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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 Fixed an algebraically closed field \(k\), the Hilbert scheme \({\mathbf H}\) parametrizing the zero-dimensional subschemes of the affine plane Spec \(k[x,y]\) is a disjoint union of its components \({\mathbf H}^l\) parametrizing the subschemes of length \(l\). The linear action of the torus \(k^*\times k^*\) on the polynomial ring \(k[x,y]\), defined by \((t_1,t_2) \cdot x^\alpha y^\beta=(t_1\cdot x)^\alpha (t_2\cdot y)^\beta\), induces an action on the Hilbert schemes \({\mathbf H}\) and \({\mathbf H}^l\). Given two relatively prime integers \(a\) and \(b\), one may consider the closed subschemes \({\mathbf H}_{ab}\subset{\mathbf H}\) and \({\mathbf H}_{ab}^l \subset{\mathbf H}^l\), parametrizing the zero-dimensional subschemes invariant under the action of the subtorus \(T_{ab}=\{(t^{-b}, t^a),\;t\in k^*\}\). Sometimes, information on \({\mathbf H}_{ab}\) may be lifted to \({\mathbf H}\). In fact \textit{G. Ellingsrud} and \textit{S. A. Strømme} [Invent. Math. 87, 343--352 (1987; Zbl 0625.14002)] have computed the Chow group of \({\mathbf H}^l\) examining the embedding \({\mathbf H}^l_{ab}\subset {\mathbf H}^l\) for general \((a,b)\), and \textit{M. Brion} [Transform. Groups 2, No. 3, 225--267 (1997; Zbl 0916.14003)] has described the equivariant cohomology of \({\mathcal H}^l\) in terms of the equivariant cohomology of all the \({\mathbf H}^l_{ab}\). This accounts for the interest in studying the schemes \({\mathbf H}_{ab}\). In the paper under review, the author determines the irreducible components of the schemes \({\mathbf H}_{ab}\) (by \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.14502)] and \textit{J. Fogarty} [Am. J. Math. 70, 511--521 (1968; Zbl 0176.18401)] one already knows that the smooth subschemes \({\mathbf H}^l\) are the irreducible components of \({\mathbf H})\). The basic and natural idea is that one may separate \({\mathbf H}_{ab}^l\) into disjoint subschemes by fixing a Hilbert function. More precisely, define the degree \(d\) of a monomial by the formula \(d(x^\alpha y^\beta)=-b\alpha +a\beta\). Then a subscheme \(Z\subset\text{Spec}\,k[x,y]\) is in \({\mathbf H}_{ab}\) if and only if its ideal \(I(Z)\) is quasi-homogeneous with respect to \(d\), i.e. \(I(Z)= \bigoplus_{n\in\mathbb{Z}}I(Z)_n\), where \(I(Z)_n=I(Z)\cap k[x,y]_n\) and \(k[x,y]_n\) denotes the vector space generated by the monomials \(m\) of degree \(d(m)=n\). One defines the numerical sequence \(H=(\text{codim} (I(Z)_n\), \(k[x,y]_n))_{n\in \mathbb{Z}}\) as the Hilbert function of \(Z\). Now, fix any sequence of integers \(H=(h_n)_{n\in \mathbb{Z}}\), and denote by \({\mathbf H}_{ab}(H)\) the (possibly empty) closed subscheme of \({\mathbf H}_{ab}\) parametrizing the subschemes \(Z\) whose Hilbert function is \(H\). It turns out that \({\mathbf H}_{ab}\) is the disjoint union of the subschemes \({\mathbf H}_{ab}(H)\), and the main result of the paper consists in proving that, when it is not empty, then \({\mathbf H}_{ab}(H)\) is smooth and irreducible. The methods involved in the proof allow the author to recover some of the results appearing in previous papers of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [loc. cit.; Invent. Math. 91, No. 2, 365--370 (1988; Zbl 1064.14500)]. Bialynicki-Birula stratification; zero-dimensional scheme; affine plane; Hilbert function Evain, L., Irreducible components of the equivariant punctual Hilbert schemes, Adv. Math., 185, 328-346, (2004) Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Group actions on varieties or schemes (quotients) Irreducible components of the equivariant punctual Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Fix a totally real number field \(F/\mathbb{Q}\) of degree \(g\) and a prime number \(p\), remaining prime in \(F\). In this paper we study the reduction mod \(p\) of Hilbert-Blumenthal varieties of level \(\Gamma_0(p)\) where \(p\) denotes a fixed prime number. This is a special case of a Shimura variety \(\text{Sh}_C (G,X)\) for which the \(p\)-primary part \(C_p\subset G(\mathbb{Q}_p)\) of the subgroup \(C\subset G(\mathbb{A}_f)\) is of parahoric type. For these Shimura varieties, one may address the two problems:
(1) Determine the local structure of the natural model \(M_C/\mathbb{Z}_{(p)}\) of this Shimura variety.
(2) Determine the global structure of the reduction mod \(p\) of this model.
Treating the second of these questions, the so called ``supersingular locus'' on \(M_C\otimes\mathbb{F}\) (where \(\mathbb{F}\) is an algebraic closure of \(\mathbb{F}_p)\) is of particular interest. It is always a closed subset of \(M_C\). The most well-known example of these Shimura varieties, given by the data \(G=\text{GL}(2)\), \(C_p=\left\{ \left(\begin{smallmatrix} a & b\\ c & d \end{smallmatrix} \right)\in\text{GL}(2,\mathbb{Z}_p) | c\equiv 0\pmod p\right\}\), is studied by \textit{P. Deligne} and \textit{M. Rapoport} [in: Modular Functions one variable. II, Proc. Int. Summer School, Antwerp 1972, Lect Notes Math. 349, 143-316 (1973; Zbl 0281.14010)] and serves as a prototype in this paper. The Shimura variety associated to these data -- the elliptic moduli curve of level \(\Gamma_0(p)\) -- possesses a model \(M_C\) over \(\mathbb{Z}_{(p)}\). Putting \(C_p'=\text{GL}(2,\mathbb{Z}_p)\) and \(C'=C^pC_p'\) and denoting by \(M_{C'}\) the model of \(\text{Sh}_{C'}(G,X)\) over \(\mathbb{Z}_{(p)}\) (classifying elliptic curves with level-\(C^p\) structure), the main results of the Deligne-Rapoport paper cited above with respect to the two questions above are: (1) The scheme \(M_C\) is regular, has the relative dimension one over \(\mathbb{Z}_{(p)}\), and possesses semi-stable reduction. (2) Over \(M_{C'} \otimes \mathbb{F}_p\) there is a section \(Fr\) to the canonical projections \(p_1:M_C \otimes\mathbb{F}_p\to M_{C'}\otimes\mathbb{F}_p\) (resp. a section \(Ver\) to \(p_2:M_C\otimes \mathbb{F}_p\to M_{C'}\otimes\mathbb{F}_p)\) being given by the Frobenius morphism (resp. the Verschiebung). The special fiber \(M_C\otimes\mathbb{F}_p\) is the union of \(Fr(M_{C'} \otimes\mathbb{F}_p)\) and \(Ver(M_{C'}\otimes\mathbb{F}_p)\), these two closed one-dimensional subschemes intersecting transversally in exactly the supersingular points.
The aim of this paper is to generalize these results to Hilbert-Blumenthal varieties. We succeed to give answers to the questions posed above in the two-dimensional case. An \(F\)-cyclic isogeny \(f: (A,i)\to (A',i')\) between abelian schemes with real multiplication in \(F\) is an isogeny \(f:(A,i)\to(A',i')\) of degree \(p^g\) satisfying \(\ker(f)\leq{}_pA\). It is shown that the moduli scheme \(M_{\Gamma_0(p)}/\mathbb{Z}_{(p)}\) classifying \(F\)-cyclic isogenies (+ additional data) has a model with semi-stable reduction.
For \(g=2\), the paper gives a global description of the supersingular locus on \(M_{\Gamma_0(p)}\) and on \(M_{\text{abs}}\) where \(M_{\text{abs}}/\mathbb{Z}_{(p)}\) denotes the moduli scheme classifying abelian schemes with real multiplication: All irreducible components \(C_i\) of \((M_{\text{abs}}\otimes\mathbb{F}_{p^2})^{ss}\) are isomorphic to \(\mathbb{P}^1\), two components meeting in at most one point and each point in \(C_i(\mathbb{F}_{p^2})\) being the intersection of \(C_i\) with another component. The irreducible components \(R_i\) of \((M_{\Gamma_0(p)} \otimes \mathbb{F}_{p^2})^{ss}\) are all smooth surfaces; the set of these components is in bijection with the components \(C_i\). Each one of the canonical projections \(p_j\), \(j=1,2\), gives \(R_i\) the structure of a rationally ruled surface over some component \(C_i\). Hilbert-Blumenthal varieties; moduli scheme; supersingular locus Stamm, H., On the reduction of the Hilbert-blumenthal-moduli scheme with \({\Gamma}\)_{0}(\textit{p})-level structure, Forum Math., 9, 4, 405-455, (1997) Algebraic moduli of abelian varieties, classification, Modular and Shimura varieties On the reduction of the Hilbert-Blumenthal-moduli scheme with \(\Gamma_0(p)\)-level structure | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0633.00007.]
Let k be an algebraically closed field. A monomial curve over k is an algebroid curve Spec(O), where O is a sub-k-algebra \(O\subseteq k[[ t_ 1]]\times k[[ t_ 2]]\) generated by elements of type \((t_ 1^{n_ 1},t_ 2^{n_ 2})\). The paper studies the semigroup \(S\subset {\mathbb{Z}}^ 2_+\) associated to a monomial curve and conversely all the monomial curves associated to a given semigroup S. semigroup associated to monomial curve; algebroid curve Special algebraic curves and curves of low genus, Singularities of curves, local rings, Power series rings, Singularities in algebraic geometry Singularities of monomial curves with more than one components | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a \(K3\) surface. The proof involves Verbitsky's theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of \(K3\) surfaces. algebraic cycles; holomorphic symplectic varieties; standard conjectures F.~Charles and E.~Markman 2013 The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of~\(K3\)~surfaces \textit{Compos. Math.}149 3 481--494 Parametrization (Chow and Hilbert schemes), Algebraic cycles, Divisors, linear systems, invertible sheaves, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, \(K3\) surfaces and Enriques surfaces The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author shows that the following Zariski theorem (Hilbert's weak Nullstellensatz) is naturally derived from fundamental properties on integral dependence and an elementary property on the polynomial ring in one variable over a field: Let \(k\) be a field and \(K=k[x_ 1,x_ 2,\ldots,x_ n]\) a finitely generated extension ring of \(k\). If \(K\) is a field, then \(K\) is algebraic over \(k\). Zariski theorem; integral dependence; polynomial ring Polynomial rings and ideals; rings of integer-valued polynomials, Integral dependence in commutative rings; going up, going down, Relevant commutative algebra On Hilbert's Nullstellensatz | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present paper is devoted to prove the following statement:
Let \(G\cong 6.\mathrm{Suz}\) be the universal perfect central extension of the Suzuki simple group and \(U\) be a \(12\)-dimensional irreducible representation of \(G\). Then the quotient singularity \(U/G\) is weakly exceptional but not exceptional, in the sense of [\textit{V. V. Shokurov}, J. Math. Sci., New York 102, No. 2, 3876--3932 (2000; Zbl 1177.14078)] and [\textit{Yu. G. Prokhorov}, Blow-ups of canonical singularities. A. G. Kurosh, Moscow, Russia, May 25-30, 1998. Berlin: Walter de Gruyter. 301--317 (2000; Zbl 1003.14005)].
As consequence of this theorem, the authors get the following classification result: let \(G\) be a sporadic group or a central extension of one with centre contained in the commutator subgroup and let \(G\hookrightarrow \mathrm{GL}(U)\) be a faithful finite-dimensional complex representation of \(G\). Then the singularity \(U/G\) is: \begin{itemize}\item[-] exceptional if and only if \(G\cong 2.\mathrm{J}_2\) is a central extension of the Hall-Janko sporadic simple group and \(U\) is a \(6\)-dimensional irreducible representation of \(G\); \item[-] weakly exceptional but not exceptional if and only if \(G\cong 6.\mathrm{Suz}\) and \(U\) is a \(12\)-dimensional irreducible representation of \(G\).
This result shows that among the sporadic simple groups, the groups \(\mathrm{J}_2\) and \(\mathrm{Suz}\) are somehow distinguished from a geometric point of view. This motivates the author to pose the following question: ``Is there a group-theoretic property that distinguishes the groups \(\mathrm{J}_2\) and \(\mathrm{Suz}\) among the sporadic simple groups?''\end{itemize} weakly exceptional singularities; log canonical threshold; sporadic simple groups Singularities in algebraic geometry, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Singularities of surfaces or higher-dimensional varieties, Minimal model program (Mori theory, extremal rays) Sporadic simple groups 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 A construction of algebraic surfaces based on two types of simple arrangements of lines, containing the prototiles of substitution tilings, has been proposed recently. The surfaces are derived with the help of polynomials obtained from the lines generating the simple arrangements. One of the arrangements gives the generalizations of the Chebyshev polynomials known as folding polynomials. The other produces a family of polynomials which generates surfaces having more real nodes, and they can also be used, in combination with Belyi polynomials, to derive hypersurfaces in the complex projective space with many \(A_j\)-singularities. In some cases explicit expressions can be obtained from the classical Jacobi polynomials. The lower bounds for the maximum possible number of \(A_j\)-singularities in certain hypersurfaces of degree \(d\) are improved for several values of \(d\) and \(j\). singularities; algebraic surfaces Escudero, JG, Hypersurfaces with many \(A_{j}\)-singularities: explicit constructions, J. Comput. Appl. Math., 259, 87-94, (2014) Singularities of surfaces or higher-dimensional varieties, Hypersurfaces and algebraic geometry Hypersurfaces with many \(A_j\)-singularities: explicit constructions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 holomorphic Euler characteristics of tautological sheaves on Hilbert schemes of points on surfaces. In particular, we establish the rationality of K-theoretic descendent series. Our approach is to control equivariant holomorphic Euler characteristics over the Hilbert scheme of points on the affine plane. To do so, we slightly modify a Macdonald polynomial identity of Mellit. Hilbert schemes; tautological bundles; Macdonald polynomials Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Symmetric functions and generalizations \(K\)-theoretic descendent series for Hilbert schemes of points on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors consider some cases of equivariant Hilbert schemes for cyclic group actions on the plane. The main theorem of the paper establishes generating series for some of these.
The authors include a discussion of the equivariant Hilbert scheme more generally. Furthermore, beyond the cases where generating series are established by the authors' main theorem, they also include a conjecture in one other case. The paper is economical and contains some useful examples. Hilbert schemes of points; equivariant Hilbert schemes; generating series Gusein-Zade, SM; Luengo, I; Melle-Hernández, A, On generating series of classes of equivariant Hilbert schemes of fat points, Mosc. Math. J., 10, 593-602, (2010) Parametrization (Chow and Hilbert schemes), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) On generating series of classes of equivariant Hilbert schemes of fat 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 These notes are based on series of seven lectures given in the combinatorics seminar at U.C. San Diego; in February and March, 2001. We discuss a series of new results in combinatorics, algebra and geometry. The main combinatorial problems we solve are
(1) we prove the positivity conjecture for Macdonald polynomials, and
(2) we prove a series of conjectures relating the diagonal harmonics to various familiar combinatorial enumerations; in particular we prove that the dimension of the space of diagonal harmonics is \((n+1)^{n-1}\).
In order to prove these results, we have to work out some new results about geometry of the Hilbert scheme of points in the plane and a certain related algebraic variety. As a technical tool for our geometric results, in turn, we need to do some commutative algebra, which although complicated, has a quite explicit and combinatorial nature. symmetric functions; positivity conjecture; diagonal harmonics; Hilbert scheme of points in the plane Mark Haiman, Notes on Macdonald polynomials and the geometry of Hilbert schemes, Symmetric functions 2001: surveys of developments and perspectives, NATO Sci. Ser. II Math. Phys. Chem., vol. 74, Kluwer Acad. Publ., Dordrecht, 2002, pp. 1 -- 64. Parametrization (Chow and Hilbert schemes), Extremal set theory Notes on MacDonald polynomials and the geometry of Hilbert schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In Gromov-Witten theory the virtual localization method is used only when the invariant curves are isolated under a torus action. In this paper, we explore a strategy to apply the localization formula to compute the Gromov-Witten invariants by carefully choosing the related cycles to circumvent the continuous families of invariant curves when there are any. For the example of the two-pointed Hilbert scheme of Hirzebruch surface \(F_{1}\), we manage to compute some Gromov-Witten invariants, and then by combining with the associativity law of (small) quantum cohomology ring, we succeed in computing all 1- and 2-pointed Gromov-Witten invariants of genus 0 of the Hilbert scheme with the help of [\textit{W.-P. Li} and \textit{Z. Qin}, Turk. J. Math. 26, No. 1, 53--68 (2002; Zbl 1054.14068)]. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Gromov-Witten invariants of the Hilbert scheme of two points on a Hirzebruch 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 Let \(R\) be a polynomial ring over a field, let \(I\subseteq R\) be a homogeneous ideal, and let \({\mathfrak a}\subseteq I\) be a homogeneous ideal generated by \(s\) forms with \(\text{codim}({\mathfrak a}:I)\geq\) s. In this paper, the authors obtain conditions for when the Hilbert function of \(R/{\mathfrak a}\) or of \(R/({\mathfrak a}:I)\) is determined by \(I\) and the degrees of the forms. This is the case for perfect ideals of codimension \(2\) or Gorenstein of codimension \(3\). The notion of ``(weakly) \(s\)-residually \({S_2}\)'' is introduced and it is shown that if \(I\) satisfies \(G_s\), i.e. for each prime ideal \(\mathfrak p\) containing \(I\) with \(\text{codim } {\mathfrak p}\leq s-1\), the minimal number of generators of \(I_{\mathfrak p}\) is at most codim \({\mathfrak p}\) , and it is weakly \((s-1)\)-residually \(S_2\), then the Hilbert function of \(R/{\mathfrak a}\) or of \(R/({\mathfrak a}:I)\) is determined, up to \(R\)-equivalences, by \(I\) and the degrees of the homogeneous generators of \(\mathfrak a\).
The main result of the paper establishes that the vanishing of local cohomology implies the residually \(S_2\)-property. At the end the authors show that general projection can create ideals that are \(s\)-residually \(S_2\) and they obtain examples that illustrate the theory. Hilbert function; Hilbert polynomial; residual intersection; residually \(S_2\) Marc Chardin, David Eisenbud, and Bernd Ulrich, Hilbert functions, residual intersections, and residually \?\(_{2}\) ideals, Compositio Math. 125 (2001), no. 2, 193 -- 219. Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Linkage, complete intersections and determinantal ideals, Linkage, Multiplicity theory and related topics Hilbert functions, residual intersections, and residually \(S_2\) ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities By results of \textit{D. Matsushita} [Topology 38, No. 1, 79--83 (1999); addendum ibid. 40, 431--432 (2001; Zbl 0932.32027)], a morphism \(f: X \rightarrow B\) from a projective symplectic \(2d\)-fold \(X\) to a normal variety \(B\), \(0 < \dim B < 2d\), is necessary an abelian fibration, i.e. a Lagrangian fibration with a generic fiber an abelian \(d\)-fold. In particular, such fibration \(f\) is given by a nef divisor \(L\) on \(X\) on which the Beauville form \(q_X\) on \(X\) vanishes. It is natural to ask whether the inverse is true, i.e. whether any nef line bundle L on a projective symplectic manifold \(X\) with \(q_X(c_1(L)) = 0\) defines an abelian fibration on \(X\) (at least in a rational sense) [see \S 21.4 of \textit{M. Gross (ed.), D. Huybrechts (ed.), D. Joyce (ed.)}, Calabi-Yau manifolds and related geometries. Lectures at a summer school in Nordfjordeid, Norway, June 2001 (2003; Zbl 1001.00028)].
