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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 Let \(H_n=\text{Hilb}^n(\mathbb{C}^2)\) be the Hilbert scheme which parametrizes the subschemes \(S\) of length \(n\) of \(\mathbb{C}^2\). To each such subscheme \(S\) corresponds a unordered \(n\)-tuple with possible repetitions \(\sigma(S)=[[P_1,...,P_n]]\) of points of \(\mathbb{C}^2\). There exists an algebraic variety \(X_n\) (called the isospectral Hilbert scheme) which is finite over \(H_n\) and which consists of all ordered \(n\)-tuples \((P_1,...,P_n)\in(\mathbb{C}^2)^n\) whose underlying unordered \(n\)-tuple is \(\sigma(S)\). The main aim of the paper under review is to study the geometry of \(X_n\), which is more complicated than the geometry of \(H_n\). For instance, a classical result of J. Fogarty asserts that \(H_n\) is irreducible and non-singular. The main result of the paper under review asserts that \(X_n\) is normal and Gorenstein (in particular, Cohen-Macaulay). Earlier work of the author indicated that there is a far-reaching correspondence between the geometry and sheaf cohomology of \(H_n\) and \(X_n\) on the one hand, and the theory of Macdonald polynomials on the other hand. The link between Macdonald polynomials and Hilbert schemes comes from some recent work [see \textit{A. M. Garsia} and \textit{M. Haiman}, Proc. Nat. Acad. Sci. USA 90, No. 8, 3607-3610 (1993; Zbl 0831.05062)].
The main result proved in this paper is expected to be an important step toward the proof of the so-called \(n!\)-conjecture and Macdonald positivity conjecture. The main theorem is based on a technical result (theorem 4.1) which asserts that the coordinate ring of a certain type of subspace arrangement is a free module over the polynomial ring generated by some of the coordinates. Macdonald polynomials; isospectral Hilbert schemes; \(n!\)-conjecture; Macdonald positivity conjecture; subspace arrangement K.B. Alkalaev and V.A. Belavin, \textit{Conformal blocks of}\( {\mathcal{W}}_n \)\textit{Minimal Models and AGT correspondence}, arXiv:1404.7094 [INSPIRE]. Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Hilbert schemes, polygraphs and the Macdonald positivity conjecture | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathbb X\subset \mathbb P^2_k\) be a general set of \(s\) distinct points, with \(k\) an algebraically closed field of characteristic \(0\). Let \(R=k[x_0,x_1,x_2]\) be the graded ring associated to \(\mathbb P^2\) and \(I_X=p_1\cap \dots \cap p_s\subset R\) be the defining ideal of \(s\). Let \(2\mathbb X\) be the scheme of \(\mathbb P^2\) with defining ideal \({p_1^2}\cap \dots\cap{p_s^2}\).
\textit{A. V. Geramita}, \textit{J. Migliore} and \textit{L. Sabourin} [J. Algebra 298, No. 2, 563--611 (2006; Zbl 1107.14048)] asked whether it is possible to find the Hilbert functions of all the non-reduced subschemes defined by ideals of the type \({p_1^2}\cap \dots\cap{p_s^2}\), whose underlying reduced subscheme was any set of \(s\) distinct points.
This difficult problem has been taken up in several papers see [\textit{A. V. Geramita}, \textit{B. Harbourne} and \textit{J. Migliore}, Collect. Math. 60, No. 2, 159--192 (2009); erratum ibid. 62, No. 1, 119--120 (2011; Zbl 1186.14008)] and [\textit{A. V. Geramita}, \textit{B. Harbourne} and \textit{J. Migliore}, ``Hilbert functions of fat point subschemes of the plane: the two-fold way'', \url{arxiv:1101.5140}] obtaining satisfying results in the case \(s\leq 9\) (in this case, the exponents of the ideals \(p_i\) can be any positive integer).
In the general case, the Hilbert function \(H_{2\mathbb X}\) has the upper bound \(H_{2\mathbb X}(t)\leq \min\{\binom{t+2}{2},3s\}\) and this bound is achieved for any \(s\neq 2,5\) and for \(X\) set of general \(s\) points having the generic Hilbert function \(H_{\mathbb X}(t)=\min\{\binom{t+2}{2},s\}.\)
Geramita, Migliore and Sabourin [Zbl 1107.14048] ask whether, in the case that the Hilbert function of \(X\) is the generic Hilbert function, there is a lower bound and wether such a bound is achieved for some \(2\mathbb X\). In their paper they give an affirmative answer in the case \(s=\binom{d}{2}\). This implies that the first open case is \(s=11\).
In this paper the authors give an affirmative answer in the case \(s=11\) and give a supportive evidence that the problem might have a positive answer for any \(s\), by proving that for any \(s\) the assertion holds for the ``first half'' of the Hilbert function. Hilbert function; fat points; infinitesimal neighborhood Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Configurations and arrangements of linear subspaces Hilbert functions of double point schemes in \(\mathbb P^{2}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a nonnegative integer \(n\) and a smooth projective surface \(S\), let \(S^{[n]}\) be the Hilbert scheme of zero-dimensional subschemes of length \(n\). It is well known that \(S^{[n]}\) is smooth projective of dimension \(2n\) and is irreducible whenever \(S\) is irreducible. The authors consider the complex cobordism ring with rational coefficients \(\Omega= \Omega^{\mathcal U}\otimes \mathbb{Q}\). For a smooth projective surface \(S\) they define an invertible power series: \(H(S)= \sum^\infty_{n=0} [S^{[n]}]z^n\) in the ring \(\Omega [[z]]\). The main result of the paper is the following:
Theorem. \(H(S)\) depends only on the cobordism class \([S]\in \Omega_2\).
Since \(\Omega\) is the polynomial ring on the classes \(CP_i\), \(i\in \mathbb{N}\), the authors are able to compute the values of genera \({\mathcal X}_y\) and \(\phi_{N,k}\) on \(H(S)\). The paper contains also some results on integrals over polynomials in Chern classes of certain tautological sheaves on \(S^{[n]}\). projective surface; Hilbert scheme; complex cobordism ring; Chern class Ellingsrud, G.; Göttsche, L.; Lehn, M., On the cobordism class of the Hilbert scheme of a surface, J. Algebr. Geom., 10, 81-100, (2001) (Equivariant) Chow groups and rings; motives, Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism), Bordism and cobordism theories and formal group laws in algebraic topology, Characteristic classes and numbers in differential topology On the cobordism class of the Hilbert scheme of a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(H_{d,g}\) \((H^0_{d,g}\) resp.) be the Hilbert scheme of locally Cohen-Macaulay (resp. smooth) curves \(C\) of degree \(d\) and genus \(g\), contained in the projective space \(\mathbb{P}^3_k\), where \(k\) is an algebraically closed field of characteristic zero. It is a classical result that if \(C\) is not contained in a plane then \(g\leq(d-2) (d-3)/2\). In the present paper the authors show that if \(d\geq 6\) and \(g\leq(d-3) (d-4)/2+1\), then \(H_{d,g}\) is not irreducible and not reduced. There exists an irreducible component which is not generically reduced and such that the underlying reduced scheme is smooth, of dimension \(3d(d-3)/2 +9-2g\). The points of this component correspond to curves \(C\) such that the values of the function \(n\to h^1(\mathbb{P}^3_k, {\mathcal I}_C(n))\) are maximal. The proof relies on a method developed by \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184-185 (1990; Zbl 0717.14017) and J. Reine Angew. Math. 439, 103-145 (1993; Zbl 0765.14017)]. The authors examine also the cases \(d<6\) and \(g>(d-3) (d-4)/2+1\). It is not known whether \(H_{d,g}\) is connected.
As a consequence, the authors exhibit examples of Hilbert schemes \(H^0_{d,g}\) which are not irreducible and not reduced. locally Cohen-Macaulay space curves; Rao module; Koszul module; Hilbert scheme Martin-Deschamps, M.; Perrin, D., Le schéma de Hilbert des courbes gauches localement Cohen-Macaulay n'est (presque) jamais réduit, Ann. Sci. École Norm. Sup. (4), 29, 757-785, (1996) Parametrization (Chow and Hilbert schemes), Plane and space curves The Hilbert scheme of locally Cohen-Macaulay space curves is (almost) never reduced | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a smooth irreducible complex projective surface \(X\), the Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) can be seen as a smooth resolution of the \(n\)-th symmetric product of \(X\). Many topological properties of \(X^{[n]}\) are known. The Betti numbers have been calculated by \textit{L. Göttsche} [Math. Ann. 286, No. 1--3, 193--207 (1990; Zbl 0679.14007)] and only depend on the Betti numbers of \(X\). This result has been clarified by \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] by constructing a representation of a Heisenberg algebra built from the rational cohomology of the surface on the direct sum \(\mathbb H := \bigoplus \mathrm H^*(X^{[n]}, \mathbb Q)\).
In the paper at hand, the author extends these results to Voisin's Hilbert schemes [\textit{C. Voisin}, Ann. Inst. Fourier 50, No. 2, 689--722 (2000; Zbl 0954.14002)] associated to compact almost-complex four-manifolds. He is able to prove both Göttsche's formula and the defining commutation relations of Nakajima's operators in this context. One main ingredient of the proof is Le Poitier's decomposition theorem for semi-small maps (following the decomposition theorem by \textit{A. A. Beilinson, J. Bernstein} and \textit{P. Deligne} [Faisceaux pervers. Astérisque 100, 172 p. (1982; Zbl 0536.14011)] without using any characteristic \(p\)-methods or étale cohomology), which is included together with a proof as it is otherwise unpublished.
Finally, tautological bundles are defined in this almost-complex setting. Hilbert scheme; Voison's Hilbert scheme; almost-complex four-manifolds; Göttsche formula; Nakajima operators J Grivaux, Topological properties of punctual Hilbert schemes of almost-complex fourfolds, to appear in Manuscripta Math. Almost complex manifolds, Parametrization (Chow and Hilbert schemes), \(4\)-folds Topological properties of Hilbert schemes of almost-complex fourfolds. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a projective surface over an algebraically closed field and \(p\in S\) a non-singular point. Denote by \(\text{H}(n)\) the reduced punctual Hilbert scheme parameterizing length \(n\) zero-dimensional subschemes of \(S\) supported at \(p\). By a result of \textit{J. Briançon} [Invent. Math. 41, 45--89 (1977; Zbl 0353.14004)], \(\text{H}(n)\) is irreducible of dimension \(n-1\). In fact, \(\text{H}(1)\) is a point, \(\text{H}(2)\cong {\mathbb P}_1\) but \(\text{H}(n)\) is singular for \(n\geq 3\) and contains as a dense open smooth subset the curvilinear schemes.
This paper is concerned with the description of the birational model for \(\text{H}(n)\) introduced by \textit{A. S. Tikhomirov} [Proc. Steklov Inst. Math. 208, 280--295 (1995; Zbl 0884.14001)]: the reduced moduli space \(\text{HF}(n)\) of length \(n\) complete flags \(\xi_1\subset \cdots\subset\xi_n\) of subschemes of \(S\) supported at \(p\) projects onto \(\text{H}(n)\), so that the closure \(\text{HF}'(n)\) in \(\text{HF}(n)\) of the inverse image of the curvilinear points is the unique component of \(\text{HF}(n)\) mapping birationally onto \(\text{H}(n)\).
In order to get a better understanding of the variety \(\text{HF}'(n)\), the author introduces the reduced moduli space \(\text{HMF}(n)\) of multiplicative complete flags \(\xi_1\subset\cdots\subset\xi_n\), that is with the property that \(I_i I_j\subset I_{i+j}\) where \(I_i\) denotes the ideal sheaf of \(\xi_i\) (\S 4) and identifies the model \(\text{HF}'(n)\) as a component of \(\text{HMF}(n)\). He shows for \(n\leq 7\) that \(\text{HMF}(n)\) is irreducible, so that \(\text{HF}'(n)=\text{HMF}(n)\) (Question 5.5). He proves further for \(n\leq 4\) that \(\text{HMF}(n)\) is smooth (Theorem 6.1), so that \(\text{HF}'(n)\) is a resolution of singularities of \(\text{H}(n)\), but that \(\text{HMF}(5)\) is singular along a curve (Theorem 6.2).
In the first sections, the author gives a detailed construction of the moduli space of (multiplicative) complete flags. punctual Hilbert scheme; complete flags; curvilinear schemes Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects) A partial resolution of the punctual Hilbert scheme of a nonsingular surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a smooth curve on a smooth surface, the Hilbert scheme of points on the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural log-resolution, namely the stratified blowup. As a consequence, the induced Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed deformations. complex surface; Hilbert scheme; stratification; normal crossings; Poisson structure Ran, Z, Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures, Int. J. Math., 27, 1650006, (2016) Parametrization (Chow and Hilbert schemes), Surfaces and higher-dimensional varieties, Compact complex surfaces, Poisson manifolds; Poisson groupoids and algebroids Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson 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 A local ring is called a local ring of simple singularity of dimension one over the real field if it is isomorphic to a ring of the form \(\mathbb R\{x,y\}/(f)\) and the number of proper ideals \(I\) of \(\mathbb R\{x,y\}\) with \(f\in I^2\) is finite. We first give a complete classification of local rings of simple singularity of dimension one over the real field. We also show that the rings have an infinitely generated prime cone unless they are isomorphic to either \(\mathbb R \{ x,y\}/(x^2\pm y^{2n})\) or \(\mathbb R\{x,y\}/(x^2y\pm y^{2n+3})\), where \(n\) is a positive integer. quadratic modules; prime cones; simple singularities Real algebra, Power series rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities in algebraic geometry, Real algebraic and real-analytic geometry On finiteness of prime cones over simple ADE-singularities of dimension one | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We show that a smooth projective curve of genus \(g\) can be reconstructed from its polarized Jacobian \((X, \Theta)\) as a certain locus in the Hilbert scheme \(\mathrm{Hilb}^{d}(X)\), for \(d = 3\) and for \(d = g+2\), defined by geometric conditions in terms of the polarization \(\Theta.\) The result is an application of the Gunning-Welters trisecant criterion and the Castelnuovo-Schottky theorem by Pareschi-Popa and Grushevsky, and its scheme theoretic extension by the authors. Schottky problem; Hilbert scheme; Jacobian; theta duality; trisecant criterion Theta functions and curves; Schottky problem, Parametrization (Chow and Hilbert schemes) Schottky via the punctual Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author shows that for any constants \(a<1/4\), and \(b,c\), there are at least \(y^{by^a+c}\) many irreducible components of the moduli space of regular surfaces of general type with \(K^2= c^2_1=y\) and fixed \(c_2\), where \(c_1\) and \(c_2\) are the Chern classes of the tangent bundle of the surface, and \(K\) is the canonical class. The same result holds true for the Hilbert scheme of surfaces in \(\mathbb{P}^4\) with \(K^2=y\) and fixed Hilbert polynomials. Similar results are given for threefolds. The idea of the proof is to look at projectively normal subvarieties of codimension two in the projective space with some very special resolution of the ideal sheaf. components of the moduli space; regular surfaces of general type; Chern classes; Hilbert scheme; surfaces; threefolds; codimension two M.-C. Chang, The number of components of Hilbert schemes. \textit{Internat. J. Math}. 7 (1996), 301-306. MR1395932 Zbl 0892.14006 Parametrization (Chow and Hilbert schemes), Low codimension problems in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The number of components 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 the interesting and well written paper under review, the author studies degeneracy loci of morphisms of the form \(\phi: \mathcal{O}_{\mathbb{P}^{n-1}}^m \to \Omega_{\mathbb{P}^{n-1}}(2)\) in the case \(2 < m < n-1\). The interest in these degeneracy loci goes back a long time ago. Indeed, many classical varieties arise as degeneracy locus of such morphisms, for instance the projected Veronese surface in \(\mathbb{P}^4\), the elliptic scroll surface of degree six, the Palatini scroll and the Segre cubic primal.
Consider an algebraically closed field \(\mathbf{k}\) of characteristic 0, and let \(U\) and \(V\) be two \(\mathbf{k}\)-vector spaces of dimension \(m\) and \(n\) with bases \(\{y_0,\ldots,y_{m-1}\}\) and \(\{x_0,\ldots,x_{n-1}\}\). Every morphism \(\phi\) can be naturally associated to a \(n \times n\) skew-symmetric matrix \(N\) whose entries \(N_{ij} = \sum_{k=0}^{m-1} a_{i,j}^k y_k\) (\(a_{i,j}^k = - a_{j,i}^k\) for all \(i,j,k\)) are linear forms in the coordinates of the projective space \(\mathbb{P}^{m-1} = \mathbb{P}(U)\) or, equivalently, to \(m\) elements of the vector space \(\bigwedge^2 V\). The linear group \(\text{GL}(U)\) acts projectively on the \(m\) elements not affecting the subspace they span, so the \(\text{GL}(U)\)-orbit of a general morphism \(\phi\) corresponds to an element of the Grassmannian \(\mathbf{Gr}(m,\bigwedge^2 V)\).
