ist.
Diese Betrachtungen werden dann angewandt auf die Untersuchung des Verhaltens einer Tangente in der Nähe des singulären Punktes. Ist für den Punkt die Entwickelung der Coordinaten gegeben durch
\[
x - x^0 = t^{\lambda}, \quad y - y^0 = c_0 t^{\mu} + \cdots ,
\]
so folgt für die Coordinaten einer Tangente
\[
u' = - \frac{\mu c_0}{\lambda} t^{\mu - \lambda} + \cdots
\]
\[
v' = \frac{\mu - \lambda}{\lambda} c_{0} t^{\mu} + \cdots ,
\]
und diese Formeln zeigen, dass die Ordnung der ersten Singularität -- im Sinne Plücker's -- identisch ist mit der Klasse der zweiten, und umgekehrt.
Schliesslich werden noch die von Cayley gegebenen Zahlen zur Ersetzung einer beliebigen Singularität durch Doppel- und Rückkehrpunkte bewiesen. Algebraic curves; singular points Plane and space curves, Singularities of curves, local rings On singular point of algebraic functions and 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 main result of this paper is the following Riemann-Hilbert correspondence in positive characteristic:
Theorem 1.0.2. Let \(R\) be a commutative \(\mathbb{F}_p\)-algebra. Then there is a fully faithful embedding of abelian categories:
\[\{\substack{p\text{-torsion étale sheaves}\\ \text{on Spec}(R)}\} \xrightarrow{RH} \{\substack{R\text{-modules }M\text{ wit a}\\ \text{Frobenius-semilinear automorpism } \varphi_M}\}.\]
Moreover, the essential image of \(RH\) consists of those \(R\)-modules \(M\) equipped with a Frobenius-semilinear automorphism \(\varphi_M: M\to M\) (i.e., \(M\) is a perfect Frobenius module) which satisfies the following condition: every element \(x\in M\) satisfies an equation of the form \[\varphi_M^nx+a_1\varphi_M^{n-1}x+\dots +a_nx=0\] for some coefficients \(a_1,\dots,a_n\in R\) (these are called algebraic Frobenius modules).
The functor \(RH\) as above is called the Riemann-Hilbert functor. The above theorem can be viewed as a positive characteristic analog of the Riemann-Hilbert correspondence in characteristic zero (that is, the equivalence between the category of perverse sheaves and the category of holonomic \(D\)-modules with regular singularities). In fact, the author also showed that the functor \(RH^c\) obtained by restricting \(RH\) to the subcategory \(Shv_{\text{ét}}^c(\text{Spec}(R), \mathbb{F}_p)\subseteq Shv_{\text{ét}}(\text{Spec}(R), \mathbb{F}_p)\) of constructible étale sheaves on Spec\((R)\), induces an equivalence of categories between \(Shv_{\text{ét}}^c(\text{Spec}(R), \mathbb{F}_p)\) and the category of {holonomic} Frobenius modules (see Definition 4.1.1, basically these are perfect Frobenius modules that satisfy certain finiteness property). The algebraic Frobenius modules are basically colimits of holonomic Frobenius modules, thus this equivalence is naturally enlarged to the equivalence in Theorem 1.0.2.
To prove Theorem 1.0.2, the author first construct the solution functor \(Sol\) that assigns each Frobenius module \(M\) the kernel of the map \(\mathrm{id}-\varphi_M: M\to M\), formed in the categroy of \(Shv_{\text{ét}}(\text{Spec}(R), \mathbb{F}_p)\) (this turns out to be the inverse of the functor \(RH^c\) when restricted to holonomic Frobenius modules). To construct the functor \(RH\), the key step is to construct a pull-back functor \(f^{\diamond}\) on perfect Frobenius modules for any morphism of \(\mathbb{F}_p\)-algebras \(A\to B\), and a functor called the \textit{compactly supported direct images of Frobenius modules}, \(f_!\), where \(f: A\to B\) is an étale morphism of \(\mathbb{F}_p\)-algebras (under \(RH\), \(f^{\diamond}\) and \(f_!\) correspond to the usual functors \(f^*\) and \(f_!\) on \(Shv_{\text{ét}}(\text{Spec}(R), \mathbb{F}_p))\). The functor \(RH\) applied to an étale sheaf \(F\) is constructed by presenting \(F\) as the cokernel of two étale sheaves of the form \(j_!\mathbb{F}_p\) (after a limiting argument), and then using the observation that the perfection of \(R\) is \(RH(\mathbb{F}_p)\), and the fact that \(RH\) is supposed to be compatible with \(j_!\). These are done in sections 5 and 6. Theorem 1.0.2 is proved in section 7, where it is shown that the functor is fully faithful and identifies its essential image.
In the second half of the paper, the authors consider several refinements of Theorem 1.0.2: in section 8 it is shown that \(RH\) is compatible with tensor product; in section 9, Theorem 1.0.2 is upgraded to an equivalence between étale sheaves with \(\mathbb{Z}/p^n\)-coefficients and Frobenius modules over \(W_n(R)\), the ring of Witt vectors of length \(n\); in section 10, Theorem 1.0.2 is generalized to an arbitrary \(\mathbb{F}_p\)-scheme \(X\) (by glueing the affines).
Finally, in section 11 and 12, the authors compare their Riemann-Hilbert correspondence with previous work of Emerton-Kisin, who estabished an anti-equivalence between the bounded derived category of locally finitely generated unit \(R[F]\)-modules \(D^b_{lfgu}(R[F])\) and the bounded derived category of constructible étale sheaves with \(\mathbb{F}_p\)-coefficients (for \(R\) smooth algebra over a field \(k\) of characteristic \(p\)). The authors show that the functor \(RH\) extends to a functor of derived categories \(D_{\text{ét}}(\text{Spec}(R, \mathbb{F}_p)\to D_{perf}(R[F])\) and identifies its essential image (the solution functor \(Sol\) also extends to the derived categories). The author further construct a duality functor \(\mathbb{D}\): \(D^b_{hol}(R[F])\to D^b_{lfgu}(R[F])^{op}\) that is compatible with the solution functors (the solution functor \(Sol\) constructed here and the solution functor constructed by Emerton-Kisin). In this way, Emerton-Kisin's Riemann-Hilbert correspondence is recovered and extended to singular \(R\) (after defining locally finitely generated unit \(R[F]\)-modules properly). Riemann-Hilbert correspondence, perfect modules, Frobenius modules, constructible étale sheaves Positive characteristic ground fields in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Étale and other Grothendieck topologies and (co)homologies A Riemann-Hilbert correspondence 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 For a finite subgroup \(\Gamma \subset \text{SL}(2, \mathbb{C})\) and for \(n \ge 1\), we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of \(n\) points on the minimal resolution \(S\) of the Kleinian singularity \(\mathbb{C}^2 / \Gamma\). It is well known that \(X := \text{Hilb}^{[n]} (S)\) is a projective, crepant resolution of the symplectic singularity \(\mathbb{C}^{2n} / \Gamma_n\), where \(\Gamma_n = \Gamma \wr \mathfrak{S}_n\) is the wreath product. We prove that every projective, crepant resolution of \(\mathbb{C}^{2n} / \Gamma_n\) can be realised as the fine moduli space of \(\theta\)-stable \(\Pi \)-modules for a fixed dimension vector, where \(\Pi\) is the framed preprojective algebra of \(\Gamma\) and \(\theta\) is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of \(\theta \)-stability conditions to birational transformations of \(X\) over \(\mathbb{C}^{2n} / \Gamma_n\). As a corollary, we describe completely the ample and movable cones of \(X\) over \(\mathbb{C}^{2n} / \Gamma_n\), and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to \(\Gamma\) by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of \textit{H. Nakajima} [Duke Math. J. 76, No. 2, 365--416 (1994; Zbl 0826.17026)] in the case where the quiver variety is smooth. Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Minimal model program (Mori theory, extremal rays), Singularities of surfaces or higher-dimensional varieties, Geometric invariant theory, Algebraic moduli problems, moduli of vector bundles Birational geometry of symplectic quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\subset {\mathbb P}_{\mathbb C}^r\) be a smooth variety of dimension \(n\) and degree \(d\). The Castelnuovo-Mumford regularity is \(\text{reg}\,(X):=\{t\in{\mathbb Z}\mid H^i({\mathcal I}_X(t-i))=0 \;\text{for}\;i>0\}\). There is a well-known conjecture concerning \(\text{reg}\,(X)\), which says that \(\text{reg}\,(X)\leq d-r+n+1\). This conjecture is true for integral curves [\textit{L. Gruson, R. Lazarsfeld, C. Peskine}, Invent. Math. 72, 491--506 (1983; Zbl 0565.14014)], and for smooth surfaces [\textit{R. Lazarsfeld}, Duke Math. J. 55, 423--429 (1987; Zbl 0646.14005)]. In higher dimension the problem is still open. For singular varieties only the case of curves has been dealt with, in the already quoted [Zbl 0565.14014]. For higher dimensional singular varieties the previous bound is easily seen to be false; hence it is meaningful to look for a bound in terms of the singularities, and this is the aim of this paper. In particular, if the singular locus is zero-dimensional and \(X\) has a good projection with respect to \(X_{\text{reg}}\). Castelnuovo-Mumford regularity; singular varieties; good general projections Projective techniques in algebraic geometry, Homological functors on modules of commutative rings (Tor, Ext, etc.), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Low codimension problems in algebraic geometry Projections of singular varieties and Castelnuovo-Mumford regularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 functorial weak factorization of projective birational morphisms of regular quasiexcellent schemes in characteristic~0 broadly based on the existing line of proof for varieties. From this general functorial statement we deduce factorization results for algebraic stacks, formal schemes, complex analytic germs, Berkovich analytic and rigid analytic spaces, answering a present need in nonarchimedean geometry. Techniques developed for this purpose include a method for functorial factorization of toric maps, variation of GIT quotients relative to general noetherian qe schemes, and a GAGA theorem for Stein compacts. birational geometry; blowing up; bimeromorphic maps Rational and birational maps, Generalizations (algebraic spaces, stacks), Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Meromorphic mappings in several complex variables Functorial factorization of birational maps for qe schemes in characteristic 0 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A projective scheme \(X\) is \textit{determinantal} if its homogeneous ideal is the ideal of \(r \times r\) minors of a homogeneous \(p \times q\) matrix, and the codimension of \(X\) is the expected value, namely \(c = (p-r+1) (q-r+1)\). If \(r = \min \{ p,q \}\) then \(X\) is \textit{standard determinantal}. If \(X\) is, in addition, a generic complete intersection then it is called \textit{good determinantal}. The degrees of the entries of the matrix can be determined by a \(p\)-tuple and a \(q\)-tuple of integers \(a_1 \leq \dots \leq a_p\) and \(b_1 \leq \dots \leq b_q\); we denote by \(W(\underline{b},\underline{a})\) the stratum in the appropriate Hilbert scheme corresponding to good determinantal schemes, and by \(W_s(\underline{b}, \underline{a})\) the stratum corresponding to standard determinantal schemes. One of the author's results is that \(W_s(\underline{b}, \underline{a})\) is irreducible, and \(W(\underline{b},\underline{a}) \neq \emptyset\) if and only if \(W_s(\underline{b},\underline{a}) \neq \emptyset\). This paper considers mainly the case of zero-dimensional schemes, and focuses on the following problems: (1) Determine when the closure of \(W(\underline{b}, \underline{a})\) is an irreducible component of the Hilbert scheme; (2) Find the codimension of \(W(\underline{b},\underline{a})\) in the Hilbert scheme if its closure is not a component; (c) Determine when the component of the Hilbert scheme is generically smooth along \(W(\underline{b},\underline{a})\). determinantal scheme; standard determinantal scheme; good determinantal scheme; Hilbert scheme; unobstructed J.O. Kleppe, Families of low dimensional determinantal schemes. J. Pure Appl. Alg., online 9.11.2010, DOI: 101016/j.jpaa.2010.10.007. Determinantal varieties, Parametrization (Chow and Hilbert schemes), Deformations and infinitesimal methods in commutative ring theory, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory Families of low dimensional determinantal 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 \(I\) be an ideal of a regular local ring and \({\mathcal J}(cI)\), \(c\in {\mathbb Q}_+\) its multiplier ideals. The jumping numbers associated to \(I\) are those \(c\in {\mathbb Q}_+\) such that \({\mathcal J}(cI)\not ={\mathcal J}((c-\epsilon)I)\) for all \(\epsilon>0\). These numbers encode much information of the singularities of the subscheme defined by \(I\), the first number, the log-canonical threshold being much studied. The purpose of this paper is to give formulas for these numbers when \(I\) is a simple complete ideal in a two dimensional regular local ring and to find relations between these numbers and other invariants of the subscheme defined by \(I\). When \(I\) defines an analytically irreducible curve \(C\) then it shows that these numbers determine the equisingularity class of \(C\). Multiplier ideals; log-canonical threshold; plane curve singularity Järvilehto, T., Jumping numbers of a simple complete ideal in a two-dimensional regular local ring, Mem. Amer. Math. Soc., vol. 214, (2011), No. 1009, viii+78 pp Regular local rings, Research exposition (monographs, survey articles) pertaining to commutative algebra, Singularities in algebraic geometry Jumping numbers of a simple complete ideal in a two-dimensional regular local ring | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(P\) be a smooth projective variety of dimension \(n\) and consider the product of the Chow variety of \(a\)-cycles on \(P\) with the Chow variety of \(b\)-cycles on \(P\), where \(a+b+1=n\). The main result of this paper is the construction of a Cartier divisor supported on the incident locus of this product, which parametrizes pairs of cycles with non-empty intersection. This then, answers in the affirmative a question posed by Barry Mazur. The author approaches the problem by constructing a line bundle on an appropriate product of Hilbert schemes and determining that this descends to the desired object on the product of Chow varieties. The paper is very detailed and contains a wealth of information on both Chow varieties and Hilbert schemes. Chow variety; Hilbert scheme Joseph Ross, The Hilbert-Chow morphism and the incidence divisor. Ph.D. Thesis, Columbia University, 2009. Parametrization (Chow and Hilbert schemes) The Hilbert-Chow morphism and the incidence divisor | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We use the Atiyah-Bott-Segal-Singer Lefschetz fixed-point formula to calculate the Hirzebruch \(\chi_y\) genus \(\chi_y(S^{[n]})\), where \(S^{[n]}\) is the Hilbert scheme of points of length \(n\) of a surface \(S\). Combinatorial interpretation of the weights of the fixed-points of the natural torus action on \((\mathbb{C}^2)^{[n]}\) is used. This is the first step to prove a conjectural formula about the elliptic genus of the Hilbert schemes. Liu, K; Yan, C; Zhou, J, Hirzebruch {\(\chi\)}\_{}\{y\} genera of the Hilbert schemes of surfaces by localization formula, Sci China Ser A-Math, 45, 420-431, (2002) Elliptic genera, Parametrization (Chow and Hilbert schemes) Hirzebruch \(\chi_y\) genera of the Hilbert schemes of surfaces by localization 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 A non-trivial involution on a complex \(K3\) surface acts non-trivially on its cohomology (i.e.~on \(H^2\)) by the Torelli theorem. It is a remarkable fact that the same does not hold true for Enriques surfaces.