A particular case of projective symplectic manifolds with vanishing Beauville form are the Hilbert powers \(S^{[d]}\) of smooth \(K3\) surfaces \(S\) with \(Pic S = {\mathbb Z}h\), \(h^2 = m^2(2d-2)\), \(m \geq 1, d \geq 2\). \textit{B. Hassett} and \textit{Yu. Tschinkel} [Int. J. Math. 11, No. 9, 1163--1176 (2000; Zbl 0982.14012)] proved that for \(m \geq 2\) and \(d = 2\) on \(S^{[d]} = S^{[2]}\) there exists a genuine (regular) abelian fibration to \(B = {\mathbb P}^2\); in addition they asked whether for \(m \geq 2\) and any \(d \geq 3\) and \(S\) as above on \(S^{[d]}\) there also exists a similar abelian fibration onto \({\mathbb P}^d\).
The present paper gives a partial answer to this question by showing that for \(m\geq 2\) and \(d \geq 3\) the Hilbert power \(X = S^{[d]}\) admits a rational map \(f\) to \({\mathbb P}^d\), which is birationally equivalent to an abelian fibration. The construction of \(f\) uses a twisted Fourier-Mukai transform which induces a birational isomorphism between \(X\) and certain abelian fibered moduli space \(M\) of twisted sheaves on another \(K3\) surface \(T\), obtained from \(S\) as a Fourier-Mukai partner. The complete positive answer to the question of Hassett and Tschinkel is given by \textit{J. Sawon} (in a collaboration with K. Yoshioka) [Lagrangian fibrations on Hilbert schemes of points on \(K3\) surfaces, \texttt{math.AG/0509224}; to appear in the J. Algebr. Geom.], by using a refinement of the same approach. Similar result for the case of generalized Kummer varieties with vanishing Beauville form has been obtained by \textit{M. Gulbrandsen} [Lagrangian fibrations on generalized Kummer varieties, \texttt{math.AG/0510145}]. Dimitri Markushevich, Rational Lagrangian fibrations on punctual Hilbert schemes of \?3 surfaces, Manuscripta Math. 120 (2006), no. 2, 131 -- 150. Vector bundles on surfaces and higher-dimensional varieties, and their moduli, \(K3\) surfaces and Enriques surfaces, Fibrations, degenerations in algebraic geometry Rational Lagrangian fibrations on punctual Hilbert schemes of \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0644.00005.]
L'A. présente dans ce texte les récents développements autour du théorème des zéros effectivs de Hilbert. Etant donnés des polynômes \(P_ 1,...,P_ m\in {\mathbb{C}}[X_ 1,...,X_ n]=:{\mathbb{C}}[X]\) de degrés \(\leq D\) et \(P\in {\mathbb{C}}[X]\) de degré \(D_ 0\) s'annulant sur l'ensemble des zéros communs de \(P_ 1,...,P_ m\) dans \({\mathbb{C}}^ n\), il s'agit de majorer un entier \(e\in {\mathbb{N}}\) et les degrés de polynômes \(A_ 1,...,A_ m\in {\mathbb{C}}[X]\) tels que \(P^ e=A_ 1P_ 1+...+A_ mP_ m\). L'A. a déjà établi de telles estimations du type \(<<_ n(D+D_ 0)^{\mu} \) où \(\mu =\min \{m,n\}\), contrastant avec les bornes précédentes de la forme \(<<_ n(D+D_ 0)^{2^{\mu}}\). Il explique dans ce texte comment obtenir la majoration plus fine \(e\leq (n+1)(\mu +1)(D_ 0+1)D^{\mu}\). Il pose également certaines questions dont la première a été récemment élucidée par \textit{C. Berenstein} et \textit{A. Yger} (``Effective Bézout identities in \({\mathbb{Q}}[X_ 1,...,X_ n]'')\), la seconde et la quatrième ont trouvé leurs réponses dans un travail de \textit{J. Kollár} [``Sharp effective Nullstellensatz'', J.Am. Math. Soc. 1, No.4, 963-975 (1988)] et une amélioration sensible en direction de la troisième a été obtenue dans le travail précité de Berenstein et Yger. Enfin, signalons qu'en direction opposée à l'exemple de Mayr et Meyer. \textit{F. Amoroso} a montré [``Tests d'appartenance'', C. R. Acad. Sci., Paris, Sér. I] que, sous des hypothèses raisonnables sur la variété des zéros de \(P_ 1,...,P_ m\) on peut effectivement écrire un polynôme \(Q\in (P_ 1,...,P_ m)\) sous la forme \(Q=A_ 1P_ 1+...+A_ mP_ m\) avec des bornes simples exponentielles pour les degrés des \(A_ i\). effective Hilbert zero theorem; effective Hilbert Nullstellensatz Relevant commutative algebra, Transcendence (general theory) Aspects of the Hilbert Nullstellensatz | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(H_{d,g} (\mathbb P^r)\) denote the Hilbert scheme parametrizing curves \(C \subset \mathbb P^r\) of degree \(d\) and arithmetic genus \(g\) and let \({\mathcal I}_{d,g,r} \subset H_{d,g} (\mathbb P^r)\) be the union of irreducible components whose general member is a smooth, irreducible, non-degenerate curve. \textit{L. Ein} showed that \({\mathcal I}_{d,g,r}\) is irreducible if \(d \geq g+r\) when \(r = 3\) [Ann. Sci. Éc. Norm. Supér. 19, 469--478 (1986; Zbl 0606.14003)] and \(r=4\) [\textit{L. Ein}, Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], but there are various examples showing reducibility of \({\mathcal I}_{d,g,r}\) when \(d \geq g+r\) and \(r > 4\), disproving a claim of Severi.
Most of these examples were constructed with families of curves that are \(m\)-fold covers of \(\mathbb P^1\) with \(m \geq 3\), but the authors gave an example with a family of curves that are double covers of irrational curves [Taiwanese J. Math. 21, 583--600 (2017; Zbl 1390.14019)].
Here the authors reconstruct their example in a more geometric way as a family \(\mathcal D\) of curves on ruled surfaces. The new construction allows them to show that \(\mathcal D\) is generically smooth of expected dimension, hence a regular component. When including the distinguished component dominating \(\mathcal M_g\), this gives the first examples of Hilbert schemes \({\mathcal I}_{d,g,r}\) satisfying \(d \geq g+r\) with \textit{two} regular components. Hilbert scheme of smooth connected curves; regular components; ruled surfaces; double covers Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) Components of the Hilbert scheme of smooth projective curves using ruled 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 \(A=k[X_ 1,\dots,X_ n]/(X_ 1^{d_ 1},\dots,X_ n^{d_ n})\). The Hilbert function of \(A\) is strictly increasing for a while, then constant, and finally strictly decreasing. This was shown by Stanley, using the hard Lefschetz theorem. In fact, if \(S=X_ 1+\cdots+X_ n\), then multiplication with \(S^{t-2i}\) gives an isomorphism of \(A_ i\) to \(A_{t-i}\), where \(t=(\sum d_ i-n)\) is the largest integer such that \(A_ t\neq 0\). Later Stanley also gave a combinatorial proof. --- The first result in this paper is an algebraic proof of this result.
An element \(s\) of degree \(d\) in a graded algebra \(A\) is called faithful, if multiplication with \(s\) gives an injection or surjection \(A_ i\to A_{i+d}\) for all \(i\), and strongly faithful if \(s^ i\) is faithful for all \(i\). The authors then discuss the existence of faithful and strongly faithful elements in a 0-dimensional graded complete intersection or Gorenstein ring. If \(\text{char}(k)=0\), then any complete intersection of embedding dimension 2 has a strongly faithful element of degree 1. It is conjectured that any complete intersection over a field of characteristic 0 has a linear strongly faithful element. Hilbert function; strongly faithful elements; graded complete intersection; Gorenstein ring \beginbarticle \bauthor\binitsL. \bsnmReid, \bauthor\binitsL. \bsnmRoberts and \bauthor\binitsM. \bsnmRoitman, \batitleOn complete intersections and their Hilbert functions, \bjtitleCanad. Math. Bull. \bvolume34 (\byear1991), no. \bissue4, page 525-\blpage535. \endbarticle \OrigBibText L. Reid, L. Roberts and M. Roitman: On Complete Intersections and their Hilbert Functions , Canad. Math. Bull. 34 (4) (1991), 525-535. \endOrigBibText \bptokstructpyb \endbibitem Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Complete intersections, Graded rings On complete intersections and their Hilbert functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The note follows a lecture given by the author during the school ``Liaison and related topics'' (Torino 2001). It refers the results already proved by the author in [Le Matematiche, LV, 517--531 (2000)] about the connectedness of Hilbert scheme of space curves of degree \(d\) and genus \((d-3)(d-4)/2 - 1\). With respect to that paper, a complete list of degrees for which smooth irreducible curves appear together with the corresponding Hartshorne-Rao modules. Plane and space curves, Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Families, moduli of curves (algebraic) A note on the Hilbert scheme of curves of degree \(d\) and genus \({d-3\choose 2}-1\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A rational elliptic surface \(X\) is a smooth rational complex projective surface admitting a map to the projective line whose general fiber is a smooth elliptic curve. The general such \(X\) is the blow-up of the plane at the nine base points of a general pencil of plane cubics. Its Hilbert scheme of \(n\)-points on \(X\), say \(X^{[n]}\) parametrizes the \(0\)-dimensional schemes of lenght \(n\) on \(X\). Observe that any curve \(C \subset X\) admitting a pencil \(g_1^n\) of degree \(n\) induces a rational curve \(C_{[n]}\) parametrizing the divisors in the pencil. The main result of the paper under review (see Thm. 1.1) provides a description of the cone of curves of \(X^{[n]}\) for \(X\) a general rational elliptic surface and \(n \geq 3\). To be precise, it is shown that this cone is spanned by the classes of the following curves: \(C_0\), the class of a curve contracted by the Hilbert-Chow morphism (from \(X^{[n]}\) to the symmetric product \(X^{(n)}\)); the classes \(E_{[n]}\) corresponding to \((-1)\)-curves \(E\subset X\); and \(F_{[n]}\), with \(F\) the class of an elliptic fiber. Using the known description of the cone of curves of \(X\), this result provides a complete description of the cone of curves of \(X^{[n]}\), which in particular implies (by duality) that its nef cone satisfies the Morrison-Kawamata conjecture (there exists a rational polyhedral fundamental domain for the action of the automorphism group of the variety on its nef cone). moduli spaces of sheaves; Hilbert schemes of points; elliptic surfaces; cone conjecture; klt Calabi-Yau pairs Parametrization (Chow and Hilbert schemes), Elliptic surfaces, elliptic or Calabi-Yau fibrations, Minimal model program (Mori theory, extremal rays), Automorphisms of surfaces and higher-dimensional varieties The nef cone of the Hilbert scheme of points on rational elliptic surfaces and the cone conjecture | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under review studies Hilbert schemes of points of the total spaces \(X_{n}\) of line bundles \(\mathcal{O}_{\mathbb{P}^{1}}(-n)\) in terms of ADHM data and realizes them as irreducible connected components of moduli spaces of quiver representations. The surfaces \(X_{n}\) are minimal resolutions of toric singularities of type \(\frac{1}{n}(1,1)\) having Hizebruch surfaces \(\Sigma_{n}\) as projective compactifications, connecting with string theory in physics.
In the paper, the authors construct ADHM data for the Hilbert schemes \(\mathrm{Hilb}^{c}(X_{n})\), by going through the description of moduli spaces of framed sheaves of Hirzebruch surfaces \(\Sigma_{n}\), \(\mathcal{M}^{n}(1,0,n)\simeq \mathrm{Hilb}^{c}(X_{n})\). This ADHM data turns out to provide a principal bundle over the Hilbert scheme (c.f. Theorem 3.1).
In section \(4\), it is shown how these Hilbert schemes are irreducible connected components of GIT quotients of representation spaces of certain quivers for a suitable choice of the stability parameter (c.f. Theorem 4.5), therefore they can be seen as embedded components into quiver varieties associated, in a natural way, with ADHM data of the Hilbert schemes.
It is worth noting that the quiver varieties of this article fall outside the notion of a Nakajima quiver variety carrying naturally a simplectic structure, while \(\mathrm{Hilb}^{c}(X_{n})\) encodes a Poisson structure in general. The authors propose to study further the wall-crossing of the stability parameters to shed light on questions in geometric representation theory, and also to study the Poisson structure of these spaces. Hilbert schemes of points; quiver varieties; HIrzebruch surfaces; ADHM data; monads; Nakajima quivers; McKay quivers; quiver varieties, moduli spaces of quiver representations Bartocci, C.; Bruzzo, U.; Lanza, V.; Rava, C.L.S., Hilbert schemes of points of \(\mathcal{O}_{\mathbb{P}^1}(- n)\) as quiver varieties, J. pure appl. algebra, 221, 2132-2155, (2017) Algebraic moduli problems, moduli of vector bundles, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Representations of quivers and partially ordered sets Hilbert schemes of points of \(\mathcal{O}_{\mathbb{P}^1}(- n)\) as quiver 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 An open question is whether the Hilbert scheme of points of a high-dimensional affine space satisfies Murphy's Law, as formulated by Vakil. In this short note, we instead consider the loci in the Hilbert scheme parameterizing punctual schemes with a given Hilbert function, and we show that these loci satisfy Murphy's Law. We also prove a related result for equivariant deformations of curve singularities with \(\mathbb{G}_m\)-action. Erman, Daniel, Murphy's law for Hilbert function strata in the Hilbert scheme of points, Math. Res. Lett., 19, 6, 1277-1281, (2012) Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Murphy's law for Hilbert function strata in the Hilbert scheme of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Among the most interesting invariants one can associate with a link \(\mathcal{L}\subset S^3\) is its HOMFLY polynomial \(P(\mathcal{L},v,s)\in\mathbb{Z}[v^{\pm 1},(s-s^{-1})^{\pm 1}]\). \textit{A. Oblomkov} et al. [Geom. Topol. 22, No. 2, 645--691 (2018; Zbl 1388.14087)] conjectured that this polynomial can be expressed in algebraic geometric terms when \(\mathcal{L}\) is obtained as the intersection of a plane curve singularity \((C,p)\subset\mathbb{C}^2\) with a small sphere centered at \(p\): if \(f=0\) is the local equation of \(C\), its Hilbert scheme \(C_p^{[n]}\) is the algebraic variety whose points are the length \(n\) subschemes of \(C\) supported at \(p\), or, equivalently, the ideals \(I\subset\mathbb{C}[[x,y]]\) containing \(f\) and such that \(\dim C[[x,y]]/I=n\). If \(m:C_p^{[n]} \to\mathbb{Z}\) is the function associating with the ideal \(I\) the minimal number \(m(I)\) of its generators, they conjecture that the generating function \(Z(C,v,s)=\sum_n s^{2n}\int_{C_p^{[n]}}(1-v^2)^{m(I)}d\chi(I)\) coincides, up to a renormalization, with \(P({\mathcal L},v,s)\). In the formula the integral is done with respect to the Euler characteristic measure \(d\chi\). A more refined version of this surprising identity, involving a ``colored'' variant of \(P(\mathcal{L},v,s)\), was conjectured to hold by E. Diaconescu, Z. Hua and Y. Soibelman [\textit{D.-E. Diaconescu} et al., Commun. Number Theory Phys. 6, No. 3, 517--600 (2012; Zbl 1276.14065)].
The seminar will illustrate the techniques used by \textit{D. Maulik} [Invent. Math. 204, No. 3, 787--831 (2016; Zbl 1353.32032)] to prove this conjecture. plane curve singularities; Hilbert scheme; stable pairs. algebraic links; HOMFLY polynomials; skein algebra Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) HOMFLY polynomials from the Hilbert schemes of a planar curve [after D. Maulik, A. Oblomkov, V. Shende, \dots] | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 singularities of theta divisors have played an important role in the study of algebraic varieties. These notes survey some of the recent progress in this subject, using as motivation some well known results, especially those for Jacobians. Casalaina-Martin, S.: Singularities of theta divisors in algebraic geometry. In: Curves and Abelian Varieties. Contemp. Math., vol. 465, pp. 25--43. Am. Math. Soc, Providence (2008) Theta functions and abelian varieties, Theta functions and curves; Schottky problem, Jacobians, Prym varieties Singularities of theta divisors in algebraic geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\text{Hilb}^P(\mathbb{P}^r)\) be the Hilbert scheme that parametrizes flat families of subschemes of \(\mathbb{P}^r\) with Hilbert polynomial \(P(t)\). It is classically constructed as subscheme of the Grassmannian \(\text{Gr}(Q(d),S_d)\) parametrizing vector subspaces of dimension \(Q(d) = \binom{r+d}{r}-P(d)\) of the homogeneous piece \(S_d\) of degree \(d\) of the coordinate ring \(S = k[x_0,\ldots,x_r]\) of \(\mathbb{P}^r\), for \(d\) sufficiently large. Hence, the Hilbert scheme can be realized as a projective scheme through the Plücker embedding of \(\text{Gr}(Q(d),S_d)\)
\[
\text{Hilb}^P(\mathbb{P}^r) \hookrightarrow \text{Gr}(Q(d),S_d) \hookrightarrow \mathbb{P}\left(\wedge^{Q(d)} S_d\right).
\]
The group of automorphisms of \(\text{Hilb}^P(\mathbb{P}^r)\) given by the action of the general linear group \(\text{GL}_{r+1}(k)\) on \(\mathbb{P}^r\) is compatible with the linear action induced by \(\text{GL}_{r+1}(k)\) on the Grassmannian \(\text{Gr}(Q(d),S_d)\) and on the wedge product \(\wedge^{Q(d)} S_d\), so that it is natural to look at the GIT quotients of the special linear group.
The author studies the quotient \(\text{Hilb}^P(\mathbb{P}^r) /\!\!/\, \text{SL}_{r+1}(k)\) using some characterization of the quotient \(\mathbb{P}\left(\wedge^{Q(d)} S_d\right) /\!\!/\, \text{SL}_{r+1}(k)\). In particular, the author tries to understand the geometry of the subschemes corresponding to unstable points. Starting from the Hesselink stratification [\textit{W.~H.~Hesselink}, J.~Reine Angew.~Math.~303/304, 74--96 (1978; Zbl 0386.20020)] of the unstable locus of \(\mathbb{P}\left(\wedge^{Q(d)} S_d\right)\)
\[
\mathbb{P}\left(\wedge^{Q(d)} S_d\right)^{\text{us}} = \mathbb{P}\left(\wedge^{Q(d)} S_d\right) \setminus \mathbb{P}\left(\wedge^{Q(d)} S_d\right)^{\text{ss}} = \coprod_{[\lambda],e} E_{[\lambda],e},
\]
where strata are indexed by the conjugacy class \([\lambda]\) of adapted 1-parameter subgroups of \(\text{SL}_{r+1}(k)\) and by the Kempf index \(e\) [\textit{G.~R.~Kempf}, Ann.~Math.~(2) 108, 299--316 (1978; Zbl 0406.14031)], the author considers the induced stratifications
\[
\text{Hilb}^P(\mathbb{P}^r)^{\text{us}} = \coprod_{[\lambda],e} \left( E_{[\lambda],e} \cap \text{Hilb}^P(\mathbb{P}^r)\right)
\]
and
\[
\text{Gr}(Q(d),S_d)^{\text{us}} = \coprod_{[\lambda],e} \Big( E_{[\lambda],e} \cap \text{Gr}\big(Q(d),S_d\big)\Big).