Moreover, the degeneracy locus of a general morphism \(\phi\) corresponds to the subscheme \(X_{\phi} \subset \mathbb{P}^{n-1} = \mathbb{P}(V)\) cut out by the maximal minors of the \(n\times m\) matrix
\[
M = \left(\begin{alignedat}{2} \sum_{i=0}^{n-1} a_{i,0}^0 x_i & \dots & \sum_{i=0}^{n-1} a_{i,0}^{m-1} x_i\\ \vdots && \vdots \\ \sum_{i=0}^{n-1} a_{i,n-1}^0 x_i & \dots & \sum_{i=0}^{n-1} a_{i,n-1}^{m-1} x_i \end{alignedat}\right).
\]
As the Hilbert polynomial of \(X_{\phi}\) is generically fixed, and the action of \(\text{GL}(U)\) induced on \(M\) does not change the ideal generated by the maximal minors defining \(X_{\phi}\), we have rational maps
\[
\text{Hom}\big(\mathcal{O}_{\mathbb{P}(V)}^m,\Omega_{\mathbb{P}(V)}(2)\big) \dashrightarrow \mathbf{Gr}(m,\bigwedge^2 V) \overset{\rho}{\dashrightarrow} \mathcal{H},
\]
where \(\mathcal{H}\) is the union of the components of the Hilbert scheme containing the degeneracy locus of a general morphism \(\phi\).
In the paper under review, the author gives a complete description of the behavior of the map \(\rho\) in the case \(2 < m < n-1\). Precisely, {\parindent=0.6cm\begin{itemize}\item[--] for \(m \geq 4\) or \((m,n)=(3,5)\), the map \(\rho\) is birational; \item[--] for \(m=3\) and \(n\neq 6\), the map \(\rho\) is generically injective and dominant on a closed subscheme \(\mathcal{H}'\subset \mathcal{H}\) of codimension \(\frac{1}{8}n(n-3)(n-5)\) if \(n\) odd or \(\frac{3}{8}(n-4)(n-6)\) if \(n\) even.
\end{itemize}} In the case \((m,n) = (3,6)\), the map \(\rho\) is not generically injective, as proved in [\textit{D. Bazan} and \textit{E. Mezzetti}, Geom. Dedicata 86, No. 1--3, 191--204 (2001; Zbl 1042.14022)]. The case \(m=2\) was treated by the same author in [Arch. Math. 105, No. 2, 109--118 (2015; Zbl 1349.14015)]. degeneracy locus; determinantal variety; Hilbert scheme; skew-symmetric matrix F. Tanturri, On the Hilbert scheme of degeneracy loci of twisted differential forms, to appear in Trans. Amer. Math. Soc. Parametrization (Chow and Hilbert schemes), Determinantal varieties, Rational and birational maps On the Hilbert scheme of degeneracy loci of twisted differential forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities ``This is the first in a series of papers which describe the action of an affine Lie algebra with central charge \(n\) on the moduli space of \(U(n)\)-instantons on a four manifold \(X\). This generalizes work of \textit{H. Nakajima} [Duke Math. J. 76, 365-416 (1995; Zbl 0826.17026)], who considered the case when \(X\) is an ALE space.''
``In the particular case of \(U(1)\)-instantons, which is essentially the subject of this present paper, the construction produces the basic representation after Frenkel-Kac''. Hilbert scheme; vertex operators; affine Lie algebras; moduli space I. Grojnowski, \textit{Instantons and affine algebras I: the Hilbert scheme and vertex operators}, alg-geom/9506020 [INSPIRE]. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Parametrization (Chow and Hilbert schemes), Vertex operators; vertex operator algebras and related structures Instantons and affine algebras. I: The Hilbert scheme and vertex operators | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 diagonal in a product of projective spaces is cut out by the ideal of \(2 \times 2\)-minors of a matrix of unknowns. The multigraded Hilbert scheme which classifies its degenerations has a unique Borel-fixed ideal. This Hilbert scheme is generally reducible, and its main component is a compactification of PGL\((d)^n\)/PGL\((d)\). For \(n = 2\), we recover the manifold of complete collineations. For projective lines, we obtain a novel space of trees that is irreducible but singular. All ideals in our Hilbert scheme are radical. We also explore connections to affine buildings and Deligne schemes. Cartwright D., Sturmfels B.: The Hilbert scheme of the diagonal in a product of projective spaces. Int. Math. Res. Not. IMRN 9, 1741--1771 (2010) Parametrization (Chow and Hilbert schemes), Compactifications; symmetric and spherical varieties The Hilbert scheme of the diagonal in a product 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 In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of \(d\) points of a complex symplectic surface \(S\) projecting onto \(\mathbb{C}\) via a surjective map \(p\) which is a submersion outside a discrete subset of \(S\). We explicitly endow the transverse Hilbert scheme \(S_{p}^{[d]}\) with a symplectic form and an endomorphism \(A\) of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians. We then provide the inverse construction, starting from a \(2d\)-dimensional holomorphic integrable system W which has an endomorphism \(A: T\mathcal{W}\rightarrow T\mathcal{W}\) satisfying the above properties and recover our initial surface \(S\) with \(\mathcal{W}\cong S_{p}^{[d]}\). holomorphic completely integrable system; symplectic geometry; Hilbert scheme Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Symplectic manifolds (general theory), Parametrization (Chow and Hilbert schemes), Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Transverse Hilbert schemes and completely integrable systems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The relation between Hilbert schemes of finite-length subschemes and crepant resolution of singularities, via the McKay correspondence, is fairly well understood now after the work of Ito and many others in the case of surfaces in characteristic zero. For higher-dimensional varieties there is progress but still much more to be done. As far as the reviewer can tell, this is the first serious attempt to understand the situation in positive characteristic. Many of the constructions are valid for subschemes of arbitrary length on arbitrary smooth varieties, but the detailed results are limited to \({\text{Hilb}}^{2}(S)\) for \(S\) a smooth surface, which is the most accessible case but quite possibly also the most interesting one.
Another, more specific use of \({\text{Hilb}}^{n}(S)\) is in Beauville's generalised Kummer construction. In this case \(S=A\) is an abelian surface: the Hilbert-Chow morphism \({\text{Hilb}}^{n}(A)\to{\text{Sym}}^{n}(A)\), composed with addition in \(A\), gives a map \({\text{Hilb}}^{n}(A)\to A\) and the fibre over \(0\) is called the generalised Kummer variety \({\text{Km}}^{n}(A)\). In characteristic zero, both \({\text{Hilb}}^{n}(A)\) and \({\text{Km}}^{n}(A)\) are smooth (and have trivial dualising sheaf), but in positive characteristic this fails for \({\text{Km}}^{n}(A)\). This phenomenon, too, is explored here in characteristic~\(2\) for \(n=2\). It turns out that \({\text{Hilb}}^{2}(A)\to A\) is a quasifibration, i.e.\ all the fibres are nonsmooth. The third strand of this paper is to study the singularities of the fibres, especially \({\text{Km}}^{2}(A)\), in this case. They depend strongly on \(A\), via the singularities of the usual Kummer variety \(A/\pm 1\), which were studied by Shioda and by Katsura in the 1970s.
Thus the author is led to revisit a large part of classical surface singularity theory, but in characteristic~\(2\). A key role is played by Artin's wild involutions on surfaces. These, and the consequences of blowing up the associated surface singularities, are studied in Section~1 of the paper. The blow-up does not resolve the singularity (it is not even normal in general) but does behave cohomologically like a resolution, and this provides a counterpart in the cases under discussion to the McKay correspondence. This relation to the Hilbert scheme is discussed in Sections~2 and~3. A byproduct is examples of nonrational canonical singularities (another positive characteristic phenomenon).
Section~4 deals with the special case of rational double points (quotient singularities need not be rational in positive characteristic), listing the cases that may arise on \(G\)-Hilbert schemes.
The next part of the paper, Sections~5--7, is concerned with the \(S=A\) and \(G=\{\pm 1\}\), the Kummer surface case. The most difficult case by far is when \(A\) is supersingular: to understand it involves an excursion into Laufer's theory of minimal elliptic singularities. Within this, the superspecial case \(A=E\times E\), where \(E=(y^2=x^3+x)\), requires its own treatment.
Section~8 is a technical one about local algebraic conditions on symmetric products of surfaces. This, and the study of symmetric products of abelian surfaces that follows in Section~9, allows us to identify the Kummer variety \(A/\pm 1\) with the closed fibre of the addition map \({\text{Sym}}^{2}(A)\to A\). Now everything fits together and the relation between \({\text{Km}}^{2}(A)\) and \(A/\pm 1\) and its singularities is fully elucidated in the final Section~10. Hilbert-Chow morphism; wild involutions; McKay correspondence doi:10.1007/s11512-007-0065-6 Singularities of surfaces or higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), McKay correspondence The Hilbert scheme of points for supersingular abelian surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give an explicit description of the Auslander algebra of the category of maximal Cohen-Macaulay modules over the ring of a simple plane curve singularity and we also compute its Hochschild dimension. Our main result is that the Auslander algebra of the category of maximal Cohen-Macaulay modules over the ring of a simple plane curve singularity is smooth of dimension two. Bilodeau, Josée, Auslander algebras and simple plane curve singularities.Representations of algebras and related topics, Fields Inst. Commun. 45, 99-107, (2005), Amer. Math. Soc., Providence, RI Cohen-Macaulay modules, Singularities in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities of surfaces or higher-dimensional varieties Auslander algebras and simple plane 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 The author give an explicit geometric description to some of H. Nakajima's quiver varieties. More precisely, let \(X = {\mathbb{C}}^2\), \(\Gamma \subset \text{SL}({\mathbb{C}}^2)\) be a finite subgroup, and \(X_\Gamma\) be a minimal resolution of \(X/\Gamma\). The main result states that \(X^{\Gamma [n]}\) (the \(\Gamma\)-equivariant Hilbert scheme of \(X\)) and \(X_\Gamma^{[n]}\) (the Hilbert scheme of \(X_\Gamma\)) are quiver varieties for the affine Dynkin graph corresponding to \(\Gamma\) via the McKay correspondence with the same dimension vectors but different parameters.
In section two, basic concepts such as the definition of quivers, quiver varieties, representation of quivers and the construction of Crawley-Boevey were reviewed. In section three, the author reproduced in a short form a geometric version of the McKay correspondence based on investigation of \(X_\Gamma\), and proved a generalization of certain result of \textit{M. Kapranov} and \textit{E. Vasserot} [Math. Ann. 316, No. 3, 565--576 (2000; Zbl 0997.14001)]. The main result mentioned above was verified in section four. In particular, it follows that the varieties \(X^{\Gamma [n]}\) and \(X_\Gamma^{[n]}\) are diffeomorphic. In section five, \(({\mathbb{C}}^* \times {\mathbb{C}}^*)\)-actions on \(X^{\Gamma [n]}\) and \(X_\Gamma^{[n]}\) for cyclic \(\Gamma \cong {\mathbb{Z}}/d {\mathbb{Z}}\) were considered. The author proved the combinatorial identity \(UCY(n, d) = CY(n, d)\) where \(UCY\) and \(CY\) denote the number of uniformly colored diagrams and the number of collections of diagrams respectively. quiver varieties; Hilbert schemes; McKay correspondence; moduli space Kuznetsov, A.: Quiver varieties and Hilbert schemes. Moscow Math. J. \textbf{7}, 673-697 (2007). arXiv:math.AG/0111092 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Quiver varieties and Hilbert schemes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let A be a one-dimensional Cohen-Macaulay local ring, (b,e,\(\rho\)) a triplet of integers. Let \(\rho_{0,b,e}=(r+1)e-\left( \begin{matrix} r+b\\ r\end{matrix} \right)\), where r is the integer such that \(\left( \begin{matrix} b+r-1\\ r\end{matrix} \right)\leq e<\left( \begin{matrix} b+r\\ r+1\end{matrix} \right)\) and \(\rho_{1,b,e}=e(e-1)/2-(b-1)(b-2)/2\). The main result of the paper is that there exists a one-dimensional Cohen-Macaulay local ring A with embedding dimension b, multiplicity e and reduction number \(\rho\) iff \(b=e=1\) and \(\rho =0\) or \(2\leq b\leq e\) and \(\rho_{0,b,e}\leq \rho \leq \rho_{1,h,e}\). In this last case, one can choose A to be the quotient of a power series ring over an algebraically closed field of characteristic zero. Using this, the Hilbert-Samuel function of one- dimensional Cohen-Macaulay rings of small multiplicity is computed. Lastly, conditions for the Cohen-Macaulayness of gr(A) are established in terms of the reduction number. curve singularities; one-dimensional Cohen-Macaulay local ring; embedding dimension; multiplicity; reduction number; Hilbert-Samuel function J. Elias, Characterization of the Hilbert-Samuel polynomial of curve singularities, Compositio Math. 74 (1990), 135--155 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Singularities of curves, local rings, Cohen-Macaulay modules Characterization of the Hilbert-Samuel polynomials 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 We prove that the Hilbert schemes of 2-points on rational surfaces are numerically rationally connected. The main idea is to show that certain 3-point genus-0 Gromov-Witten invariant of the Hilbert scheme of two points on the complex projective plane is positive and can be calculated enumeratively. Rationally connected varieties, Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On the numerical rational connectedness of the Hilbert schemes of 2-points on rational 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 main purpose of the paper under review is to prove that the cotangent complex of the Hilbert scheme is precisely the Fourier-Mukai transform of the cotangent complex of the curve or the family. planar curves; Hilbert scheme; Fourier-Mukai transform Projective and enumerative algebraic geometry, Plane and space curves, Parametrization (Chow and Hilbert schemes) A note on cotangent sheaves of Hilbert schemes of families of planar curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a nonsingular projective surface, \(H_ d\) the Hilbert scheme of 0-dimensional subschemes of length \(d\) of \(S\), \(Z_ d \subset S \times H_ d\) the universal family, and \(\pi, \tau\) the projections of \(Z_ d\) on \(H_ d\) and \(S\), respectively. If \(D\) is a divisor on \(S\), \(\pi_ * \tau^* {\mathcal O}_ S(D)\) is a rank \(d\) vector bundle on \(H_ d\). The paper is concerned with the computation of the degree \(\delta_ d\) of the top Segre class \(s_{2d}\) of this vector bundle in terms of the following numerical invariants: \(x = (D^ 2)\), \(y = (D \cdot K_ S)\), \(z = (K^ 2_ S) - \chi_{\text{top}} (S)\), \(w = (K^ 2_ S)\). -- The interest of the author in this computation stems mainly from its applications to the description of the smooth structure of the 4-manifold underlying \(S\), emphasized by recent work of \textit{V. Ya. Pidstrigach} and \textit{A. N. Tyurin} [Russ. Acad. Sci., Izv., Math. 40, No. 2, 267-351 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 2, 279- 371 (1992; Zbl 0796.14024)].
The computation reduces to an enumerative problem: \(\delta_ d\) equals the number of \(d\)-secant \((d - 2)\)-planes to a convenient embedding of \(S\) into \(\mathbb{P}^{3d-2}\). The main result of the paper is that \(\delta_ d\) is a polynomial in \(x,y,z,w\) with rational coefficients. An explicit formula for \(\delta_ 2\) is essentially the classical formula of double points of surfaces in \(\mathbb{P}^ 4\). An explicit formula for \(\delta_ 3\) is also obtained. It is equivalent to the formula for the number of trisecant lines of a surface in \(\mathbb{P}^ 7\) due to \textit{P. Le Barz} [Enseign. Math., II. Sér. 33, 1-66 (1987; Zbl 0629.14037)]. standard bundles; number of \(d\)-secant \((d - 2)\)-planes; Hilbert scheme; Segre class; embedding; double points of surfaces; number of trisecant lines Tikhomirov, A; Tikhomirov, A (ed.); Tyurin, A (ed.), Standard bundles on a Hilbert scheme of points on a surface, 183-203, (1994), Wiesbaden Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Enumerative problems (combinatorial problems) in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Standard bundles on a Hilbert scheme of points on a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X \subset \mathbb{P}^N\) be a smooth complex quasiprojective variety. A line \(L \subset \mathbb{P}^N\) is said \(k\)-secant to \(X\) if the scheme \(L \cap X\) is finite of degree \(\geq k\). The locus of \(k\)-secant lines to \(X \subset \mathbb{P}^N\) is a classical object of study in projective geometry, naturally in relation with the study of the projections \(\pi:X \to X_1 \subset \mathbb{P}^{N-1}\) into \(\mathbb{P}^{N-1}\) from a general point of \(\mathbb{P}^N\). In particular, it is of interest to study the subscheme \(X_{\{k_1, \dots, k_r\}} \subset X_1\) formed by the points \(x \in X_1\) such that \(\pi^{-1}(x)\) contains \(r\) points \(\{x_1,\dots, x_r\}\) (possibly equal) with multiplicity greater than or equal to \(k_i\) in \(x_i\). In the paper under review it is proven (see \textit{General Projection Theorem}, Theorem 1.1) that this scheme is pure dimensional of dimension \(N-1-\sum (k_i(N-n)-1)\). Moreover its singular locus is \(X_{\{k_1, \dots, k_r,1\}}\) and its normalization is smooth.