In this context one distinguishes two kinds of involutions \(\imath\) on an Enriques surface \(S\).
1) If \(\imath^*\) acts trivially on \(H^2(S,\mathbb{Z})\), then \(\imath\) is called cohomologically trivial.
2) If \(\imath^*\) acts trivially on \(H^2(S,\mathbb{Q})\), then \(\imath\) is called numerically trivial.
In [\textit{S. Mukai, Y. Namikawa}, Invent. Math. 73, 383--411 (1983; Zbl 0518.14023)] \textit{W. Barth} and \textit{C. Peters} exhibited a two-dimensional family of Enriques surfaces with a cohomologically trivial involution. Here the covering \(K3\) surfaces are generally not of Kummer type. In [\textit{S. Mukai, Y. Namikawa}, Invent. Math. 77, 383--397 (1984; Zbl 0559.14038)] it was claimed that a second family arises from Kummer surfaces of product type as Liebermann involutions were claimed to be cohomologically trivial. This is corrected in the present paper: arguing with some natural elliptic fibrations, it is shown that involutions of Liebermann type are numerically trivial, but not cohomologically trivial.
The author then continues by classifying Enriques surfaces of Kummer type with numerically trivial involution. Next to the above family, there is one further construction that was discovered by \textit{S. Kondo} (see [Jap. J. Math., New Ser. 12, 191--282 (1986; Zbl 0616.14031)]) after going unnoticed in [Zbl 0559.14038]. The present paper derives a numerically trivial involution from standard Cremona involutions of the quadric surface. This also reveals relations to quartic del Pezzo surfaces.
The main result of the present paper is the following: any Enriques surface of Kummer type with a numerically trivial involution arises through one of the above two constructions. Enriques surface; involution; cohomology; Kummer surface Mukai, [Mukai 10] S., Numerically trivial involutions of Kummer type of an Enriques surface., \textit{Kyoto J. Math.}, 50, 889-902, (2010) \(K3\) surfaces and Enriques surfaces, Families, moduli, classification: algebraic theory, Torelli problem, Algebraic moduli of abelian varieties, classification Numerically trivial involutions of Kummer type of an Enriques surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a resolution of an isolated singularity of an algebraic variety with an exceptional divisor, having normal intersection, one can construct a cell complex dual to the configuration of exceptional divisors and their mutual intersections. The author's main result states that the homotopy type of the dual cell complex is an invariant of the given singularity. It is derived from the Abramovich-Karu-Matsuki-Włodarczyk factorization of birational maps in the logarithmic category [\textit{D. Abramovich}, \textit{K. Karu}, \textit{K. Matsuki} and \textit{J. Włodarczyk}, J. Am. Math. Soc. 15, No. 3, 531--572 (2002; Zbl 1032.14003)]. Another result states that the dual complex of an isolated singularity of a toric variety is contractible. D. A. Stepanov, A remark on the dual complex of a resolution of singularities, Uspekhi Mat. Nauk 61 (2006), no. 1(367), 185 -- 186 (Russian); English transl., Russian Math. Surveys 61 (2006), no. 1, 181 -- 183. Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Toric varieties, Newton polyhedra, Okounkov bodies, Singularities of surfaces or higher-dimensional varieties A note on the dual complex of a 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 Abstracting common features and properties of Noetherian schemes and complex spaces, Binge\-ner-Kosarew [\textit{J. Bingener}, ``Local moduli spaces in analytic geometry'', Aspekte der Mathematik, D2, D3 (1987; Zbl 0644.32001)] arrived at the notion of an admissible pair of categories. These are pairs \(({\mathcal C},{\mathcal M})\) where \({\mathcal C}\) is a suitable category of algebras over a field \({\mathbb K}\) and \({\mathcal M}\) is a category of modules over algebras in \({\mathcal C}\) with a tensor product. An admissible pair is meant to describe a category of ``affine'' spaces.
The theme of the article under review is Hochschild cohomology for admissible pairs of categories. The author starts discussing in this framework the suitable notion of free algebras and the (graded) simplicial context. For a simplicial complex of indices, seen as a category \({\mathcal N}\), denote by \(\text{gr}({\mathcal C})^{\mathcal N}\) the associated category of simplicial (non-positively) graded algebras. For a morphism \(A\to B\) in \(\text{gr}({\mathcal C})^{\mathcal N}\), a resolvent of \(B\) over \(A\) is a free DG algebra \(R\) over \(A\) in \(\text{ gr}({\mathcal C})^{\mathcal N}\) together with a morphism \(R\to B\) which is a surjective quasi-isomorphism (qis) on each \(\alpha\in{\mathcal N}\). While existence of resolvents follows from loc. cit., the author proves that any two of them are homotopically equivalent.
Then he introduces the appropriate notion of a regular sequence \(X\subset R\) of an algebra \(R\). As expected, the Koszul complex \(K(X)\) is a DG resolvent of \(R/(X)\) over \(R\). It is clearly pointed out at which place restrictions on the characteristic of \({\mathbb K}\) come into play.
In the second section, the author introduces Hochschild cohomology associated to a morphism \(k\to a\) in \(\text{gr}({\mathcal C})^{\mathcal N}\) such that there exists a resolvent \(A\) of \(a\) over \(k\). It is defined via the simplicial Hochschild complex \({\mathbb H}_*(a/k)\) which is the object represented by \(S\otimes_Ra\) in the homotopy category \(K^-({\mathcal M}^{\mathcal N}(a))\). Here \(S\) is a resolvent of the multiplication map \(\mu:A\otimes A\to A\). The Hochschild complex \({\mathbb H}(a/k)\) is then the object represented by \(\check{C}^{\bullet}({\mathbb H}_*(a/k))\) in the derived category \(D({\mathbb K}\text{-Mod})\) for a Čech-construction \(\check{C}^{\bullet}(-)\) (transforming a simplicial graded object into a total complex) applied to \({\mathbb H}_*(a/k)\). The difficulty circumvented here resides in the fact that in the analytic context the usual bar complex \(C^{\text{bar}}(a)\) is not a complex of projective \(a\)-modules.
The main result is a HKR-type theorem for free DG algebras \(A\) in \(\text{gr}'({\mathcal C})^{\mathcal N}\). The author shows under these hypotheses the existence of a homotopy equivalence
\[
\Lambda\Omega_{A/k}\to {\mathbb H}(A/k),
\]
where \(\text{gr}'({\mathcal C})\) is the full subcategory of \(\text{gr}({\mathcal C})\) generated by graded algebras \(A\) such that \(A^i\) is a finite \(A^0\)-module. Recall that for an algebra \(a\) over \(k\) in \({\mathcal C}^{\mathcal N}\) with resolvent \(A\) in \(\text{gr}'({\mathcal C})\), the cotangent complex \({\mathbb L}(a/k)\) is defined as the class of \(\Omega_A\otimes_Aa\) in \(K^-({\mathcal M}^{\mathcal N}(a))\). An immediate consequence of the above is a homotopy equivalence
\[
\Lambda{\mathbb L}(a/k)\to{\mathbb H}(a/k)
\]
in \(\text{gr}'({\mathcal C})^{\mathcal N}\) over \(a\), provided \({\mathbb Q}\subset k\). The main result is then carefully translated in section \(4\) into a qis
\[
{\mathbb H}(X/Y)\approx\Lambda{\mathbb L}(X/Y)
\]
for a morphism \(X\to Y\) of paracompact, separated complex spaces or a separated morphism of finite type of Noetherian schemes in characteristic zero.
Further results include several decomposition theorems for Hochschild cohomology: one with respect to tangent cohomology
\[
\text{HH}^n(X/Y,{\mathcal M})\cong\coprod_{i+j=n}\text{Ext}^i (\Lambda^j{\mathbb L}(X/Y),{\mathcal M}),
\]
one with respect to sheaf cohomology with values in exterior powers of the tangent sheaf
\[
\text{HH}^n(X)\cong\coprod_{i+j=n}H^i(X,\Lambda^j{\mathcal T}_X),
\]
and the Hodge decomposition
\[
\text{HH}_n(X) \cong\prod_{i-j=n}H^j(X,\Lambda^i\Omega_X)
\]
thereby reproving many theorems spread over the literature in a unified way. Hochschild cohomology; Noetherian scheme; analytic space; admissible pair of categories; cotangent complex; HKR theorem; Hodge decomposition Schuhmacher, F.: Hochschild cohomology of complex spaces and noetherian schemes. Homology homotopy appl. 6, No. 1, 299-340 (2004) Other (co)homology theories (cyclic, dihedral, etc.) [See also 19D55, 46L80, 58B30, 58G12], Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Relations of \(K\)-theory with cohomology theories, Simplicial sets, simplicial objects (in a category) [See also 55U10], Analytic sheaves and cohomology groups, Derived categories, triangulated categories, Differential graded algebras and applications (associative algebraic aspects), Deformations of complex structures Hochschild cohomology of complex spaces and Noetherian 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 Various useful versions of Hilbert Nullstellensatz for the countably infinite dimensional complex affine space are given. Hilbert Nullstellensatz; infinite dimensional complex affine space Relevant commutative algebra, Polynomials in real and complex fields: location of zeros (algebraic theorems) Hilbert Nullstellensatz for infinite dimensional complex affine space | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0688.00008.]