\]
As the Kemps index is a measure of instability, the author defines the worst unstable point of \(\text{Hilb}^P(\mathbb{P}^r)\) (resp.~\(\text{Gr}(Q(d),S_d)\)) as a point contained in the non-empty stratum \(E_{[\lambda],e} \cap \text{Hilb}^P(\mathbb{P}^r)\) (resp.~\(E_{[\lambda],e} \cap \text{Gr}(Q(d),S_d))\)) with highest Kempf index.
In the paper, the author focuses on
\begin{itemize}
\item the behavior of the worst unstable points of \(\text{Hilb}^P(\mathbb{P}^r)\) depending on the degree \(d\) chosen for the embedding in the Grassmannian \(\text{Gr}(Q(d),S_d)\);
\item the relation between the worst unstable points of the Grassmannian and of the Hilbert scheme;
\item the special case of constant Hilbert polynomials.
\end{itemize} Hilbert scheme; geometric invariant theory; instability; Hesselink stratification; Kempf index; Castelnuovo-Mumford regularity Parametrization (Chow and Hilbert schemes), Geometric invariant theory Worst unstable points of a 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 Let \((Y,0)\) be the germ of an isolated singulariy. The singularity is called small if it has a resolution \(\varphi: X\to Y\) whose exceptional set \(E=\varphi^{-1}(0)\) has dimension less than \(\dim (Y)-1\). It is called simple small if additionally \(E\) is irreducible. As an example the hypersurface singularity \((V(z_1z_2-z_3z_4),0)\subseteq (\mathbb{C}^4,0)\) is simple small. It has a resolution \(\varphi: X\to V(z_1z_2-z_3z_4)\) such that \(\varphi^{-1}(0)\simeq \mathbb{P}^1\). Some examples of simple small singularities are given. One of them is the \(E_6\)--singulariy defined by \(w^2-z^3+xy^3-3x^2yz-x^5+xzw-x^4y=0\) in \(\mathbb{C}^4\). small singularities Ando, T., Some examples of simple small singularities, Comm. Algebra, 41, 6, 2193-2204, (2013) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Local complex singularities, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Some examples of simple small 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 introductory exposition of the theory of complex plane curve singularities, including relevant commutative algebra, Puiseux series, the Milnor number, resolution of singularities and the Gauß-Manin connection. conductor; Gauß-Manin connection; isolated singularity; milnor number; nullstellensatz; Puiseux series Singularities of curves, local rings, Plane and space curves, Milnor fibration; relations with knot theory Singularities of plane algebraic curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework for various multidimensional spinning tops, as well as Beauville systems, we prove that if the spectral curve is nodal, then all singularities on the corresponding fiber of the system are non-degenerate. We also show that the type of a non-degenerate singularity can be read off from the behavior of double points on the spectral curve under an appropriately defined antiholomorphic involution. Our analysis is based on linearization of integrable flows on generalized Jacobian varieties, as well as on the possibility to express the eigenvalues of an integrable vector field linearized at a singular point in terms of residues of certain meromorphic differentials. Relationships between algebraic curves and integrable systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures Singularities of integrable systems and algebraic curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove that the complex cobordism class of any hyper-Kähler manifold of dimension \(2n\) is a unique combination with rational coefficients of classes of products of punctual Hilbert schemes of K3 surfaces. We also prove a similar result using the generalized Kummer varieties instead of punctual Hilbert schemes. As a key step, we establish a closed formula for the top Chern character of their tangent bundles. Chern numbers; hyper-Kähler manifolds; Hilbert scheme of points; generalized Kummer manifold; cobordism ring; Chern character Surfaces and higher-dimensional varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Hilbert schemes of K3 surfaces, generalized Kummer, and cobordism classes of hyper-Kähler manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author uses the compactness theorem of Model Theory to prove a generalization of the Nullstellensatz to complex polynomial rings of infinite transcendence degree. Generalizations (algebraic spaces, stacks) A generalization of Hilbert's Nullstellensatz | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(p\in V\) be a normal complex surface singularity and \(\Gamma\) the resolution graph, namely, the weighted dual graph of the exceptional set on the minimal good resolution \(M\). Let \(Z\) denote the fundamental cycle on \(M\). \textit{M. Artin} [Am. J. Math. 88, 129--136 (1966; Zbl 0142.18602)] proved that rational surface singularities are characterized by their resolution graph and that a rational surface singularity \((V,p)\) is a hypersurface singularity if and only if \(-Z^2=2\), and classified the resolution graphs of the rational hypersurface singularities. \textit{H. B. Laufer} [Am. J. Math. 99, 1257--1295 (1977; Zbl 0384.32003)] introduced the minimally elliptic singularities, which are characterized by their resolution graph, and proved that a minimally elliptic singularity \((V,p)\) is a hypersurface (resp. complete intersection) singularity if and only if \(-Z^2\leq 3\) (resp. \(-Z^2\leq 4\)), and classified the resolution graphs of all minimally elliptic hypersurface singularities. On the other hand, \textit{H. B. Laufer} [Rice Univ. Studies 59, No. 1, 53--96 (1973; Zbl 0281.32009)] also proved that, if \(\Gamma\) does not correspond to rational or minimally elliptic singularities, then the general surface singularities with resolution graph \(\Gamma\) are not Gorenstein. Note that a complete intersection singularity is Gorenstein.
In the paper under review, the authors classify the resolution graphs of ``simplest Gorenstein non-complete intersection surface singularities'', that is, the minimally elliptic singularities with \(-Z^2=5\). The methods are the same as that in [\textit{F. Chung} et al., Trans. Am. Math. Soc. 361, No. 7, 3535--3596 (2009; Zbl 1171.32016)] in which the resolution graphs of the minimally elliptic singularities with \(-Z^2=4\) are classified. normal complex surface singularities; minimally elliptic singularities; weighted dual graph Complex surface and hypersurface singularities, Singularities in algebraic geometry, Topological invariants on manifolds Topological classification of simplest Gorenstein non-complete intersection singularities of dimension 2 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider an analytic germ \(f:(X,x) \to (\mathbb{C},0)\) defined on a normal surface singularity \((X,x)\). Let \(T_\lambda\) be the algebraic monodromy of \(f\) restricted to its generalized \(\lambda\)-eigenspace. With some restrictions for \(f\) and \(\lambda\), we establish criteria for the finiteness of \(T_\lambda\). Examples show that these restrictions are natural. normal surface singularity; algebraic monodromy; finiteness Némethi, A.; Steenbrink, J. H. M.: On the monodromy of curve singularities, Math. Z. 223, 587-593 (1996) Complex surface and hypersurface singularities, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Singularities in algebraic geometry, Singularities of curves, local rings On the monodromy of curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X_d := H_{d,g}\) be the Hilbert scheme of locally Cohen-Macaulay, equidimensional curves in \({\mathbb P}^3_k\). The curves are not assumed to be smooth or reduced, and \(k\) is an algebraically closed field. This paper considers the case \(g = (d-3)(d-4)/2\), with the goal of describing the families of such curves, such that none of these families is a specialization of any other. As was done initially by \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184-185 (1990; Zbl 0717.14017)] and subsequently by several authors, the work centers around a study of the Rao module \(M_C := \bigoplus_{n \in {\mathbb Z}} \text{H}^1({\mathcal I}_C (n))\) and a stratification of the Hilbert scheme using this module.
The case \(g = (d-3)(d-4)\) is the first non-trivial and interesting case, since \(d\leq 3\) is understood as is \(g > (d-3)(d-4)/2\). The author describes the possible Rao modules for curves in \(X_d\), the general element of each irreducible component and the dimension of these components. A classification is given for when the curves can be smooth and connected, and finally a proof is given that \(X_d\) is connected for \(d \geq 4\). An important ingredient is the notion of \textit{extremal} curves, namely those whose Rao modules have maximum possible dimension. This has been studied by several authors, and a connection to the connectness of the Hilbert scheme was also made by \textit{S. Nollet} [Commun. Algebra 28, 5745-5747 (2000; Zbl 0991.14002)] when \(g > (d-3)(d-4)/2\). space curves; Hilbert scheme; parametrization; Rao module; triad; extremal curve; connectedness Plane and space curves, Linkage, Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) Sur le schéma de Hilbert des courbes gauches de degré \(d\) et genre \(g=(d-3)(d-4)/2\). (On the Hilbert scheme of space curves of degree \(d\) and genus \(g=(d-3)(d-4)/2\)) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article studies relations between ADE-Lie theory and ADE-singularities of surfaces.
Let \(X\) denote a compact complex surface with a rational double point and \(\pi :Y \to X\) the minimal resolution of. If \(C_1 , \dots , C_n\) in \(Y\) are the irreducible components of the exceptional locus, then the dual graph of the exceptional divisor \(\sum_{i=1}^n C_i\) is a Dynkin diagram of one of the types A, D, E.
For the integer homology of \(Y\), there is a natural decomposition \(H^2(Y, {\mathbb{Z}} )= H^2(X, {\mathbb{Z}} ) \oplus \Lambda \), where \(\Lambda = \{ \sum _i a_i[C_i] | a_i \in {\mathbb{Z}} \} \). The subset \(\Phi := \{ \alpha \in \Lambda | \alpha^2 =-2 \}\) is an ADE root system of a simple Lie-algebra \(\mathbf g \). Its associated Lie algebra bundle over \(Y\) is defined as
\[
{\mathcal E}_0^{\mathbf g}:= {\mathcal O_Y}^n \oplus \{ \bigoplus_{\alpha \in \Phi}{\mathcal O_Y} (\alpha ) \} .
\]
This bundle does not descend to the original surface \(X\). The authors show, if \(p_g(X)=0\) then \( {\mathcal E}_0^{\mathbf g}\) has a deformation to a bundle which can descend to \(X\).
Their result generalizes the work of Friedman-Morgan for \(E_n\)-bundles over del Pezzo surfaces [\textit{R. Friedman} and \textit{J. W. Morgan}, Contemp. Math. 312, 101--115 (2002; Zbl 1080.14533)].
Furthermore, the authors describe the minuscule representation bundles of these Lie algebra bundles in terms of configurations of (reducible) \((-1)\)-curves in \(Y\). ADE bundle; ADE singularity; singularities of surfaces; simple Lie algebra; Lie algebra bundle; minimal resolution of an ADE singularity Chen, YX; Leung, NC, ADE bundles over surfaces with ADE singularities, Int. Math. Res. Not., 15, 4049-4084, (2014) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Special surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Structure theory for Lie algebras and superalgebras \(ADE\) bundles over surfaces with \(ADE\) 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 Using the methods developed in an earlier paper [Math. Ann. 324, No.1, 105--133 (2002; Zbl 1024.14001)], the authors obtain a second set of generators for the cohomology ring of the Hilbert scheme of points on an arbitrary smooth projective surface \(X\) over the field of complex numbers. These generators have clear and simple geometric as well as algebraic descriptions. Li, W.; Qin, Z.; Wang, W.: Generators for the cohomology ring of Hilbert schemes of points on surfaces. Int. math. Res. not. 2001, No. 20, 1057-1074 (2001) Parametrization (Chow and Hilbert schemes), Classical real and complex (co)homology in algebraic geometry Generators for the cohomology ring of Hilbert schemes of points on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\subseteq \mathbb P^m\) be a smooth algebraic variety of dimension \(r\) and \(\pi: X\to \mathbb P^m\) a generic projection, \(r<m\leq 2r\), and \(X'=\pi(X)\). Assume that \(X\) is analytically irreducible at \(y=\pi(x)\) and let \(q\) be the multiplicity of \(\mathcal O_{X', y}\). Let \(\pi^\ast: \widehat{\mathcal O}_{\mathbb P^m, y}\to \widehat{\mathcal O}_{X,x}\) be the corresponding homomorphism of the corresponding complete local rings. In the article under review, the ideals \(I\) and \(I'\) in \(\widehat{\mathcal O}_{\mathbb P^m, y}\) defining \(X'\) resp. the singular locus \(\text{Sing}(X')\) at \(y\) as ideals of maximal and sub-maximal minors of a generalized Sylvester matrix are computed. Gröbner basis techniques are used in the proof. elimination ideal; fitting ideal; singularities of generic projections P. Salmon, R. Zaare-Nahandi, Ideals of minors defining generic singularities and their Gröbner bases, preprint, 2003. Singularities in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Ideals of minors defining generic singularities and their Gröbner bases | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is a continuation of authors' previous work [Math. Ann. 324, No. 1, 105--133 (2002; Zbl 1024.14001); Int. Math. Res. Not. 2001, No. 20, 1057--1074 (2001; Zbl 1069.14004)] on the cohomology ring of the Hilbert scheme \(X^{[n]}\) of \(n\)-points on a smooth complex projective surface. In particular the authors prove a stability result which says that the cup product of certain cohomology classes in \(H^{*}(X^{[n]})\) with large \(n\) can be recovered from the cup-product of cohomology classes in \(H^{*}(X^{[m]})\) with small \(m\). This allows the authors to construct a super-commutative associative ring \({\mathfrak H}_X\), called the Hilbert ring of \(X\) and to determine its structure. Hilbert scheme; cohomology; cup product Li, W. -P; Qin, Z.; Wang, W.: Stability of the cohomology rings of Hilbert schemes of points on surfaces. J. reine angew. Math. 554, 217-234 (2003) Parametrization (Chow and Hilbert schemes), Classical real and complex (co)homology in algebraic geometry Stability of the cohomology rings of Hilbert schemes of points on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies vector bundles on Hilbert schemes of points on a surface \(S\) coming from line bundles (and more generally vector bundles) on the base \(S\). For a rank \(r\) vector bundle \(E\) on \(S\), the so-called associated \textit{tautological bundle} \(E^{[n]}\) is defined as the rank \(rn\) vector bundle whose fiber at the point \([Z]\in S^{[n]}\) is \(H^0(S,E\otimes\mathcal O_Z)\). Tautological bundles have been studied by numerous authors; to name a couple, \textit{M. Lehn} [Invent. Math. 136, No. 1, 157--207 (1999; Zbl 0919.14001)] computed their cohomology, and \textit{G. Ellingsrud} and \textit{S. A. Strømme} [J. Reine Angew. Math. 441, 33--44 (1993; Zbl 0814.14003)] showed that the Chern classes of \(\mathcal O_{\mathbb P^2}^{[n]},\mathcal O_{\mathbb P^2}(1)^{[n]},\mathcal O_{\mathbb P^2}(2)^{[n]}\) generate the cohomology of \((\mathbb P^2)^{[n]}\).
In the present work, the author proves that for a smooth projective surface \(S\) and any vector bundle \(E\ncong \mathcal O_S\) on \(S\) that is slope-stable with respect to an ample divisor, \(E^{[n]}\) is slope-stable with respect to the corresponding (nef and big) Chow divisor, completely generalizing preliminary results of \textit{U. Schlickewei} [Rend. Semin. Mat. Univ. Padova 124, 127--138 (2010; Zbl 1208.14036)] and \textit{M. Wandel} [Osaka J. Math. 53, No. 4, 889--910 (2016; Zbl 1360.14037); Nagoya Math. J. 214, 79--94 (2014; Zbl 1319.14049)] for Hilbert schemes of 2 or 3 points on a \(K3\) or abelian surface. Another interesting result of the author is about the tautological bundle associated to the tangent bundle \(T_S\): \(T_S^{[n]}\) admits an injection into \(T_{S^{[n]}}\), identifying it with \(\mathrm{Der}_{\mathbb C}(-\log B_n)\), the sheaf of vector fields on \(S^{[n]}\) that are tangent to the divisor \(B_n\) of nonreduced subschemes.
Finally, using the spectral curves of \textit{A. Beauville} et al. [J. Reine Angew. Math. 398, 169--179 (1989; Zbl 0666.14015)], the author proves the following interesting result: for any rank \(n\) semistable vector bundle \(E\) of sufficiently large degree on a smooth projective curve \(C\), there exists an embedding \(C\to(\mathbb P^2)^{[n]}\) such that \(E\cong\mathcal O_{\mathbb P^2}(1)^{[n]}|_C\). In other words, tautological bundles on surfaces capture most of the geometry of semistable sheaves on surfaces. Hilbert schemes of points; vector bundles on surfaces; slope-stability; spectral curves; log tangent bundle; tautological bundles Stapleton, D, Geometry and stability of tautological bundles on Hilbert schemes of points, Algebra Numb. Theory, 10, 1173-1190, (2016) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Parametrization (Chow and Hilbert schemes), Algebraic cycles Geometry and stability of tautological bundles on Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Fix a complex projective elliptic curve \(E\). The choice of diffeomorphism \(\Phi\) from \(Y = \mathbb C^* \times \mathbb C^*\) to \(X = E \times \mathbb C\) (note that both are diffeomorphic to \(S^1 \times S^2 \times \mathbb R \times \mathbb R\)) induces an isomorphism \(\phi: H^*(X,\mathbb Q) \to H^*(Y,\mathbb Q)\) by pulling back. For \(n \geq 0\), the authors show that \(\phi\) extends to an isomorphism \(\phi^{[n]}: H^*(X^{[n]}, \mathbb Q) \to H^*(Y^{[n]}, \mathbb Q)\) on the \(\mathbb Q\)-cohomology of Hilbert schemes of \(n\) points. While work of \textit{C. Voisin} suggests that \(Y^{[n]} \cong X^{[n]}\) [Ann. Inst. Fourier 50, No. 2, 689--722 (2000; Zbl 0954.14002)], the authors do not use this but instead obtain \(\phi^{[n]}\) by the decomposition of the Hilbert scheme given by partitions \(\nu = (\nu_1, \nu_2, \dots, \nu_l)\) of \(n\); in particular \(\phi^{[n]}\) need not be an isomorphism of mixed Hodge structures.