In order to be precise about the scheme structure of \(X_{\{k_1, \dots, k_r\}}\), the General Projection Theorem is presented as a consequence of a result stated in the language of Hilbert schemes of aligned points, naturally equipped with a map to the Grassmannian of lines \(G(1,N)\). This is Theorem 1.3, named \textit{Aligned Hilbert Scheme Theorem}, where the schemes \(X_{\{k_1, \dots, k_r\}}\) can be related naturally with fibers of natural projections of incidence varieties constructed by means of the map from the Hilbert scheme to the Grassmannian (see Theorem 1.3 for details).
The Aligned Hilbert Scheme Theorem results to be a consequence of the \textit{Aligned ordered Hilbert Scheme Theorem} where Hilbert schemes are substituted by ordered Hilbert schemes, parameterizing finite aligned subschemes supported in an ordered set of pints \((x_1,\dots, x_r)\) and with multiplicity \(k_i\) at \(x_i\). Being this ordered Hilbert scheme finite and flat over the non-ordered one, its smoothness implies smoothness of the non-ordered one.
An interesting section (Section 5) of examples, questions and conjectures (specially about the irreducibility of the locus of points of order \(k\) of a general projection) is also provided. \(k\)-secants; general projections; Hilbert schemes Gruson, L.; Peskine, C., On the smooth locus of aligned Hilbert schemes. the \textit{k}-secant lemma and the general projection theorem, Duke Math. J., 162, 553-578, (2013) Low codimension problems in algebraic geometry, Projective techniques in algebraic geometry On the smooth locus of aligned Hilbert schemes, the \(k\)-secant lemma and the general projection theorem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 extends the results of \textit{E. Shustin} and \textit{E. Westenberger} [J. Lond. Math. Soc., II. Ser. 70, No. 3, 609--624 (2004; Zbl 1075.14034)] on the existence of algebraic hypersurfaces with prescribed simple singularities to the real case. The main result states that, given a collection of real simple singularities (with repetitions) of finitely many types satisfies the condition that the total Milnor number does not exceed \(d^n/n!+O(d^{n-1})\), there exists a real projective \((n-1)\)-dimensional hypersurface of degree \(d\), whose real singularity set coincides with the given collection, and, furthermore, the germ of the corresponding equisingular stratum in the space of hypersurfaces of degree \(d\) is smooth of expected dimension. The proof reduces to an application of the patchworking construction along the lines of the above cited article. simple singularities; real hypersurfaces; patchworking construction Singularities of surfaces or higher-dimensional varieties, Topology of real algebraic varieties, Hypersurfaces and algebraic geometry Real hypersurfaces with many simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main result of the paper is a sufficient existence condition for an algebraic hypersurface of degree \(d\) in the projective \(n\)-space, \(n\geq 3\), having prescribed simple singularities. The criterion is expressed as an upper bound for the total Milnor number of the given singularities by a polynomial in \(d\) variables of degree \(n\) with coefficients depending only on \(n\). The construction goes as follows. First, for any simple singularity, the author finds a hypersurface of a relatively small degree, having just one given singularity, then he constructs a hypersurface of degree \(d\) with some ordinary singularities in general position, which finally are deformed into prescribed simple singularities along a procedure suggested by the reviewer [\textit{E. Shustin}, Algebra Anal. 10, 221-249 (2000; Zbl 0967.14002)]. simple hypersurface singularities; cohomology vanishing Westenberger E.: Existence of hypersurfaces with prescribed simple singularities. Commun. Algebra 31(1), 335--356 (2003) Singularities of surfaces or higher-dimensional varieties, Hypersurfaces and algebraic geometry Existence of hypersurfaces with prescribed simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The starting point of this paper is \textit{R. Hartshorne}'s theory of reflexive sheaves and the Serre correspondence between rank 2 reflexive sheaves on \(\mathbb P^3\) and curves in \(\mathbb P^3\) [Math. Ann. 238, 229--280 (1978; Zbl 0411.14002); ibid. 254, 121--176 (1980; Zbl 0431.14004)]. In fact the author continues the study he began in his previous works [Pac. J. Math. 219, No. 2, 391--398 (2005; Zbl 1107.14032); J. Pure Appl. Algebra 211, no. 3, 622--632 (2007; Zbl 1128.14009); An effective bound for reflexive sheaves on canonically trivial 3-folds, to appear], investigating some cohomological properties of rank 2 reflexive sheaves on smooth projective threefolds (on which one can easily extend the Serre correspondence), with interesting consequences for the structure of the sheaves themselves. In particular, he finds conditions for these sheaves to be locally free and he studies a case in which the Riemann-Roch formula becomes quite simple. Finally the author applies these results to the relation between the moduli space of torsion free sheaves and the Hilbert scheme of elliptic curves on a particular class of threefolds (e.g. Fano threefolds). reflexive sheaf; Fano threefold; elliptic curve Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Fano varieties The Hilbert scheme of elliptic curves and reflexive sheaves on Fano 3-folds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Études Sci. 29, 5-48 (1966; Zbl 0171.41502)] proved that the Hilbert scheme \(\text{Hilb}^{p(z)}_{\mathbb{P}^n}\) parametrizing closed subschemes of \(\mathbb{P}^n\) with Hilbert polynomial \(p(z)\), is connected (\(\mathbb{P}^n=n\)-dimensional projective space over a fixed algebraically closed field \(k\) of characteristic zero). By using \textit{A. Galligo}'s theorem [Ann. Inst. Fourier 29, 107-184 (1979; Zbl 0412.32011)], in the present paper the author improves and reduces Hartshorne's proof. The main theorem of the paper claims that the radius of the incidence graph of irreducible components of the Hilbert scheme is at most \(d+1\), where \(d\) is the dimension of the subschemes parametrized. From this the author recovers another known result which states that the lexicographic ideal has the worst regularity of any ideal on the Hilbert scheme [see \textit{G. Gotzmann}, Math. Z. 158, 61-70 (1978; Zbl 0352.13009)] and \textit{Bayer} [``The division algorithm and the Hilbert scheme'' (Thesis), Harvard University, Order number 82-22588, Univ. Microfilms International, 300N (1982)].
The central notion in the proofs of previous theorems is that of Borel-fixed ideal, i.e. a monomial ideal fixed by the action of the group of upper triangular matrices on the polynomial ring \(k[x_0,\dots, x_n]\). To this purpose, the author presents a method of partitioning Borel fixed ideals into classes, each of which must lie in a single component of the Hilbert scheme. fan; incidence graph of irreducible components of the Hilbert scheme; Borel fixed ideal Reeves, AA, The radius of the Hilbert scheme, J. Algebraic Geom., 4, 639-657, (1995) Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Actions of groups on commutative rings; invariant theory The radius 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 In the paper under review, the author studies the rational cohomology rings of Voisin's Hilbert schemes \(X^{[n]}\) associated to a symplectic compact four-manifold \(X\). The author proves that these rings can be universally constructed from \(H^*(X, \mathbf{C})\) and \(c_1(X)\), that Ruan's cohomological crepant resolution conjecture holds for \(X^{[n]}\) if \(c_1(X)\) is a torsion class, and that the complex cobordism class of \(X^{[n]}\) depends only on the complex cobordism class of \(X\).
These results are straight-forward generalizations of known results when \(X\) is a smooth projective algebraic surface [\textit{G. Ellingsrud, L. Göttsche} and \textit{M. Lehn}, J. Algebr. Geom. 10, No. 1, 81--100 (2001; Zbl 0976.14002); \textit{W.-P. Li, Z. Qin} and \textit{W. Wang}, J. Reine Angew. Math. 554, 217--234 (2003; Zbl 1092.14007)]. Hilbert schemes; almost-complex four-manifolds; complex cobordism class Grivaux, J.: Topological properties of Hilbert schemes of almost-complex four-manifolds (II). Geom. Topol. (2011) arxiv:1001.0119 Parametrization (Chow and Hilbert schemes), \(4\)-folds Topological properties of Hilbert schemes of almost-complex four-manifolds. II. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this article we construct a bridge between two objects that are unrelated at first sight. One is the infinite-dimensional Heisenberg algebra which plays a fundamental role in the representation theory of affine Lie algebras. The other is the Hilbert schemes of points on a complex surface appearing in algebraic geometry.
This article is an abbreviated version of the author's lecture series [Lectures on Hilbert schemes of points on surfaces. University Lecture Series. 18. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0949.14001)]. Four items are added in the bibliography in the translation. Nakajima, Hiraku: Jack polynomials and Hilbert schemes of points on surfaces Parametrization (Chow and Hilbert schemes), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras The Heisenberg algebra and 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 This paper is one in a long list of papers by several specialists concerning the following question, called the Modified Severi Conjecture. Fix integers \(g\ge 0\), \(r\ge 3\), \(d\ge r\) such that \((g,r,d)\) is in the Brill-Noether range \(\rho(g,r,d)\ge 0\). Let \(H^L_{d,g,r}\) denote the Hilbert scheme of all smooth and linearly normal curves \(C\subset \mathbb{P}^r\) of degree \(d\) and genus \(g\). Is \(H^L_{d,g,r}\) non-empty? Is it irreducible? Has it the expected dimension? The restriction to components whose general member is linearly normal is essential. The paper gives an answer in several cases, e.g. to all 3 question under for some \(r, d, g\). For instance if \(d=g+r-2\) if \(g\le r+3\) all answers are Yes, while \(H^L_{g-r+2,g,r} =\emptyset\) if \(g\le r+2\). For \(d=g+r-3\) the paper proves several existence and irreducibility statements. The paper also lists the results previously known and so it may be used as an up to date survey. Hilbert scheme; algebraic curves; linearly normal; linear series Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) Existence and reducibility of the Hilbert scheme of smooth and linearly normal curves in \(\mathbb{P}^r\) of relatively high degrees | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Curves of degree one (lines) in a fixed projective space are parameterized by Grassmannians, which are well understood. In higher degree they are parameterized by Hilbert schemes. This paper primarily considers the Hilbert scheme of a rope in projective space, that is, a closed, locally Cohen-Macaulay projective subscheme of pure dimension one that is non-reduced, supported on a line, and is contained in the first infinitesimal neighborhood of the line.
Not every point of the Hilbert scheme of a rope actually corresponds to a rope, so in this paper the authors construct smooth parameter spaces for ropes and use them to study Hilbert schemes that contain ropes. They show that all ropes of fixed degree and genus lie in the same component of their Hilbert scheme. Furthermore, when the genus is small enough (compared to the degree) and the degree is at least 3, then the ropes actually form a component, which is generically smooth, and they give its dimension. This is independent of the characteristic of the ground field. The case of degree 2 is slightly different: when the genus is \(\leq -2\), it is still true that the ropes comprise their component of their Hilbert scheme, and the dimension is again computed. However, it is generically smooth if the characteristic is not 2 or if the genus is \(-2\); otherwise it is generically non-reduced. curves; ropes; Hilbert scheme; minimal free resolution; normal sheaf U. Nagel, R. Notari and M. L. Spreafico, The Hilbert scheme of degree two curves and certain ropes, Internat. J. Math. 17 (2006), no. 7, 835--867. Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Syzygies, resolutions, complexes and commutative rings The Hilbert scheme of degree two curves and certain ropes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a projective scheme over a field \(k\) with \(k=\overline{k}\). This paper studies the spaces of configurations \(F(X,n)=X^n-\Delta\), where \(\Delta\) is the union of the diagonals of \(X^n\). In particular the aim of this work is to furnish a class of compactifications of \(F(X,n)\) and of its quotients by subgroups of the symmetric group. Detailed results are given for the case \(n=3\).
The main idea is to give morphisms \(f_{\eta}: F(X,n) \rightarrow P_1\times \dots \times P_t\), where each \(P_i\) is of the form: \(\text{ Hilb}^{i_1}(\text{ Hilb}^{i_2}(\dots\text{ Hilb}^{i_l}(X)))\), and \(\eta\) is formed by a collection of subsets of \(\{1,\dots,n\}\). Via the combinatorial data of \(\eta\) and intersection relations of families, a certain subscheme \(R_{\eta}\subset P_1\times\dots\times P_t\) is defined, and the compactifications considered here are those for which \(\overline {f_{\eta}(F(X,n))}=R_{\eta}\). This procedure realizes a ``triple way'' to view such compactifications (functorially, via intersections of families inside the product of Hilbert schemes, and as a closure) which allows to use it more easily. Such \(R_{\eta}\)-s are called ``varieties of enriched \(n\)-tuples''.
In particular, the case \(n=3\) is studied: it is shown that, even if there are infinitely many choices for constructing such \(R_{\eta}\)'s, there are only 11 different isomorphism classes of them; moreover in any of those 11 cases the factors in \(P_1\times \dots \times P_t\) are either \(\text{ Hilb}^i(X)\) or \(\text{ Hilb}^i(\text{ Hilb}^j(X))\).
When studying quotients via subgroups of the symmetric group on 3 elements, it is shown that if \(R_{\beta}\) is such a quotient of an \(R_{\alpha}\), then (modulo isomorphisms) we get that the quotient \(R_{\alpha} \rightarrow R_{\beta}\) is a ``forgetting morphism''.
Applications for these constructions and a comparison with more ``classical'' ones are given. Hilbert schemes; compactifications; collision Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory Compactifications of configuration spaces inside 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 this book the authors indicate a general procedure for obtaining multiple point formulas for morphisms \(f:V\to W\) of non-singular complex varieties, under quite general conditions. Intuitively a \(k\)-tuple point of \(f\) is a point \(x\) such that \(f^{-1}f(x)\) consists of at least \(k\) points, and the \(k\)-tuple locus is the subvariety of \(V\) consisting of all such points. A \(k\)-tuple point formula is a formula that expresses the class of the \(k\)-tuple locus in a suitable intersection theory for \(V\), in terms of polynomials in the Chern classes of the virtual normal bundle \(\nu(f) =f^\ast TW-TV\). For the formulas to be meaningful certain relations between the numbers \(\dim(V)\), \(\dim(W)\) and \(k\) must be satisfied. To obtain \(k\)-tuple formulas it is necessary to have a manageable parameter space for \(k\)-tuple points in \(V\). In this space one should be able to define a class of \(k\)-tuple points of \(f\), that, at least formally, represents the subvariety consisting of \(k\)-tuple points of \(V\) that are \(k\)-tuple points of the morphism \(f\). Unfortunately the variety consisting of \(k\)-tuple points of \(V\) that are \(k\)-tuple points of \(f\) often contain excess components that correspond to improper \(k\)-tuple points of \(f\). In order that the \(k\)-tuple formulas should have a geometric meaning it is necessary to give procedures for removing the improper ones. The authors give parameter spaces for the \(k\)-tuple points of \(V\) for \(k=2,3,4\), and define the class of \(k\)-tuple points of \(f\) in the Chow ring of these spaces. They also perform the necessary calculations in the Chow ring to obtain double- and triple-point formulas under quite general conditions on \(f\). We give some details of the construction of the parameter spaces used for double and triple points. To this end we denote by \(H^i(V)\) the Hilbert scheme of colength \(k\) points of a scheme \(V\). For the double points the authors use the two-sheeted cover \(\widehat{H^2(V)}\) of \(H^2(V)\). They use this space to obtain the well known double point formula [see, e.g., \textit{W. Fulton}, ``Intersection theory'' (1998; Zbl 0885.14002)]. The construction of the parameter space and the calculations to obtain the formula are performed in chapters 1 and 2. For the triple points the situation is much more complicated. The parameter space \(\widehat {H^3(V)}\) of triple points of \(V\) is the locus in \(V^3\times H^2(V)^3\times H^3(V)\) consisting of points of the form \((p_1, p_2, p_3, d_{12}, d_{23}, d_{13}, t)\) satisfying the relations:
\[
\begin{cases} p_i\subset d_{ij} \subset t\\ p_j=\text{Res}(p_i,d_{ij})\\ p_k=\text{Res}(d_{ij},t) \quad \text{with} \quad \{i,j,k\} =\{1,2,3\}, \end{cases}
\]
where \(\text{Res}(\eta,\xi)\) denote the residual closed point of the \((k-1)\)-tuple \(\eta\) contained in the \(k\)-tuple \(\xi\). The authors follow a method of \textit{Z. Ran} [Acta Math. 155, 81-101 (1985; Zbl 0578.14046)] and \textit{T. Gaffney} [Math. Ann. 295, 269-289 (1993; Zbl 0841.14002)], and define a class that formally represents the triple points of \(f\). In chapter 3 they obtain the well known triple points formula of \textit{S. L. Kleiman} [Acta Math. 147, 13-49 (1993; Zbl 0479.14004)] and \textit{F. Ronga} [Compos. Math. 53, 211-223 (1984; Zbl 0563.57014)] under quite general conditions (see also \textit{W. Fulton}, loc. cit.). Many of the computations are tedious and are performed in chapter 4. The intersections of the locus of triple points corresponding to the class of triple points have excess intersections in the presence of \(S_2\)-singularities, and are difficult to handle. The authors indicate how this can be done, by treating the case \(\dim(V)=2\), \(\dim(W)=3\), when \(f\) has \(S_2\) singularities. A similar treatment of the triple points have been given by \textit{P. le Barz} [Duke Math. 5, 57, 925-946 (1988; Zbl 0687.14042) and Bull. Soc. Math. Fr. 112, 303-324 (1984; Zbl 0561.14021)]. The novel part of the book is the case of quadruple points. Here there is a real leap in difficulty. The second part of the book is devoted to this case. We shall not even indicate the construction of the parameter space. The reader who enjoys concrete, detailed geometry, and who likes to draw pictures representing ideals of the same colength, having various properties, will enjoy this part of the work. multiple point formulas; Hilbert scheme; Chow ring; Chern class of virtual normal bundle Danielle, D.; Le Barz, P.: Configuration spaces over Hilbert schemes and applications. Lecture notes in math. 1647 (1996) Parametrization (Chow and Hilbert schemes), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Special surfaces Configuration spaces over Hilbert 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 Fundamental and deep connections have been developed in recent years between the geometry of Hilbert schemes \(X^{[n]}\) of points on a (quasi-)projective surface \(X\) and combinatorics of symmetric functions. Among distinguished classes of symmetric functions, let us mention the monomial symmetric functions, Schur polynomials, Jack polynomials (which depend on a Jack parameter), and Macdonald polynomials, etc. The monomial symmetric functions can be realized as certain ordinary cohomology classes of the Hilbert schemes associated to an embedded curve in a surface. Nakajima further showed that the Jack polynomials whose Jack parameter is a positive integer can be realized as certain \(\mathbb{T}\)-equivariant cohomology classes of the Hilbert schemes of points on the surface \(X(\gamma)\) which is the total space of the line bundle \({\mathcal{O}}_{\mathbb{P}^{1}}(-\gamma)\) over the complex projective line \({\mathbb{P}^{1}}\).