Every cusp s of a Hilbert modular variety belongs to a pair (M,V), where M is a lattice of rank n and V a group of rank n-1 consisting of totally positive units in a totally real number field F of degree n over \({\mathbb{Q}}\), where V acts on M by multiplication, \(VM=M\). In a fundamental paper [Enseign. Math., II. Ser. 19, 183-281 (1973; Zbl 0285.14007)] \textit{F. Hirzebruch} defined a rational number \(\phi (s)=\phi (M,V)\) called the signature defect of s, and proved in the case that \(n=2\),
\[
(1)\quad \phi (M,V)=-\frac{1}{3}\sum_{k}(b_ k-3),
\]
where the \(b_ k\) are integers arising from the continued fraction expansion of a quadratic irrationality associated with M. On the other hand, the pair (M,V) determines a Hecke L-function
\[
L(M,V,s)=\sum '_{m\in M/V}\frac{sign N(m)}{| N(m)|^ s},\quad Re(s)>1.
\]
Using results of Hecke and C. Meyer, Hirzebruch proved the relation
\[
(2)\quad \phi (M,V)=C\cdot L(M,V,1)\quad (C\text{ is a constant } C=C(M))
\]
for \(n=2\) and conjectured it for all \(n>2\), which has been proved by \textit{M. F. Atiyah, H. Donnelly} and \textit{I. M. Singer} in [Ann. Math., II. Ser. 118, 131-177 (1983; Zbl 0531.58048)].
The author gives an explicit formula for \(\phi\) (M,V) in terms of the triangulation of \({\mathbb{R}}^{n-1}/V\) generalizing Hirzebruch's formula (1) in the case \(n=2\). He also explains his idea, how one could give a ``natural'', elementary and nice proof of Hirzebruch's conjecture (2) for \(n>2\), but he has not solved all arising difficulties to do it completely. Hilbert modular variety; signature defect; explicit formula; Hirzebruch's conjecture [Sc2] Sczech, R.: Cusps on Hilbert modular varieties and values of L-functions. In: Hashimoto, V. Namikawa, Y. (eds) Automorphic Forms and Geometry of Algebraic Varieties. (Adv. Stud. Pure Math.,15, vol. pp. 29-40) Amsterdam: North-Holland and Tokyo: Kinokuniya 1989 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Arithmetic ground fields for surfaces or higher-dimensional varieties, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols Cusps on Hilbert modular varieties and values of L-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 This paper continues the investigation, begun in the authors' previous paper [Boll. Unione Mat. Ital., VII. Ser., B 1, 633-647 (1987; Zbl 0635.14001)], of perfect codimension 2 varieties, which are defined in \({\mathbb{P}}^ n\) by maximal minors of suitable matrices of homogeneous polynomials. The authors obtain extensions of the inequalities of \textit{P. Maroscia} and \textit{W. Vogel} [Math. Ann. 269, 183-189 (1984; Zbl 0533.14022)] and construct some nonsmoothable curves via Tim Sauer's criterion [\textit{T. Sauer}, Math. Ann. 272, 83-90 (1985; Zbl 0546.14023)]. perfect codimension 2 varieties Ciliberto, C.; Geramita, A. V.; Orecchia, F.: Remarks on a theorem of Hilbert--burch. Boll. unione. Math. ital. 7, No. 2-B, 463-483 (1988) Determinantal varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Low codimension problems in algebraic geometry Remarks on a theorem of Hilbert-Burch | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 desingularization of quasi-excellent schemes as conjectured by \textit{A. Grothendieck} [Publ. Math., Inst. Hautes Étud. Sci. 20, 101--355 (1964; Zbl 0136.15901); ibid. 24, 1--231 (1965; Zbl 0135.39701); ibid. 28, 1--255 (1966; Zbl 0144.19904); ibid. 32, 1--361 (1967; Zbl 0153.22301)] was shown in the author's work [Adv. Math. 219, No. 2, 488--522 (2008; Zbl 1146.14009)] in 2008. Compared to the analogue for varieties, that result had the following disadvantages: Centers of the necessary blowups in the resolution procedure could be non-regular, and functoriality was not satisfied for the given construction.
The article under review is the first of two papers strengthening the results previously obtained: A desingularization is given by only blowing up regular centers and such that the resulting sequence of blowing ups gives an object which is functorial for regular morphisms. The case treated here is the non-embedded desingularization, whereas for the embedded one the author refers to his forthcoming paper [``Functorial desingularization over \(\mathbb{Q}\): boundaries and the embedded case'', \url{arXiv:0912.2570}].
Main result of the article is
Theorem 1.2.1: For any Noetherian quasi-excellent generically reduced scheme \(X=X_0\) over \(\text{Spec} (\mathbb{Q})\) there exists a blow-up sequence \({\mathcal F} (X): X_n \dashrightarrow X_0\) such that the following conditions are satisfied: {\parindent=8mm \begin{itemize}\item[(i)] the centers of the blowups are disjoint from the preimages of the regular locus \(X_{\mathrm{reg}}\); \item[(ii)] the centers of the blowups are regular; \item[(iii)] \(X_n\) is regular; \item[(iv)] the blow-up sequence \({\mathcal F} (X)\) is functorial with respect to all regular morphisms \(X' \to X\), in the sense that \({\mathcal F} (X')\) is obtained from \({\mathcal F} (X)\times_X X'\) by omitting all empty blowups.
\end{itemize}}
The Construction of \({\mathcal F} \) is done starting with any algorithm \({\mathcal F}_{\mathrm{Var}}\) giving desingularizations for varieties in characteristic 0 and which is functorial for regular morphisms in the sense of (i), (iii) and (iv). Furthermore, \({\mathcal F} \) will be found to satisfy the above condition (ii) if this is the case for the algorithm \({\mathcal F}_{\mathrm{Var}}\). This algorithm is extended to pairs \((X,Z)\) of quasi-excellent schemes \(X\) and Cartier divisors \(Z\) in \(X\) containing the singular locus and isomorphic to a disjoint union of varieties, such that \({\mathcal F}_{\mathrm{Var}} (X,Z) \) desingularizes \(X\). Now the formal completion \({\mathcal X} := \hat{X}_Z\) is algebraized by some \(X'\), and \({\mathcal F}_{\mathrm{Var}} (X')\) gives rise to desingularizations on \(\mathcal X\) (and on \(X\)). The main work remaining now is to show that \({\mathcal F}_{\mathrm{Var}} (\mathcal X) = \widehat{{\mathcal F}_{\mathrm{Var}} (X')}\) is canonically defined by \(X_n\), where \(X_n\subseteq \mathcal X\) is some sufficiently large nilpotent neighborhood of the closed fibre. Algebraization is done using the classical approximation results of \textit{R. Elkik} [Ann. Sci. Éc. Norm. Supér. (4) 6, 553--603 (1973; Zbl 0327.14001)].
From the author's abstract: ``As a main application, we deduce that any reduced formal variety of characteristic zero admits a strong functorial desingularization. Also, we show that as an easy formal consequence of our main result one obtains strong functorial desingularization for many other spaces of characteristic zero including quasi-excellent stacks, formal schemes, and complex or nonarchimedean analytic spaces. Moreover, these functors easily generalize to noncompact settings by use of generalized convergent blow-up sequences with regular centers.'' desingularization of quasi-excellent schemes; nonembedded desingularization; functorial desingularization Temkin, M., Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case, Duke Mathematical Journal, 161, 2207-2254, (2012) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Valuation rings Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 1978, \textit{S. S. Abhyankar} [cf. Am. J. Math. 101, 267-300 (1979; Zbl 0425.14009)] suggested the following characterization. An irreducible analytic curve \(F(X,Y)=0\) has one characteristic pair if, and only if, there exist non-units A, B of k[[T,U]] without non-unit common factors such that F(A,B) can be factorized in k[[T,U]] in the form \(\prod^{s}_{r=1}(L_ r(T,U))^{a(r)}\quad\), where the \(L_ r\) have degree 1 and are distinct forms of degree 1. The author establishes this characterization in the case \(k={\mathbb{C}}\) by showing that the existence of such a factorization implies that the fundamental group of the link of the singularity F has non-trivial centre. torus knot; irreducible analytic curve; characteristic pair Singularities of curves, local rings, Coverings of curves, fundamental group, Knots and links in the 3-sphere A characterization of plane curve singularities with one characteristic pair | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 Hilb\(^p\) be the Hilbert scheme parametrizing the closed subschemes, with fixed Hilbert polynomial \(p=p(t)\in \mathbb Q[t]\), of the projective space \(\mathbb P_K^n\) over an algebraically closed field \(K\) of characteristic \(0\). For each point \(x\in\) Hilb\(^p\), and any \(i\in \mathbb N\), consider the \(i\)-th cohomological Hilbert function \(h_x^i:\mathbb Z\to \mathbb N\), \(m \to \dim_{k(x)}H^i(\mathbb P_{k(x)}^n, \mathcal I^{(x)}(m))\), where \(k(x)\) and \(\mathcal I^{(x)}\subset \mathcal O_{\mathbb P_{k(x)}^n}\) denote the residue field and the ideal sheaf of the subscheme corresponding to \(x\). Now fix a sequence \((f_i)_{i\in\mathbb N}\) of numerical functions \(f_i:\mathbb Z\to \mathbb N\), and consider the sets \(H^{\geq}:=\{x\in\) Hilb\(^p \mid h_x^i\geq f_i \,\,\forall i\in \mathbb N\}\) and \(H^{=}:=\{x\in\) Hilb\(^p\mid h_x^0=f_0, h_x^i\geq f_i \,\,\forall i\geq 1\}\). Semicontinuity implies that \(H^{\geq}\) is closed in Hilb\(^p\), and that \(H^{=}\) is locally closed.
In the case \(f_i\equiv 0\) for \(i\geq 1\), many authors gave proofs that \(H^{\geq}\) and \(H^{=}\) are connected, i.e. that the subsets of Hilb\(^p\) which are defined lower bounding the functions \(h_x^0\) are connected. For instance, \textit{R. Hartshorne} [Publ. Math. Inst. Hautes Études Sci. 29, 5--48 (1966; Zbl 0171.41502)] proved that Hilb\(^p\) is connected (see also \textit{G. Gotzmann} [Comment. Math. Helv. 63, No. 1, 114--149 (1988; Zbl 0656.14004)], \textit{D. Mall} [J. Pure Appl. Algebra 150, No. 2, 175--205 (2000; Zbl 0986.14002)], \textit{I. Peeva} and \textit{M. Stillman} [J. Algebr. Geom. 14, No. 2, 193--211 (2005; Zbl 1078.14007)]).
Without assumption on the lower bounding functions \(f_i\), in the paper under review the author proves that \(H^{\geq}\) and \(H^{=}\) are connected.
The line of the proof is the following. Let \(S:= K[X_0,\dots,X_n]\) be the polynomial ring, and, for any homogeneous ideal \(\mathfrak a\subset S\) and any \(i\in\mathbb N\), consider the \(i\)-th locally cohomological Hilbert function \(h_{\mathfrak a}^i:\mathbb Z\to \mathbb N\), \(m\to \dim_KH^i_{S_+}(\mathfrak a)_m\), where \(H^i_{S_+}(\mathfrak a)\) denotes the graded \(i\)-th local cohomology module of \(\mathfrak a\) with respect to the irrelevant ideal \(S_+\). And denote by \(h_{\mathfrak a}\) the Hilbert function of \(\mathfrak a\).