The main result of the paper is the equality of two natural filtrations under \(\phi^{[n]}\). The weight filtration \(W_k \subset H^*(Y^{[n]}, \mathbb Q)\) has the property that \(W_{2k} = W_{2k+1}\) for each \(k\), so one may define the \textit{halved} filtration by \({}_{\frac{1}{2}}W_{Y^{[n]},k}=W_{Y^{[n]},2k}\). On the other hand the projection \(p:X \to \mathbb C\) induces a map \(h_n: X^{[n]} \to X^n/S_n \to \mathbb C^n/S_n\) which is projective and flat of relative dimension \(n\). Associated to the map \(h_n\) is the perverse Leray filtration \(P_{X^{[n]}}\) via the complex \({h_n}_* \mathbb Q_{X^{[n]}} [n]\). With this notation, the main theorem says that \(\phi^{[n]}(P_{X^{[n]}}) = {}_{\frac{1}{2}} W_{Y^{[n]}}\). This result gives more evidence of the general ``\(P=W\)'' conjecture. The authors have proved this for moduli space of rank 2 degree 1 stable Higgs bundles [\textit{M. A. A. De Cataldo} et al., Ann. Math. (2) 175, No. 3, 1329--1407 (2012; Zbl 1375.14047)]. A second result is an analog of the hard Lefschetz theorem for \(Y^{[n]}\). The closed 2-form \(\alpha_Y = \frac{1}{(2 \pi i)^2} \frac{dz \wedge dw}{zw}\) is an integral class in \(H^2(Y, \mathbb Q) \cap H^{2,2} (Y)\). Summing over the pull-backs of the projections \(p_i: Y^n \to Y\), quotienting by the action of \(S_n\) and pulling back by the Hilbert-Chow maps gives the cohomology class \(\alpha_{Y^{[n]}} \in H^2(Y^{[n]}, \mathbb Q) \cap H^{2,2} (Y^{[n]})\). Cup product with \(\alpha_{Y^{[n]}}\) gives isomorphisms
\[
\text{Gr}^{W_{Y^{[n]}}}_{2n-2k} H^*(Y^{[n]},\mathbb Q) \cong \text{Gr}^{W_{Y^{[n]}}}_{2n+2k} H^{*+2k} (Y^{[n]},\mathbb Q)
\]
which are analogs of the ``curious hard Lefschetz'' theorem of \textit{T. Hausel} and \textit{F. Rodriguez-Villegas} [Invent. Math. 174, No. 3, 555--624 (2008; Zbl 1213.14020)]. Under the identification given by the map \(\phi^{[n]}\), the authors show that these isomorphisms become the relative hard Lefschetz theorems on \(X^{[n]}\) found in work on perverse sheaves of \textit{A. A. Beilinson} et al. [Astérisque 100, 172 p. (1982; Zbl 0536.14011)]. perverse filtration; weight filtration; Hilbert scheme of points Cataldo, MA; Hausel, T; Migliorini, L, Exchange between perverse and weight filtration for the Hilbert schemes of points of two surfaces, J. Singul., 7, 23-38, (2013) Parametrization (Chow and Hilbert schemes), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Exchange between perverse and weight filtration for the Hilbert schemes of points of two 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 describe the components of the Hilbert scheme \(H_{d,\tilde g}\) of locally Cohen-Macaulay curves of degree \(d\) and arithmetic genus \(\tilde g=\binom{d-3}{2}-1\). We show that \(H_{d,\tilde g}\) is connected thanks to the irreducible component of extremal curves to which every curve can be connected. Sabadini, I.: On the Hilbert scheme of curves of degree \(d\) and genus \(\left(\begin{array}{c}{d-3}\{2\)\end{array}\right)-1\(. Le Mate. {\mathbf 45}, 517-531 (2000) (volume dedicated to Silvio Greco in occasion of his sixtieth birthday)\) Special algebraic curves and curves of low genus On the Hilbert scheme of curves of degree \(d\) and genus \(\binom{d-3}{2}-1\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A=k[[ z_ 1,...,z_ r]]\) be the formal power series ring in the variables \(z_ i\) over an algebraically closed field k of arbitrary characteristic. We show that if the local ring of an isolated singularity \(g(z_ 1,...,z_ i)\) in \(k[[ z_ 1,...,z_ i]]\) in dimension \(1\quad or^ 2\) is of finite Cohen-Macaulay type, then the local rings of the singularities \(f=g(z_ 1,...,z_ i)+z_{i+1}z_{i+2}+...+z_{r-1}z_ r\) in A also are of finite Cohen-Macaulay type. We also describe their Auslander-Reiten quivers. formal power series ring; isolated singularity; finite Cohen-Macaulay type; Auslander-Reiten quivers; almost split sequences O. Solberg, Hypersurface singularities of finite Cohen-Macaulay type, Proc. London Math. Soc. (3) 58 (1989), no. 2, 258-280. Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Representation theory of associative rings and algebras, Singularities of surfaces or higher-dimensional varieties, Formal power series rings Hypersurface singularities of finite Cohen-Macaulay type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article can be seen as a sequel to the first author's article [J. Algebr. Geom. 14, No. 4, 761-787 (2005; Zbl 1120.14002)], where he calculates the total Chern class of the Hilbert schemes of points on the affine plane by proving a result on the existence of certain universal formulas expressing characteristic classes on the Hilbert schemes in term of Nakajima's creation operators. The purpose of this work is (at least) two-fold. First of all, we clarify the notion of ``universality'' of certain formulas about the cohomology of the Hilbert schemes by defining a universal algebra of creation operators. This helps us to reformulate and extend a lot of the first author's previous results in a very precise manner.
Secondly, we are able to extend the previously found results by showing how to calculate any characteristic class of the Hilbert scheme of points on the affine plane in terms of the creation operators. In particular, we have included the calculation of the total Segre class and the square root of the Todd class. Using these methods, we have also found a way to calculate any characteristic class of any tautological sheaf on the Hilbert scheme of points on the affine plane. This in fact gives another complete description of the ring structure of the cohomology spaces of the Hilbert schemes of points on the affine plane. Samuel Boissière and Marc A. Nieper-Wißkirchen, Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces, J. Algebra 315 (2007), no. 2, 924 -- 953. Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Cycles and subschemes Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In [Ann. Sci. Éc. Norm. Supér. (4) 15, 401--418 (1982; Zbl 0517.14007)], \textit{L. Gruson} and \textit{C. Peskine} completed the classical work of Halphen, characterizing all the pairs $(d,g)$ that can be obtained as degree and genus of a smooth connected projective curve in $\mathbb P^3$.
In particular, they proved that all pairs with \(g\leq {\frac{1}{6}}d(d-3)+1\) can be obtained from curves contained either in a smooth cubic surface or in a quartic surface with a double line. Also to give a tool to continue the study of smooth curves contained in singular non-normal surfaces, \textit{R. Hartshorne} [\(K\)-Theory 8, No. 3, 287--339 (1994; Zbl 0826.14005)] developed the theory of generalized divisors on Gorenstein schemes.
In the article under review, based on this theory, the Author tackles the problem of computing the invariants of smooth curves contained in a surface of \(\mathbb P^3\) with ordinary singularities, in terms of the cohomology groups of the divisors on its normalization. He applies his results to smooth curves that are projections in \(\mathbb P^3\) of curves contained in rational normal scrolls in spaces of higher dimension, and in particular to smooth curves contained in a ruled cubic surface with a double line. He gets a complete characterization of all curves of maximal rank on such a surface. This allows him to give a negative answer to a question posed by \textit{R. Hartshorne} [Math. Ann. 238, 229--280 (1978; Zbl 0411.14002)], stating the existence of smooth arithmetically Cohen-Macaulay curves on a rational normal scroll \(S(1,2)\) whose general projection in \(\mathbb P^3\) is not of maximal rank.
Another interesting result of the article regards the biliaison class of a smooth curve on a surface of \(\mathbb P^3\) with ordinary singularities, saying that two smooth curves in such a class are linearly equivalent. general projection; biliaison; maximal rank curve; ruled cubic surface; rational normal scroll Plane and space curves, Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Linkage Space curves on surfaces with ordinary singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic zero and let \(A\) be a finitely generated \(k\)-algebra. The Nori-Hilbert scheme of \(A\), \(\mathrm{Hilb}^n_A\), parameterizes left ideals of codimension \(n\) in \(A\). It is well known that \(\mathrm{Hilb}^n_A\) is smooth when \(A\) is formally smooth. {
}In this paper we will study \(\mathrm{Hilb}^n_A\) for 2-Calabi-Yau algebras. Important examples include the group algebra of the fundamental group of a compact orientable surface of genus \(g\), and preprojective algebras. For the former, we show that the Nori-Hilbert scheme is smooth only for \(n=1\), while for the latter we show that a component of \(\mathrm{Hilb}^n_A\) containing a simple representation is smooth if and only if it only contains simple representations. Under certain conditions, we generalize this last statement to arbitrary 2-Calabi-Yau algebras. representation theory; Calabi-Yau algebras; Nori-Hilbert scheme Parametrization (Chow and Hilbert schemes), Noncommutative algebraic geometry, Representations of quivers and partially ordered sets, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) The Nori-Hilbert scheme is not smooth for 2-Calabi-Yau algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The multi-graded Hilbert scheme parametrizes all ideals with a given Hilbert function. An important subclass is the class of all toric Hilbert schemes. This paper details the main component of a toric Hilbert scheme in case it contains a point corresponding to an affine toric variety. Here the component can be viewed as the set of all flat limits of this variety.
The main tool used in the paper is a computation of the fan of the toric variety. The treatment is very self-contained and thorough. Moreover, using a Chow morphism, a connection with certain GIT quotients of the toric variety is included. Some helpful examples are provided as well. toric Hilbert scheme; fiber polytope; toric Chow quotient Chuvashova O.V., The main component of the toric Hilbert scheme, Tôhoku Math. J., 2008, 60(3), 365--382 Parametrization (Chow and Hilbert schemes), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies The main component of the toric Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Abhyankar defined the index of a monomial in a matrix of indeterminates X to be the maximal size of any minor of X whose principal diagonal divides the given monomial. Using this concept, he characterized a free basis for a general type of determinantal ideals formed by the minors coming from a saturated subset of X.
In this paper, to a monomial in X of \(index\quad p\) we associate a combinatorial object called a superskeleton of \(latitude\quad p,\) which can loosely be described as a p-tuple of ``almost nonintersecting paths'' in a rectangular lattice of points. Using this map, we prove that the ideal generated by the \(p\quad by\quad p\) minors of a saturated set in X is hilbertian, i.e., the Hilbert polynomial of this ideal coincides with its Hilbert function for all nonnegative integers. hilbertian determinant ideals; index of a monomial; matrix of indeterminates; Hilbert polynomial; Hilbert function S. S. Abhyankar and D. M. Kulkarni, On Hilbertian ideals, in ''Linear Algebra and Its Applications,'' in press. Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Determinantal varieties On Hilbertian ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors present a number of interesting results on singularities of complex algebraic varieties. Their main result says that given a germ \((Y,y)\) of complex isolated singularity, there is a complex simply connected projective variety \(X\) having a single singular point \(x\), such that the germs \((X,x)\) and \((Y,y)\) are isomorphic. The fact that a projective variety \(X\) with such a singularity \(x\) exists is an easy consequence of a celebrated theorem of M. Artin and the theory of resolution of singularities. The extra condition ``\(\pi _1 (X) = 0\)'' is more difficult to obtain. The authors verify first that one can get a variety \(X\) as above, but with \(\pi _1(X)\) abelian, not necessarily trivial. To accomplish this they work with suitable complete intersections inside certain symmetric powers, exploiting the fact that, for variety \(V\), \(\pi _1(S^n V)\) is isomorphic to \(H_1(V,\mathbb{Z})\) (hence abelian), for \(n>1\). Then they ``reduce'' the abelian fundamental group.
They conjecture that given a germ \((Y,y)\) as before, there is a projective variety \(X\) as above, but such that moreover a resolution \(X'\) of \(X\) is simply connected. The authors prove other results. For instance:
(a) In their main result, in the case of surfaces, \(X\) cannot have Kodaira dimension equal to \(- \, \infty\).
(b) Conditions on a set of two-dimensional rational double points so that there is a rational surface \(X\) whose singularities are (up to isomorphism) the given ones.
(c) A refined version of the classical local parametrization theorem for germs of locally irreducible analytic varieties.
(d) Using (b) they obtain a new proof of the algebraicity of a normal two-dimensional analytic singularity. isolated singularity; fundamental group; local parametrization; algebraization Singularities in algebraic geometry, Local complex singularities, Projective techniques in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects) Singularities on complete 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 For a given finite subgroup \(G\) of \(PGL(n)\) one can consider the Hilbert scheme of \(G\)-orbits in projective \(n\)-space, referred to as the \(G\)-Hilbert scheme. This is a natural generalization to the standard Hilbert scheme of Grothendieck and is similar to the \(G\)-Hilbert scheme introduced by Ito and Nakamura.
In this paper the author considers when \(G\) is a subgroup of the maximal torus. The author proves that if \(I\) is a \(G\) invariant ideal then the toric variety corresponding to the fan consisting of the images in the quotient lattice of the Gröbner cones for \(I\) is isomorphic to the normalization of the \(G\)-Hilbert scheme.
Before proving this main theorem the author reviews the relevant facts concerning Gröbner bases and fans. The proof of the theorem is broken down into three major steps and subsequently the author devotes a section to detailed computed examples. Gröbner fan; G-Hilbert scheme; toric singularity Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Hilbert schemes of finite abelian group orbits and Gröbner fans | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a connected semisimple complex algebraic group, let \(V\) be a simple \(G\)-module, and let \({\mathbb P}(V)\) be the projective space of all hyperplanes in \(V\) endowed with the natural action of \(G\). The space \({\mathbb P}(V)\) contains the unique closed \(G\)-orbit \(X\). Using Weyl's dimension formula, the authors compute the Hilbert polynomial, the Hilbert series, the dimension, and the degree of \(X\), and consider several examples.
Reviewer's remark. These results are the special cases of the known results about Hilbert polynomials, degrees, and dimensions of arbitrary normal spherical varieties; see [\textit{M. Brion}, Duke Math. J. 58, No. 2, 397--424 (1989; Zbl 0701.14052); Lect. Notes Math. 1296, 177--192 (1987; Zbl 0667.58012); \textit{A. Yu. Okounkov}, Funct. Anal. Appl. 31, No. 2, 138--140 (1997); translation from Funkts. Anal. Prilozh. 31, No. 2, 82--85 (1997; Zbl 0928.14032); and also \textit{D. Panyushev}, Transform. Groups 2, No. 1, 91--115 (1997; Zbl 0891.22013)]. semisimple group; homogeneous projective variety; Hilbert polynomial Gross (B. H.); Wallach (N. R.).-- On the Hilbert polynomials and Hilbert series of homogeneous projective varieties, In: Arithmetic geometry and automorphic forms, Adv. Lect. Math. 19 p. 253-263 (2011). Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Semisimple Lie groups and their representations, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the Hilbert polynomials and Hilbert series of homogeneous projective 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 Using a new technique to determine obstructions for deforming space curves, we exhibit non-reduced components of the Hilbert scheme of space curves such that
(1) the general curve \(C\) in a component is smooth,
(2) the imbedding codimension of a component at a general point may be arbitrarily large, and
(3) the degree of the minimal degree surface containing \(C\) may be arbitrarily large.
In contrast all earlier examples of non-reduced componenents have consisted of curves lying on cubic surfaces. --- The non-reducedness of a component is established by calculating the obstructions to lifting first order deformations of a general curve \(C\) in the component. The obstructions are calculated by using some recent theory by Walter showing how to calculate coboundaries of some spectral sequences associated to space curves.
As a part of the deformation theory we develop, we also study deformations of coherent sheaves on \(\mathbb{P}^ n\) where parts of its cohomology vary flatly. non-reduced components of the Hilbert scheme of space curves; deformations of coherent sheaves Fløystad, G., Determining obstructions for space curves, with applications to nonreduced components of the Hilbert scheme, J. Reine Angew. Math., 439, 11-44, (1993) Plane and space curves, Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Formal methods and deformations in algebraic geometry Determining obstructions for space curves, with applications to nonreduced components of the Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors look for the minimal possible Castelnuovo-Mumford regularity \(m_{p(z)}\) of schemes with a given Hilbert polynomial \(p(z)\) in characteristic \(0\), providing a sharp lower bound. Many authors have focused their efforts in finding bounds for the Castelnuovo-Mumford regularity \(\mathrm{reg}(X)\) of a scheme \(X\). A very famous upper bound is due to \textit{G. Gotzmann} [Math. Z. 158, 61--70 (1978; Zbl 0352.13009)] and coincides with the regularity of the saturated lexicographical ideal with a given Hilbert polynomial \(H(z)\). The authors involve the regularity \(\varrho\) of the Hilbert function \(H(z)\) of a scheme \(X\) and the general hyperplane section \(Z\) of \(X\), they obtain that if \(H(\rho -1) > p \,(\rho -1)\) then \(\mathrm{reg}(X) >\rho +1\), by using the relation \(\mathrm{reg}(X)=\max\{\mathrm{reg}(Z),\varrho+1\}\) proved in [\textit{F. Cioffi} et al., Collect. Math. 60, No. 1, 89--100 (2009; Zbl 1188.14020)]. Moreover the authors introduce a well-suited notion of minimal functions exploiting an idea of \textit{L. Roberts} [in: Curves Semin. at Queen's, Vol. 2, Kingston/ Can. 1981--82, Queen's Pap. Pure Appl. Math. 61, Exp. F, 21 p. (1982; Zbl 0593.13009)]. So they show that there is a scheme \(X\) having Hilbert polynomial \(p(z)\) and the minimal Hilbert function with the smallest possible \(\varrho\), which achieves the Castelnuovo-Mumford regularity \(m_{p(z)}\) and obtain a formula for \(m_{p(z)}\) depending on the Hilbert function of the hyperplane section. Their proofs are based on two new constructive methods,the first is called ``ideal graft'' of two schemes and the second ``expanded lifting''. Both these methods exploit the notion of growth-height lexicographic Borel sets introduced by \textit{D. Mall} [J. Pure Appl. Algebra 150, No. 2, 175--205 (2000; Zbl 0986.14002)], and developed also in [\textit{F. Cioffi} et al., Discrete Math. 311, No. 20, 2238--2252 (2011; Zbl 1243.14007)]. In an Appendix the authors give an algorithm and its implementation to compute \(m_{p(z)}\) and a Borel ideal defining a scheme with regularity \(m_{p(z)}\). Hilbert function; Hilbert polynomial; Castelnuovo-Mumford regularity of a projective scheme; regularity of a Hilbert function; Borel ideal Cioffi, F., Lella, P., Marinari, M.G., Roggero, M.: Minimal Castelnuovo-Mumford regularity fixing the Hilbert polynomial. arXiv:1307.2707 [math.AG] Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Symbolic computation and algebraic computation, Calculation of integer sequences, Computational aspects in algebraic geometry Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 natural question arising in the study of algebraic curves is whether the Hilbert scheme of locally Cohen-Macaulay projective space curves of given degree and arithmetic genus is connected. The answer is unknown at present. In the paper under review, the author gives a nice survey of the current state of this question. projective space curve; degree; extremal curves; liaison; Rao module; triad Hartshorne, R.: Questions of connectedness of the Hilbert scheme of curves in \$\$\{\(\backslash\)mathbb\{P\}\^3\}\$\$ . In: Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), pp. 487--495. Springer, Berlin (2004) Parametrization (Chow and Hilbert schemes), Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Questions of connectedness of the Hilbert scheme of curves in \(\mathbb P^3\) | 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 The first counterexample to Kodaira vanishing in positive characteristic was constructed by \textit{M. Raynaud} [in: C.P. Ramanujam. - A tribute. Collect. Publ. of C.P. Ramanujam and Pap. in his Mem., Tata Inst. fundam. Res., Stud. Math. 8, 273--278 (1978; Zbl 0441.14006)], many other counter-examples have been found satisfying various prescribed properties. In this paper, let \(k\) be a field of charactristic 2. The author shows that there exists a Fano variety \(X\) of dimension six over \(k\) such that anticanonical sheaf \(\omega_X^{-1}\) is very ample and \(\omega_X^{-2}\) violates Kodaira vanishing. By taking the cone over \(X\) given the emedding induced by the global sections of \(\omega_X^{-1}\), there exists a variety \(Z\) of dimension \(7\) with a single isolated canoncial singularity over \(k\), but \(Z\) is not Cohen-Macaulay. Fano variety; Kodaira vanishing theorem; positive characteristic Singularities in algebraic geometry, Rational and birational maps, Fano varieties Non-Cohen-Macaulay 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 construct an explicit, multiplicative Chow-Künneth decomposition for the Hilbert scheme of points of a \(K3\) surface. We further refine this decomposition with respect to the action of the Looijenga-Lunts-Verbitsky Lie algebra. (Equivariant) Chow groups and rings; motives, Parametrization (Chow and Hilbert schemes), Algebraic cycles, \(K3\) surfaces and Enriques surfaces Motivic decompositions for the Hilbert scheme of points of a \(K3\) surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field and \(I \subset k[x_0, \dots, x_n]\) a homogeneous ideal. \textit{Galligo} constructed the generic initial ideal \(\text{Gin}_{\prec} (I)\) with respect to a monomial ordering \(\prec\), which is roughly the monomial ideal generated by the initial terms of \(I\) after a generic coordinate change [\textit{A. Galligo}, Lect. Notes Math. 409, 543--579 (1974; Zbl 0297.32003)]. \textit{D. Bayer} and \textit{M. Stillman} [Duke Math. J. 55, 321--328 (1987; Zbl 0638.06003)] and \textit{K. Pardue} [Non-standard Borel-fixed ideals. Ann Arbor, MI: Brandeis University (PhD thesis) (1994)] worked out the theory in more generality. Generic initial ideals appear in work on Hilbert schemes, Castelnuovo-Mumford regularity and complexity of Gröbner basis calculations.