In the paper under review, generalizing earlier work of Nakajima and Vasserot, the authors study the (equivariant) cohomology rings of Hilbert schemes of certain toric surfaces and establish their connections to Fock space and Jack polynomials. In particular, they describe the equivariant and ordinary cohomology rings of the Hilbert schemes of points on the surface \(X(\gamma)\). They first show that the Jack polynomials can be realized in terms of certain \(\mathbb{T}\)-equivariant cohomology classes of the Hilbert schemes of points on the affine plane, and the Jack parameter comes from the ratio of the \(\mathbb{T}\)-weights on the two affine lines preserved by the \(\mathbb{T}\)-action. In addition, the authors study the \(\mathbb{T}\)-equivariant cohomology ring of \(X(\gamma)^{[n]}\) with respect to a certain \(\mathbb{T}\)-action. Finally, they note that the ordinary cohomology ring of the Hilbert scheme \(X(\gamma)^{[n]}\) can be shown to be isomorphic to the graded ring associated to a natural filtration on the ring \(H^{2n}_{\mathbb{T}}(X(\gamma)^{[n]})\). In this way, they obtain an algorithm for computing the ordinary cup product of cohomology classes in \(X(\gamma)^{[n]}\). Jack polynomials; Hilbert schemes; Heisenberg algebras; equivariant cohomology Wei-Ping Li, Zhenbo Qin, and Weiqiang Wang, The cohomology rings of Hilbert schemes via Jack polynomials, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, vol. 38, Amer. Math. Soc., Providence, RI, 2004, pp. 249 -- 258. Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) The cohomology rings of Hilbert schemes via Jack polynomials | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Białynicki-Birula strata on the Hilbert scheme \(H^n(\mathbb{A}^d)\) are smooth in dimension \(d=2\). We prove that there is a schematic structure in higher dimensions, the Białynicki-Birula scheme, which is natural in the sense that it represents a functor. Let \(\rho_i:H^n(\mathbb{A}^d)\to\text{Sym}^n(\mathbb{A}^1)\) be the Hilbert-Chow morphism of the \(i\)th coordinate. We prove that a Białynicki-Birula scheme associated with an action of a torus \(T\) is schematically included in the fiber \(\rho_i^{-1}(0)\) if the \(i\)th weight of \(T\) is non-positive. We prove that the monic functors parametrizing families of ideals with a prescribed initial ideal are representable. Hilbert scheme of points; standard sets; Białynicki-Birula decomposition; stratification; torus action; representable functor; Gröbner basis Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Computational aspects in algebraic geometry Białynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 develop the methods of integral operators on the cohomology of Hilbert schemes of points on surfaces. These are used to obtain integral bases for the cohomology of Hilbert schemes \(X^{[n]}\) of \(0\)-cycles of length \(n\), where \(X\) is a complex projective surface.
The first part of the paper is devoted to study the integral operators and integral classes coming from the creation Heisenberg operators \({\mathfrak a}_{-r}(1)\) and \({\mathfrak a}_{-r}(x)\) associated to the identity cohomology class and the point cohomology class \(x.\) \noindent In the second part the authors develop the integral operators and integral classes coming from the creation Heisenberg operators \({\mathfrak a}_{-i}(\alpha),\) where \(\alpha \in H^2(X,\mathbb Q).\) Hilbert scheme; integral operator; integral basis Qin, Z.; Wang, W., Integral operators and integral cohomology classes of Hilbert schemes, Math. Ann., 331, 3, 669-692, (2005) Parametrization (Chow and Hilbert schemes), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Vertex operators; vertex operator algebras and related structures, Classical real and complex (co)homology in algebraic geometry Integral operators and integral cohomology classes 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 There is a notion, due to Nakumura, of \(G\)-Hilbert scheme \(\mathrm{Hilb}^G{\mathbb C}^n\) for any finite, abelian subgroup \(G\) of \(\mathrm{GL}(n,{\mathbb C})\). The \(G\)-Hilbert scheme can be described in terms of \(G\)-sets. In the article under review the author describes the \(G\)-Hilbert scheme when \(G\) is a finite cyclic group generated by a \((3\times 3)\) matrix of a particular form. After giving an classification of all possible \(G\)-sets, the author also obtains a formula for the number of different \(G\)-sets that appear for these groups. \(G\)-sets; \(G\)-Hilbert scheme O. Kȩdzierski, The G-Hilbert scheme for \(\frac{1}{r}\)(1,a,r-a), Glasg. Math. J. 53 (2010), 115 -129. McKay correspondence, Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry The \(G\)-Hilbert scheme for \(\frac 1{r} (1,a, r-a)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X^{[n]}\) denote the Hilbert scheme of \(n\) points on a nonsingular projective variety \(X\).
An isomorphism \(g:X \to Y\) of varieties induces an isomorphism \(g^{[n]}: X^{[n]} \to Y^{[n]}\).
\textit{S. Boissière} defined an isomorphism \(\sigma: X^{[n]} \to Y^{[n]}\) to be \textit{natural} if \(\sigma = g^{[n]}\) for some isomorphism \(g:X \to Y\) [Can. J. Math. 64, No. 1, 3--23 (2012; Zbl 1276.14006)].
Using work of \textit{A. Beauville} [J. Differ. Geom. 18, 755--782 (1983; Zbl 0537.53056)], \textit{R. Zuffetti} recently exhibited isomorphisms \(X^{[2]} \cong Y^{[2]}\) that are not natural, with \(X\) and \(Y\) (non-isomorphic) \(K3\) surfaces [Rend. Semin. Mat., Univ. Politec. Torino 77, No. 1, 113--130 (2019; Zbl 1440.14183)].
Defining a variety \(X\) to be \textit{weak Fano} if \(\omega_X^\vee\) is nef and big, the authors prove that if \(X\) is a smooth projective surface that is weak Fano or of general type and \(n\) is an integer, then every automorphism of \(X^{[n]}\) is natural unless \(n=2\) and \(X = C \times D\) is a product of two curves. In the latter case, if \(C,D\) are both rational or both of genus \(g \geq 2\), then there is a unique nonnatural isomorphism of \(X = C \times D\) up to composition with natural automorphisms.
They also prove that every automorphism of \((\mathbb P^n)^{[2]}\) is natural.
The authors also show that if \(X,Y\) are smooth projective surfaces with \(Y\) weak Fano or of general type, then every isomorphism \(X^{[n]} \to Y^{[n]}\) is natural (they assume \(n \geq 3\) if \(Y\) is a product of curves). This can be thought of as an analog to a theorem of \textit{A. Bondal} and \textit{D. Orlov} [Compos. Math. 125, No. 3, 327--344 (2001; Zbl 0994.18007)] and extends results of \textit{T. Hayashi}, who proved this for rational surfaces whose Iitaka dimension of \(\omega_X^\vee\) is at least one [Geom. Dedicata 207, 395--407 (2020; Zbl 1444.14012)]. Hilbert scheme of points; natural isomorphisms; smooth surfaces; automorphisms Parametrization (Chow and Hilbert schemes), Automorphisms of surfaces and higher-dimensional varieties Automorphisms of Hilbert schemes of points on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper reproduces, with the addition of an introduction and some notes to make it more readable and up to date, a 1986 letter from the author to \textit{W. Messing}, in which he proves, as an application of the comparison theorem between crystalline cohomology and \(p\)-adic étale cohomology [cf. \textit{J.-M. Fontaine} and \textit{W. Messing}, in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata, Contemp. Math. 67, 179-207 (1987; Zbl 0632.14016)], the following:
Theorem 1. Let \({\mathcal X}\) be a proper and smooth scheme over \(\mathbb{Z}\) and let \(X = {\mathcal X} \otimes \mathbb{Q}\) (or, slightly more generally, let \(X\) be a proper and smooth scheme over \(\mathbb{Q}\) with good reduction everywhere). Then \(H^j (X, \Omega^i_X) = 0\) whenever \(i,j \in \mathbb{N}\) satisfy \(i \neq j\) and \(i + j \leq 3\).
This theorem generalizes a previous result of the author on the non- existence of abelian varieties over \(\mathbb{Z}\) to higher dimensions [\textit{J.-M. Fontaine}, Invent. Math. 81, No. 3, 515-538 (1985; Zbl 0612.14043)], and, as was the case in that paper, the author begins by giving a bound for the different of certain field extensions. He starts with \(K\), a field of characteristic 0, complete for a discrete valuation, with perfect residue field of characteristic \(p > 0\) and absolutely unramified. If we take \(\overline K\) to be an algebraic closure of \(K\) and \(G_K = \text{Gal} (\overline K/K)\), he proves the following.
Theorem 2. Let \(V\) be a crystalline \(p\)-adic representation of \(G_K\) whose Hodge-Tate weights are in \([0,r]\), where \(r\) is an integer, \(0 < r < p - 1\). Let \(U\) be a subquotient of \(V\), stable under \(G_K\) and killed by \(p\). Let \(H\) be the kernel of the action of \(G_K\) on \(U\) and \(L = \overline K^H\). If \(v_0\) denotes the valuation on \(L\) normalized so that \(v_0 (p) = 1\), and if \({\mathcal D}_{L/K}\) is the different of the extension \(L/K\), then \(v_0 ({\mathcal D}_{L/K}) \leq 1 + r/(p - 1)\).
In the proof of this theorem a key role is played by the comparison functor between filtered Dieudonné modules and crystalline Galois representations. -- The author uses this bound, together with the general lower bounds for the \(n\)-th root of the discriminant of a field extension of degree \(n\) found by Diaz y Diaz, using the method of Odlyzko-Poitou- Serre, to study the category of 7-adic finite-dimensional representations of \(G =\text{Gal} (\overline \mathbb{Q}/ \mathbb{Q})\), unramified outside 7 and such that, when seen as representations of \(\text{Gal} (\overline \mathbb{Q}_7/ \mathbb{Q}_7)\), they are crystalline with Hodge-Tate weights between 0 and 3. He proves in particular the following proposition: Let \(V\) be such a representation and \(\chi\) be the cyclotomic character. -- Put \(V_4 = 0\) and for \(i = 0,1,2,3\), \(V_i = \{v \in V |gv - \chi^i (g)v \in V_{i + 1}\) for all \(g \in G\}\). Then \(V_0 = V\).
Theorem 1 follows from this proposition, applied to the representation dual to the one given by the action of \(G\) on \(H^m_{\text{ét}} (X \otimes \overline \mathbb{Q}, \mathbb{Q}_7)\), \(m \in \{1,2,3\}\), which is explicitly related to de Rham cohomology through the comparison theorem of Fontaine-Messing. Note that, as we have pointed out, the relation between filtered modules and Galois representations figures prominently in the proofs of the two main results of the paper. -- The author mentions that results similar to those in this paper, and in particular a generalization of theorem 2, have been obtained independently by \textit{V. A. Abrashkin} [see Invent. Math. 101, No. 3, 631-640 (1990; Zbl 0761.14006) and references therein]. comparison theorem between crystalline cohomology and \(p\)-adic étale cohomology; vanishing theorem; 7-adic finite-dimensional representations Jean-Marc Fontaine, Schémas propres et lisses sur \({\mathbf Z}\), Proceedings of the Indo-French Conference on Geometry (Bombay, 1989), Hindustan Book Agency, Delhi, 1993, pp. 43-56 (French). \(p\)-adic cohomology, crystalline cohomology, Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry Schemes which are proper and smooth over \(\mathbb{Z}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 algebraic geometry, the \(n\)th Hilbert scheme of a projective variety \(S\) is a projective variety Hilb\(^n(S)\) that can be thought of as a smoothed version of the \(n\)th symmetric product of \(S\). The \(n\)th symmetric product of a manifold \(M\) admits a simple combinatorial interpretation: outside of a negligible subset, the symmetric product is the collection of subsets of \(M\) of size \(n\) assembled as a manifold in its own right. Interestingly, the Hodge numbers of a complex projective surface \(S\) determine the Hodge numbers of Hilb\(^n(S)\) for all \(S\) in a very pleasing combinatorial way. In this paper, the authors consider the circle method to obtain exact formulae for these Hodge numbers for a certain class of complex projective surfaces. This work is a generalization of [\textit{N. Gillman} et al., Res. Number Theory 4, No. 4, Paper No. 39, 26 p. (2018; Zbl 1441.14022)]. partitions; Hodge numbers; Hilbert schemes Analytic theory of partitions, Parametrization (Chow and Hilbert schemes) From partitions to Hodge numbers of Hilbert 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 article we characterize the classification of stably simple curve singularities given by V. I. Arnold, in terms of invariants. On the basis of this characterization we describe an implementation of a classifier for stably simple curve singularities in the computer algebra system SINGULAR. stably equivalent; irreducible singularities; stably simple curve singularities; singular 4. M. A. Binyamin, J. A. Khan, F. K. Janjua and N. Hussain, Characterization of stably simple curve Singularities, Studia Sci. Math. Hungarica, accepted for publication. Computational aspects of algebraic curves, Singularities of curves, local rings Characterization of stably 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 This paper studies the nef cone of the Hilbert schemes of points on smooth projective surfaces with irregularity \(0\). The main idea is to utilize the Bayer-Macri construction of ample line bundles over the moduli spaces of Bridgeland semistable objects.
The ample cone and the nef cone are fundamental invariants of algebraic varieties, but their computations for Hilbert schemes of points on surfaces lack a general principle. The recent breakthrough by \textit{A. Bayer} and \textit{E. Macrì} [J. Am. Math. Soc. 27, No. 3, 707--752 (2014; Zbl 1314.14020)] on the study of Bridgeland stability conditions implies a possible approach to this problem, as the Hilbert schemes of points on surfaces are the examples of moduli spaces of Bridgeland semistable objects. In this sense, the motivation of this paper is a natural one.
Stability manifolds, namely the spaces of stability conditions, are still mysterious objects and remain lots of things to study. One of the known facts is the concrete description of the slice in the stability manifold parametrized by a half-plane where the stability condition is given by a pair of am ample divisor and an effective divisor. On this slice one has an explicit chart of the wall-chamber structure. The computation method of nef divisor is based on such a description.