Via Serre-Grothendieck correspondence, one sees that the set of closed points of \(H^{\geq}\) (\(H^{=}\) resp.) is equal to the set \(\mathbb I^{\geq}\) (\(\mathbb I^{=}\) resp.) of saturated homogeneous ideals \(\mathfrak a\subset S\), with Hilbert polynomial \(q(t)={\binom{t+n}{n}}-p(t)\), such that \(h_{\mathfrak a}\geq f_0\) (\(h_{\mathfrak a}= f_0\) resp.) and \(h_{\mathfrak a}^i\geq f_{i-1}\) for all \(i\geq 2\). So to prove that \(H^{\geq}\) and \(H^{=}\) are connected amounts to prove that \(\mathbb I^{\geq}\) and \(\mathbb I^{=}\) are connected. To this aim it suffices to prove that \(\mathbb I^{\geq}\) and \(\mathbb I^{=}\) are connected by Gröbner deformations, i.e. that for any two ideals \(\mathfrak a\), \(\mathfrak b\) \(\in \mathbb I^{\geq}\) (\(\mathbb I^{=}\) resp.), there exists a sequence of ideals \(\mathfrak a=\mathfrak c_1,\dots,\mathfrak c_r=\mathfrak b\) in \(\mathbb I^{\geq}\) (\(\mathbb I^{=}\) resp.) such that \(\mathfrak c_i\) is the saturation of the initial ideal or of the generic initial ideal of \(\mathfrak c_{i+1}\) with respect to some term order or vice versa for all \(i\in\{1,\dots,r-1\}\).
To prove that \(\mathbb I^{\geq}\) and \(\mathbb I^{=}\) are connected by Gröbner deformations, the author makes use of certain ideals constructed by \textit{D. Mall} [loc. cit.]. The author proves that a Mall ideal \(\mathfrak c\) is sequentially Cohen-Macaulay, i.e. \(h^i_{\mathfrak c}=h^i_{\text{Gin}_{\text{rlex}}}\mathfrak c\) for all \(i\in \mathbb N\), where \(\text{Gin}_{\text{rlex}}\mathfrak c\) denotes the generic initial ideal of \(\mathfrak c\) with respect to the homogeneous reverse lexicographic order, and that \(\text{Gin}_{\text{rlex}}\mathfrak c=\text{in}_{\text{rlex}}\mathfrak c\). Combining these properties with the general fact that \(h^i_{\mathfrak a}\leq h^i_{\text{in}\mathfrak a}\) for any homogeneous ideal \(\mathfrak a\), any \(i\in \mathbb N\) and any term order, the author is able to conclude the proof of the quoted connectedness property. Hilbert scheme; local cohomology; Mall ideals; sequentially Cohen-Macaulayness S. Fumasoli, Connectedness of Hilbert scheme strata defined by bounding cohomology, PhD thesis, Universität Zürich (2005). arXiv: math.AC/0509123 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Parametrization (Chow and Hilbert schemes) Hilbert scheme strata defined by bounding cohomology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\) be a standard graded domain of dimension two over an algebraically closed field \(k\) of characteristic 0. Let \(I \subset R\) be a homogeneous ideal of finite colength. Then there exists a finitely generated \(\mathbb{Z}\)-algebra \(A \subseteq k\), a finitely generated \(\mathbb{N}\)-graded algebra \(R_A\) over \(A\), and a homogeneous ideal \(I_A \subset R_A\) such that \(R_A \otimes_A k=R\) and for any closed point \(s \in \text{Spec} A\) the ring \(R_s=R_A \otimes_A k(s)\) is a finitely generated \(\mathbb{N}\)-graded two dimensional domain over \(k(s)\) and the ideal \(I_s=\text{Im}(I_A \otimes_A k(s) \to R_s)\) is a homogeneous ideal of finite colength. The author proves that the limit
\[
\lim_{s \to s_0} {e_{HK}} (R_s, I_s)
\]
exists and is a rational number, where \({e_{HK}}\) is the Hilbert-Kunz multiplicity, \(s_0\) is the generic point of \( \text{Spec} A\), and the limit is taken over the closed points \(s \in \text{Spec} A\) . Hilbert-Kunz multiplicity; spread Trivedi, V., Hilbert-Kunz multiplicity and reduction mod \textit{p}, Nagoya Math. J., 185, 123-141, (2007) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Vector bundles on curves and their moduli, Multiplicity theory and related topics Hilbert-Kunz multiplicity and reduction mod \(p\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathbf{k}\) be an infinite field, \(\Delta\) a simplicial complex of dimension \(r\) with vertex set \(\{1, 2, \dots, n\}\) and \(S=\mathbf{k}[x_1, x_2, \dots, x_n]\) the polynomial ring with \(n\) variables. We consider the Stanley-Reisner ring (the face ring) \(\mathbf{k}[\Delta]\) as the quotient ring of \(S\) by the Stanley-Reisner ideal \(I_\Delta\).
It is known that \(\mathbf{k}[\Delta]\) is Cohen-Macaulay if and only if \(\tilde H^i(\text{link} F;\mathbf{k})=0\) for any \(i<\dim(\text{link} F)\) and \(F\in\Delta\), where \(\tilde H^\bullet(-;-)\) denotes the reduced cohomology. And \(\mathbf{k}[\Delta]\) is Buchsbaum if and only if \(\Delta\) is pure and \(\tilde H^i(\text{link} F;\mathbf{k})=0\) for any \(i<\dim(\text{link} F)\) and \(F\in\Delta\setminus\{\emptyset\}\). Recall that a Stanley-Reisner ring is Buchsbaum if and only if it is a ring with finite local cohomology. Also note that if \(\tilde H^i(\text{link} F;\mathbf{k})=0\) for any \(i<r-|F|\) and \(F\in\Delta\setminus\{\emptyset\}\), then \(\Delta\) is pure, where \(|F|\) denotes the cardinality of \(F\). Therefore, \(\mathbf{k}[\Delta]\) is a ring with finite local cohomology if and only if \(\tilde H^i(\text{link} F;\mathbf{k})=0\) for any \(i<r-|F|\) and \(F\in \Delta\) with \(|F|>0\).
In this excellent paper, the authors generalize the above result and show the following Theorem: Let \(m\) be an integer with \(0\leq m\leq r\) and \(\theta_1, \dots, \theta_m\) generic linear forms of \(S\). Then \(\mathbf{k}[\Delta]/(\theta_1,\dots, \theta_m)\) is a ring with finite local cohomology if and only if \(\tilde H^i(\text{link} F;\mathbf{k})=0\) for any \(i<r-|F|\) and \(F\in \Delta\) with \(|F|>m\). The authors prove this result in two ways, first by careful enumeration of the dimensions of graded pieces of local cohomology, next by analysis of Ext modules. The enumerative result is generalized to squarefree modules introduced by Yanagawa.
The paper also contains some results on simplicial complexes with isolated singularities: \(\Delta\) is said to have isolated singularities if \(\tilde H^i(\text{link} F;\mathbf{k})=0\) for any \(i<r-|F|\) and \(F\in \Delta\) with \(|F|>1\). face ring; Stanley-Reisner ring; Buchsbaum ring; ring with finite local cohomology; FLC Miller, E.; Novik, I.; Swartz, E., Face rings of simplicial complexes with singularities, \textit{Math. Ann.}, 351, 857-875, (2011) Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Combinatorial aspects of commutative algebra, Combinatorial aspects of simplicial complexes, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Local cohomology and commutative rings, Cohen-Macaulay modules Face rings of simplicial complexes with 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 Die einfachen algebraischen Gruppen über einem algebraisch abgeschlossenen Körper werden bis auf Überlagerungen durch ihre Wurzelsysteme \(A_n\), \(B_n\), \(C_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\), \(F_4\) und \(G_2\) klassifiziert. In \(A_n\), \(D_n\) und \(E_n\) haben alle Wurzeln die gleiche Länge, man nennt diese Systeme homogen, die übrigen inhomogen.
Die einfachen Hyperflächensingularitäten lassen sich unter allen Singularitäten durch gewisse Schnittdiagramme charakterisieren, die den Coxeter-Dynkin-Witt-Diagrammen der homogenen Wurzelsysteme entsprechen. Im Falle der Charakteristik 0 haben Grothendieck und Brieskorn eine Konstruktion angegeben, die aus einem homogenen Wurzelsystem eine einfache Singularität vom entsprechenden Typ liefert. 1987 fand Knop noch eine weitere Konstruktion (für beliebige Charakteristik).
In der vorliegenden Dissertation wird nun untersucht, welche Singularitäten sich bei dieser Konstruktion im inhomogenen Fall ergeben und deren Zusammenhang mit den in diesem Fall bei der Brieskorn- Grothendieck-Slodowy-Konstruktion auftretenden Singularitäten geklärt. Schließlich wird noch auf die Nahmsche Konstruktion modular invarianter Partitionsfunktionen aus Wurzelsystemen eingegangen, wobei diese im inhomogenen Falle so modifiziert wird, daß sich auch hier nicht-triviale Partitionsfunktionen ergeben. (Deren Klassifikation im homogenen Fall, die man in der konformen Quantenfeldtheorie benötigt, wurde von Cappelli, Itzykson und Zuber bzw. Kato 1987 gegeben). orbits; simple singularities; modular invariant partition functions; root systems; hypersurface singularities Lie algebras of Lie groups, Singularities in algebraic geometry, Simple, semisimple, reductive (super)algebras, Spinor and twistor methods applied to problems in quantum theory On nilpotent orbits, simple singularities and modular invariant partition functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study a natural Hodge module on the Hilbert scheme of four points on affine three-space, which categorifies the Donaldson--Thomas invariant of the Hilbert scheme. We determine the weight filtration on the Hodge module explicitly in terms of intersection cohomology complexes, and compute the E-polynomial of its cohomology. The computations make essential use of a description of the singularity of the Hilbert scheme as the degeneracy locus of the Pfaffian function. Alexandru Dimca and Balázs Szendrői, The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on \(\mathbb C^3\), Math. Res. Lett. 16 (2009), no. 6, 1037-1055. Transcendental methods, Hodge theory (algebro-geometric aspects), Classical real and complex (co)homology in algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Milnor fibration; relations with knot theory, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on \(\mathbb{C}^3\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a survey of recent works of several people on the relation between the Waring's problem (concerning the representation of the \(j\)-forms of \(R=K[X_1,\ldots, X_n]\) as a sum of \(j\)-powers of linear forms), families PGOR\((T)\) of graded Artin algebra quotients of \(R\) having Hilbert function \(T\) and the Hilbert scheme parametrizing locally Gorenstein punctual subschemes of projective space [see also \textit{A. Iarrobino} and \textit{V. Kanev}, ``Power sums, Gorenstein algebras, and determinantal loci'', Lect. Notes Math. 1721 (1999; Zbl 0942.14026)]. The author studies the dimension, the closure and the irreducibility problems on the families PS\((s,j,n)\subset {\mathbb P}^N\), \(N={n+j\choose n}\) parametrizing the degree \(j\) homogeneous polynomials \(f\) of \(R\) which can be written as \(f=L_1^j+\cdots +L_s^j\) for some linear forms \(L_i\), the determinantal varieties of the catalecticant matrices Cat\(_F(u,v,n)\), \(u+v=j\) (the catalecticant matrices extend the well known Hankel matrices of case \(n=2\)), PGOR\((T)\), the quoted Hilbert scheme, etc.. Hilbert scheme; Waring's problem; catalecticant matrices; Hankel matrices; Hilbert function Iarrobino, A. 1996. Gorenstein Artin Algebras, Additive Decompositions of Forms and the Punctual Hilbert scheme. Proceedings of Hanoi conference in Commutative Algebra. 1996. Edited by: Eisenbud, D. and Tuan Hoa, Le. Springer-Verlag. to appear Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Parametrization (Chow and Hilbert schemes), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Determinantal varieties Gorenstein Artin algebras, additive decompositions of forms and the punctual Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0588.00015.]