In the paper under review, the authors further develop the theory of generic initial ideals starting from the geometric perspective of \textit{M. L. Green} [Prog. Math. 166, 119--186 (1998; Zbl 0933.13002)]. Letting \(p(t)\) be the Hilbert polynomial of the closed subscheme \(X \subset \mathbb P^n\) defined by the ideal \(I\), \textit{G. Gotzmann} [Math. Z. 158, 61--70 (1978; Zbl 0352.13009)] proved that there is a integer \(m_0\) depending only on \(p(t)\) such that \(I\) is \(m\)-regular for all \(m \geq m_0\) so that there is an exact sequence \[ 0 \to I_m \to H^0(\mathbb P^n, {\mathcal O}_{\mathbb P^n} (m)) \to H^0 (X, {\mathcal O}_X (m)) \to 0 \] and the corresponding subvector space \(I_m \subset W=H^0(\mathbb P^n, {\mathcal O}_{\mathbb P^n} (m))\) defines the \textit{\(m\)th Hilbert point of \(I\)} in the Grassmann variety \(\text{Gr}_r (W)\), where \(r = \dim_k I_m\). This defines the Grothendieck-Plücker embedding \(\phi_m: \text{Hilb}^{p(t)} \mathbb P^n \hookrightarrow \text{Gr}_r (W)\). The authors use Schubert cells in \(\text{Gr}_r (W)\) to define a locally closed stratification of \(\text{Hilb}^{p(t)} \mathbb P^n\) and in the process recover some results of \textit{A. Conca} and \textit{J. Sidman} [J. Symb. Comput. 40, No. 3, 1023--1038 (2005; Zbl 1120.14050)] and of \textit{R. Notari} and \textit{M. L. Spreafico} [Manuscr. Math. 101, No. 4, 429--448 (2000; Zbl 0985.13006)] with shorter proofs. They also give short proofs for the existence of the generic initial ideal and its Borel fixedness. The authors clarify the distinction between their stratification and similar decompositions of the Hilbert scheme studied by Notari and Spreafico [loc. cit.] and \textit{C. Bertone} et al. [J. Symb. Comput. 53, 119--135 (2013; Zbl 1312.14011)]. At the end they prove that the Grothendieck-Plucker embedding \(\phi_m: \text{Hilb}^{p(t)} \mathbb P^n \hookrightarrow \text{Gr}_r (W)\) has degenerate image for \(m > m_0\) if \(p(t)\) is nonconstant, explaining title of paper. The paper is self-contained and well written. generic initial ideals; Schubert cells in Grassmann variety Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Grothendieck-Plücker images of Hilbert schemes are degenerate | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a \(K3\) surface of Picard number \(2\), and denote by \(\mathrm{Hilb}^2(S)\) the Hilbert scheme of \(0\)-dimensional closed subschemes of length two on \(S\). In his previous paper [J. Alg. Geom. 23, No. 4, 775--795 (2014; Zbl 1304.14051)], the author proved that the group of automorphisms of \(S\) is of finite order. In the paper under review the author gives a sufficient condition in geometric terms for \(\mathrm{Hilb}^2(S)\) to have automorphism group of infinite order. He also finds a concrete example of \(K3\) surfaces \(S\) with \(|\mathrm{Aut}(\mathrm{Hilb}^2(S))|=\infty\). In his example, the Néron-Severi group of \(S\) is isomorphic to a certain even hyperbolic lattice of rank \(2\) and of discriminant \(17\).
As an interesting application for Mori dream space, the author shows that for a \(K3\) surface \(S\) in his example, the Hilbert-Chow morphism \(\mathrm{Hilb}^2(S)\to \mathrm{Sym}^2(S)\) is an extremal crepant resolution such that the source \(\mathrm{Hilb}^2(S)\) is not a Mori dream space but the target \(\mathrm{Sym}^2(S)\) is a Mori dream space. hyperkähler manifolds; automorphisms; Mori dream space; crepant resolution Oguiso, K., On automorphisms of the punctual Hilbert schemes of K3 surfaces, Eur. J. Math., 2, no. 1, 246-261, (2016) Automorphisms of surfaces and higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), \(4\)-folds, \(K3\) surfaces and Enriques surfaces On automorphisms of the punctual Hilbert schemes of \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we study the Gromov-Witten theory of the Hilbert schemes \(X^{[n]}\) of points on a smooth projective surface \(X\) with positive geometric genus \(p_g\). For fixed distinct points \(x_1,\dots , x_{n-1} \in X\), let \(\beta_n\) be the homology class of the curve \(\{ \xi +x_2+\cdots + x_{n-1} \in X^{[n]}| \operatorname{Supp}(\xi)=\{x_1\}\}\), and let \(\beta_{K_X}\) be the homology class of \(\{x+x_1+\cdots + x_{n-1}\in X^{[n]}| x \in K_X\}\). Using cosection localization technique due to [\textit{Y. Kiem} and \textit{J. Li}, ``Gromov-Witten invariants of varieties with holomorphic 2-forms'', Preprint, \url{arXiv:0707.2986}], we prove that if \(X\) is a simply connected surface admitting a holomorphic differential two-form with irreducible zero divisor, then all the Gromov-Witten invariants of \(X^{[n]}\) defined via the moduli space \(\overline{\mathfrak{M}}_{g,r}(X^{[X]},\beta)\) of stable maps vanish except possibly when \(\beta\) is a linear combination of \(\beta_n\) and \(\beta_{K_X}\). When \(n=2\), the exceptional cases can be further reduced to the Gromov-Witten invariants: \(\langle 1\rangle^{X^{[2]}}_{0,\beta_{K_X}-d\beta_2}\) with \(K^2_X=1\) and \(d \leq 3\), and \(\langle 1\rangle ^{X^{(2)}}_{1,d\beta_2}\) with \(d \geq 1\). When \(K^2_X=1\), we show that \(\langle 1\rangle^{X^{[2]}}_{0,\beta_{K_X}-3\beta_2}=(-1)^{\chi(\mathcal{O}_x)}\) which is consistent with a well-known formula of \textit{C. H. Taubes} [J. Differ. Geom. 52, No. 3, 453--609 (1999; Zbl 1040.53096)]. In addition, for an arbitrary surface \(X\) and \(d \geq 1\), we verify that \(\langle 1\rangle^{X^{[2]}}_{1,d\beta_2}=K^2_X/(12d)\). Gromov-Witten invariants; Hilbert schemes; cosection localization Hu, J.; Li, W. -P.; Qin, Z.: The Gromov-Witten invariants of the Hilbert schemes of points on surfaces with pg>0 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) The Gromov-Witten invariants of the Hilbert schemes of points on surfaces with \(p_g>0\) | 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 In the paper under review, the authors study schemes supported on the singular loci of hyperplane arrangements in \(\mathbb{P}^{n}_{\Bbbk}\), where \(\Bbbk\) is a field of characteristic zero. Let \(\mathcal{A}\) be a hyperplane arrangement in \(\mathbb{P}^{n} := \mathbb{P}^{n}_{\Bbbk}\) containing \(d\) hyperplanes and let \(F\) be a product of linear forms defining hyperplanes in \(\mathcal{A}\), and \(J\) be the Jacobian ideal of \(F\). The ideal \(J\) has height two. Let \(E\) be the syzygy module of \(J\), so we have an exact sequence \[0\rightarrow E \rightarrow \bigoplus_{i=1}^{n+1}R(1-d) \rightarrow J \rightarrow 0,\] where \(R = \Bbbk[x_{0},\dots,x_{n}]\), and let \[J = \mathfrak{q}_{1} \cap \dots \cap \mathfrak{q}_{r}\] be a primary decomposition of \(J\). Denote by \(\mathfrak{p}_{i}\) the associated prime of \(\mathfrak{q}_{i}\). When all the \(\mathfrak{p}_{i}\) have the same height, we say that \(J\) is unmixed. In the present paper, the authors are interested in three unmixed ideals that arise somehow naturally from \(J\). First of all, the authors consider the intersection of the codimension two ideals in a primary decomposition of \(J\), which will be denote by \(J^{\mathrm{un}}\). The second ideal that the authors study is the radical of \(J\), denoted by \(\sqrt{J}\), which is just the intersection of the associated primes of \(J\). The third ideal is less natural, i.e., the authors consider fatten components of \(\sqrt{J}\). The first main result of the paper under review can be formulated as follows.
Theorem A. Let \(\mathcal{A}\) be a hyperplane arrangement in \(\mathbb{P}^{n}\) defined by a product of linear forms \(F\). Let \(J, \sqrt{J}, J^{\mathrm{un}}\) be the ideals defined above. Assume that no linear factor of \(F\) is in the associated prime for any two non-reduced components of \(J^{\mathrm{un}}\). Then both \(R/J^{\mathrm{un}}\) and \(R/\sqrt{J}\) are Cohen-Macaulay.
As an application, the authors translate this result to graphic arrangements, namely let \(G\) be a graph and we assume that no two \(3\)-cycles of \(G\) share an edge, then for the associated graphic arrangement \(\mathcal{A}_{G}\) and the Jacobian ideal \(J\), \(R/\sqrt{J}\) and \(R/J^{\mathrm{un}}\) are Cohen-Macaulay.
From now on we assume that \(n=3\). In the case of a curve \(C \in \mathbb{P}^{3}\) there is a natural way to measure the failure of \(R/I_{C}\) to be Cohen-Macaulay degree by degree, and this is the so-called Hartshorne-Rao module of \(C\), denote here by \(M(C)\). We denote by \(C_{F}^{\mathrm{un}}\) the scheme defined by \(J^{\mathrm{un}}\), invoking the form \(F\) defining the arrangement of \(\mathcal{A}\). This scheme \(C_{F}^{\mathrm{un}}\) is the equidimensional top dimensional part of \(C_{F}\), removing components of higher codimension. Similarly, one denotes by \(C_{F}^{\mathrm{red}}\) the support of \(C_{F}^{\mathrm{un}}\) which is defined by \(\sqrt{J}\). In this setting, the authors show the following theorem.
Theorem B. Let \(r\geq 1\) be a positive integer. Then
1) there exists a positive integer \(N\) and a product of linear forms \(F\), defining an arrangement \(\mathcal{A}_{F}\) in \(\mathbb{P}^{3}\) such that \[\dim \, M(C_{F}^{\mathrm{un}})_{N} = r\] and all other components of \(M(C)_{F}^{\mathrm{un}}\) are zero;
1) there exists a positive integer \(N'\) and a product of linear forms \(F'\), defining an arrangement \(\mathcal{A}_{F'}\) in \(\mathbb{P}^{3}\) such that \[\dim \, M(C_{F'}^{\mathrm{red}})_{N'} = r\] and all other components of \(M(C_{F'}^{\mathrm{red}})\) are zero;
3) for each \(h\geq 1\) we can replace \(N\) by \(N+h\) and find a polynomial \(G\) so that \[\dim \, M(C_{G}^{\mathrm{un}})_{N+h} = r\] and all other components of \(M(C_{G}^{\mathrm{un}})\) are zero. The curve \(C_{G}^{\mathrm{un}}\) is in the same even liaison class as \(C_{F}^{\mathrm{un}}\). The analogous result for \(C^{\mathrm{red}}\) also holds. hyperplane arrangements; ideals; freeness; Cohen-Macaulay algebras Configurations and arrangements of linear subspaces, Polynomial rings and ideals; rings of integer-valued polynomials Schemes supported on the singular locus of a hyperplane arrangement in \(\mathbb{P}^n\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author constructs interesting new variations of Hodge structures (VHS) on smooth projective varieties \(Y\) which are obtained as follows.
First one takes a generic \(m\)-dimensional linear section \(U_m\) of the open set \(U\) parametrizing the smooth hyperplane sections of a given odd dimensional projective smooth variety \(X\). Then, for \(3 \leq m \leq 6\), \(Y\) can be realized as the smooth compactification of the total space of a finite Galois covering of \(U_m\).
A rational, polarizable VHS on \(X\) gives then rise to a VSH on \(Y\) using standard pull-back and push-forward operations.
This construction is applied to checking the Carlson-Toledo conjecture for classes of examples of dimension \(m\). variation of Hodge structure; hyperplane section; simple singularities Damien Mégy, Sections hyperplanes à singularités simples et exemples de variations de structure de Hodge, Ph. D. Thesis, Institut Fourier (Grenoble), available at , 2010 Variation of Hodge structures (algebro-geometric aspects), Structure of families (Picard-Lefschetz, monodromy, etc.), Intersection homology and cohomology in algebraic topology Hyperplane sections for simple singularities and examples of variations of Hodge structures | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 integrals over Hilbert schemes of points involving tautological bundles and certain ``geometric'' subsets of these Hilbert schemes. For a smooth projective complex variety \(X\) of dimension \(d\), the Hilbert scheme \(X^{[n]}\) parametrizes length-\(n\) \(0\)-dimensional closed subschemes of \(X\). A vector bundle on \(X\) induces a tautological bundle \(E^{[n]}\) on \(X^{[n]}\). Roughly speaking, a geometric subset of \(X^{[n]}\) is a constructible subset \(P\) such that if \(Z, Z' \in X^{[n]}\) satisfying \(Z \cong Z'\) as \(\mathbb C\)-schemes, then either \(Z, Z' \in P\) or \(Z, Z' \not \in P\).
The main theorem of the paper states that the integral over \(X^{[n]}\) involving the Chern classes of \(E^{[n]}\) and the fundamental class (respectively, the Chern-Mather class, the Chern-Schwartz-MacPherson class) of a geometric subset \(P\) can be written as a universal polynomial, depending on the type of \(P\), in the Chern numbers involving \(E\) and the tangent bundle \(T_X\) of \(X\). When \(X^{[n]}\) is smooth, the integral is also allowed to contain the Chern classes of \(T_{X^{[n]}}\).
The main idea in proving this theorem is to use Jun Li's concept of Hilbert scheme \(X^{[[\alpha]]}\) of \(\alpha\)-points [\textit{J. Li}, Geom. Topol. 10, 2117--2171 (2006; Zbl 1140.14012)]. The main theorem generalizes many known results when \(X\) is a surface.
As an application, the author obtains a generalized Göttsche's conjecture for all isolated singularity types and in all dimensions. More precisely, if \(L\) is a sufficiently ample line bundle on a smooth projective variety \(X\), then in a general subsystem \(\mathbb P^m \subset |L|\) of appropriate dimension \(m\), the number of hypersurfaces with given isolated singularity types is a polynomial in the Chern numbers involving \(T_X\) and \(L\). Another application is to obtain similar results, when \(X\) is a surface, for the locus of curves with fixed ``BPS spectrum'' in the sense of stable pairs theory.
Section~2 is devoted to the preliminaries such as the definition of the tautological bundle \(E^{[n]}\) on the Hilbert scheme \(X^{[n]}\), the construction of the Chern-Mather and Chern-Schwartz-MacPherson classes, the Hilbert scheme \(X^{[[\alpha]]}\) of \(\alpha\)-points, and the definition of geometric subsets in \(X^{[n]}\) and \(X^{[[\alpha]]}\). Section~3 contains an outline of the proof of the main theorem, while the formal proof of the main theorem is presented in Section~4. In Section~5, the author verifies a technical lemma which is used in Section~4. Section~6 deals with the generating series of the above-mentioned integrals over all the Hilbert schemes \(X^{[n]}\), \(n \geq 0\). In Section~7, The precise definition of sufficiently ample is given, and the main theorem is applied to the problem of counting geometric objects with prescribed singularities. Hilbert schemes; tautological bundles; Göttsche's conjecture; counting singular divisors; BPS spectrum J. V. Rennemo, Universal polynomials for tautological integrals on Hilbert schemes , preprint, [math.AG]. arXiv:1205.1851v1 Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Universal polynomials for tautological integrals on Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If \(A\) is a differential graded (DG) algebra and \(D(A)\) its derived category, a categorical resolution of \(A\) is a pair \((B,M)\) consisting of a smooth DG algebra \(B\) and a DG \(A^{\text{op}}\otimes B\)-module \(M\) such that the derived tensor functor from \(D(A)\) to \(D(B)\) associated with \(M\) is full and faithful on the subcategory of perfect DG \(A\)-modules. Here, a DG algebra \(B\) is smooth if \(B\) is perfect as a DG \(B^{\text{op}}\otimes B\)-module. In particular, since the derived category \(D(X)\) of quasi-coherent sheaves on a quasi-compact and separated scheme \(X\) is equivalent to \(D(A)\) for some DG algebra \(A\), it makes sense to speak of a categorical resolution of \(X\). Note that \(X\) is smooth if and only if \(D(X)\) is smooth in the above sense.
The main result of the paper under review states that if \(X\) is a reduced separated scheme of finite type over a field of characteristic zero, then it has a categorical resolution by a smooth poset scheme if and only if \(X\) has Du Bois singularities.
A poset scheme is, roughly speaking, defined as follows. Fix a finite partially ordered set (poset) \(S\). An \(S\)-scheme \(\mathcal{X}\) is the datum of a scheme \(X_\alpha\) for every \(\alpha\in S\) and morphisms \(f_{\alpha\beta}: X_\alpha \rightarrow X_\beta\) for \(\alpha\geq \beta\) satisfying the obvious compatibility condition. The usual notions for schemes, for example, regularity, are defined componentwise. Similarly, a quasi-coherent sheaf on a poset scheme is a collection of sheaves \(F_\alpha\) on the \(X_\alpha\) and maps \(\phi_{\alpha\beta}: f^*_{\alpha\beta}F_\beta\rightarrow F_\alpha\) satisfying the usual cocycle conditions. Quasi-coherent sheaves on a poset scheme \(\mathcal{X}\) form an abelian Grothendieck category and one can consider its derived category \(D(\mathcal{X})\). Among many other properties relevant in this context the author shows that \(D(\mathcal{X})\) admits a semi-orthogonal decomposition into the derived categories of the \(X_\alpha\), that \(D(\mathcal{X})\) admits a compact generator and that it is smooth if \(\mathcal{X}\) is a regular \(S\)-scheme essentially of finite type over a perfect field.