This paper gives good presentation together with brief preliminaries in Section 2 and explicit computations in Section 3. The reviewer recommend it for those interested in applications of the works by Bayer and Macri [loc. cit.; Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011)]. Hilbert schemes; surfaces; nef cone; ample cone; birational geometry; Bridgeland stability B. Bolognese, J. Huizenga, Y. Lin, E. Riedl, B. Schmidt, M. Woolf, and X. Zhao, Nef cones of Hilbert schemes of points on surfaces, Algebra Number Theory 10 (2016), 907--930. Parametrization (Chow and Hilbert schemes), Minimal model program (Mori theory, extremal rays), Surfaces of general type, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Nef cones 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 In this paper, by using the \(K\)-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds, the author studies the \(K\)-theoretic integrals of characteristic classes of tautological bundles on the Hilbert scheme of points on a surface.
Let \(S\) be a complex smooth quasi-projective surface, and let \(S^{[n]}\) denote the Hilbert scheme of \(n\) points on \(S\).
For a line bundle \(\mathcal L\) on \(S\), let \({\mathcal L}^{[n]}\) be the rank-\(n\) tautological bundle over \(S^{[n]}\).
When \(S = \mathbb C^2\), define \(F(z, m_1, m_2, m_3, y)(t_1, t_2)\) to be the series of torus-equivariant Euler characteristics
\[
\sum_{n \ge 0} (-z)^n \, \chi\big ((\mathbb C^2)^{[n]}, \Lambda_{m_1}^\bullet(\mathcal O^{[n]})^\vee \otimes \Lambda_{m_2}^\bullet(\mathcal O^{[n]})^\vee \otimes \Lambda_{m_3y}^\bullet \mathcal O^{[n]} \otimes \mathrm{Sym}_y^\bullet \mathcal O^{[n]} \otimes \Lambda^n \mathcal O^{[n]} \big )
\]
where for a vector bundle \(\mathcal V\) on a scheme \(X\),
\[
\Lambda_m^\bullet \mathcal V = \sum_{i \ge 0} (-m)^i \, \Lambda^i \mathcal V \in K(X)[[m]], \quad \mathrm{Sym}_y^\bullet \mathcal V = \sum_{i \ge 0} y^i \, \mathrm{Sym}^i \mathcal V \in K(X)[[y]].
\]
The main theorem of the paper asserts that
\[
\frac{F(z, m_1, m_2, m_3, y)(t_1, t_2)}{F(z, m_1, m_2, m_3, 0)(t_1, t_2)} =\frac{F(z, m_1, m_3, m_2, y)(t_1, t_2)}{F(z, m_1, m_3, m_2, 0)(t_1, t_2)} =\frac{F(y, m_1, m_2, m_3, z)(t_1, t_2)}{F(y, m_1, m_2, m_3, 0)(t_1, t_2)},
\]
that is, the series \(F\) enjoys two types of symmetries: one between the ``box-counting variable'' \(z\) and the ``Segre variable'' \(y\), and another among the ``Chern variables'' \(m_1, m_2, m_3\).
Moreover, the denominator \(F(z, m_1, m_2, m_3, 0)(t_1, t_2)\) can be characterized via the plethystic exponential:
\[
F(z, m_1, m_2, m_3, 0)(t_1, t_2) = \exp \left (\sum_{n \ge 1} -\frac{z^n}{n} \frac{(1-m_1^n)(1-m_2^n)}{(1-t_1^{-n})(1-t_2^{-n})} \right ).
\]
As an application, it is verified that
\[
\chi\big (S^{[n]}, \mathrm{Sym}^k({\mathcal L}^{[n]}) \big ) = \binom{\chi(\mathcal L) + k -1}{k}
\]
for a line bundle \(\mathcal L\) on a general surface \(S\) with \(\chi(\mathcal O_S) = 1\) and for \(n \ge k\).
The idea in proving the main results is to use equalities of particular limits of K-theoretic Donaldson-Thomas partition functions of certain Calabi-Yau threefolds.
Section~2 defines \(K\)-theoretic Donaldson-Thomas invariants of Calabi-Yau threefolds, the plethystic exponential and equivariant localization. Section~3 proves a general relationship between noncompact directions in a moduli space and the limits of Euler characteristics of coherent sheaves under slopes.
Section~4 recalls the descriptions of the vertex and edge contributions to K-theoretic Donaldson-Thomas invariants of toric Calabi-Yau threefolds in terms of partitions, and contains explicit combinatorial descriptions of their limits under preferred slopes. The author proves the main theorems in Section~5, and deduces the above formula of \(\chi\big (S^{[n]}, \mathrm{Sym}^k({\mathcal L}^{[n]}) \big )\) in Section~6. Hilbert schemes; complex surfaces; Donaldson-Thomas theory; tautological bundles; Calabi-Yau threefolds; plethystic exponential; localization Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry \(K\)-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The rational Chow ring \(A^*(S[n],\mathbb{Q})\) of the Hilbert scheme \(S[n]\) parametrising the length \(n\) zero-dimensional subschemes of a toric surface \(S\) can be described with the help of equivariant techniques. In this paper, we explain the general method and we illustrate it through many examples. In the last section, we present results on the intersection theory of graded Hilbert schemes. Chow ring; equivariant Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Intersection theory on punctual Hilbert schemes and graded 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 \(H^N\) be the Hilbert scheme of \(N\) points in the projective plane. \textit{G. Gotzmann} [Math. Z. 199, 539--547 (1988; Zbl 0637.14003)] observed that \(H^N\) is naturally stratified by the Hilbert function, and he showed the irreducibility and dimension of the strata. The current paper, instead, studies the nested Hilbert scheme \(H_{N-i,N}\) parameterizing pairs \((Z,W)\) such that \(Z \in H^{N-i}, W \in H^{N}\) and \(Z \subset W\). In the case \(i=1\) this is known to be smooth and irreducible. In this paper, \(H_{N-1,N}\) is shown to again be stratified by irreducible subvarieties of the type \(H_{\phi, \psi} = \{ (Z,W) \in H_{N-1,N} | Z \in H_\phi, W \in H_\psi \}\), where \(H_\phi\) is the locally closed subscheme of the Hilbert scheme parameterizing zero-dimensional schemes with Hilbert function \(\phi\). The dimension of these strata is also computed.
On the other hand, when \(i>1\), it is shown that \(H_{N-i,N}\) is never smooth, and the corresponding strata may be reducible. However, if the Hilbert functions \(\phi\) and \(\psi\) are very close to each other (in a sense made precise in the paper), then the strata are irreducible, and the dimensions are computed. The authors also show that \(H_{N-2,N}\) is irreducible.
An important ingredient in the proof is the classical notion that algebraic liaison can be used (for codimension two arithmetically Cohen-Macaulay subschemes) to pass from any scheme to a simpler one. Here the authors reduce a statement about a specific nested scheme \(H_{\phi,\psi}\) to a corresponding one about a ``simpler'' nested scheme \(H_{\psi^*,\phi^*}\), via liaison. As a byproduct they find a new proof of Gotzmann's results. The results of this paper are motivated by the authors' study of globally generated and very ample Hilbert functions [Math. Z. 245, 155--181 (2003; Zbl 1079.14057)], and has applications to the classification of globally generated line bundles on the blow-up of \(\mathbb P^2\) at points, and to the study of Cayley-Bacharach schemes in \(\mathbb P^2\). globally generated line bundles; Cayley-Bacharach schemes; Hilbert function Parametrization (Chow and Hilbert schemes), Linkage, Projective techniques in algebraic geometry On the stratification of nested 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 For a smooth algebraic surface \(S\) over the complex numbers let \(S^{[n]}\) be the Hilbert scheme of zero-dimensional subschemes of length \(n\). The authors compute the singular cohomology \(H^*(S^{[n]}, \mathbb{Q})\). For the proof they use perverse sheaves. The result has been shown in special cases by \textit{G. Ellingsrud} and \textit{S. A. Strømme} in Invent. Math. 87, 343-352 (1987; Zbl 0625.14002) and for the case of a projective surface by \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193- 207 (1990; Zbl 0686.14022)]. Using the mixed Hodge modules of Saito, the authors also describe the Hodge structure of \(H^*(S^{[n]}, \mathbb{Q})\), proving a conjecture of the first author.
With the same methods they also compute the cohomology and its Hodge structure for higher order Kummer varieties, which were defined by \textit{A. Beauville} [J. Differ. Geom. 18, 755-782 (1983; Zbl 0537.53056)]. Hilbert scheme; singular cohomology; perverse sheaves; Kummer varieties Göttsche, L.; Soergel, W., Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann., 296, 235-245, (1993) Parametrization (Chow and Hilbert schemes), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, (Co)homology theory in algebraic geometry, Surfaces and higher-dimensional varieties Perverse sheaves and the cohomology of Hilbert schemes of smooth 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 aim of this paper is to investigate components and singularities of Quot schemes and varieties of commuting matrices. The authors classify its components for any number of matrices of size at most \(7\). They prove that starting from quadruples of \(8\times 8\) matrices, this scheme has generically non reduced components, while up to degree \(7\) it is generically reduced.
Their approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and Bialynicki-Birula decompositions to this setup. The authors include a thorough review of their methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. The results obtained in this paper give the corresponding statements for the Quot schemes of points, in particular the authors classify the components of Quot\(_d(\mathcal{O}_{\mathbb{A}^n}^{\oplus r})\) for \(d\leq7\) and all \(r,n\).
This paper is organized as follows. The first Section is an introduction to the subject and statement of the results. Section 2 deals with notation and Section 3 with some preliminaries. Section 4 concerns structural results on the variety \(C_n(\mathbf{M}_d)\) of \(n\)-tuples of commuting \(d\times d\) matrices and Quot\(^d_r\). Section 5 is devoted to Bialynicki-Birula decompositions and components of Quot\(^d_r\) and Section 6 to some results specific for degree at most eight. The paper is supported by an appendix concerning a functorial approach to comparison between \(C_n(\mathbf{M}_d)\) and Quot\(^d_r\). Quot schemes; varieties of commuting matrices; components and singularities Parametrization (Chow and Hilbert schemes), Relationships between algebraic curves and integrable systems Components and singularities of Quot schemes and varieties of commuting matrices | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Related to a Coxeter group are certain sets of tangents of the deltoid with evenly distributed orientations forming simplicial line configurations. These configurations are used to construct curves and surfaces with \(ADE\) singularities. Other surfaces associated with invariants of exceptional complex reflection groups are considered. A new lower bound for the maximal number of \(D_4\) singularities in a sextic surface is obtained. Several Calabi-Yau threefolds defined as double coverings of the complex projective 3-space branched along nodal octic surfaces and Calabi-Yau quintic threefolds are analyzed. The Hodge numbers of a small resolution of all the nodes of the singular threefolds are obtained. algebraic varieties; singularities; Calabi-Yau threefolds Calabi-Yau manifolds (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Algebraic varieties with simple singularities related to some reflection groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert-Kunz multiplicity \(e_{HK}(I)\) [see \textit{E. Kunz}, Am. J. Math. 91, 772-784 (1969; Zbl 0188.33702)], respectively \textit{P. Monsky} [Math. Ann. 263, 43-49 (1983; Zbl 0509.13023)], of an \(\mathfrak m\)-primary ideal \(I\) in a local ring \((A, \mathfrak m)\) of characteristic \(p > 0\) defined by the Frobenius morphism is known to be a rather mysterious invariant. The notion of a good ideal was introduced by \textit{S. Goto, S.-I. Iai} and \textit{K.-I. Watanabe} [see Trans. Am. Math. Soc. 353, No. 6, 2309-2346 (2001; Zbl 0966.13002)], as an ideal \(I\) in a local ring such that its associated graded ring \(\text{gr}_I(A)\) is a Gorenstein ring with \(a\)-invariant \(a(\text{gr}_I(A)) = 1-d.\) In the two-dimensional case it follows that \(I\) is good in \(A\) if and only if \(I\) is an integrally closed ideal represented on the minimal resolution \(f : X \to \text{Spec} A.\) The main result of the paper is a nice formula for the Hilbert-Kunz multiplicity for certain ideals \(I:\)
Let \(A\) be a two dimensional F-finite rational Gorenstein local ring such that \(A\) is a pure subring of a regular ring \(B\) which is a finite \(A\)-module of rank \(N.\) Let \(I = H^0(X, \mathcal O_X(-Z))\) be a good ideal such that \(I\mathcal O_X = \mathcal O_X(-Z)\) with \(Z = \sum_{i=1}^r a_iE_i,\) where \(E_1, \ldots, E_r\) are irreducible exceptional curves on a minimal resolution of the singularity. Then
\[
e_{HK}(I) - \text{Length}_A (A/I) = \left(\sum_{i=1}^r a_in_i\right) \Biggl/N,
\]
where the \(n_i\) is determined by the fundamental cycle \(Z_0 = \sum_{i=1}^r n_iE_i\) on \(X.\) Moreover
\[
e_{HK}(I) - \text{Length}_A (A/I) = e_{HK}(\widetilde{I}) - \text{Length}_A (A/\widetilde{I}),
\]
where \(\widetilde{I}\) denotes the good closure of \(I,\) that is the minimal good ideal containing \(I.\) In the case \(A\) the completion of \(k[x,y]^G\), \(G\) a finite subgroup of \(SL(2,k),\) one can apply the result. So there is a description of \(e_{HK}(I)\) of any good ideal in terms of the dual graph of \(X.\) For the proof of their main results the authors use the McKay correspondance and the Riemann-Roch formula. The paper is completed by several lists of good ideals in two-dimensional rational singularities. By virtue of their main result there are calculations of \(e_{HK}(I)\) using its dual graph. So they provide a lot of new samples of the Hilbert-Kunz multiplicities of certain ideal. Hilbert Kunz multiplicity; McKay correspondance; good ideals; two-dimensional F-finite rational ring; Frobenius morphism; characteristic \(p\) Watanabe, K.-I., Yoshida, K.-I.: Hilbert-Kunz multiplicity, McKay correspondence and good ideals in two-dimensional rational singularity. Manuscr. Math. 104(3), 275--294 (2001) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Singularities in algebraic geometry, Riemann-Roch theorems Hilbert-Kunz multiplicity, McKay correspondence and good ideals in two-dimensional rational singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Denote by \(\mathfrak{M}_\Lambda\) the coarse moduli space of marked pairs \((S^{[n]},\,\eta)\) of the Hilbert scheme of a \(K3\) surface \(S\) of length \(n\) (resp. of marked pairs \((K^{[n]}(S),\,\eta)\) of the generalized Kummer variety associated to a two-dimensional compact complex torus \(S\)), for \(\Lambda\) the orthogonal sum of the \(K3\) lattice and the rank-one lattice \(\langle 2-2n \rangle\) (resp. the \(H^{\oplus 3}\oplus\langle -2-2n \rangle\), where \(H\) is the hyperbolic lattice of rank \(2\)).
The aim of the paper under review is to show that the locus of marked pairs \((X,\, \eta)\) of an irreducible holomorphic symplectic manifold \(X\) which is isomorphic to the Hilbert scheme of a projective \(K3\) surface, or to the generalized Kummer variety associated to an abelian surface, is dense in a connected component of the respective moduli space \(\mathfrak{M}_\Lambda\).