For every compact smooth algebraic surface X the Chern numbers \(c_ 2 = Euler\) number of X and \(c^ 2_ 1 = self\) intersection number of a canonical divisor of X are defined. If X is a surface of general type, then \(c_ 2>0\) and \(c_ 1,c_ 2\) satisfy the Miyaoka-Yau inequality \(c^ 2_ 1\leq 3c_ 2\). It is known that \(c^ 2_ 1=3c_ 2\) if and only if the universal cover of X is the ball \(B=\{z\in {\mathbb{C}}^ 2: | z_ 1|^ 2+| z_ 2|^ 2<1\}.\)
The author reports on the (separate) work of R. Kobayashi and Y. Miyaoka, who obtain a lower bound for \(3c_ 2-c^ 2_ 1\), in case the surface contains rational or elliptic curves. This result has many geometric applications. The author presents several examples to explain the results and applications, including material from the work of \textit{Th. Höfer} [''Ballquotienten als verzweigte Überlagerungen der projektiven Ebene'' (Dissertation, Bonn 1985)] and \textit{K. Ivinskis} [''Normale Flächen und die Miyaoka-Kobayashi Ungleichung'' (Diplomarbeit, Bonn 1986)]. singularity; rational curve; algebraic surface; Chern numbers; elliptic curves F. HIRZEBRUCH , On the characteristic classes of differentiable manifolds (Colloque H. Poincaré, octobre 1954 ). Families, moduli, classification: algebraic theory, Singularities of surfaces or higher-dimensional varieties, Characteristic classes and numbers in differential topology Singularities of algebraic surfaces and characteristic numbers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0559.00004.]
Generalizing a classical vanishing theorem of Kodaira-Akizuki-Nakano [see \textit{Y. Akizuki} and \textit{Sh. Nakano}, Proc. Japan Acad. 30, 266-272 (1954; Zbl 0059.147)], \textit{H. Grauert} and \textit{O. Riemenschneider} [Invent. Math. 11, 263-292 (1970; Zbl 0202.076)] proved, for \(p=n,\) a) \(H^ q(X,\pi_*\Omega^ p_{\tilde X}\otimes L)=0\) for \(q>0,\)
b) \(R^ q\pi_*\Omega^ p_{\tilde X}=0\) for \(q>0,\)
where X is a compact complex space of dimension n, L an ample line bundle on X, and \(\pi: \tilde X\to X\) a proper birational morphism with \(\tilde X\) smooth. It is known that a) may fail for \(p\neq n\). Guillen, Navarro Aznar, and Puerta [Barcelona Notes (1982)] showed that, for X and L as above,
\[
a)\quad H^ m(X,Gr^ p_ FK_{\dot X}\otimes L)=0\text{ for } m>n,
\]
\[
b)\quad {\mathcal H}^ m(Gr^ p_ FK_{\dot X})=0\;text{ for } m<p \text{ or } m>n,
\]
where \((K^._{X},F)\) is the filtered de Rham complex of X, and H denotes hypercohomology and \({\mathcal H}\) denotes cohomology sheaf.
The author proves the following vanishing theorem, from which he derives a') and b'): Let X be an n-dimensional complex projective variety, \(\Sigma\) \(\subset X\) such that \(X\setminus \Sigma\) is nonsingular, L an ample line bundle on X and \(\pi: \tilde X\to X\) a proper birational mapping such that \(\tilde X\) is nonsingular, \(E=\pi^{-1}(\Sigma)\) is a divisor with normal crossings on \(\tilde X\) and \(\pi\) maps \(X\setminus E\) isomorphically to \(X\setminus \Sigma\). Then
\[
a)\quad H^ q(\tilde X,J_ E \Omega^ P_{\tilde X}(\log E)\otimes \pi L)=0\quad for\quad p+q>n,\quad b)\quad R^ q\pi_*J_ E\Omega^ P_{\tilde X}(\log E)=0\quad for\quad p+q>n.
\]
Here \(\Omega^._{\tilde X}(\log E)\) is the logarithmic de Rham complex and \(J_ E\) is the ideal sheaf of the divisor E. vanishing theorem; de Rham complex J. H. M. Steenbrink, ``Vanishing theorems on singular spaces'' in Differential Systems and Singularities (Luminy, France, 1983) , Astérisque 130 , Soc. Math. France, Montrouge, 1985, 330-341. Vanishing theorems, Analytic sheaves and cohomology groups, de Rham cohomology and algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Compact analytic spaces Vanishing theorems on singular spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a projective curve over an algebraically closed field \(k\) of positive characteristic \(p\), and let \(\mathcal L\) be a base point free line bundle on \(X\). Set \(B=\bigoplus_{n\geq 0} H^{0}(X,{\mathcal L}^{\otimes n})\) and \(B_1= H^{0}(X,{\mathcal L})\). Then the Hilbert-Kunz (for short HK) multiplicity of the section ring \(B\) with respect to the ideal \(B_{1}B\) is denoted by \(\text{HKM}(X,\mathcal L)\). The HK multiplicity of \(B\) with respect to the ideal generated by \(W\subseteq H^0(X,\mathcal L)\) where \(W\) is a base point free linear system, is denoted by \(\text{HKM}(X,\mathcal L,W)\). Let \(V_{\mathcal L}(W)\) denote a vector bundle of rank \(r=\) vector space dimension of \(W -1\) and be the kernel of the surjective map \(W\times {\mathcal O}_X \rightarrow \mathcal L\). If \(W=H^0(X,\mathcal L)\), then \(V_{\mathcal L}(W)\) is denoted by \(V_{\mathcal L}\,.\) The author proves that if \(V_{\mathcal L}\) is strongly semistable then \(\text{HKM}(X,\mathcal L)\) is equal to the HK multiplicity of the section ring with respect to its graded maximal ideal (it may not be true in general). For an arbitrary base-point free line bundle \(\mathcal L\) on a nonsingular curve \(X\) of genus \(g\), the author finds an expression of \(\text{HKM}(X,\mathcal L, W)\) in terms of the ranks and degrees of the vector bundles occuring in a strongly stable Harder-Narasimham filtration. Even if this seems difficult to use, this result implies that the HK multiplicity of an irreducible projective curve is a rational number [\textit{H. Brenner}, Math. Ann. 334, No. 1, 91--110 (2006; Zbl 1098.13017) got the same result independently]. Section \(5\) is devoted to plane curves. Theorem \(5.3\) gives a formula for the HK multiplicity of an arbitrary plane curve \(C\) of degree \(d\) over a field of characteristic \(p>0\); corollary \(5.4\) gives a formula in the case that \(X\) is a nonsingular plane curve of degree \(d\). At the end of the paper, the author recalls some results of Monsky about nonsingular quartics of a certain type. Hilbert-Kunz function; projective curves; vector bundles Trivedi V, Semistability and Hilbert-Kunz multiplicities for curves, J. Algebra 284 (2005) 627--644 Vector bundles on curves and their moduli, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Singularities of curves, local rings Semistability and Hilbert-Kunz multiplicities 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 The paper under preview proves the following theorem.
Theorem 5.2. Consider the set of \(n\)-dimensional varieties \(X\) with Gorenstein canonical singularities and trivial canonical class which can be written as a connected component of a scheme-theoretic intersection of base point free divisors \(F_i \in |D_i |\) in an \((n + k)\)-dimensional bpf-big Fano variety \(P\) with \(\sum_i D_i = -K_P\). Then this set is birationally bounded for any \(n\) independent of \(k\).
Here a variety is bpf-big Fano if it is Gorenstein, normal, with base point free, big anticanonical divisor. The motivation of this result comes from string theory and the Batyrev-Borisov construction of Calabi-Yau complete intersections in toric Fano varieties. As a corollary of the main theorem, the construction of Batyrev-Borisov only gives a birationally bounded family. For example the stringy Hodge numbers are bounded.
The proof uses the deep results of McKernan-Hacon-Xu [\textit{C. D. Hacon} et al., Ann. Math. (2) 180, No. 2, 523--571 (2014; Zbl 1320.14023)]. Fano varieties; Calabi-Yau varieties; Hodge numbers; Batyrev-Borisov construction; complete intersections Fano varieties, Complete intersections, Calabi-Yau manifolds (algebro-geometric aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves On complete intersections with trivial canonical class | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a noetherian normal scheme X over an algebraically closed field k and a finite group G of k-automorphisms of X, it is well-known that the quotient scheme \(Y=X/G\) has structure sheaf \({\mathcal O}_ Y\) given by the invariant subsheaf (\({\mathcal O}_ X)^ G\) of \({\mathcal O}_ X\). This paper examines the analogous relationship for the dualizing sheaves \(\omega\) in the case that X and Y are Cohen-Macaulay. In particular, we show that the natural inclusion \(\omega_ Y\hookrightarrow(\omega_ X)^ G\) is always an isomorphism when k is a field of characteristic zero, but that for characteristic \(p\neq 0\) it is an isomorphism if and only if the quotient map \(X\to Y\) contains no wild ramification in codimension one. The proof is based on duality theory and involves a careful examination of the trace map \({\mathcal O}_ X\to {\mathcal O}_ Y.\) quotient scheme; invariant differentials; group of automorphisms; dualizing sheaves Peskin, BR, On the dualizing sheaf of a quotient scheme, Commun. Algebra, 12, 1855-1869, (1984) Group actions on varieties or schemes (quotients), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials On the dualizing sheaf of a quotient scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let k be an algebraically closed field of characteristic 0. By \(B(\cdot)\) we denote the Brauer group functor. Let A be a regular local ring which is a k-algebra essentially of finite type. Let G be a finite group of k- automorphisms of A. Suppose no height 1 prime of A ramifies over \(A^ G\). Let P be a prime ideal of height \(\geq 2\) in \(A^ G\) and let R be the local ring \((A^ G)_ P\). Set \(S=A\otimes_{A^ G}R\). We say that the ring R has ``quotient singularities'' if S is a local ring. It is shown that if R is a local ring with quotient singularities and quotient field K, the natural map \(B(R)\to B(K)\) is injective. Brauer group functor; regular local ring; quotient singularities Brauer groups of schemes, Regular local rings On the Brauer group and quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the preview in Zbl 0714.13003. Brauer group functor; regular local ring; quotient singularities Brauer groups of schemes, Regular local rings On the Brauer group and quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \({\mathcal L} \subseteq {\mathbb Z}^n\) be a sublattice of dimension \(r.\) Let \(G = \mathbb Z^n/\mathcal L\) denote the factor group. Let \(S = k[x_1,\ldots,x_n]\) denote the polynomial ring in \(n\) variables over the field \(k.\) Then \(S\) becomes \(G\)-graded by setting \(\deg(x_i) = e_i + \mathcal L\) for all \(i = 1,\ldots, n,\) where \(e_i\) denotes the \(i\)-th unit vector in \(\mathbb Z^n.\) A homogeneous ideal \(I \subset S\) is called \(\mathcal L\)-graded, whenever the Hilbert function satisfies \(\dim_k ((S/I)_g) = 1\) for all \(g \in \mathbb N^n/\mathcal L.\) The toric Hilbert scheme of the lattice \(\mathcal L,\) denoted by \(H_{\mathcal L},\) is the multigraded scheme that parametrizes all \(\mathcal L\)-graded ideals in \(S.\) The authors prove the following theorem: Let \(\mathcal L\) be a two-dimensional lattice contained in \(\mathbb Z^n.\) Then the toric Hilbert scheme is smooth and irreducible.