If \(S\) and \(S'\) are posets and \(\tau\) is an order preserving map, one can define a map from an \(S\)-scheme \(\mathcal{X}\) to an \(S'\)-scheme \(\mathcal{X}'\). Note that any scheme \(Y\) can be considered as an \(S'\)-scheme for \(S'\) a point. Given a smooth \(S\)-scheme \(\mathcal{X}\), a map \(\pi: \mathcal{X}\rightarrow Y\) is a categorical resolution if the derived pullback functor is one. In Section 7 the author establishes, in particular, that \(\pi\) is a categorical resolution if and only if the adjunction morphism \(\mathcal{O}_Y\rightarrow R\pi_*\mathcal{O}_\mathcal{X}\) is a quasi-isomorphism. Now, in characteristic zero any reduced separated scheme of finite type \(X\) has a so-called cubical hyperresolution \(\mathcal{Z}\) and if \(X\) has Du Bois singularities, the result from Section 7 precisely gives that \(\mathcal{Z}\) provides a categorical resolution. For the other direction the author shows that the adjunction map \(\mathcal{O}_X\rightarrow R\pi_*\mathcal{O}_\mathcal{Z}\) of a hyperresolution \(\pi: \mathcal{Z}\rightarrow X\) has a left inverse, which is sufficient to conclude that \(X\) is Du Bois.
The author also proves three results on degeneration of spectral sequences for smooth projective poset schemes. These are, in particular, used to prove that the de Rham-Du Bois complex can be defined by means of any smooth projective poset scheme which satisfies descent in the classical topology. derived categories; differential graded algebras; resolutions of singularities; poset schemes; categorical resolution; Du Bois singularities; cubical hyperresolution; degeneration of spectral sequence Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Grothendieck categories, Spectral sequences, hypercohomology Categorical resolutions, poset schemes, and Du Bois 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 investigates hypersurface sections in four-dimensional affine toric varieties.
Let a non-degenerate function \(f\in\mathbb{C}[\check\sigma\cap\mathbb{Z}^ 4]\) be given \((\sigma\subset\mathbb{R}^ 4\) is a polyhedral cone, \(\check\sigma\) denotes its dual). If \(f\) defines a simple K3 singularity \([f=0]\subseteq\text{Spec} \mathbb{C}[\check\sigma\cap\mathbb{Z}^ 4]\) then \(\sigma\) has to be Gorenstein and the Newton polyhedron \(\Gamma_ +(f)\subset\check\sigma\) of \(f\) will meet a very special condition: It has to be contained in some suitable affine hyperplane (defined by an element \(u_ 0\in int(\sigma))\) meeting special conditions. --- Finally, it is proved that for such pairs \((\sigma,u_ 0)\) there are (up to automorphisms) only finitely many possibilities to occur. An algorithm to determine these pairs is given also. Tsuchihashi singularity; hypersurface sections in four-dimensional affine toric varieties; K3 singularity; Newton polyhedron Tsuchihashi, H.: Simple K3 singularities which are hypersurface sections of toric singularities. preprint. Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, \(3\)-folds, \(n\)-folds (\(n>4\)), Singularities of surfaces or higher-dimensional varieties Simple K3 singularities which are hypersurface sections of toric singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For \(d\geq 3g\) and \(1\leq s\leq [g/2]\), we study the strata \(N_{d,g}(s)\) of degree \(d\) genus \(g\) space curves \(C\) whose normal bundle \(N_C\) is stable with stability degree (integer of Lange-Narasimhan) \(\sigma(N_C)=2s\). We prove that \(N_{d, g}(s)\) has an irreducible component of the right dimension whose general curve has a normal bundle with the right number of maximal subbundles. We consider also the semi-stable case (\(s=0\)), obtaining similar results. We prove our results by studying the normal bundles of reducible curves and their deformations. Parametrization (Chow and Hilbert schemes), Plane and space curves, Families, moduli of curves (algebraic) Stratification of the Hilbert scheme of space curves with a stable normal bundle | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A Sol-3 manifold is a quotient space of the form \(\mathbb R^2 \times \mathbb R^+ / \Gamma\), where Sol is the space \(\mathbb R^2 \times \mathbb R^+\) equipped with the group operation \((x_1, y_1, t_1) \cdot (x_2, y_2, t_2) = (x_1 + e^{t_1} x_2, y_1 + e^{t_2} y_2, t_1 + t_2)\) and \(\Gamma\) is a discrete subgroup of Aff(Sol) such that \(\text{Sol} / \Gamma\) is compact and \([\Gamma: \Gamma \cap \text{Sol}] < \infty\). In this paper, motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, the author determines a nessary and sufficient condition for a manifold to be diffeomorphic to such a cusp cross-section. In particular, this proves that every Sol-3 manifold is diffeomorphic to a cusp cross-section of a Hilbert modular surface. He also deduces an obstruction to geometric bounding in this setting. As a consequence, there exists Sol-3 manifolds that cannot arise as a cusp cross-section of a 1-cusped nonsingular Hilbert modular surface. Hilbert modular varieties; Sol-3 manifolds; cusp cross-sections Modular and Shimura varieties, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces Cusps of Hilbert modular varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A \textit{Rees algebra} over a ring \(B\) is a finitely generated graded \(B\)-subalgebra of the polynomial ring \(B[W]\). This concept globalizes to give the notion of Rees algebra (or \textit{Rees sheaf of algebras}) over a base scheme \(V\), say \(\mathcal G \subset \mathcal O _V[W]\). \textit{O. Villamayor} [Ann. Sci. Éc. Norm. Supér. (4) 22, No. 1, 1--32 (1989; Zbl 0675.14003); Ann. Sci. Éc. Norm. Supér. (4) 25, No. 6, 629--677 (1992; Zbl 0782.14009)] introduced the concept several years ago, with the intention to apply it to the theory of resolution of singularities of algebraic varieties. For this purpose, the most interesting situation is that where \(B\) is a regular, finitely generated algebra over a perfect field \(k\), or \(V\) is a smooth algebraic variety over such \(k\). Henceforth, we suppose these assumptions valid.
In Part I, the authors review the basic theory and present some new results. Among other things, given a Rees algebra \(\mathcal G \subset {\mathcal O}_V[W]\), with \(V\) a smooth variety over a perfect field \(k\), they recall the notions of order of \(\mathcal G\) at \(x \in V\), zero set and singular locus of \(\mathcal G\), and integral closure \(\overline {\mathcal G}\) of \(\mathcal G\) in \({\mathcal O}_V[W]\). They define \textit{differential Rees algebra}, i.e., one closed under the action of differential operators, and recall the construction of the minimal differential Rees containing a given \(\mathcal G\), denoted by \({\mathbb D}(\mathcal G)\). They also define the concept of \textit{weakly equivalent} (w.e.) Rees algebras: essentially, Rees algebras (over \(V\)) \(\mathcal G\) and \(\mathcal K\) are w.e. if Sing(\(\mathcal G\)) = Sing(\(\mathcal K\)) and this equality is preserved when we take, successively, suitable ``transforms'', in a sense explained in the text. They relate Rees algebras to \textit{pairs}, that is ordered couples \((I,b)\) where \(I\) is a coherent sheaf of ideals of \(\mathcal O _V\) and \(b \geq 0\) an integer. One may associate to a pair \((I,b)\) a Rees algebra \({\mathcal O}_V[IW^b]\), and any Rees algebra is closely related to one of this type.
Their main new result (Theorem 3.11, called the \textit{canonicity principle}) says that Rees algebras (both over \(V\)) \(\mathcal G\) and \(\mathcal K\) are w.e. if and only if \(\overline{{\mathbb D}(\mathcal G)}=\overline{{\mathbb D}(\mathcal K)}\). Theorem 3.11 is a an immediate consequence of a more general result (Theorem 3.10), expressed in terms of certain inclusions, whose proof is rather involved.
In Part II they discuss some applications of Theorem 3.11. It has been known for a long time (thanks to Hironaka's efforts) that resolution of singularities of an algebraic variety \(X\) (say, embedded in a smooth \(V\)) follows if we can resolve, in a suitable sense, certain pair \((I,b)\) associated to \(X\). But this is a local process: \(I\) is not defined on the whole \(V\), but on an étale neighborhood of a point \(x \in V\). Moreover, the pair \((I,b)\) is not unique. To verify that this process globalizes has been traditionally a hard gluing problem, requiring complicated methods. The authors reinterpret the theory in terms of Rees algebras, and use their main result to give a simple solution to the mentioned gluing problem.
They also review a method to resolve Rees algebras (or, equivalently, pairs), valid in characteristic zero, which yields partial results in positive characteristic. Again one has to face a ``gluing problem'', to verify that certain procedures are well-defined. The authors once more use the canonicity principle to deal with this question.
They conclude the paper by discussing an example showing that an inductive process to lower the multiplicity of a hypersurface, valid over fields of characteristic zero, may fail in positive characteristic. Rees algebra; equivalence; integral closure; differential operators; resolution of singularities Bravo, A.; García-Escamilla, M. L.; Villamayor U., O. E., On Rees algebras and invariants for singularities over perfect fields, Indiana Univ. Math. J., 61, 3, 1201-1251, (2012) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Integral closure of commutative rings and ideals, Derivations and commutative rings On Rees algebras and invariants for singularities over perfect 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 We review some basics on the theory of generic singularities in algebraic geometry. We state certain approaches towards the explicit local equations of such singularities. The main point of view is the constructive approach in commutative algebra and algebraic geometry. Singularities in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc. Local defining ideals of ordinary singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S_{0}({\mathcal V}_{\mathbb C})\) be the Grothendieck semiring of complex quasi-projective varieties. The generators of the additive semigroup are isomorphism classes \([X]\) of such varieties subject to the relation \([X]=[X-Y]+[Y]\) if \(Y\subset X\) is a closed subvariety of \(X\). Multiplication is given by the cartesian product. A power structure over a semiring \(R\) is a map
\[
(1+T\cdot R[[T]]) \times R \rightarrow 1+T\cdot R[[T]] : (A(T), m) \rightarrow (A(T))^m,
\]
where \(A(T)= 1+ a_{1}T + a_{2}T^2 +\cdots , a_{i}\in R, m\in R, \) satisfying all the usual properties of the exponential function. The main result of the paper is the computation of the generating series (in \(S_{0}({\mathcal V}_{\mathbb C})\)) of Hilbert schemes of zero-dimensional subschemes on a smooth quasi-projective variety of dimension \(d\) as an exponent of such series for the complex affine space \({\mathbb A}^d.\) Grothendieck semiring; Hilbert scheme; generating series Gusein-Zade, SM; Luengo, I; Melle-Hernández, A, Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points, Mich. Math. J., 54, 353-359, (2006) Parametrization (Chow and Hilbert schemes), Grothendieck groups (category-theoretic aspects) Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a complex smooth connected projective surface with effective curve class \(\beta \in H_2 (X, \mathbb Z)\). For any \(N > 0\), the Quot scheme \(\text{Quot}_X (\mathbb C^N, \beta, n)\) parametrizes quotients \({\mathcal O}^N \to Q\) with \(c_1 (Q) = \beta\) and \(\chi (Q)=n\), or equivalently short exact sequences \(0 \to S \to \mathbb C^N \otimes \mathcal{O}_X \to Q \to 0\). The virtual fundamental class \([\text{Quot}_X (\mathbb C^N, \beta, n)]^{\text{vir}}\) was used by Marian, Oprea and Pandiharipande [\textit{A. Marian} et al., Ann. Sci. Éc. Norm. Sup. 50, 239--267 (2017; Zbl 1453.14016)] to prove a conjecture of \textit{M. Lehn} [Invent. Math. 136, 157--207 (1999; Zbl 0919.14001)] for \(K3\) surfaces. The virtual Euler characteristic is defined by integrating the total Chern class of the virtual tangent complex of the canonical obstruction theory (see [\textit{B. Fantechi} and \textit{L. Göttsche}, Geom. Topol. 14, 83--115 (2010; Zbl 1194.14017)], namely
\[
\text{e}^{\text{vir}} (\text{Quot}_X (\mathbb C^N, \beta, n)) = \int_{[\text{Quot}_X (\mathbb C^N, \beta, n)]^{\text{vir}}} c(T^{\text{vir}} \text{Quot}) \in \mathbb Z
\]
where \(T^{\text{vir}} \text{Quot} = \text{Ext}^\bullet_X (S, Q)\) is evaluated at the exact sequence above. The second two named authors conjectured that the corresponding generating series
\[
Z_{X,N.\beta} = \sum_{n \in \mathbb Z} \text{e}^{\text{vir}} (\text{Quot}_X (\mathbb C^N, \beta, n)) q^n
\]
is the Laurent expansion of a rational function in \(q\). The authors prove the conjecture when \(N=1\), when \(\text{Quot}_X (\mathbb C^1, \beta, n)\) is a Hilbert scheme of curves on \(X\).
The authors further consider descendent series. Letting \(\pi_1, \pi_2\) be the projections from \(\text{Quot}_X (\mathbb C^N, \beta, n) \times X\) onto its two factors, let \(\mathcal Q\) be the universal quotient on \(\text{Quot}_X (\mathbb C^N, \beta, n) \times X\). A given \(K\)-theory class \(\alpha \in K^0 (X)\) yields the \(K\)-theory class \(\alpha^{[n]} =\mathbf{R} \pi_{1 *} (\mathcal Q \otimes \pi_2^* \alpha) \in K^0 (\text{Quot}_X (\mathbb C^N, \beta, n))\) and one can generalize the series of virtual Euler characteristics by defining the descendent series \(Z_{X,N,\beta}(\alpha_1, \dots, \alpha_{\ell} \ k_1, \dots, k_{\ell})\) by
\[
\sum_{n \in \mathbb Z} \int_{[\text{Quot}_X (\mathbb C^N, \beta, n)]^{\text{vir}}} \text{ch}_{k_1} (\alpha_1^{[n]}) \dots \text{ch}_{k_{\ell}} (\alpha_{\ell}^{[n]}) c(T^{\text{vir}} \text{Quot}) q^n.
\]
The authors conjecture that \(Z_{X,N,\beta}(\alpha_1, \dots, \alpha_{\ell} \ k_1, \dots, k_{\ell})\) is the Laurent expansion of a rational function in \(q\) in general and prove it in the special cases \(\beta=0\) and \(N=1\). \textit{E. Carlsson} has studied descendent integrals as above against the (non-virtual) fundamental class of the Hilbert scheme of points and proven the descendent series to be quasi-modular [Adv. Math. 229, 2888--2907 (2012; Zbl 1255.14005)]. Hilbert scheme of points on a surface; obstruction theory; generating series of virtual Euler characteristics; tautological sheaf Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Rationality of descendent series for Hilbert and Quot schemes of surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper we are interested in quotients of Calabi-Yau threefolds with isolated singularities. In particular, we analyze the case when \(X/G\) has terminal singularities. We prove that, if \(G\) is cyclic of prime order and \(X/G\) has terminal singularities, then \(G\) has order lower than or equal to 5. Calabi-Yau manifolds (algebro-geometric aspects), Automorphisms of surfaces and higher-dimensional varieties, Singularities of surfaces or higher-dimensional varieties Calabi-Yau quotients with terminal singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In J. Am. Math. Soc. 15, No. 3, 599--615 (2002; Zbl 0998.14009), \textit{M. Mustaţă} provides characterisations of KLT and LC pairs \((X, qY)\) with \(X\) smooth via dimension of jet schemes of \(Y\). The aim of the paper is to extend this result to the case \(X\) is normal and \(\mathbb{Q}\)-Gorenstein. KLT singularities; LC singularities Yasuda, T.: Dimensions of jet schemes of log singularities. Amer. J. Math. 125, No. 5, 1137-1145 (2003) Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Dimensions of jet schemes of log singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a survey article about the study of the links of some complex hypersurface singularities in \(\mathbb{C}^3\). We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group \(\mathrm{SU}(2)\), \(\mathrm{Nil}^3\) or \(\mathrm{Sol}^3\), respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of \(\mathrm{S}^5\). Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in \(\mathrm{S}^5\) as codimension two contact submanifolds. links of complex hypersurface singularities, contact structures Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Symplectic and contact topology in high or arbitrary dimension On the links of simple singularities, simple elliptic singularities and cusp singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review studies deformations of schemes with normal crossing singularities, a kind of singularities appearing naturally in many different problems in algebraic geometry.
The well-known Mumford's semi-stable reduction theorem states that, after a finite base change and a birational modification, any flat projective morphism \(f: \mathcal X \rightarrow C\) from a variety \(\mathcal X\) to a curve \(C\) can be brought to \(f': \mathcal X' \rightarrow C'\), where \(\mathcal X'\) is smooth and the special fibers are simple normal crossing varieties.
In the compactification of the moduli space of varieties of general type, stable varieties appear in the boundary. Recall that a stable variety is a proper reduced scheme \(X\) such that \(X\) has only semi-log-canonical singularities and \(\omega_X^{[k]}\) is locally free and ample for some \(k>0\). The simplest class of non-normal semi-log-canonical singularities are the normal crossing singularities.
In the Minimal Model Program, normal crossing singularities also play an important role. One of the outcomes of the Minimal Model Program starting with a smooth \(n\)-dimensional projective variety \(X\) is a Mori fiber space \(f: Y \rightarrow Z\), where \(f\) is a projective morphism, \(Y\) is a \(\mathbb Q\)-factorial terminal projective variety such that \(-K_Y\) is \(f\)-ample, and \(Z\) is normal with \(\dim Z \leq n-1\). If \(Z\) is a curve, then \(Y_z=f^{-1} (z)\) for \(z \in Z\) is a Fano variety of dimension \(n-1\) and \(Y\) is a \(\mathbb Q\)-Gorenstein smoothing of \(Y_z\). In general \(Y_z\) has non-isolated singularities and may not even be normal. It is difficult to describe the singularities of the special fibers but normal crossing singularities naturally occur and are the simplest possible non-normal singularities.
Motivated by the above different problems, the author investigates the deformation spaces of varieties with normal crossing singularities. Let \(X\) be a reduced scheme with normal crossing singularities defined over a field \(k\) and let \(T^1 (X)\) denote the sheaf of first order deformations of \(X\). Then \(T^1 (X)\) is an invertible sheaf supported on the singular locus \(D\) of \(X\). In general, \(D\) is not smooth, and may not even be Cohen-Macaulay. So the author works in a suitable log resolution of \((X, D)\) and obtains explicit formulas for \(T^1 (X)\) in this setting.
The paper under review is well written. In section \(1\) the author explains the motivation to study deformations of schemes with normal crossing singularities and states the main theorem. In section \(2\) the author recalls some technical results for future use. He proves the first part of the main theorem in section \(3\), where the case that \(X\) has only double point singularities is treated. The general case that \(X\) has higher multiplicity singularities is done in section \(4\). The log resolution \((X', D')\) of \((X, D)\) is obtained by successively blowing ups the singular locus of highest multiplicity of \(X\). This works in all dimensions but is not unique. In dimension at most three, the author obtains a unique log resolution \((\tilde{X}, \tilde{D})\) by running an explicit minimal model program on \((X', D')\) and proves the second part of the main theorem. At the end of the paper, the author constructs a Fano \(3\)-fold with normal crossing singularities and show that it is not smoothable. deformations of schemes; normal crossing singularities Formal methods and deformations in algebraic geometry, Fibrations, degenerations in algebraic geometry First order deformations of schemes with normal crossing singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For an abelian surface \(A\) with a symplectic action by a finite group \(G\), one can define the partition function for \(G\)-invariant Hilbert schemes
\[
Z_{A,G}(q)=\sum\limits_{d=0}^{\infty}e(\mathrm{Hilb}^d(A)^G)q^d.