The key is to realize the moduli space in terms of lattices. For the existence of such symplectic manifolds, they construct a parallel-transport operator between the second cohomology groups with integer coefficients, and use Verbitsky's Torelli-type theorem for irreducible holomorphic symplectic manifolds to be bimeromorphic. holomorphic symplectic manifolds; Hilbert schemes; Torelli theorem; monodromy group Markman, E.; Mehrotra, S., Hilbert schemes of \(k\)3 surfaces are dense in moduli, Math. Nach., 290, 5-6, 876-884, (2017) Families, moduli, classification: algebraic theory, \(K3\) surfaces and Enriques surfaces Hilbert schemes of \(K3\) surfaces are dense in moduli | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be an infinite field and let \(E\) be the exterior algebra over \(K\). In the paper under review the authors study the tangent space at a monomial point \(M\) in the Hilbert scheme, that parameterizes all ideals with the same Hibert function as \(M\) over \(E\). They introduce the notion of flips and show that the basic flips form a basis of the tangent space at \(M\). This implies that the tangent space at \(M\) has a basis consisting of directions tangent to deformations built using Gröbner basis. Peeva I., Stillman M.: Flips and the Hilbert scheme over an exterior algebra. Math. Ann. 339, 545--557 (2007) Formal methods and deformations in algebraic geometry, Rational and birational maps, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes Flips and the Hilbert scheme over an exterior algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X^{[n]}\) be the Hilbert scheme of \(n\) points on a smooth projective surface \(X\) over the complex numbers. In these lectures we describe the action of the Heisenberg algebra on the direct sum of the cohomologies of all the \(X^{[n]}\), which has been constructed by \textit{H. Nakajima} [Ann. Math. (2) 145, 379--388 (1997; Zbl 0915.14001)]. In the second half of the lectures we study the relation of the Heisenberg algebra action and the ring structures of the cohomologies of the \(X^{[n]}\), following recent work of \textit{M. Lehn} [Invent. Math. 136, 157--207 (1999; Zbl 0919.14001)]. In particular we study the Chern and Segre classes of tautological vector bundles on the Hilbert schemes \(X^{[n]}\). G. Ellingsrud and L. Göttsche, Hilbert schemes of points and Heisenberg algebras, Moduli spaces in Algebraic Geometry, 1999. Parametrization (Chow and Hilbert schemes), Infinite-dimensional Lie (super)algebras, Algebraic cycles, Relationships between surfaces, higher-dimensional varieties, and physics Hilbert schemes of points and Heisenberg 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 By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces \(X\) the Hilbert schemes \(\text{Hilb}^n(X)\) can be identified for all \(n\geq 1\) with moduli spaces of Gieseker stable vector bundles on \(X\). We also introduce a new Fourier-Mukai type transform for such surfaces. Fourier-Mukai transform; sheaves on K3 surfaces; Hilbert schemes; Gieseker stable vector bundles Bruzzo-Maciocia U.~Bruzzo and A.~Maciocia, Hilbert schemes of points on some \(K3\) surfaces and Gieseker stable bundles, Math.\ Proc. Cambridge Philos.\ Soc., 120 (1996), 255--261. Parametrization (Chow and Hilbert schemes), \(K3\) surfaces and Enriques surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Hilbert schemes of points on some K3 surfaces and Gieseker stable bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main theorem of this paper is that the Hilbert scheme compactification of the space of twisted cubic curves is smooth. This is shown by an explicit computation of the universal deformation of the worst possible flat degeneration of a twisted cubic. Along the way the authors prove the following result, now commonly known as the Piene- Schlessinger comparison theorem: Suppose X in \({\mathbb{P}}^ n\) is defined by homogeneous polynomials \(f_ 1,...,f_ r\) of degrees \(d_ 1,...,d_ r\), respectively, and that the linear systems cut out by hypersurfaces of degrees \(d_ 1,...,d_ r\) on X are complete. Then any infinitesimal deformation of X is induced by a unique deformation of the affine cone over X. Hilbert scheme compactification of the space of twisted cubic; curves; Piene-Schlessinger comparison theorem; infinitesimal deformation; Hilbert scheme compactification of the space of twisted cubic curves R. Piene and M. Schlessinger, On the Hilbert scheme compactification of the space of twisted cubics, Amer. J. Math. 107 (1985), no. 4, 761-774. Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Formal methods and deformations in algebraic geometry, Special algebraic curves and curves of low genus On the Hilbert scheme compactification of the space of twisted cubics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 lectures was to review several results on Hilbert schemes of points which were obtained after author's lecture note [Lectures on Hilbert schemes of points on surfaces. RI: American Mathematical Society (AMS). xi, 132 p. (1999; Zbl 0949.14001)] was written. Among many results, we choose those which are about equivariant homology groups \(H^T_*(X^{[n]})\) of Hilbert schemes of points on the affine plane \(X=\mathbb{C}^2\) with respect to the torus action. Study of equivariant homology groups increases its importance recently.
In particular, it is a basis of the AGT correspondence between instanton moduli spaces on \(\mathbb{C}^2\) and the representation theory of \(W\)-algebras, which is a very hot topic now (see e.g., \textit{D. Maulik} and \textit{A. Okounkov} [Quantum groups and quantum cohomology, \url{arxiv:1211.1287}]). H. Nakajima, \textit{More lectures on Hilbert schemes of points on surfaces}, arXiv:1401.6782. Parametrization (Chow and Hilbert schemes), 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 More lectures on 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 Hilbert scheme \({\text{Hilb}}^n_{X/S}\) parametrizing length \(n\) closed subschemes of \(X\) over \(S\) continues to draw great interest from algebraic geometers. The case \(X = \mathbb P^N\) is already interesting, as \({\text{Hilb}}^n_{\mathbb P^N/k}\) is smooth and irreducible for \(N=2\), but reducible for \(N = 3\) and \(n\) large [\textit{A. Iarrobino}, Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)] and hence singular by \textit{R. Hartshorne}'s connectedness theorem [Publ. Math. Inst. Hautes Études Sci. 29, 5--48 (1966; Zbl 0171.41502)]. Motivated by \textit{Haiman's} construction of \({\text{Hilb}}^n_{\mathbb A^2/\mathbb C}\) as the blow-up of \(\text{Sym}^n (\mathbb A^2)\) at a concrete ideal [Discrete Math. 193, No. 1--3, 201--224 (1998; Zbl 1061.05509)], the authors construct the \textit{good component} \(G^n_{X/S} \subset {\text{Hilb}}^n_{X/S}\), the closure of subschemes consisting of \(n\) distinct points, as a concrete blow-up of a symmetric product for separated morphisms \(f:X \to S\) of algebraic spaces.
Working at this level of generality, the authors need to show existence of the Hilbert scheme as an algebraic space, extending \textit{M. Artin's} result [in: Global Analysis, Papers in Honor of K. Kodaira 21--71 (1969; Zbl 0205.50402)] for \(f\) locally of finite presentation. In this context, the \(n\)th symmetric product need not exist, so instead they use the \(n\)th divided power product \(\Gamma^n_{X/S}\) (the affine model is due to \textit{N. Roby} [C. R. Acad. Sci., Paris, Sér. A 290, 869--871 (1980; Zbl 0471.13008)]), which is homeomorphic to \(\text{Sym}^n X\) in general and isomorphic when \(f\) is flat [\textit{D. Rydh}, ``Families of zero cycles and divided powers: I. Representability'', \url{arXiv:0803.0618}]. When \(X\) and \(S\) are affine, the authors define the \textit{ideal of norms} \(I\) in the ring corresponding to \(\Gamma^n_{X/S}\) and show that these patch together to define a closed subscheme \(\Delta_X \subset \Gamma^n_{X/S}\). With this machinery in place, the authors prove that if \(f: X \to S\) is a separated morphism of algebraic spaces, then \(G^n_{X/S} \subset {\text{Hilb}}^n_{X/S}\) is isomorphic to the blow-up of \(\Gamma^n_{X/S}\) along the closed subspace \(\Delta_X\). In the important case that \(f\) is flat, \(G^n_{X/S}\) is obtained by blowing up the geometric quotient \(X^n_S/S_n\). As a byproduct of their method, they show that \(G^n_{X/S} = {\text{Hilb}}^n_{X/S}\) is smooth for \(f\) smooth and separated of relative dimension two, extending the result of \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. Hilbert schemes; algebraic spaces Torsten Ekedahl and Roy Skjelnes, Recovering the good component of the Hilbert scheme, Preprint, May 2004, arXiv:math.AG/0405073. Parametrization (Chow and Hilbert schemes), Generalizations (algebraic spaces, stacks) Recovering the good component 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 This article provides the summary of \textit{A. Gholampour}, the author and \textit{S.-T. Yau} [``Nested Hilbert schemes on surfaces: Virtual fundamental class'', Preprint, \url{arXiv:1701.08899}; ``Localized Donaldson-Thomas theory of surfaces'', Preprint, \url{arXiv:1701.08902}] where the authors studied the enumerative geometry of \textit{``nested Hilbert schemes''} of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theories. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Mirror symmetry (algebro-geometric aspects) Hilbert schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In studying projective space curves, a natural question is to ask whether the Hilbert scheme \(H_{d,g}\) parametrizing locally Cohen-Macaulay curves in \(\mathbb{P}^3\) of fixed degree \(d\) and genus \(g\), is connected. This question is open.
The purpose of the present paper is to review what is known, to prove some partial results, and to identify some particular cases that will require new methods and may be potential counterexamples. -- For instance, the author proves that disjoint unions of lines, smooth rational curves, smooth curves with \(d\geq g+3\), arithmetically Cohen-Macaulay curves, curves in the biliaison class of two skew lines, and integral curves in the biliaison class of a double line with arithmetic genus \(\leq-2\), are connected to each other by flat families. extremal curve; biliaison class; Rao module; locally Cohen-Macaulay space curves; connectedness of Hilbert scheme [H2]\textsc{R. Hartshorne},\textit{On the connectedness of the Hilbert Scheme of Curves in}\textbf{P}\^{}\{3\}, Communications in Algebra,\textbf{28} (12), pp. 6059-6077. Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Plane and space curves On the 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 purpose of this paper is to throw a bridge between two seemingly unrelated subjects. One is the Hilbert scheme of points on projective surfaces, which has been intensively studied by various people. The other is the infinite dimensional Heisenberg algebra which is closely related to affine Lie algebras.
We shall construct a representation of the Heisenberg algebra on the homology group of the Hilbert scheme. In other words, the homology group will become a Fock space. The basic idea is to introduce certain ``correspondences'' in the product of the Hilbert scheme. Then they define operators on the homology group by a well-known procedure. They give generators of the Heisenberg algebra, and the only thing we must check is that they satisfy the defining relation. Here we remark that the components of the Hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components. Our correspondences live in the product of the different components. Thus it is quite essential to study all components together. -- Our construction has the same spirit as the author's earlier construction [\textit{H. Nakajima}, Duke Math. J. 76, No. 2, 365-416 (1994; Zbl 0826.17026) and Int. Math. Res. Not. 1994, No. 2, 61-74 (1994; Zbl 0832.58007)] of representations of affine Lie algebras on homology groups of moduli spaces of ``instantons'' on ALE spaces which are minimal resolutions of simple singularities. Certain correspondences, called Hecke correspondences, were used to define operators. These twist instantons along curves (irreducible components of the exceptional set), while ours twist ideals around points. In fact, the Hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles, or more precisely torsion free sheaves. Our construction should be considered as a first step to extend the two papers cited above to general 4-manifolds.
Another motivation of our study is the conjecture about the generating function of the Euler number of the moduli spaces of instantons, which was recently proposed by \textit{C. Vafa} and \textit{E. Witten} [Nucl. Phys., B 431, No. 1-2, 3-77 (1994)]. They conjectured that it is a modular form under certain conditions. If \(X^{[n]}\) is the Hilbert scheme parameterizing \(n\)-points in \(X\), then the generating function of the Poincaré polynomials is given by
\[
\sum^\infty_{n= 0} q^nP_t(X^{[n]}) = \prod^\infty_{m=1} {(1+t^{2m-1} q^m)^{b_1 (X)} (1+t^{2m+1} q^m)^{b_3(X)} \over(1-t^{2m-2} q^m)^{b_0(X)} (1-t^{2m}q^m)^{b_2 (X)} (1-t^{2m+2} q^m)^{b_4(X)}},\tag{1}
\]
where \(b_i(X)\) is the Betti number of \(X\).
The paper is organized as follows. In section 2 we give preliminaries. We recall the definition of the convolution product in \S 2(i) with some modifications and describe some properties of the Hilbert scheme \(X^{[n]}\) and the infinite Heisenberg algebra and its representations in \S\S 2(ii), 2(iii). The definition of correspondences and the statement of the main result are given in section 3. The proof will be given in section 4.
While the author was preparing this manuscript, he learned that a similar result was announced by \textit{T. Grojnowski} [Math. Res. Lett. 3, No. 2, 275-291 (1996; Zbl 0879.17011)] who introduced exactly the same correspondence. Heisenberg algebra; Hilbert scheme of points Nakajima, H.: Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. Math. (2). \textbf{145}(2), 379-388 (1997). arXiv:alg-geom/9507012. http://dx.doi.org/10.2307/2951818 Parametrization (Chow and Hilbert schemes), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Heisenberg algebra and Hilbert schemes of points on projective surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective variety. We show that the map that sends a codimension one distribution on \(X\) to its singular scheme is a morphism from the moduli space of distributions into a Hilbert scheme. We describe its fibers and, when \(X = \mathbb{P}^n\), compute them via syzygies. As an application, we describe the moduli spaces of degree 1 distributions on \({\mathbb{P}^3}\). We also give the minimal graded free resolution for the ideal of the singular scheme of a generic distribution on \(\mathbb{P}^3\). distributions; Hilbert scheme; singular scheme; syzygy Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Fine and coarse moduli spaces, Sheaves in algebraic geometry, Syzygies, resolutions, complexes and commutative rings Moduli of distributions via 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 We provide a framework for working with Gröbner bases over arbitrary rings \(k\) with a prescribed finite standard set \(\Delta\). We show that the functor associating to a \(k\)-algebra \(B\) the set of all reduced Gröbner bases with standard set \(\Delta\) is representable and that the representing scheme is a locally closed stratum in the Hilbert scheme of points. We cover the Hilbert scheme of points by open affine subschemes which represent the functor associating to a \(k\)-algebra \(B\) the set of all border bases with standard set \(\Delta\) and give reasonably small sets of equations defining these schemes. We show that the schemes parametrizing Gröbner bases are connected; give a connectedness criterion for the schemes parametrizing border bases; and prove that the decomposition of the Hilbert scheme of points into the locally closed strata parametrizing Gröbner bases is not a stratification. Hilbert scheme of points; Gröbner bases; standard sets; locally closed strata Mathias Lederer, Gröbner strata in the Hilbert scheme of points , J. Comm. Alg. 3 (2011), 349-404. Parametrization (Chow and Hilbert schemes), Polynomial rings and ideals; rings of integer-valued polynomials, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Gröbner 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 Motivated by applications in computer vision, the authors study multiview varieties. These are subvarieties of \((\mathbb P^2)^n\) that contain the images of \(n\) linear projections from \(\mathbb P^3\) to \(\mathbb P^2\) (``cameras''). Knowledge about multiview varieties and their ideals is important for the numeric reconstruction of points from their images. The authors' findings go well-beyond these immediate applications and reveal a fascinating algebraic and combinatorial geometry of the varieties under scrutiny.
A first important result is that a known generating system of generic multiview ideals is in fact a universal Gröbner basis. This observation is followed by a thorough investigation of multiview ideals and, in particular, a distinguished monomial subideal. An explicit description is given and interesting results on its Hilbert function in the \(\mathbb Z^n\)-grading are provided. The article's main result states that the Hilbert scheme that parametrizes the \(\mathbb Z^n\)-homogeneous ideals with this Hilbert function contains the space of camera positions as a special component.
The above statements are true in generic cases. Other results pertain to collinear and infinitesimally neighbouring cameras or less than five cameras with a toric multiview variety. multigraded Hilbert scheme; computer vision; monomial ideal; Groebner basis; generic initial ideal C. Aholt, B. Sturmfels, and R. Thomas, \textit{A Hilbert scheme in computer vision}, Canad. J. Math., 65 (2013), pp. 961--988, . Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Toric varieties, Newton polyhedra, Okounkov bodies, Projective techniques in algebraic geometry, Projective analytic geometry, Computational issues in computer and robotic vision A Hilbert scheme in computer vision | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 is constructed the Hilbert scheme of the surfaces of degree \(g+1\) in \({\mathbb{P}}^{g+2}\) and some properties are analyzed, like the dimension and irreducibility of this scheme, for which Brieskorn's results [\textit{E. Brieskorn}, Math. Ann. 157, 343-357 (1965; Zbl 0128.170)] about \(\Sigma\)-varieties and of \textit{M. Nagata} [Mem. Coll. Sci., Univ., Kyoto, Ser. A 32, 351-370 (1960; Zbl 0100.167)] about rational surfaces are applied. Hilbert scheme of the surfaces of degree \(g+1\); dimension; irreducibility Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry, Special surfaces The Hilbert scheme of surfaces of degreee \(g+1\) in \({\mathbb{P}}^{g+2}\) | 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 analyze the Gorenstein locus of the Hilbert scheme of \(d\) points on \(\mathbb{P}^n\) i.e., the open subscheme parameterizing zero-dimensional Gorenstein subschemes of \(\mathbb{P}^n\) of degree \(d\). We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when \(d\leq 13\) and find its components when \(d = 14\). The proof is relatively self-contained and it does not rely on a computer algebra system. As a by-product, we give equations of the fourth secant variety to the \(d\)-th Veronese reembedding of \(\mathbb{P}^n\) for \(d\geq 4\). Hilbert scheme of points; smoothability; Gorenstein algebra; secant variety Casnati, G; Jelisiejew, J; Roberto, N, Irreducibility of the Gorenstein loci of Hilbert schemes via ray families, Algebr. Number Theory, 9, 1525-1570, (2015) Parametrization (Chow and Hilbert schemes), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Formal methods and deformations in algebraic geometry Irreducibility of the Gorenstein loci of Hilbert schemes via ray families | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a \(K3\) surface \(X\). We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on \(X\). The resulting recursions are then solved explicitly. The formula proves the \(K\)-trivial case of a conjecture of \textit{M. Lehn} [Invent. Math. 136, No. 1, 157--207 (1999; Zbl 0919.14001)].