This result extends the following two cases:
(1) \(\mathcal L = \ker_{\mathbb Z}(A),\) where \(A\) is an integer matrix of corank two, shown by \textit{V. Gasharov} and \textit{I. Peeva} [Compos. Math. 123, 225--241 (2000; Zbl 0987.14034)], resp. \textit{I. Peeva} and \textit{M. Stillman} [Duke Math. J. 111, 419--449 (2002; Zbl 1067.14005)].
(2) The \(G\)-Hibert scheme of an abelian subgroup of \(GL(2)\), see \textit{R. Kidoh} [Hokkaido Math. J. 30, 91--103 (2001; Zbl 1015.14004)]. This applies for a two-dimensional lattice \({\mathcal L} \subseteq {\mathbb Z}^2\) where \(G\) is a finite abelian group \(G \subseteq \text{GL}(2)\) and the description of Nakamura's \(G\)-Hilbert scheme. toric variety; connectedness; codimension two; triangulations Maclagan D., Thomas R.R.: The toric Hilbert scheme of a rank two lattice is smooth and irreducible. J. Comb. Theory Ser. A 104, 29--48 (2003) Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies The toric Hilbert scheme of a rank two lattice is smooth and irreducible. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0756.00007.]
The author describes his adaptation, to the case of singular curves, of Viro's ``gluing'' method for the construction of plane real algebraic curves [\textit{O. Viro}, in Topology, general and algebraic topology, and applications, Proc. int. Conf., Leningrad 1982, Lect. Notes Math. 1060, 187-200 (1984; Zbl 0576.14031)]. He constructs in this way curves of degree \(d\) with any combination of double points (real nodes, real isolated points, conjugate imaginary nodes) up to the bound \((d-1)(d- 2)/2\). He also constructs curves of degree \(d\) with any number of real cusps up to \(2[d/3]^ 2-[d/3]\). gluing; singular curves; construction of plane real algebraic curves E.I. Shustin : Real Plane Algebraic Curves with Many Singularities . Preprint Samara State University, 1991. Topology of real algebraic varieties, Singularities of curves, local rings Real plane algebraic curves with many 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 under review proves an extension of the Akizuki-Kodaira-Nakano vanishing theorem for singular varieties over the complex numbers. Namely, for a complete complex variety \(X\), \(Z\subset X\) a closed subset with \(X\backslash Z\) dense in \(X\) and a bounded complex \({\mathcal F}^\bullet\) of locally free sheaves, we have
\[
H^i(X, \underline{\Omega}_X^j(\log Z)\otimes {\mathcal F}^\bullet)=0 \qquad\text{for}\quad i+j > \dim X + \phi({\mathcal F}^\bullet)
\]
(Corollary 4.3). Here, we use as the replacement for differential forms in the Hodge theory for smooth varieties the filtered complex \(\underline{\Omega}_X^\bullet\) of sheaves introduced by \textit{P. Du Bois} [Bull. Soc. Math. Fr. 109, 41--81 (1981; Zbl 0465.14009)] and \(\phi({\mathcal F}^\bullet)\) is the Frobenius-amplitude, which is the crucial notion for extending the cerebrated Deligne-Illusie's theorem for smooth varieties to singular varieties through Frobenius splitting of the complex (Theorem 3.4).
The idea of Frobenius-amplitude is as follows: in order to describe reduction with regard to almost all primes we use ultrafilter \({\mathcal U}\) on the set \(\Sigma\) of all prime numbers and the ultra-fiber \(X_{\mathcal U}\), which roughly denotes the whole bunch of reductions with regard to almost all primes specified by \({\mathcal U}\). Similarly, for a locally free sheaf \({\mathcal F}\) on \(X\), we define the sheaf \({\mathcal F}_{\mathcal U}\) on \(X_{\mathcal U}\). Then we define
\[
\phi({\mathcal F}) = \min\left\{i \mid H^i(X_{\mathcal U}, {Fr^N}^*({\mathcal F}_{\mathcal U})\otimes {\mathcal E})=0 \text{ for any }{\mathcal U}, \text{ coherent }{\mathcal E} \text{ and }N\gg 0 \right\}-1
\]
where \(Fr: X_{\mathcal U}\to X_{\mathcal U}\) is the Frobenius. Moreover, by replacing the tensor product \({Fr^N}^*({\mathcal F}_{\mathcal U})\otimes {\mathcal E}\) by derived tensor \(\otimes^{\mathbb L}\) we can also define \(\phi({\mathcal F}^\bullet)\). Akizuki-Kodaira-Nakano vanishing; singular variety; Frobenius splitting; Frobenius amplitude; ultrafilter Arapura, Donu, Frobenius amplitude, ultraproducts, and vanishing on singular spaces, Illinois J. Math., 55, 4, 1367-1384 (2013), (2011) Vanishing theorems in algebraic geometry, Ultraproducts and related constructions Frobenius amplitude, ultraproducts, and vanishing on singular spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The compound Du Val singular points (cDV points, in short) are expected to play important roles in a theory of minimal models of varieties of dimension 3. The author tried to provide a complete list of their normal forms. This article is a report on his attempt. In fact, he gave some preparatory results and gave a rigorous foundation for Arnold's theory of spectral sequences for reduction of functions. However, he gave only a few examples of normal forms. Because of an immense increase in the amount of elementary computation, he could not give the list.
In spite of the very sound frame of his theory, it is strange that we can find a lot of misunderstandings and misprints even in the Russian version of this article.
Now, a three-dimensional hypersurface singular point on a complex variety is called a cDV point if the intersection with a general hyperplane passing through the point is a rational double point of dimension 2. A wider class of singular points is called canonical singularities. In section 1 a characterization of hypersurface canonical singularities with respect to the Newton diagrams and the diagonal point (1,1,...,1) is given. In section 2, after a characterization for normal forms of functions with 4 variables to define cDV points, the author begins the preparation to give normal forms for cDV points with a special form. Section 3 is devoted to the explanations of Arnold's theory of spectral sequences for reduction of functions. In section 4 the author carries out the computation for the reduction of functions for a few cases by using spectral sequences. Let us have a look on part I of the proof of theorem 8, for example. Here the author has forgotten to treat the monomial \(y^ 2z^{\ell}\) appearing in \(\pi\) when \(2\alpha_ 1\geq \alpha_ 2>\alpha_ 1\) and \(\alpha_ 3| 2\alpha_ 1-\alpha_ 2\). Thus theorem 8 is not verified. Moreover, in the proof of lemma 2' in section 3, the Taylor expansion is strange, and we cannot understand why the author uses condition A but does not use the following coordinate change associated with the vector field \(s=\sum v_ i\partial /\partial x_ i:\) \((*)\quad y_ i=x_ i+\sum^{\infty}_{j=1}(j!)^{-1}s^ j(x_ i)\)\ (1\(\leq i\leq n).\) The condition A is very hard to check, while (*) has a meaning if \(\phi (s)>0\). The following equality also holds under the coordinate change (*): \(f(y_ 1,...,y_ n)=f(x_ 1,...,x_ n)+\sum^{\infty}_{j=1} (j!)^{-1}s^ j(f(x_ 1,..., x_ n)).\)\ I could not give a counter-example for the author's results themselves. Thus these misunderstandings and misprints may have been made when he wrote the final version of the manuscript. Perhaps he had forgotten the exact arguments which he had used at the earlier stage. compound Du Val singular points; canonical singularities; Newton diagrams Markushevich, DG, Canonical singularities of three-dimensional hypersurfaces, Math. USSR-Izv., 26, 315-345, (1986) Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, Singularities in algebraic geometry Canonical singularities of three-dimensional 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 aim of this paper is to present some of the old conjectures and problems related to the classification of irreducible projective plane curves \(C\) in \(\mathbb P^2\) up to the action of the automorphism group \(\text{PGL}(3,\mathbb C)\), together with some results and new conjectures from recent works of the authors, focusing on the case of rational curves. For instance, the classification problem is related to the Nagata-Coolidge problem (whether every rational cuspidal curve can be transformed into a line by a Cremona transformation), and to the determination of the maximal number of cusps for a rational cuspidal plane curve; on this point, the maximal number of cusps known by the authors is 4 and recently \textit{K. Tono} [Math. Nachr. 278, No.~1--2, 216--221 (2005; Zbl 1069.14029)] proved that it is less than 9. The classification problem is also related to the theory of open surfaces; for example, the open surface \({\mathbb P^2 } \backslash C\) is \(\mathbb Q\)-acyclic if and only if \(C\) is a rational cuspidal curve, but according to the \textit{H. Flenner-M. Zaidenberg} rigidity conjecture [see Contemp. Math. 162, 143--208 (1994; Zbl 0838.14027)], if such a surface has logarithmic Kodaira dimension 2, then it should be rigid; this would imply that every equisingular deformation of \(C\) in \(\mathbb P^2\) would be projectively equivalent to \(C\).
Among the classification results, the authors show in particular the classification of rational unicuspidal curves, whose cusp has one Puiseux pair, see authors' paper [in: Real and complex singularities, São Carlos workshop 2004. Papers of the 8th workshop, Marseille, France, July 19--23, 2004. Trends in Mathematics, 31--45 (2007; Zbl 1120.14019)]. From the characterization problem, of the realization of prescribed topological types of singularities, many compatibility properties arise, connecting local invariants of the singular germs \((C,p)\) with some global invariant of the curve \(C\). The authors present their compatibility condition for the characterization problem, [see Proc. Lond. Math. Soc. (3) 92, No.~1, 99--138 (2006; Zbl 1115.14021)], together with some equivalent reformulation; in particular they deal with the semigroup distribution property (connecting the semigroup of the local singularity and the degree of the curve) and they explain its relations (for the unicuspidal case) with the Seiberg-Witten invariant conjecture, formulated by the forth author and \textit{L. I. Nicolaescu} [Sel. Math., New Ser. 11, No. 3--4, 399--451 (2005; Zbl 1110.14006)], also in connection with the Heegaard-Floer homology, introduced by \textit{P. Ozsváth} and \textit{Z. Szabó} [in: Floer homology, Gauge theory, and low-dimensional topology. Proc. Clay Math. Inst. 2004 summer school, Budapest, Hungary, June 5--26, 2004. Clay Math. Proc. 5, 3--27 (2006; Zbl 1107.57022)]. rational curves; logarithmic Kodaira dimension; open surfaces; graded roots J. Fernández de Bobadilla, I. Luengo, A. Melle-Hernández and A. Némethi, On rational cuspidal plane curves, open surfaces and local singularities, Singularity theory, World Scientific, Hackensack (2007), 411-442. Singularities of curves, local rings, Singularities in algebraic geometry On rational cuspidal plane curves, open surfaces and local singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We compute generating functions for elliptic genera with values in line bundles on Hilbert schemes of points on surfaces. As an application we also compute generating functions for elliptic genera with values in determinant line bundles on moduli spaces of sheaves on \(K3\) surfaces. Hilbert schemes; moduli spaces of sheaves; elliptic genus; Verlinde formula Parametrization (Chow and Hilbert schemes), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Holomorphic symplectic varieties, hyper-Kähler varieties, Elliptic genera Refined Verlinde formulas for Hilbert schemes of points and moduli spaces of sheaves on \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a fixed projective scheme which is flat over a base scheme \(S\). The association taking a quasi-projective \(S\)-scheme \(Y\) to the scheme parametrizing \(S\)-morphisms from \(X\) to \(Y\) is functorial. We prove that this functor preserves limits, and both open and closed immersions. As an application, we determine a partition of schemes parametrizing rational curves on the blow-ups of projective spaces at finitely many points. We compute the dimensions of its components containing rational curves outside the exceptional divisor and the ones strictly contained in it. Furthermore, we provide an upper bound for the dimension of the irreducible components intersecting the exceptional divisors properly. rational curves; moduli spaces; blow-ups Fine and coarse moduli spaces, Families, moduli of curves (algebraic), Rational and birational maps, Rational and unirational varieties Properties of schemes of morphisms and applications to blow-ups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the degeneracy loci of general bundle morphisms of the form \({\mathcal O}_{\mathbb{P}^n}^{\oplus m}\to \Omega_{\mathbb{P}^n}(2)\), also from the point of view of the classical geometrical interpretation of the sections of \(\Omega_{\mathbb{P}^n}(2)\) as linear line complexes in \(\mathbb{P}^n\). We consider in particular the case of \(\mathbb{P}^5\) with \(m=2,3\). For \(n=5\) and \(m=3\) we give an explicit description of the Hilbert scheme \({\mathcal H}\) of elliptic normal scrolls in \(\mathbb{P}^5\)
by defining a natural rational map \(\rho:\mathbb{G} (2,14) \to {\mathcal H}\), which turns out to be dominant with general fibre of degree four. locally free resolution of ideals; cohomology groups of ideal sheaf; degree; unirational varieties; degeneracy loci; general bundle morphisms; Hilbert scheme; elliptic normal scrolls Dolores Bazan and Emilia Mezzetti, On the construction of some Buchsbaum varieties and the Hilbert scheme of elliptic scrolls in \Bbb P\(^{5}\), Geom. Dedicata 86 (2001), no. 1-3, 191 -- 204. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Parametrization (Chow and Hilbert schemes), Elliptic curves, Complete intersections, Projective techniques in algebraic geometry On the construction of some Buchsbaum varieties and the Hilbert scheme of elliptic scrolls in \(\mathbb{P}^5\). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Unter einem \((n,m)\) Punkt einer Curve \((n >m)\) versteht der Verfasser einen singulären Punkt, für welchem die Potenzentwickelung der Coordinaten die Form hat:
\[
y= Nx^{ \frac{n}{m}} +N_1 x^{ \frac{n+1}{m}} + \cdots
\]
Man kann diese Gleichung durch die zwei anderen ersetzen:
\[
x= \alpha^m; \quad y= N\alpha^n +N_1 \alpha^{n+1} +\cdots,
\]
und somit die Curve in der Nähe des singulären Punktes mit einer unicursalen vertauschen, für welche die Parameterdarstellung der Coordinaten in den ersten Gliedern mit dieser übereinstimmt.