\]
We prove the reciprocal \(Z_{A,G}^{-1}\) is a modular form of weight \(\frac{1}{2}e(A/G)\) for the congruence subgroup \(\Gamma_0(|G|)\) and give explicit expressions in terms of eta products. Refined formulas for the \(\chi_y\)-genera of \(\mathrm{Hilb}(A)^G\) are also given. For the group generated by the standard involution \(\tau:A\rightarrow A\), our formulas arise from the enumerative geometry of the orbifold Kummer surface \([A/\tau]\). We prove that a virtual count of curves in the stack is governed by \(\chi_y(\mathrm{Hilb}(A)^{\tau})\). Moreover, the coefficients of \(Z_{A,\tau}\) are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin. Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, \(K3\) surfaces and Enriques surfaces \(G\)-invariant Hilbert schemes on abelian surfaces and enumerative geometry of the orbifold Kummer 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 Similar to the Siegel case the author formulates results without details or proofs of an action \(f_{\bullet}\) of the Hecke ring HR(\(\Gamma\) (N), \(G_ 2^+({\mathcal O}_ K))\) of a totally real algebraic number field K of degree g with ring of integers \({\mathcal O}_ K\) on the singular homology \(H_{\bullet}(M(N)_{{\mathbb{C}}},{\mathbb{Z}})\) of a smooth toroidal compactification \(M(N)_{{\mathbb{C}}}\) of the Hilbert modular variety \(\Gamma (N)\setminus {\mathbb{H}}^ g\), \(\Gamma\) (N) being the principal congruence subgroup of level \(N\geq 3\) of the Hilbert modular group of K as well as an \(\ell\)-adic version \(f^{\bullet}\) for the \(\ell\)-adic cohomology of \(M(N)_{{\mathbb{C}}}\) resp. D(N). Here D(N) is a suitable proper smooth Hilbert modular variety over the field \(\bar F_ p\), \(p\nmid N\) prime. The main results are
(1) an estimation of the eigenvalues \(\lambda_ p\) of the Hecke operator \(f^ n(Tp\) \({\mathcal O}_ K):\)
\[
| \lambda_ p| \leq p^{n/2}+p^{(2g-n)/2}\text{ for } 0\leq n\leq 2g,
\]
(2) an expression of the local zeta function Z(D(N),X) in terms of the above Hecke operators:
\[
Z(D(N),X)=\sqrt{\prod_{n}P^*_ n(M(N)_{{\mathbb{C}}},X)^{(-1)^{n+1}}}
\]
where the polynomial \(P^*_ n\) equals
\[
\det [1-(f_ n(Tp\quad {\mathcal O}_ K)\otimes id)X+(f_ n(\Gamma (N)\sigma_ p\Gamma (N))\otimes id)p^ gX^ 2].
\]
Hilbert modular variety; Hilbert modular group; Hecke operator; local zeta function K. Hatada: On the local zeta functions of the Hilbert modular schemes. Proc. Japan Acad., 66A, 195-200 (1990). 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, \(p\)-adic theory, local fields, Hecke-Petersson operators, differential operators (several variables) On the local zeta functions of the Hilbert modular 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 \(F\) be a graded module over an AS-regular algebra of dimension 3, generated in degree 1 with 3 quadratic relations. The author verifies that the Hilbert scheme, in the sense of \textit{M. Artin} and \textit{J. J. Zhang} [Algebr. Represent. Theory 4, No. 4, 305--394 (2001; Zbl 1030.14003)], parametrizing quotients of \(F\) with a fixed Hilbert polynomial, is projective. Hilbert scheme; quantum plane Algebraic moduli problems, moduli of vector bundles, Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry Hilbert schemes for quantum planes are projective | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is devoted to the rationality of the Gieseker moduli space \(M_{\mathbb{P}^2}(r,\chi)\) of semistable sheaves \(\mathcal{F}\) such that their Hilbert polynomial is \(P_{\mathcal{F}}(m)=rm+\chi\). In particular, \(M_{\mathbb{P}^2}(n+r,n)\times \mathbb{P}^{r-1}\) is a rational variety for pairs \((n-r,r)\) of the form \(R^k(1,3)\) where \(R(m,n)=(n,3n-m)\), or \(2\leq r<\frac{\sqrt{5}-1}2n\). As a consequence, the author concludes that \(M_{\mathbb{P}^2}(r,\pm1)\) is a rational variety.
The key of this proof is to establish certain relations between \(M_{\mathbb{P}^2}(r,\chi)\) and the flag Hilbert scheme of pairs \((Z,C)\), denoted by \(\text{Hilb}_{\mathbb{P}^2}(\ell,d)\), where \(C\subset \mathbb{P}^2\) is a curve of degree~\(d\), and \(Z\subset C\) is a zero-dimensional subscheme of length~\(\ell\). Such a relation is expressed by the map \(\eta:\text{Hilb}_{\mathbb{P}^2}(\ell,d)\to M_{\mathbb{P}^2}(d,-d(d-3)/2-\ell)\) defined by \(\eta(C,Z)=[\mathcal{J}_Z]\) the ideal sheaf of \(Z\) in \(\mathcal{O}_C\). Once that it is proved to be a well-defined map, two open subspaces \(\text{Hilb}^0_{\mathbb{P}^2}(\ell,d)\subset \text{Hilb}_{\mathbb{P}^2}(\ell,d)\) and \(H_0(\ell,d)\subset M_{\mathbb{P}^2}(n+2,n+1)\) are defined such that \(\eta\) restricted to these two subspaces is an isomorphism. This leads to conclude the rationality of \(M_{\mathbb{P}^2}(n+1,n)\) for \(n\geq 2\) integer.
However to prove his main result of this article, Maican redefines the map \(\eta\) by means of understand the elements of these moduli spaces as a maps between sheaves. In particular, \(\varphi:r\mathcal{O}(-2)\oplus (n-r)\mathcal{O}(-1)\to n\mathcal{O}\) defines a point \([\text{coker}(\varphi)]\in M_{\mathbb{P}^2}(n+r,n)\), and \(\psi:(r-1)\mathcal{O}(-2)\oplus (n-r)\mathcal{O}(-1)\to n\mathcal{O}\) produces an element of \((C,Z)\in \text{Hilb}_{\mathbb{P}^2}(\ell,d)\), where \(Z\) is determined by the cokernel of \(\psi\). Following this description, Maican proves the main result of this paper mentioned in the first paragraph of this review.
The last two results of the paper is concerned about the map \(\eta\). More concretely, the author gives some conditions on which such a map is an isomorphism.
The author points out that the main result is proved by [\textit{Le Potier}, Rev. Roum. Math. Pures Appl. 38, No. 7--8, 635--678 (1993; Zbl 0815.14029)] by a different method. Gieseker moduli space; semi-stable sheaves; moduli of sheaves; Hilbert schemes Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Rationality questions in algebraic geometry Flag Hilbert schemes and moduli spaces of torsion plane 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 note gives a sketch of the proof of the following theorem: \(Let\quad X\) be a 3-dimensional normal complex algebraic variety with at most canonical singularities. Then the ring \(\oplus_{m\geq 0}{\mathcal O}_ X(m.D)\quad is\) finitely generated, where D is a Weil divisor on X.
This result is then applied to prove the existence of minimal models for on-parameter-families of surfaces of non-negative Kodaira-dimension whose degenerate members are reduced and have only simple normal crossings. blowing-ups; threefold; canonical singularities; minimal models for on- parameter-families of surfaces Y. Kawamata, Crepant blowing-ups of \(3\)-dimensional canonical singularities and its application to degenerations of surfaces , Tokyo University, preprint. JSTOR: Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), \(3\)-folds, Divisors, linear systems, invertible sheaves, Singularities in algebraic geometry, Families, moduli, classification: algebraic theory On the crepant blowing-ups of canonical singularities and its application to degenerations of surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities How the nature of the base space affects the jet schemes is a natural question. This paper is motivated by a question: if a scheme is defined by a monomial ideal, then are its jet schemes also defined by monomial ideals? The answer is no. The paper gives a simple counterexample. But if one thinks of the reduced subschemes of the jet schemes instead of the jet schemes themselves, then the answer is yes. This paper proves it by describing the generators of the ideal of the reduced subschemes of the jet schemes concretely. Goward, Russell A. Jr. and Smith, Karen E. The jet scheme of a monomial scheme \textit{Comm. Algebra}34 (2006) 1591--1598 Math Reviews MR2229478 Computational aspects in algebraic geometry, Computational aspects and applications of commutative rings The jet scheme of a monomial 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 \(\alpha\) be a regular local two-dimensional ring, and let \(m = (x, y)\) be its maximal ideal. Let \(m > n > 1\) be coprime integers, and let \(p\) be the integral closure of the ideal \((x^m , y^n )\). Then \(p\) is a simple complete \(m\)-primary ideal, and its value semigroup is generated by \(m, n\).
We construct a minimal system of generators \(\{z_0 ,\dots , z_n \}\) of \(p\), and from this we get a minimal system of generators of the polar ideal \(p'\) of \(p\), consisting of \(n =\theta\) elements. In particular, we show that \(p\) and \(p'\) are monomial ideals. When \(\alpha = k[[x, y]]\), a ring of formal power series over an algebraically closed field \(k\) of characteristic zero, this implies the existence of some relevant property. Greco, S. and Kiyek, K.: The polar ideal of a simple complete ideal having one characteristic pair. Preprint, Politecnico di Torino, Rapporto interno N. 32. Regular local rings, Singularities of curves, local rings Some results on simple complete ideals having 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 \textit{J. Riou} [J. Topol. 3, No. 2, 229--264 (2010; Zbl 1202.19004)] showed that operations on the higher \(K\)-theory of smooth schemes are completely determined by operations in degree zero. More precisely, if we denote by \(K\) a simplicial presheaf representing \(K\)-theory in the unstable motivic homotopy category of smooth schemes (over a fixed base), and if we view \(K_0\) as a presheaf on smooth schemes (over the same base), then there is a canonical bijection between operations \(K^n\to K\) and operations \(K_0^n\to K_0\).
In the present paper, the author extends this result in several directions. His main results extend Riou's theorem to (possibly singular) divisorial schemes by establishing natural bijections between the following sets of operations:
\begin{itemize}
\item operations \(K_0^n \to K_0\), where \(K_0\) is viewed as presheaf on the category of \emph{divisorial} schemes
\item operations \(K^n \to K\), where \(K\) is a representing simplicial presheaf in the unstable motivic homotopy category of smooth divisorial schemes
\item operations \(K^n \to K\), where \(K\) is viewed as an object in a homotopy category of simplicial presheaves on divisiorial schemes
\item operations \(K^n \to K\), where \(K\) is viewed as an object in a homotopy category of simplical presheaves on noetherian schemes
\end{itemize}
In the last two cases, the homotopy categories are defined with respect to Zariski local model structures.
While Riou assumes all schemes to be smooth and separated, the author, as a first step, observes that the arguments remain unaffected if we more generally consider smooth divisorial schemes. A key imput in the further generalizations are his previous results on embedding divisorial schemes into smooth ones [\textit{F. Zanchetta}, J. Algebra 552, 86--106 (2020; Zbl 1441.14005)].
The author moreover extends Riou's theorem to symplectic \(K\)-theory (the variant \(GW^{[2]}\) of hermitian \(K\)-theory in the notation of [\textit{M. Schlichting}, J. Pure Appl. Algebra 221, No. 7, 1729--1844 (2017; Zbl 1360.19008)]). The arguments in this case are very similar. \(K\)-theory; operations; divisorial schemes; motivic homotopy theory; hermitian \(K\)-theory; symplectic \(K\)-theory \(K\)-theory of schemes, Motivic cohomology; motivic homotopy theory, Homotopy theory Unstable operations on \(K\)-theory for singular schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of this paper is to prove a criterion for projectively Cohen- Macaulay-two-codimensional subschemes \(Y\) of \(\mathbb{P}_ k^ N\) (\(k\) algebraically closed of characteristic 0) to be smoothable. The considered schemes are determinantal, i.e. the homogeneous ideal \(I(Y)\), defining \(Y\), is generated by the \(t\times t\)-minors of a homogeneous \(m\times n\)-matrix \(M\) with \(\text{ht}(I(Y))= (m-t+1) (n-t+1)\). Determinantal schemes are equidimensional with a Cohen-Macaulay coordinate ring. It is easy to see that a closed subscheme \(Y\subset \mathbb{P}_ k^ N\) of equicodimension two with a Cohen-Macaulay coordinate ring is necessarily determinantal. The author determines first the codimension of the singular locus of \(Y\) (see propositions 1 and 2 in section 2.1) under certain conditions on the entries of \(M\). Then she proves the existence of a deformation \(f:X\to Z\) of \(Y\) (determinantal and of codimension 2) over an irreducible \(k\)-scheme \(Z\) (i.e. \(Y=f^{- 1}(z_ 0)\) over a closed point \(z_ 0\)); and \(Y\) is then smoothable iff \(2\leq N\leq 5\) (see theorem in section 2.2). This result generalizes a corresponding criterion of \textit{T. Sauer} [Math. Ann. 272, 83-90 (1985; Zbl 0546.14023)] for curves in \(\mathbb{P}_ k^ 3\). determinantal schemes; smoothability Frauke, S.:Generic determinantal schemes and the smoothability of determinantal schemes of codumension 2, Manuscripta Math.82 (1994) 417--431. Determinantal varieties, Linkage, complete intersections and determinantal ideals, Deformations and infinitesimal methods in commutative ring theory Generic determinantal schemes and the smoothability of determinantal schemes of codimension 2 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author defines the \(\omega\)-multiplier ideals on a normal variety and use it to study rational singularities. One main result is a characterization of rational singularity via cores of ideals. The author also investigate this \(\omega\)-multiplier ideals in details for two-dimensional singularities: such as some formulas relating the \(\omega\)-multiplier ideal, the core of ideal, and the multiplicity, and a subadditivity formula. It is also proved that in a two-dimensional normal local ring, every integrally closed ideal is an \(\omega\)-multiplier ideal. rational singularities; multiplier ideals; cores of ideals; integral closure Singularities in algebraic geometry, Integral closure of commutative rings and ideals Rational singularities, \(\omega\)-multiplier ideals, and cores of ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the review in Zbl 0602.14004. weighted graphs of exceptional divisors for desingularizations; log- terminal singularities Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Log-terminal singularities of algebraic surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author classifies the weighted graphs of exceptional divisors for desingularizations of algebraic surfaces with log-terminal singularities (in the sense of V. V. Shokurov and Y. Kawamata). weighted graphs of exceptional divisors for desingularizations; log- terminal singularities A. I. Iliev, ''Log-terminal singularities of algebraic surfaces,'' Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 3, 38--44 (1986) [in Russian]. Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Log-terminal singularities of algebraic surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic \(T\)-spectra, especially the motivic cobordism spectrum. When the base field \(k\) admits resolution of singularities and \(X\) is a scheme of finite type over \(k\), we show that Voevodsky's slice filtration leads to a spectral sequence for \(\mathrm{MGL}_X\) whose terms are the motivic cohomology groups of \(X\) defined using the cdh-hypercohomology. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of \(X\).
A similar spectral sequence for the connective \(K\)-theory leads to a cycle class map from the motivic cohomology to the homotopy invariant \(K\)-theory of \(X\). We show that this cycle class map is injective for a large class of projective schemes. We also deduce applications to the torsion in the motivic cohomology of singular schemes. algebraic cobordism; Milnor \(K\)-theory; motivic homotopy theory; motivic spectral sequence; \(K\)-theory; slice filtration; singular schemes Motivic cohomology; motivic homotopy theory, Algebraic cycles, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) The slice spectral sequence for singular schemes 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 Denote by \(E\) an Enriques surface, and by \(E^{[n]}\) for \(n\geq 1\), the Hilbert scheme of \(n\) points on \(E\).
Contrary to the case \(n=1\), the first result is to show that the small deformations of \(E^{[n]}\) for \(n\geq 2\) are induced from those of its universal covering space, by computing the dimensions of these deformation spaces.
In the second result, an equivalent condition for an automorphism \(f\) of \(E^{[n]}\) to be natural is given, namely, there exists an automorphism \(g\) of \(E\) such that \(f\) is naturally induced by \(g\). This is analogous to the case of the Hilbert scheme of \(n\) points on \(K3\) surfaces, that is, the automorphism \(f\) is natural if and only if it preserves the exceptional divisor of the Hilbert-Chow morphism of \(E^{[n]}\).
Contrary to the case of \(n=1\) (where the universal covering space being a \(K3\) surface), it is proved that there exists exactly one Enriques surface type quotient for the universal covering space \(X\) of \(E^{[n]}\). The proof is done by a classification of all involutions of \(X\) acting identically on \(H^2(X,\,\mathbb{C})\). Calabi-Yau manifld; Enriques surfaces; Hilbert scheme Calabi-Yau manifolds (algebro-geometric aspects), \(K3\) surfaces and Enriques surfaces, Parametrization (Chow and Hilbert schemes) Universal covering Calabi-Yau manifolds of the Hilbert schemes of \(n\) points of Enriques surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper contains a complete list of all finite Auslander-Reiten quivers of local Gorenstein orders over a complete Dedekind domain of finite lattice type. For each translation quiver \(\Gamma\) in this list, a Gorenstein order \(\Lambda\) is explicitly indicated, with \(\Gamma\) as its Auslander-Reiten quiver. In each case, indecomposable \(\Lambda\)-lattices are described. finite Auslander-Reiten quivers; local Gorenstein orders; finite lattice type; indecomposable \(\Lambda \) -lattices Wiedemann, A.: Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve singularities. J. pure appl. Algebra 41, No. 2-3, 305-329 (1986) Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Representation theory of associative rings and algebras, Singularities in algebraic geometry Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give explicitly in the closed formulae the genus zero primary potentials of the three \(6\)-dimensional FJRW theories of the simple-elliptic singularity \(\tilde{E}_7\) with the non-maximal symmetry groups. For each of these FJRW theories we establish the CY/LG correspondence to the Gromov-Witten theory of the elliptic orbifold \([\mathcal{E} / (\mathbb{Z}/2\mathbb{Z})]\) -- the orbifold quotient of the elliptic curve by the hyperelliptic involution. Namely, we give explicitly the Givental's group elements, whose actions on the partition function of the Gromov-Witten theory of \([\mathcal{E} / (\mathbb{Z}/2\mathbb{Z})]\) give up to a linear change of the variables the partition functions of the FJRW theories mentioned. We keep track of the linear changes of the variables needed. We show that using only the axioms of Fan-Jarvis-Ruan, the genus zero potential can only be reconstructed up to a scaling. FJRW theories; Landau-Ginzburg models; mirror symmetry Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Mirror symmetry (algebro-geometric aspects) \(6\)-dimensional FJRW theories of the simple-elliptic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article contains the details of the following two announcements: C. R. Acad. Sci., Soc. R. Can. 3, 273-278 (1981; Zbl 0495.14015) by these authors and Can. J. Math. 34, 169-180 (1982; Zbl 0477.14019) by the second author. The authors construct the coordinate ring A of a curve with one singular point P via Cartesian squares \(A\to^{f}\bar A\to^{\pi}\prod^{s}_{i=1}k[t_ i]/t^ n_ i;\quad A\to D\to^{g}\prod^{s}_{i=1}k[t_ i]| t^ n_ i\) (f and g are inclusions and \(\pi\) is onto), where \(\bar A\) is the normalization of A and k is a field. They compute several invariants of the local ring \(A_ P\), which only depend on g. In particular, there is shown an algorithm for computing the Hilbert function H of P. Thus they decided in several cases when H can have a temporary decrease. In particular, there is a generic homogeneous ordinary singular point such that H can decrease and \(H(1)=4\). Furthermore, the authors discuss the Cohen-Macaulay type of \(A_ P\) and its relationship to the Cohen-Macaulay type of its reduced tangent cone. In the case of generic homogeneous ordinary singularities both are the same, but in other cases, they may differ. This answers some questions posed by \textit{A. V. Geramita} and \textit{F. Orecchia} in J. Algebr 78, 36-57 (1982; Zbl 0502.14001) and by \textit{A. V. Geramita} and \textit{P. Maroscia} in C. R. Math. Acad. Sci., Soc. R. Can. 4, 179-184 (1982; Zbl 0493.14001). Furthermore the authors show that the K-theory of A, \(K_ i(A)\), \(i\leq 1\), depends on the number of irreducible components of Spec A (and on the inclusion). ordinary singularities of curves; coordinate ring of a curve; Hilbert function; Cohen-Macaulay type; K-theory Gupta, S. K.; Roberts, L. G., Cartesian squares and ordinary singularities of curves, \textit{Commun. Algebra}, 11, 2, 127-182, (1983) Singularities of curves, local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Grothendieck groups, \(K\)-theory and commutative rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry Cartesian squares and ordinary singularities of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Rees algebras seem to be a useful tool in the context of resolution of singularities. The notions of Rees algebras over smooth schemes, and that of Rees algebras closed by higher order differentials, already appear in the literature [\textit{H. Hironaka}, J. Korean Math. Soc. 40, No. 5, 901--920 (2003; Zbl 1055.14013); in: Société Mathématique de France. Séminaires et Congrés 10, 87--126 (2005; Zbl 1093.14021)], and more recently by \textit{H. Kawanoue} [Publ. Res. Inst. Math. Sci. 43, No. 3, 819--909 (2007; Zbl 1170.14012)] and the previous work by \textit{O. Villamayor} [Rev. Mat. Iberoam., 24, No. 1, 213--242 (2008; Zbl 1151.14013); Rev. Unión Mat. Argent. 46, No. 2, 1--18 (2005; Zbl 1112.14015)].