The relations determining the Segre classes fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes on surfaces, we obtain new systems of relations among tautological classes on moduli spaces of surfaces and their relative Hilbert schemes of points. For the moduli of \(K3\) surfaces, we produce relations intertwining the \(\kappa\) classes and the Noether-Lefschetz loci. Conjectures are proposed. Hilbert schemes of points; Quot schemes; moduli of \(K3\) surfaces; tautological classes Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory, \(K3\) surfaces and Enriques surfaces, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Segre classes and Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We generalize the algebraic results of \textit{R. M. Skjelnes} and \textit{D. Laksov} [Compos. Math. 126, 323--334 (2001; Zbl 1056.14500)] and \textit{R. M. Skjelnes} [Ark. Mat. 40, No. 1, 189--200 (2002; Zbl 1022.14003)], and obtain easy and transparent proofs of the representability of the Hilbert functor of points on the affine scheme whose coordinate ring is any localization of the polynomial ring in one variable over an arbitrary base ring. The coordinate ring of the Hilbert scheme is determined. We also make explicit the relation between our methods and the beautiful treatment of the Hilbert scheme of curves via norms, indicated by \textit{A. Grothendieck} [Sem. Bourbaki 13(1960/61), No. 221 (1961; Zbl 0236.14003)], and performed by \textit{P. Deligne} [in: Sémin. Géométrie algébrique, Bois-Marie 1963/64 SGA4, Tome 3, exposé XVII, Lect. Notes Math. 305, Springer-Verlag, New York, 250--480 (1973; Zbl 0255.14011)]. Laksov, D., Skjelnes, R.M., Thorup, A.: Norm on rings and the Hilbert scheme of points on a line. Q. J. Math. 56, 367--375 (2005) Parametrization (Chow and Hilbert schemes), Ideals and multiplicative ideal theory in commutative rings Norms on rings and the Hilbert scheme of points on the line | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \({\text{ Hilb}}^{s}({X})\) be the Hilbert scheme parameterizing 0-dimensional subschemes of \(X\) of length \(s\). In the case \(X \subset {\mathbb{P}}^r\) is an \(n\)-dimensional projective cone over a projective variety \(Y\subset {\mathbb{P}}^{r-1}\) of dimension \(n-1 \geq 1\), the authors prove that there exists an integer \(s_0\) such that \({\text{Hilb}}^{s}({X})\) is reducible for every \(s \geq s_0\) provided the degree of \(X\) is larger than \(4\). This improves \textit{A. Iarrobino}'s result in [Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)]. Indeed since \({\text{ Hilb}}^{s}({X})\) contains an irreducible component of dimension \(ns\), the authors achieve their result by explicitly constructing irreducible families of dimension larger than \(ns\). Hilbert scheme; punctual Hilbert scheme R.M. Miró-Roig, J. Pons-Llopis, Reducibility of punctual Hilbert schemes of cone varieties, Comm. Algebra, in press. Parametrization (Chow and Hilbert schemes) Reducibility of punctual Hilbert schemes of cone 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 We denote by \(\hbox{Hilb}^d \mathbb P^n\) the Hilbert scheme of closed zero-dimensional subschemes of \(\mathbb P^n\) of degree \(d\). It was constructed by \textit{A. Grothendieck} [Sem. Bourbaki 14(1961/62), No. 232, 19 p. (1962; Zbl 0238.14014)] and shown by \textit{R. Hartshorne} to be connected [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)]. Other important work and conjectures have been put forward in the intervening years, especially about the tangent space. According to the authors, ``the case of \(\hbox{Hilb}^d \mathbb P^3\) is of particular interest, since it lies at the boundary between the smooth cases \(n \leq 2\) and the cases \(n \geq 4\) which are believed to be wildly pathological.''
\textit{J. Briancon} and \textit{A. Iarrobino} [J. Algebra 55, 536--544 (1978; Zbl 0402.14003)] established an upper bound for the dimension of \(\hbox{Hilb}^d \mathbb P^n\) and formed two conjectures on the subject. Recall that a fat point subscheme of \(\mathbb P^n\) is one defined by (after change of variables) the ideal \((x_1,\dots,x_n)^r\) for some \(r\). In the first conjecture, Briançon and Iarrobino posited that a fat point subscheme yields the most singular point in its own Hilbert scheme, with an explicit conjectural bound. The second conjecture is again a bound.
In this paper, for points parametrizing monomial subschemes, the authors decompose the tangent space into six distinguished subspaces and show that a fat point exhibits an extremal behavior in this respect. They also characterize smooth monomial points on the Hilbert scheme using their decomposition. They prove the conjecture of Briançon and Iarrobino up to a factor of \(\frac{4}{3}\) and they improve the known asymptotic bound on the dimension of \(\hbox{Hilb}^d \mathbb P^3\). They construct infinitely many counterexamples to the second conjecture of Briançon and Iarrobino, and they also settle a weaker conjecture of Sturmfels in the negative. Briançon-Iarrobino conjecture; Haiman theory; Borel-fixed point; monomial ideal; duality Parametrization (Chow and Hilbert schemes), Homological functors on modules of commutative rings (Tor, Ext, etc.), Combinatorial aspects of commutative algebra On the tangent space to the Hilbert scheme of points in \(\mathbf{P}^3\) | 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 Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of \(n\) points of an algebraic surface is algebraic at a CM point \(\tau\) and rational numbers \(z_1\) and \(z_2\). Our result gives a refinement of the algebraicity on Betti numbers. Hodge number; algebraicity; Hilbert scheme Parametrization (Chow and Hilbert schemes), Transcendental methods, Hodge theory (algebro-geometric aspects), Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Dedekind eta function, Dedekind sums On the algebraicity about the Hodge numbers of the Hilbert schemes 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 study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.
For Part I, see [the first two authors, J. Eur. Math. Soc. (JEMS) 15, No. 6, 2353--2367 (2013; Zbl 1303.14019)]. compactified Jacobians; Hilbert scheme of points; reduced curves with locally planar singularities; perverse filtration; decomposition theorem; support theorem Jacobians, Prym varieties, Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) A support theorem for Hilbert schemes of planar curves. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the author studies the deformations of the Hilbert schemes of points on a del Pezzo surface. Let $S$ be a smooth del Pezzo surface over $\mathbb C$ obtained from $\mathbb P^2$ by blowing up at $k$ ($1 \le k < 9$) points in general position, and $\text{Hilb}^n(S)$ be the Hilbert scheme parametrizing length-$n$ $0$-dimensional closed subschemes of $S$. Let $A$ be a noetherian ring such that $\text{Spec}(A)$ is a smooth curve over $\mathbb C$. Let $(\mu_A, p_i)$ be a flat family of non-commutative del Pezzo surfaces such that one fiber is the surface $S$, where $\mu_A: U_A \otimes V_A \to W_A$ is a non-degenerate morphism of $A$-modules with $U_A, V_A, W_A$ being rank-$3$ projective $A$-modules and $p_1, \ldots, p_k$ are certain degenerate elements in $U_A$. The author constructs the relative Hilbert scheme of $n$ points $\mathcal M^s_{\mu_A, p_i}(n) \to \text{Spec}(A)$, and proves that $\mathcal M^s_{\mu_A, p_i}(n) \to \text{Spec}(A)$ is a smooth family of non-commutative deformations of the commutative Hilbert scheme $\text{Hilb}^n(S)$. Moreover, if $k \in \{1, 7, 8\}$, then $\mathcal M^s_{\mu_A, p_i}(n)$ admits a Poisson structure which is generically symplectic. The main idea in the proof of the smoothness of $\mathcal M^s_{\mu_A, p_i}(n) \to \text{Spec}(A)$ is to translate the problem of whether the Jacobian matrix has full rank to the exactness of global sections of a complex of sheaves on an elliptic curve.
Section 2 recalls exceptional sheaves on del Pezzo surfaces, and introduces non-commutative version of del Pezzo surfaces and their deformations. Section 3 gives a description of the Hilbert scheme of a commutative del Pezzo surface via geometric invariant theory. The deformation space $\mathcal M^s_{\mu_A, p_i}(n) \to \text{Spec}(A)$ is constructed in Section 4, and its smoothness is verified in Sections 5 and 6. In Section 7, the author investigates the Poisson structure on $\mathcal M^s_{\mu_A, p_i}(n)$. Hilbert scheme; exceptional collection; geometric invariant theory; holomorphic Poisson structure Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes), Derived categories and associative algebras Deformations of the Hilbert scheme of points on a del Pezzo 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 We compute certain 1-point genus-0 Gromov-Witten invariants of the Hilbert scheme \(X^{[n]}\) of points on a simply-connected smooth projective surface.
The Hilbert scheme can seen as the desingularization of the \(n\)-th symmetric product \(X^{(n)}\) of \(X\). In fact the Hilbert-Chow map \(\rho:X^{[n]}\to X^{(n)}\), sending an element in \(X^{[n]}\) to its support in \(X^{(n)}\) is a crepant resolution of the orbifold \(X^{(n)}\). Recently, \textit{Y. Ruan} [Cohomology ring of crepant resolutions of orbifolds, Preprint, \texttt{http://arxiv.org/abs/math.AG/0108195}] formulated some conjecture on the relation between the cohomology rings of crepant resolutions of orbifolds and the orbifold cohomology rings of the orbifolds themselves. It turns out that the Gromov-Witten invariants of the crepant resolutions appear in a very interesting way in Ruan's conjecture. In this paper, we shall compute the 1-point Gromov-Witten invariants of \(X^{[n]}\) with respect to some special degree-2 homology cycles on \(X^{[n]}\). Our result partially verifies Ruan's conjecture for the crepant resolution \(\rho:X^{[n]}\to X^{(n)}\). crepant resolution; orbifold; orbifold cohomology Li W.-P., Qin Z.B.: On 1-point Gromov-Witten invariants of the Hilbert schemes of points on surfaces. Turkish J. Math. 26(1), 53--68 (2002) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) On 1-point Gromov-Witten invariants of 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 Let \(H_{n,g,r}\) be the Hilbert scheme of smooth irreducible curves in \(\mathbb P^r\) of degree \(n\) and genus \(g\). An irreducible component \(W\) of \(H_{n,g,r}\) is said to be regular if \(H^1(N_C)=0\) for \(C\) a general curve in \(W\). \(W\) is said to have the expected number of moduli, if its natural image in the moduli space of curves of genus \(g\) has dimension \(\min\{3g-3, 3g-3+\rho \}\), where \(\rho\) is the Brill-Noether number \(\rho(g,n,r)\).
The problem of the existence of regular components with the expected number of moduli, for negative \(\rho\), makes sense for \(\rho\geq -3g+3\). Partial results on this problem have been obtained by several authors, since the article of \textit{E. Sernesi} [Invent. Math. 75, 25-57 (1984; Zbl 0541.14024)], who constructed curves of the required type with smoothing techniques, starting from reducible curves.
Here \textit{A. Lopez} improves the results of his previous paper [Math. Ann. 289, No. 3, 517-528 (1991; Zbl 0702.14018)], proving the following theorem for all \(r\geq 4\):
There exists a regular component of \(H_{n,g,r}\) with the expected number of moduli, for all \(n,g\) satisfying the relation
\[
0\geq \rho \geq -\max \Biggl\{{1\over r}g-{{r^2+3r+2}\over r}, g-2r^2-2r-2, \biggl(2-{6\over{r+3}} \biggr)g-h(r) \Biggr\}
\]
where \(h(r)={{4r^3+8r^2-9r+3}\over{r+3}}\). This bound can be slightly ameliorated if \(r=4\) or \(r=6\).
Also this result is based on smoothing techniques. The required curves are obtained by an inductive procedure, smoothing reducible curves, having a canonical curve as an irreducible component. Hilbert scheme; moduli space of curves; expected number of moduli; smoothing reducible curves; Brill-Noether number Lopez, AF, On the existence of components of the Hilbert scheme with the expected number of moduli II, Commun. Algebr., 27, 3485-3493, (1999) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) On the existence of components of the Hilbert scheme with the expected number of moduli. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider the Hirzebruch surface \(\mathbb{F}_1\) and denote, as usual, by \(C_0\) the minimal section and by \(f\) the fiber of the ruling. Now consider a very ample, rank two vector bundle \(\mathcal{E}\) over \(\mathbb{F}_1\) whose first Chern class is numerically equivalent to \(3C_0+bf\). This vector bundle results to appear as an extension of two line bundles \(A\) and \(B\) whose numerical class are known. This description as an extension allows the authors to embed the projective bundle associated to \(\mathcal{E}\), say \(X\), in a projective space \(\mathbb{P}^n\), where its degree \(d\) is known. Then \([X]\) is a point in \(\mathcal{X}\), an irreducible component of the Hilbert scheme of 3-dimensional varieties of degree \(d\) in \(\mathbb{P}^n\) and it can be shown, see Prop. 5, that is a smooth point in \(\mathcal{X}\) whose dimension is computed. Moreover, see Theorem 5.1, a count of parameters shows that the general point of \(\mathcal{X}\) parametrizes one of these scrolls. Hilbert schemes; special threefolds; vector bundles; ruled varieties G. M. Besana, M. L. Fania, F. Flamini, Hilbert scheme of some threefold scrolls over the Hirzebruch surface F_{\textit{1}}. \textit{J. Math. Soc. Japan} 65 (2013), 1243-1272. MR3127823 Zbl 1284.14050 \(3\)-folds, Low codimension problems in algebraic geometry, Varieties of low degree, Adjunction problems Hilbert scheme of some threefold scrolls over the Hirzebruch surface \({\mathbb F}_1\) | 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 \(S\to C\) be a smooth quasi-projective surface properly fibered onto a smooth curve. We prove that the multiplicativity of the perverse filtration on \(H^*(S^{[n]},\mathbb{Q})\) associated with the natural map \(S^{[n]}\to C^{(n)}\) implies that \(S\to C\) is an elliptic fibration. The converse is also true when \(S\to C\) is a Hitchin-type elliptic fibration. perverse filtration; \(P = W\) conjecture; Hilbert schemes of points; Nakajima Heisenberg operators Topological properties in algebraic geometry Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper extends previous work by \textit{R. M. Skjelnes} and \textit{D. Laksov} [Compos. Math. 126, 323-334 (2001; Zbl 1056.14500)] and gives a description of the Hilbert scheme of \(n\) points on the affine scheme \(C:=\text{Spec}(K[x]_U)\), where \(K\) is a field and \(K[x]_U\) is a fraction ring of the polynomial ring in one variable.
The Hilbert functor of \(n\) points on \(C\), \(\text{Hilb}^n\), associates to a \(K\)-algebra \(A\) the set \(\text{Hilb}^n(A)\), formed by the ideals \(I\) of \(A\otimes{_KK}[x]_U\) such that the residue class ring \(A\otimes{_KK}[x]_U/I\) is locally free of finite rank \(n\) (as \(A\)-module). The main result is that \(\text{Hilb}^n\) is represented by the fraction ring \(H=K[s_1,...,s_n]_{U(n)}\), where \(s_1,...,s_n\) are the elementary symmetric functions in the variables \(t_1,...,t_n\) and \(U(n)=\{f(t_1)\dots f(t_n)\mid f\in U\}\). Moreover, the universal family of \(n\)-points on \(C\) is isomorphic to \(\text{Sym}_K^{n-1}(C)\times_K C\), as in the case where \(C\) is a smooth curve.
The result relies on a characterization of the characteristic polynomial of the multiplication by the residue of a polynomial in \(A[x]/(F)\), where \(A\) is any commutative ring and \(F\) a monic polynomial in \(A[x]\). Hilbert scheme; resultants; points on the line; characteristic polynomial of the multiplication Skjelnes, R. M.: Resultants and the Hilbert scheme of the line. Ark. mat. 40, No. 1, 189-200 (2002) Parametrization (Chow and Hilbert schemes), Algebraic functions and function fields in algebraic geometry, Fine and coarse moduli spaces, Polynomial rings and ideals; rings of integer-valued polynomials Resultants and the Hilbert scheme of points on the line | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies the family A of all ideals in \(K[x_ 1,...,x_ n]\) that have reduced Gröbner basis with respect to a sequential term ordering \(<_{\sigma}\) on the set T of all monomials in \(x_ 1,...,x_ n\). We obtain that all ideals are of the same dimension and that they allow a parametrization by an affine scheme \(V_ A\) over K. Moreover, if the ordering preserves degrees, then all such ideals have the same Hilbert function. \(V_ A\) is connected; the set of all prime ideals in A, the set of all smooth ideals in A are one to one with the open subsets in \(V_ A.\)
For different J in A, Top(J) can have different associated monomial ideals. Finally, since one can find Top of the monomial ideal associated to J, then it is possible to decide whether this ideal is the same as the monomial ideal of Top(J). reduced Gröbner basis; Hilbert function Ferro, G. Carrà: Gröbner bases and Hilbert schemes. I. J. symb. Comput. 6, No. 2-3, 219-230 (1988) Polynomial rings and ideals; rings of integer-valued polynomials, Parametrization (Chow and Hilbert schemes) Gröbner bases and Hilbert schemes. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a nice survey on the connectedness problem for the Hilbert scheme of curves in the \(3\)-dimensional projective space, with fixed degree \(d\) and genus \(g\).