Im Anschluss an diese Auffassung behandelt der Verfasser die Fragen: Welche Singularität entspricht einem \((n,m)\) Punkt der Originalcurve 1) in der Evolute der Curve, 2) in einer Parallelcurve zu derselben, 3) in einer Curve, die aus ihr durch quadratische Transformation hervorgegangen ist. Der Einfachheit wegen werden die Coefficienten \(N, N_1, \ldots\) von Null verschieden angenommen. Das Verhalten der Evolute ergiebt sich aus den Ausdrücken für die (homogenen) Liniencoordinaten:
\[
u: v: w= \frac{ dx}{ d\alpha} :\frac{ dy}{ d\alpha} : -\left( x \frac{dx}{ d\alpha} +y \frac{ dy}{d \alpha} \right),
\]
indem man bemerkt, dass ein \((n,m)\) Punkt zu einem \((n, n-m)\) Punkt dualistisch reciprok ist. Dabei sind jedoch vier Fälle zu unterscheiden, je nachdem der gegebene singuläre Punkt im Endlichen oder unendlich weit gelegen ist, und eine Tangente durch einen der Kreispunkte im Unendlichen geht oder nicht. Zu weiteren Unterscheidungen geben die Zahlen \(n,m\) Veranlassung, je nachdem \(n \gtreqqless 2m\) ist, u. s. w. Für alle diese Fälle wird das Verhalten des Evolutenpunktes untersucht und seine Gestalt und Lage gegen den singulären Punkt der ersten Curve durch Zeichnungen veranschaulicht. Der Einfluss des letzteren auf Grad und Classe der Evolute lässt sich durch allgemeine Formeln nicht darstellen; für Curven von der \(3^{\text{ten}}\), \(4^{\text{ten}}\), \(5^{\text{ten}}\), \(6^{\text{ten}}\) Ordnung kann die Ordnung der Evolute bis 4 sinken. Für die Liniencoordinaten der Parallelcurve im Abstande \(k\) hat man:
\[
u': v': w' = u: v: w \pm k \sqrt{ u^2 +v^2},
\]
wo \(u, v, w\) die Liniencoordinaten der gegebenen Curve sind. Vermöge dieser Formeln bestimmen sich die Charactere des einem \((n,m)\) Punkt entsprechenden singulären Punktes ähnlich wie oben, wobei die nämlichen Fälle zu unterscheiden sind. Die im Falle der quadratischen Transformation vorzunehmenden Unterscheidungen beziehen sich auf die Lage des Punktes gegen die Seiten und Ecken des Transformationsdreiecks.
Manche Bemerkungen des Verfassers findet man in den theilweise gleichzeitig erschienenen Arbeiten über Singularitäten von Nöther, Halphén und dem Referenten weiter ausgeführt. plane curves; singularities Plane and space curves, Singularities in algebraic geometry On corresponding singularities in algebraic plane curves. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is well known that every smooth complex projective curve \(C\) is birational (via a generic linear projection) to a plane curve with no worse than ordinary double points. Similarly, if \(Y\) is a smooth projective variety of dimension \(n\), a suitable linear projection yields a hypersurface \(X\) that is birational to \(Y\). This paper examines the type of singularities lying on \(X\).
The main result of the paper states that if \(Y\) is embedded in \(\mathbb{P}^N\) via the \(d\)-uple embedding, where \(d \geq 3n\) and if \(\pi \colon \mathbb{P}^N \to \mathbb{P}^{n+1}\) denotes a generic linear projection, then \(X=\pi(Y)\) has no worse than semi log canonical singularities for \(n\leq 5\). More precisely, the author proves that \(X\) has Du Bois singularities, and that in this setting Du Bois singularities are semi log canonical. Du Bois singularities are interescting in their own right: by definition, their cohomology is easy to determine and they seem to provide the natural setting for vanishing theorems by Kollár's principle [see \textit{K. Schwede}, Compos. Math. 143, No. 4, 813--828 (2007; Zbl 1125.14002) and \textit{S. Kovács, K. Schwede} and \textit{K. Smith}, ``Cohen-Macaulay semi-log canonical singularities are Du Bois'', \url{arXiv:0801.1541}]. The proof relies on the direct analysis of the local analytic type of singularities that arise in such generic projections in \textit{J. Roberts} [Trans. Am. Math. Soc. 212, 229--268 (1975; Zbl 0314.14003)].
Finally, the author presents a counter example for \(n = 30\); a similar statement cannot therefore be expected to hold for arbitrarily large dimension. linear projection; singularities Doherty, DC, Singularities of generic projection hypersurfaces, Proc. Am. Math. Soc., 136, 2407-2415, (2008) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Singularities of generic projection 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 Let \(S\) be a smooth quasiprojective variety over an algebraically closed field \(k\) of characteristic \(0\).
In this note we determine the class \([S^{[n]}]\) of the Hilbert scheme \(S^{[n]}\) of subschemes of length \(n\) on \(S\) in the Grothendieck ring
\(K_0(V_k)\) of \(k\)-varieties. The result expresses \([S^{[n]}]\) in terms of the classes of the symmetric powers
\(S^{(l)}=S^l/GG_l\). Here \(GG_l\) is the symmetric group acting by permutation of the factors of \(S^l\).
Theorem.
\[
[S^{[n]}]=\sum_{\alpha\in P(n)} [S^{(\alpha)}\times
A^{n-|\alpha|}].
\]
Here \(P(n)\) is the set of partitions of \(n\). We write a partition \(\alpha\in P(n)\) as \((1^{a_1},2^{a_2},\ldots,n^{a_n})\),
where \(a_i\) is the number of occurences of \(i\) in \(\alpha\). Then the length \(|\alpha|\) of \(\alpha\) is the sum of the \(a_i\)
and \(S^{(\alpha)}=S^{(a_1)}\times \ldots\times S^{(a_n)}.\) L. Göttsche, 'On the motive of the Hilbert scheme of points on a surface', \textit{Math. Res. Lett.}8 (2001) 613-627. Parametrization (Chow and Hilbert schemes), Motivic cohomology; motivic homotopy theory, (Equivariant) Chow groups and rings; motives, Algebraic cycles On the motive of the Hilbert scheme of points on a surface. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For part I see ibid. 38, 321-341 (1986; Zbl 0618.14004), for part II see the preceding review.]
In this last part of the paper the author applies the concept of universally one-equicodimensional schemes (defined and studied in the second part of the paper) to the problem of finite generation of k- subalgebras of a k-algebra of finite type, with k a field. This approach is different from the methods developed by Zariski and Nagata in connection with the Hilbert's fourteenth problem. The problem can be translated into more geometric terms as follows: if X is an algebraic variety over k and if Y is a k-scheme dominated by X, find natural conditions ensuring that Y is also an algebraic variety. For example, if Y is quasi-compact and integral, then one of the main results of the paper says that Y is an algebraic variety iff Y is a universally one- equicodimensional scheme. As applications, the author considers the low- dimensional cases and gives new proofs to some results due to Zariski and Nagata. finite generation of subalgebras; universally one-equicodimensional schemes; algebra of finite type Adrian Constantinescu. Proper morphisms and finite generation of subalgebras. III. Schemes dominated by algebraic varieties. Stud. Cerc. Mat., 38(6):477--510, 1986. Schemes and morphisms, Commutative Artinian rings and modules, finite-dimensional algebras Proper morphisms and finite generation of subalgebras. III: Schemes dominated by 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 this well written paper, the author examines the following question of Abhyankar:
(Q) Given a dominant, generically finite morphism \(f: Y\to X\) of complete \(k\)-varieties (where \(k\) is a field), does there exist a commutative diagram
\[
\begin{tikzcd} Y_1 \ar[r,"f_1"]\ar[d,"\tau" '] & X_1\ar[d,"\sigma"]\\ Y\ar[r,"f" '] & X\end{tikzcd}
\]
of morphisms of complete \(k\)-varieties with \(Y_1\) nonsingular and \(X_1\) normal and such that \(f_1\) is finite and \(\sigma\) and \(\tau\) are proper and birational?
This is weaker than the question on simultaneous resolutions (requiring \(X_1\) also to be nonsingular) which is known to have a negative answer. A local form of (Q) has a positive answer in dimension 2, as shown by Abhyankar.
The author gives a counterexample to the question (Q) already for nonsingular projective surfaces, but also proves some positive results by weakening the requirements on \(f_1\), \(\sigma\) and \(\tau\) or by assuming additional conditions on \(f\).
Theorem 1 (Counterexample to (Q)). Let \(k\) be an algebraically closed field with \(\text{char}(k)\neq 2\). Then there exists a generically finite morphism \(f: Y\to X\) of nonsingular projective \(k\)-surfaces for which the answer to question (Q) is in the negative.
Theorem 2. Let \(f\) be as in question (Q). Assume that \(\text{char}(k)= 0\). Then there exists a diagram as asserted in (Q) with \(X_1\) having (normal) toric singularities but with \(f_1\) only quasi-finite and the properness of \(\sigma\) and \(\tau\) replaced by the following weaker requirements: Every \(k\)-valuation ring of \(k(X)\) has a center on \(X_1\) and every \(k\)-valuation ring of \(k(Y)\) has a center on \(Y_1\). (This means that separatedness may be lost in patching up the local solutions.)