In this paper, the authors use Rees algebras to show that algorithms of desingularization define the same Log-resolution for two ideals having the same integral closure. To prove this fact they use the form of induction introduced by \textit{J. Włodarczyk} [J. Am. Math. Soc. 18, No. 4, 779--822 (2005; Zbl 1084.14018)].
They have done a remarkable work to extend the notion of log-resolution of ideals over a smooth scheme \(V\) to that of Rees algebras over \(V\), and to prove that two Rees algebras with the same integral closure undergo the same constructive resolution. The key point is the good behavior of Rees algebras under integral closure and differential operators.
In the first sections the authors carry out the extension of Log-resolution theorems of ideals over fields of characteristic zero to the case of Rees algebras.
They introduce the notion of Rees algebra and start the extension of all Hironaka style definitions on idealistic pairs and smooth schemes (such as permissible transformations, singular locus, etc) to the context of Rees algebras. They define the notion of \textit{integrally equivalent} Rees algebras (those that have the same integral closure), and prove that it is equivalent to the notion of \textit{equivalence} of idealistic pairs. They study the behavior of Rees algebras under differential operators and the notion of integral closure of these algebras.
In these first 4 sections they include all the necessary theoretical background to make the article self contained, and refer to \textit{O. Villamayor} [loc. cit.], for details.
The definition of the order of a Rees algebra at a point provides an upper semi-continuous function in the context of Rees algebras. This function leads to the construction of a resolution function and therefore to an algorithm of resolution of Rees algebras. If two Rees algebras are integrally equivalent then the algorithm defines the same resolution for both Rees algebras. resolution of singularities; Rees algebras; desingularization Encinas, S., Villamayor, O.: Rees algebras and resolution of singularities. Proceedings of the XVIth Latin American algebra colloquium (Spanish), (Colonia 2005). Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, pp. 63-85 (2007) Global theory and resolution of singularities (algebro-geometric aspects), Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Rees algebras and 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 Given a projective scheme \(X\) over an Artin commutative ring and a coherent sheaf \(F\) on that scheme, one defines the \(i\)th cohomological Hilbert function of an integer argument \(n\) measuring the length of the \(i\)th cohomology module of \(X\) with coefficients in the \(n\)th twist of \(F\). Because of the various vanishing theorems, like those of Grothendieck, Castelnuovo-Serre, and Severi-Enriques-Zariski-Serre, there are obvious constraints on the values of the cohomological Hilbert functions. There are also constraints on the rate of growth of those functions. The goal of the paper is to find further bounds on those functions, which would only depend on their diagonal values. The authors find that such bounds naturally break into two groups: bounds of Castelnuovo type and bounds of Severi type. Those bounds are given in terms of recursively defined bounding functions. The results are first established for finitely generated graded modules and then transferred to sheaves using the Serre - Grothendieck correspondence. cohomological Hilbert functions; Castelnuovo bounds; Severi bounds Brodmann, M.; Matteotti, C.; Minh, N. D.: Bounds for cohomological Hilbert-functions of projective schemes over Artinian rings. Vietnam J. Math. 28, No. 4, 341-380 (2000) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Local cohomology and algebraic geometry, Local cohomology and commutative rings, Commutative Artinian rings and modules, finite-dimensional algebras, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Bounds for cohomological Hilbert-functions of projective schemes over Artinian 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 Let \(S\) be an irreducible smooth projective surface defined over an algebraically closed field \(k\). For a positive integer \(d\), let \(\text{Hilb}^d(S)\) be the Hilbert scheme parametrizing the zero-dimensional subschemes of \(S\) of length \(d\). For a vector bundle \(E\) on \(S\), let \(\mathcal H(E) \rightarrow \mathrm{Hilb}^d(S)\) be its Fourier-Mukai transform constructed using the structure sheaf of the universal subscheme of \(S \times \mathrm{Hilb}^d (S)\) as the kernel. We prove that two vector bundles \(E\) and \(F\) on \(S\) are isomorphic if the vector bundles \(\mathcal H(E)\) and \(\mathcal H(F)\) are isomorphic. Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Parametrization (Chow and Hilbert schemes) Fourier-Mukai transform of vector bundles on surfaces to 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 A commutative ring \(A\) with identity is called decent if its total quotient ring is absolutely flat. For example, integral domains, reduced Noetherian rings, and reduced nontrivial positively graded rings with \(\text{Min} (A)\) compact are all decent rings. The author defines a reduced scheme \(X\) to be decent if there is an open affine base \((U_\lambda)\) of \(X\) such that every ring of sections \(\Gamma(U_\lambda,{\mathcal O}_X)\) is a decent ring. For a positively graded ring \(A\), the projective scheme \(X=\text{Proj}(A)\) is said to be canonically decent if \(\Gamma(D_+(f),{\mathcal O}_X)=A_{(f)}\) is decent for each homogeneous \(f\in A\) of positive degree. For such schemes, the author defines total quotient schemes and uses them to define the normalization, seminormalization, and \(t\)-closure of a decent scheme. The author also investigates seminormal and \(t\)-closed schemes and Rees rings. (Recall that \(A\) is seminormal if whenever \(a^3=b^2\) for \(a,b\in A\), there is a \(t\in A\) such that \(a=t^2\) and \(b=t^3\); and that \(A\) is \(t\)-closed if the annihilator of each element of \(A\) is generated by an idempotent and whenever \(a^3+rab-b^2=0\) for \(r,a,b\in A\), then \(a=t^2-rt\) and \(b=t^3-rt^2\) for some \(t\in A.)\) For example, if \(A\) is a decent ring and \(F=(I_k)\) is a regular filtration on \(A\), then the associated Rees ring \(R=\bigoplus_{k\in\mathbb{N}}I_kX^k\) is decent and \(\text{Proj}(R)\) is canonically decent. absolutely flat total quotient ring; seminormal ring; \(t\)-closed ring; decent ring; decent scheme; Rees ring 34.Picavet, G.: Seminormal or t-closed schemes and Rees rings. Algebra Represent. Theory 1, 255-309 (1998) Schemes and morphisms, Seminormal rings, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Rings of fractions and localization for commutative rings Seminormal or \(t\)-closed schemes and Rees 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 For any connected reductive group \(G\) over \(\mathbb{C}\), we revisit Goresky-Kottwitz-MacPherson's description of the torus equivariant Borel-Moore homology of affine Springer fibers \(\mathrm{Sp}_\gamma\subset \mathrm{Gr}_G\), where \(\gamma =zt^d\) and \(z\) is a regular semisimple element in the Lie algebra of \(G\). In the case \(G=\mathrm{GL}_n\), we relate the equivariant cohomology of \(\mathrm{Sp}_\gamma\) to Haiman's work on the isospectral Hilbert scheme of points on the plane. We also explain the connection to the HOMFLY homology of \((n,dn)\)-torus links, and formulate a conjecture describing the homology of the Hilbert scheme of points on the curve \(\{x^n=y^{dn}\}\). Homological methods in group theory, Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Lie algebras of Lie groups, Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) Unramified affine Springer fibers and isospectral Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A_n\) be the ring of generic \(n\times n\) matrices over an algebraically closed field \(K\), modulo the characteristic polynomial. It is shown that the codimension of the singular locus of \(A_n\) is 3. Similarly, the singular locus of the Galois splitting is considered. Artin's method is used to construct smooth splittings \(Y\to X\) of Azumaya algebras over a smooth quasi-projective surface \(X\) over \(K\), having a smooth irreducible Galois closure. smooth quasi-projective surfaces; Azumaya algebras; flat morphisms; Galois groups; singular loci Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Surfaces of general type, Brauer groups of schemes, Singularities of surfaces or higher-dimensional varieties Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over 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 Every determinantal ring is the dehomogenization of a Schubert cycle. In a previous paper [J. Pure Appl. Algebra 90, No. 2, 105-113 (1993; Zbl 0801.14003)], \textit{Z. Boudhraa} resolved the singularities of a Schubert cycle \(\sigma\) by blowing up the Schubert cycles contained in \(\sigma\). The analogue of this resolution theorem is proved here for determinantal rings. resolution of singularities; determinantal ring; singularities of a Schubert cycle Linkage, complete intersections and determinantal ideals, Global theory and resolution of singularities (algebro-geometric aspects), Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds Resolution of singularities of determinantal 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 The motivation of this research was the desire to understand the complexity of the moduli space of the surfaces of general type (minimal, complete and smooth over \(\mathbb{C})\) \(S\) such that their numerical invariants \(\chi ({\mathcal O}_ S)\), \(K^ 2_ S\) take some given integer values \(\chi ({\mathcal O}_ S) = x\), \(K^ 2_ S = y\). It is well known that these two invariants must take positive values and, conversely, that if these two numerical invariants are positive, then the surface (now not necessarily assumed to be minimal) is of general type except possibly if \(\chi ({\mathcal O}_ S) = 1\), \(K^ 2_ S \leq 9\).
The purpose of this paper is on the one hand to remark that one can give effective upper bounds for the number \(\iota(x,y)\) of irreducible components of \({\mathcal M}_{x,y}\), and on the other hand to give, more generally, effective estimates for Chow and Hilbert varieties; it seems, though, that some deeper techniques are needed in order to attack the problem of determining the precise asymptotic growth of \(\iota(x,y)\). Main results:
Theorem A. The number \(\iota(y)\) of irreducible components of the moduli space of surfaces of general type with \(K^ 2_ S = y\) satisfies for \(y \geq 3\): \(\iota(y) \leq 6^{(y+ 5/9)^{15}}\). If one restricts to regular surfaces \((q = 0)\), then one has for \(x,y \geq 3\) the better estimate \(\iota^ 0(y) \leq y^ 3 \cdot (440y)^{76y^ 2}\).
Theorem B. Let \({\mathcal H}^ 0\) (resp. \({\mathcal H}^*)\) be the open subset of the Hilbert scheme parametrizing smooth (resp. normal) irreducible subvarieties of dimension \(k\) and degree \(d\) in \(\mathbb{P}^ n\), then the natural morphism of \({\mathcal H}^ 0_{\text{red}}\) (resp. \({\mathcal H}^*_{\text{red}})\) to the Chow variety is an isomorphism (resp. a homeomorphism).
Theorem B is based on:
Theorem (1.14). Let \(V\) be an irreducible subvariety in \(\mathbb{P}^ n\), let \(F = F_ V\) be its Chow form, and let \(W = W_ F\) be the subscheme of \(\mathbb{P}^ n\) canonically associated to \(F\). Then \(V = W\) if \(V\) is a hypersurface, otherwise the equality \(W = V\) holds exactly at the smooth points of \(V\).
Several other results are given concerning the complexity of Chow varieties and Hilbert schemes. We should remark that, thanks to \textit{T. Ekedahl}'s extension [Publ. Math., Inst. Hautes Étud. Sci. 67, 97-144 (1988; Zbl 0674.14028)] of \textit{E. Bombieri}'s work [ibid. 42(1972), 171- 219 (1973; Zbl 0259.14005)] in positive characteristic, our results are also valid over an algebraically closed field of arbitrary characteristic. number of irreducible components of the moduli space; moduli space of the surfaces of general type; Hilbert varieties; Chow varieties F. Catanese, Chow varieties, Hilbert schemes, and moduli spaces of surfaces of general type,J. Algebraic Geom. 1 (1992), 561--595. Algebraic moduli problems, moduli of vector bundles, Surfaces of general type, Parametrization (Chow and Hilbert schemes) Chow varieties, Hilbert schemes, and moduli spaces of surfaces of general type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities ``Let \({\mathcal O} = {\mathcal O}_ n\) denote the local ring of germs of analytic functions \(f:(\mathbb{C}^ n,0) \to \mathbb{C}\) and \(m\) its maximal ideal. For an analytic germ \(f \in {\mathcal O}\) we denote by \(J_ f\) its Jacobi ideal, namely \(J_ f = ({\partial f \over \partial z_ 1}, \dots, {\partial f \over \partial z_ n})\). For an ideal \(I \subset {\mathcal O}\) we consider
-- the primitive ideal \(\int I\), defined by \(\int I = \{f \in {\mathcal O} |\;(f) + J_ f \subset I\}\); we have \(I^ 2 \subset \int I \subset I\);
-- the group \({\mathcal D}_ I\) of local analytic isomorphisms \(h: (\mathbb{C}^ n,0) \to (\mathbb{C}^ n,0)\) such that \(h^*(I) = I\); it is a subgroup of the group of all germs of local analytic isomorphisms of \((\mathbb{C},0)\).
\({\mathcal D}_ I\) acts on \(\int I\) and we shall consider the \({\mathcal R}_ I\) (right-equivalence) relation on \(\int I\).
We prove the following.
Theorem 1. Let \(I \subset {\mathcal O}\) be a radical ideal defining a germ of a quasihomogeneous complete intersection in \((\mathbb{C}^ n,0)\) with isolated singularity. Suppose that there exist \({\mathcal R}_ I\)-simple germs in \(\int I\). Then in some coordinates \((z_ 1, \dots, z_ n)\) of \((\mathbb{C}^ n,0)\) we have either
a) there exists \(k \in \{1, \dots, n\}\) such that \(I = (z_ 1, \dots, z_ k)\), or
b) there exists \(k \in \{1, \dots, n\}\) and a quasihomogeneous isolated singularity \(g = g(z_ 1, \dots, z_ k) \in {\mathcal O}_ k\) such that \(I = (g,z_{k + 1}, \dots, z_ n)\).
In the last section we derive the list of \({\mathcal R}_ I\)-simple germs for \(I = (z_ 1,z_ 2)\)''. germs of analytic functions; isolated singularity A. Zaharia, ''On simple germs with non-isolated singularities,'' Math. Scand., 68, 187--192 (1991). Local complex singularities, Complete intersections, Germs of analytic sets, local parametrization On simple germs with non-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 Here we show the existence of strong restrictions for the Hilbert function of zero-dimensional curvilinear subschemes of \(\mathbb P^n\) with one point as support and with high regularity index. Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the Hilbert function of curvilinear zero-dimensional subschemes of projective 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 normal integral surface over an algebraically closed field \(k\) with field of rational functions \(K\). \textit{F. Demeyer, T. Ford} and \textit{R. Miranda} [J. Algebra 162, No. 2, 287--294 (1993; Zbl 0809.13001)] have studied the kernel of the natural map \(B(X)\to B(K)\) between the Brauer groups. The problem to construct examples where this map is not injective goes back to \textit{A. Grothendieck} [Adv. Stud. Pure Math. 3, 46--66 (1968; Zbl 0193.21503)]. For an isolated singularity \(x\) of \(X\), there is an exact sequence
\[
0\to \mathrm{Cl}(\mathcal O_{X,x})\to \mathrm{Cl}(\mathcal O^h_{X,x})\to B'(\mathcal O_{X,x})\to B(K),
\]
where \(B'(\mathcal O_{X,x})\) denotes the étale Brauer group \(H^2(\mathcal O_{X,x},\mathbb G_m)\). If \(g\) denotes the genus of the exceptional divisor, the authors use this exact sequence to construct explicit examples where \(B(\mathcal O_{X,x})\) contains a subgroup isomorphic to the direct sum of \(2g\) copies of \(\mathbb Q/\mathbb Z\). Azumaya algebras; Brauer group; class group; non rational singularity; rational surface Brauer groups of schemes, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Singularities of surfaces or higher-dimensional varieties, Rational and ruled surfaces Generically trivial Azumaya algebras on a rational surface with a non rational 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 \textit{H. Tsuchihashi} [Tôhoku J. Math., II. Ser. 35, 607--639 (1983; Zbl 0585.14004)] introduced a class of cusp singularities extending the class of Hilbert modular variety cusps. Here, a necessary and sufficient condition for a cusp in this sense to be a Hilbert modular variety cusp is proved. Some cusps occur as finite quotients of Hilbert modular variety cusps: these are described, and so, more generally, are all the germs of automorphisms at a Hilbert modular variety cusp. cusp singularities; Hilbert modular variety cusp Sankaran, G.K.: Tsuchihashi's cusp singularities and automorphisms of Hilbert modular variety cusps. Math. Ann.274, 691-698 (1986) Singularities of surfaces or higher-dimensional varieties, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces Tsuchihashi's cusp singularities and automorphisms of Hilbert modular variety cusps | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the study of the rational cohomology of Hilbert schemes of points on a smooth surface, it is particularly interesting to understand the characteristic classes of the tautological bundles and the tangent bundle. In this note we pursue this study. We first collect all results appearing separately in the literature and prove some new formulas using Ohmoto's results on orbifold Chern classes on Hilbert schemes. We also explain the algorithmic counterpart of the topic: the cohomology space is governed by a vertex algebra that can be used to compute characteristic classes. We present an implementation of the vertex operators in the rewriting logic system \texttt{MAUDE}, and address observations and conjectures obtained after symbolic computations. S. Boissière and M. A. Nieper-Wisskirchen, Generating series in the cohomology of Hilbert schemes of points on surfaces , LMS J. Comput. Math. 10 (2007), 254-270. Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Generating series in the cohomology of Hilbert schemes of points on surfaces | 0 |
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