Theorem 3. Let \(f\) be as in question (Q). Assume that \(k\) is algebraically closed of characteristic zero, \(X\) and \(Y\) are projective and \(f\) is generically finite Galois. Then there exists a diagram as asserted in (Q) with \(X_1\) having (normal) toric singularities. dominant morphism; resolution of singularities S.D. Cutkosky, Generically finite morphisms and simultaneous resolution of singularities, Contemp. Math., AG/0109003, to appear. Rational and birational maps, Global theory and resolution of singularities (algebro-geometric aspects), Local structure of morphisms in algebraic geometry: étale, flat, etc. Generically finite morphisms and simultaneous 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 There are various classical and new results concerning irreducibility, smoothness and local structure of the variety of all irreducible plane projective algebraic curves (over an algebraically closed field of characteristic 0) of a given degree with prescribed singularities (nodes and cusps). The present article and some earlier work by the author [Funct. Anal. Appl. 21, 82-84 (1987); translation from Funkts. Anal. Prilozh. 21, No. 1, 90-91 (1987; Zbl 0627.14023) and Sel. Math. Sov. 10, No. 1, 27-37 (1991); translation from: Methods of the qualitative theory of differential equations, Gorkij, 148-163 (1983; Zbl 0804.14011)] give sufficient conditions for more complicated singularities. Let \(V=V(d,m_1, \dots, m_r)\) be the set of irreducible plane curves of degree \(d\) with \(r\) ordinary singularities (multiple points with distinct tangents) of multiplicities \(m_1, \dots, m_r\). Then, the variety \(V\) is a smooth \(T\)-variety (i.e. locally a transversal intersection of germs of equisingular strata) if \(\sum^r_{i=1} (m_i-1)\) \((3m_i-4)\leq \alpha_0d^2\) and is irreducible if this number is \(\leq\alpha_1d^2\). The coefficients \(\alpha_0\) and \(\alpha_1\) depend on \(m=\max \{m_1, \dots, m_r\}\), and the number on the left is the sum of the total Milnor number of a curve \(F\in V\) and the total \(\delta\)-invariant of a corresponding generic polar curve. The proof is based on non-speciality of certain new linear systems of plane curves. variety of plane projective algebraic curves; \(\delta\)-invariant; ordinary singularities; Milnor number; linear systems of plane curves Eugenii Shustin, Smoothness and irreducibility of families of plane algebraic curves with ordinary singularities, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 393 -- 416. Singularities of curves, local rings, Enumerative problems (combinatorial problems) in algebraic geometry, Families, moduli of curves (algebraic) Smoothness and irreducibility of families of plane algebraic curves with ordinary singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We introduce a new fundamental domain \(\mathscr{R}_n\) for a cusp stabilizer of a Hilbert modular group \(\Gamma\) over a real quadratic field \(K=\mathbb{Q}(\sqrt{n})\). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of \(\mathcal{H}^2 \times \mathcal{H}^2\). The region \(\mathscr{R}_n\) is the product of \(\mathbb{R}^+\) with a 3-dimensional tower \(\mathcal{T}_n\) formed by deformations of lattices in the ring of integers \(\mathbb{Z}_K\), and makes explicit the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism. We include computer generated images and data illustrating various examples. Hilbert modular surfaces; topological manifolds; geometric structures on manifolds; algebraic numbers; rings of algebraic integers; real and complex geometry; geometric constructions Modular and Shimura varieties Cusp shapes of Hilbert-Blumenthal surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we discuss the singularities of rational space curves. Two methods are provided to compute the singularities of arbitrary degree curves. These methods are a generalization of the paper [the second author et al., J. Symb. Comput. 43, No. 2, 92--117 (2008; Zbl 1130.14039)], which are based on the \(\mu\)-basis of the rational space curve and on random technique. The \(\mu\)-basis induces a matrix \(M\) which contains all the information about the singularities including the parameter values corresponding to the singularities, multiplicities and infinitely near singularities. These information can be obtained by computing the Smith form of the matrix \(M\). We compare our methods with previous approaches such as generalized resultants, and provide some examples to illustrate the effectiveness of our methods. \(\mu\)-basis; rational space curve; singularities Shi, X.; Chen, F., Computing the singularities of rational space curves, (ISSAC 2010 - Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, (2010), ACM New York), 171-178 Computational aspects of algebraic curves, Singularities of curves, local rings, Plane and space curves, Symbolic computation and algebraic computation Computing the singularities of rational 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 article is devoted to algebraic geometry related with a Noetherian ring. Dimension and codimension functions are studied, examples are given. Chains of irreducible closed subsets are investigated. Specific features of algebraic schemes over a ring containing an infinite set of elements are elucidated. In particular, the spectrum of the Noetherian ring is considered and topology is used. biequidimensionality; spectrum of a Noetherian ring; dimension formula; codimension function Dimension theory, depth, related commutative rings (catenary, etc.), Commutative Noetherian rings and modules, Relevant commutative algebra, Elementary questions in algebraic geometry Some remarks on biequidimensionality of topological spaces and Noetherian schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The rational surface singularities have a lot of characterizations. See for example: \textit{J. Lipman}, Publ. Math., Inst. Hautes Étud. Sci. 36, 195-279 (1970; Zbl 0181.489). The main new theorem is the converse of a result of J. Lipman. Let (R,\({\mathfrak m})\) be an excellent normal local domain of dimension 2 such that \(k=R/{\mathfrak m}\) is algebraically closed. Then R has a rational singularity if and only if, for any \({\mathfrak m}\)- primary complete ideal I in R, \(I^ 2\) is a complete ideal. The author also proves that if k is not algebraically closed, the theorem fails (the example is a simple elliptic singularity over \({\mathbb{Q}})\). rational surface singularities; excellent normal local domain Cutkosky S.D.: A new characterization of rational surface singularities. Invent. math. 102, 157--177 (1990) Singularities of surfaces or higher-dimensional varieties, Excellent rings A new characterization of rational surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper discusses a conjecture on vanishing theorems of higher direct image sheaves in complex algebraic geometry, as for example mentioned in [\textit{R. Lazarsfeld}, Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Berlin: Springer (2004; Zbl 1093.14500)]. It also has some impacts from Hodge theory of singularities in the form studied by \textit{J. H. M. Steenbrink} [Proc. Symp. Pure Math. 40, 513--536 (1983; Zbl 0515.14003)].
Let \(Z\) be a complex variety of dimension \(n \geq 2\). Assume \(f:Y \to Z\) is a blow up of singularities with \(E\) the exceptional locus.
The authors concern the vanishing conjecture of the form
\[
R^{n-1}f_*\Omega_Y(\log E)=0\tag{1}
\]
where they prove the result under mild assumptions. They prove the conjecture (1) in two cases, the first when \(Z\) has isolated singularities (Theorem B), and the second when it has toric singularities (Theorem C).
Some of the lemmas in use are classical and similar results to them already exists in the literature, however the authors prove these lemmas by their own method on their former results. Examples are Lemma 1.1 and Lemma 4.1. A relevant reference is probably the classical text \textit{P. Deligne} [Publ. Math., Inst. Hautes Étud. Sci. 40, 5--57 (1971; Zbl 0219.14007)]. There in section 3, he concerns similar results on the independence of the MHS on compactifying divisor which correspond to the exceptional divisor in the blow up. Some versions of Lemma 4.1 also exists in the literature for instance in [\textit{C. A. M. Peters} and \textit{J. H. M. Steenbrink}, Mixed Hodge structures. Berlin: Springer (2008; Zbl 1138.14002)]; and also in the above reference by Deligne. But the proofs of the authors are independent and somewhat different.
The vanishing question (1) can be discussed in a more solid language which is the approach in [\textit{M. A. A. de Cataldo} and \textit{L. Migliorini}, Ann. Sci. Éc. Norm. Supér. (4) 38, No. 5, 693--750 (2005; Zbl 1094.14005)] and [\textit{M. A. A. de Cataldo} and \textit{L. Migliorini}, Lond. Math. Soc. Lect. Note Ser. 343, 102--137 (2007; Zbl 1131.14015)]. In this approach they can be studied from a motivic view and the perverse sheaves. The terminology in use by the authors is self dependent and concerns their former results in previous articles, [\textit{M. Mustaţă} and \textit{M. Popa}, Hodge ideals. Providence, RI: American Mathematical Society (AMS) (2019; Zbl 1442.14004); Int. Math. Res. Not. 2018, No. 11, 3587--3605 (2018; Zbl 1408.14074)].
In proposition 2.2 they give a reformulation of the question by standard arguments of algebraic geometry. The proof of the conjecture (in the special cases that the authors are considering) is an spectral sequence argument of Steenbrink type spectral sequences, (see [\textit{J. H. M. Steenbrink}, Proc. Symp. Pure Math. 40, 513--536 (1983; Zbl 0515.14003)] and the references there).
A variant of the conjecture (1), namely Theorem E, is given in terms of the Hodge filtration of mixed Hodge modules (MHM) of [\textit{M. Saito}, Publ. Res. Inst. Math. Sci. 26, No. 2, 221--333 (1990; Zbl 0727.14004); Mosc. Math. J. 9, No. 1, 151--181 (2009; Zbl 1196.14015)]. By the work of Saito; a MHM on a base variety \(X\), can be identified with the VMHS on a Zariski open dense subset of the variety. In case, the Hodge filtration is given by the pole filtration. They mention a relation concerning the compatibility of pole filtration and the degree filtration on the \(D\)-module \(D_X\) of differential operators on \(X\) that, the Hodge (pole) filtration satisfies what is called a good filtration. At the level of MHM, \textit{M. Saito} says [Mosc. Math. J. 9, No. 1, 151--181 (2009; Zbl 1196.14015)] the Hodge filtration is generated at level \(k\) if
\[
F_lD_X.F_k\mathcal{O}_X(*D)=F_{k+l}\mathcal{O}_X(-D), \qquad k \gg 0 , \text{ for all } l\tag{2}
\]
The smallest integer \(k\) with this property is called the generating level.
There are descriptions of the property in the hypersurface isolated singularities in the literature [\textit{M. Saito}, Math. Ann. 295, No. 1, 51--74 (1993; Zbl 0788.32025)], and also in the normal crossing case by Saito himself, [loc. cit.].
In the last section, another formulation of the conjecture D in the paper is given in terms of multiplier ideal sheaves and Hodge ideals. For example in the isolated singularity case, the Hodge filtration is closely related to the multiplier ideals defined by the singularity and the roots of \(b\)-function of the fibration. rational singularities; vanishing theorems; Hodge ideals Singularities of surfaces or higher-dimensional varieties, Vanishing theorems in algebraic geometry, Complex surface and hypersurface singularities, Mixed Hodge theory of singular varieties (complex-analytic aspects) Local vanishing and Hodge filtration for rational 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 studies a certain divisor, \(V\), of a moduli space for abelian surfaces with real multiplication by a real quadratic field \(L\). To be precise: It studies the complement of the ordinary locus, \(V\), in Hilbert-Blumenthal surfaces, in positive characteristic \(p\). If \(p\) is not split, \(V\) is equal to the supersingular locus. If \(p\) is not split, an explicit parameterization of \(V\) is given, yielding that it consists of rational curves, whose total number is essentially \(C\zeta_L(-1)\) \((C=1\) for \(p\) inert and \(1/2\) for \(p\) ramified).
If \(p\) is inert, the singularities of \(V\) are ordinary with two branches. If \(p\) is ramified it follows from the work by \textit{P. Deligne} and \textit{G. Pappas} [Compos. Math. 90, No. 1, 59-79 (1994; Zbl 0826.14027)] that the modular surfaces is singular. The authors determine in the case at hand the exact parameterization and nature of the singularities \((p+1\) branches). For \(p\) split, it is shown that the non-ordinary locus consists of curves intersecting transversely, and that the Hecke orbit of every nonsingular point in it is dense. singular modular surface; divisor; moduli space for abelian surfaces with real multiplication; Hilbert-Blumenthal surfaces; positive characteristic; non-ordinary locus Bachmat, E.; Goren, E. Z., \textit{on the non-ordinary locus in Hilbert-blumenthal surfaces}, Math. Ann., 313, 475-506, (1999) Modular and Shimura varieties, Singularities in algebraic geometry, Algebraic moduli of abelian varieties, classification, Finite ground fields in algebraic geometry, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces On the non ordinary locus in Hilbert-Blumenthal surfaces | 0 |
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