After recalling the connectedness theorem of \textit{R. Hartshorne} [Publ. Math. Inst. Hautes Études Sci. 29, 5--48 (1966; Zbl 0171.41502)] and the results of \textit{J. Harris} [``Curves in projective space'', Semin. Math. Super. 85, Les Presses de l'Université de Montreal (1982; Zbl 0511.14014)] and \textit{L. Ein} [Ann. Sci. Éc. Norm. Supér., IV. Sér., 19, No. 4, 469--478 (1986; Zbl 0606.14003)] about the open subset of smooth connected curves, the author focuses on \(H_{d,g}\), the Hilbert scheme of equidimensional locally Cohen-Macaulay curves, a particularly interesting object in view of liaison theory. He revises the conditions for \(H_{d,g}\) being non-empty and the notions of extremal and subextremal curve. This is related to an approach to the conjecture on the connectedness of \(H_{d,g}\), which looks for a deformation of a given curve to an extremal one, without passing through curves with embedded points. This strategy results to be winning in several special cases. The conjecture is now proved to be true under the condition \(g\geq {{d-3}\choose{2}}-1\), or for \(d\leq 4\), thanks to the results of several authors.
In the last part of the article, a complete description of the example \(H_{4,-99}\) is given. In particular all families of subcanonical curves in this Hilbert scheme are explicitly determined. Plane and space curves, Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Deformations of space curves: connectedness 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 The results of \textit{S. Boissiére} et al. [Prog. Math. 315, 1--15 (2016; Zbl 1375.14015)] about automorphism of Hilbert schemes of generic
(that is, Picard rank \(1\)) \(K3\) surfaces are here extended from the length
\(2\) case to arbitrary length \(n\). The main result, that there is at most
one nontrivial such automorphism, remains the same. The conditions
under which the automorphism group is nontrivial are determined
precisely. Both a geometric condition and a numerical one are given:
the latter, involving solutions to Pell's equation, is quite
complicated to state, but the geometric condition is just that there
should exist a primitive class in the Néron-Severi group of square \(2\),
or of square \(2(n-1)\) and divisibility \(n-1\). automorphisms; cones of divisors; Hilbert schemes of points; irreducible holomorphic symplectic manifolds; Torelli theorem Automorphisms of surfaces and higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Picard groups, Torelli problem Automorphisms of Hilbert schemes of points on a generic projective \(K3\) 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 We show that the cohomology ring of Hilbert scheme of \(n\)-points in the affine plane is isomorphic to the coordinate ring of a \(\mathbb G_m\)-fixed point scheme of the \(n\)-th symmetric product of \(\mathbb C^2\) for a natural \(\mathbb G_m\)-action on it. This result can be seen as an analogue of a theorem of \textit{C. De Concini} and \textit{C. Procesi} [Invent. Math. 64, 203--219 (1981; Zbl 0475.14041)] and \textit{T. Tanisaki} [Tohoku Math. J. (2) 34, 575--585 (1982; Zbl 0544.14030)] on a description of the cohomology ring of Springer fiber of type \(A\). We also prove similar results for type \(A\) S3-varieties and smooth hypertoric varieties. These results can be formulated in terms of the symplectic duality of \textit{T. Braden} et al. [``Quantizations of conical symplectic resolutions II: category \(\mathcal O\) and symplectic duality'', Preprint, \url{arXiv:1407.0964}]. T. Hikita, \textit{An algebro-geometric realization of the cohomology ring of Hilbert scheme of points in the affine plane}, arXiv:1501.02430. Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients) An algebro-geometric realization of the cohomology ring of Hilbert scheme of points in the affine plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a complex, complete algebraic curve \(C\), one can form the Hilbert schemes \(C^{[m]}\) parametrizing subschemes \(Z \subset C\) of length \(m\) and nested Hilbert schemes \(C^{[m,m+1]}\) parametrizing pairs \(\{(Y,Z): Y \in C^{[m]}, Z \in C^{[m+1]}, Y \subset Z\}\). If \(\pi: \mathcal C \to B\) is a proper flat family of curves, there are relative versions \(\pi^{[m]}: {\mathcal C}^{[m]} \to B\) and \(\pi^{[m,m+1]}: {\mathcal C}^{[m,m+1]} \to B\). For families of curves that are reduced and locally planar over a smooth base \textit{V. Shende} showed that if \(\mathcal C \to B\) is a locally versal family of reduced, locally planar curves over a smooth base \(B\), then the total space \({\mathcal C}^{[m]}\) is smooth over sufficiently small analytic open sets in \(B\) provided \(m \leq \dim B\) [Compos. Math. 148, 531--547 (2012; Zbl 1312.14015)]. The author uses similar methods to prove an analogous smoothness result for \({\mathcal C}^{[m,m+1]}\).
For a smooth family \(\pi^{[m]}: {\mathcal C}^{[m]} \to B\), the decomposition theorem of \textit{Beilinson, Bernstein and Deligne} [Astérisque 100, (1982; Zbl 0536.14011)] says that the complex \(R \pi_*^{[m]} \mathbb C\) decomposes as a direct sum of shifted intersection complexes. In this setting, \textit{L. Migliorini} and \textit{V. Shende} determined the decomposition of \(R \pi^{[m]}_* \mathbb Q [m + \dim B]\) [J. Eur. Math. Soc. (JEMS) 15, 2353--2367 (2013; Zbl 1303.14019)], showing that none of the summands have proper support in \(B\). The author determines the analogous decomposition of \(R \pi_*^{[m,m+1]} \mathbb Q [m+1+\dim B]\) assuming that \({\mathcal C}^{[m,m+1]}\) is smooth, again showing that none of the summands has proper support in \(B\). The proof uses the theory of higher discriminants introduced by \textit{L. Migliorini} and \textit{V. Shende} [Algebr. Geom. 5, No. 1, 114--130 (2018; Zbl 1406.14005)]. locally planar curves; nested Hilbert schemes; versal deformations Parametrization (Chow and Hilbert schemes), Plane and space curves, Algebraic moduli problems, moduli of vector bundles, Transcendental methods, Hodge theory (algebro-geometric aspects) A support theorem for nested Hilbert schemes of planar curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main subject is the Hilbert scheme \(X^{[n]}\) of \(n\) points on a smooth quasi-projective algebraic surface \(X\). The tautological sheaves on the Hilbert scheme are defined by means of the Fourier--Mukai functor. The Bridgeland--King--Reid transform of a tautological sheaf is compute as well as the same for the tensor product of tautological sheaves.
Also Brion--Danila's formulas for the derived direct images of a tensor product of tautological sheaves are proven for the Hilbert-to-Chow morphism. The author obtains general formulas for the derived direct image of a tautological sheaf or for a tensor product of two of them.
As an application, author gives the explicit formulas for cohomology of \(X^{[n]}\) with value in any tautological sheaf or in the tensor product of torsion-free tautological sheaves with disjoint singular loci.
Relevant papers: \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)]; \textit{G. Danila} [J. Algebr. Geom. 10, No. 2, 247--280 (2001; Zbl 0988.14011)]; \textit{M. Haiman} [J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)]; \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)]. Hilbert scheme; tautological sheaves; cohomology; smooth quasiprojective surface Scala, L, Some remarks on tautological sheaves on Hilbert schemes of points on a surface, Geom. Dedicata, 139, 313-329, (2009) Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Some remarks on tautological sheaves on Hilbert schemes of points on a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A=k\langle x_1,\dots,x_m\rangle\) be the free associative \(k\)-algebra, \(k\) algebraically closed. A representation of \(A\) on a finite dimensional vector space \(W\) of dimension \(d\) consists of a tuple \(\varphi_\ast\in\text{End}_k(W)^m.\) The orbits of \(\text{GL}(W)\) in \(\text{End}(W)^m\) correspond bijectively to the isomorphism classes of \(d\)-dimensional representations of \(A\).
Fix another \(k\)-vector space \(V\) of dimension \(n\), together with a basis \(v_1,\dots,v_n.\) Then \(X=\text{Hom}(V,W)\oplus\text{End}(W)^m\) parametrizes \(d\)-dimensional representations of \(A\), together with a fixed linear map from \(V\) to \(W\), and again the group \(\text{GL}(W)\) acts on \(X\) via base change in \(W\).
Define a tuple \((f,\varphi_\ast)\) to be stable if the image of \(f\) generates \(W\) as a representation of \(A\), and let \(X^s\) denote the subset of \(X\) consisting of stable tuples. Then \(H_{d,n}^{(m)}=X^s/\text{GL}(W)\) is the quotient variety of \(X^s\) by \(\text{GL}(W).\) It is then known that \(H_{d,n}^{(m)}\) is smooth and irreducible, of dimension \(N=nd+(m-1)d^2,\) and that the set \(X^s\) of stable tuples is a principal \(\text{GL}(W)\) bundle over \(H_{d,n}^{(m)}.\)
The variety \(H_{d,n}^{(m)}\) has several interpretations, that is, the \(k\)-points parameterize each of the following sets:
(1) Equivalence classes of \(d\)-dimensional representations \(W\) of \(A\) together with an \(n\)-tuple of vectors generating \(W\) as a representation of \(A.\)
(2) Equivalence classes of \(d\)-dimensional representations \(W\) of \(A,\) together with a surjective \(A\)-homomorphism from the free representation \(A^n\) to \(W\).
(3) \(A\)-subrepresentations of codimension \(d\) of the free representation \(A^n.\)
(4) Isomorphism classes of stable representations of the quiver \(Q_n^{(m)}.\)
In particular, the variety \(H_{d,1}^{m}\) parametrizes left ideals of codimension \(d\) in \(A\). Thus it can be viewed as a noncommutative Hilbert scheme for the free algebra in \(m\) generators, in the same way as the Hilbert scheme \(\text{Hilb}^d(\mathbb{A}^m)\) parametrizes ideals of codimension \(d\) in the polynomial ring \(k[x_1,\dots,x_m].\) Denote the quotient variety \(\text{End}(W)^m//\text{GL}(W)\) by \(V_d^{(m)}.\) Then \(V_d^{(m)}=\text{Spec}(k[\text{End}(W)^m]^{\text{GL}(W)}),\) such that its \(k\)-points are in bijection with the semisimple representations of \(A\).
By a result of C. Procesi, the ring \(R=k[\text{End}(W)^m]^{\text{GL}(W)}\) is generated by the functions \((\varphi_1,\dots,\varphi_m)\mapsto\text{tr}(\varphi_{i_1},\dots,\varphi_{i_s})\) for sequences \((i_1,\dots,i_s)\in \{1,\dots,m\}.\) In fact, \(R\) is already generated by such functions for \(s\leq d^2+1.\)
One of the problems in dealing with the varieties \(V_d^{(m)}\) is that they are highly singular, except in the cases \(m=1,\) and \(d=2=m\). Although no explicit desingularizations of the varieties \(V_d^{(m)}\) are known (except in case \(d=2\)), the noncommutative Hilbert schemes \(H_{d,n}^{(m)}\) form a class of closely related smooth varieties: The obvious map \(X^s\rightarrow\text{End}(W)^m,\) is \(\text{GL}(W)\)-equivariant. Thus it descends to a projective morphism \(p:H_{d,n}^{(m)}\rightarrow V_d^{(m)}\) on the level of quotients by \(\text{GL}(W).\) The fibres of \(p\) are difficult to determine in general, but they are at least tractable using the Luna stratification of \(V_d^{(m)}\) and the theory of nullcones of quiver representations. The morphism \(p\) extends the canonical map from the Hilbert scheme \(\text{Hilb}^d(\mathbb{A}^m)\) to the \(d\)th symmetric power \((\mathbb{A}^m)^d/S_d.\)
The main aim of this paper is to prove the following results on the geometry of the varieties \(H_{d,n}^{(m)}\)
The variety \(H_{d,n}^{(m)}\) has a cell decomposition, whose cells are parametrized by \(m\)-ary forests with \(n\) roots and \(d\) nodes.
As a consequence, the Betti numbers in the cohomology of \(H_{d,n}^{(m)}(\mathbb{C})\) can be described in a compact form by assembling the Poincaré polynomials into a generating function. This leads to the result that the cohomological Euler characteristic of \(H_{d,n}^{(m)}\) is given by
\[
\chi(H_{d,n}^{(m)})=\frac{n}{(m-1)d+n}\left(\begin{matrix} md+n-1\\d\end{matrix}\right).
\]
The last main result of the article considers the asymptotic behavior of both the Euler characteristic and the Poincaré polynomials:
After a suitable normalization, the distribution of the Betti numbers of \(H_{d,1}^{(m)}\) for large \(d\) has the Airy distribution as a limit law.
The author thoroughly defines words and forests and uses this to construct a cell decomposition. This is very well written, and the applications that follows are then easy to understand. That is, the normal forms for representations and submodules, and in some sense the intersection theory of \(H_{d,n}^{(m)}\). Also, the generating functions are well treated. The article ends with the final result about the asymptotics of the Euler characteristic of \(H_{d,n}^{(m)}\). representations of free algebras; moduli of representations; Hilbert schemes; Betti numbers Reineke, M.: Cohomology of non-commutative Hilbert schemes. Algebras Represent. Theory \textbf{8}, 541-561 (2005) Algebraic moduli problems, moduli of vector bundles, Representations of quivers and partially ordered sets, Noncommutative algebraic geometry Cohomology of noncommutative 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 \({\mathbb P}^{2[n]}\) be the Hilbert scheme parametrizing zero dimensional subschemes of \({\mathbb P}^2_{\mathbb C}\) of length \(n\). \({\mathbb P}^{2[n]}\) is a smooth irreducible projective variety of dimension \(2n\). In this paper, the authors study the birational geometry of \({\mathbb P}^{2[n]}\). They show that \({\mathbb P}^{2[n]}\) is a Mori-Dream space (in particular \(R(D):=\bigoplus _{m\geq 0}H^0(\mathcal O (mD))\) is finitely generated for any integral divisor \(D\) on \({\mathbb P}^{2[n]}\)). They characterize the effective cone (for many values of \(n\)), and investigate its stable base locus decomposition (into finitely many rational polyhedral cones) and the birational models (corresponding to \({\text{Proj}}(R(D))\) for \(D\) in the big cone). For \(n\leq 9\) they determine the Mori cone decomposition of the cone of big divisors corresponding to different birational models \({\text{Proj}}(R(D))\) and the birational maps (flips and divisorial contractions) between models of adjacent chambers (wall crossings). They also give a modular interpretation in terms of the moduli spaces of Bridgeland semi-stable objects and a description as a moduli space of quiver representations using G.I.T. Hilbert scheme; minimal model program; quiver representations; Bridgeland stability conditions Arcara, D.; Bertram, A.; Coskun, I.; Huizenga, J., The minimal model program for the Hilbert scheme of points on \(\mathbb{P}^2\) and Bridgeland stability, Adv. Math., 235, 580-626, (2013) Minimal model program (Mori theory, extremal rays), Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Stacks and moduli problems The minimal model program for the Hilbert scheme of points on \(\mathbb P^2\) and Bridgeland stability | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 take a new look at the curvilinear Hilbert scheme of points on a smooth projective variety \(X\) as a projective completion of the nonreductive quotient of holomorphic map germs from the complex line into \(X\) by polynomial reparametrisations. Using an algebraic model of this quotient coming from global singularity theory we develop an iterated residue formula for tautological integrals over curvilinear Hilbert schemes. Hilbert scheme of points; curve counting; Göttsche formula; tautological integrals; nonreductive quotients; equivariant localisation; iterated residue Bérczi, G., Tautological integrals on curvilinear Hilbert schemes, Geom. Topol., 2897-2944, (2017) Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Equivariant homology and cohomology in algebraic topology Tautological integrals on curvilinear 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 \(W=m_1P_1+\cdots+m_sP_s\) be a scheme of fat points in \(\mathbb P^n_K\), where \(K\) is a field of characteristic zero.
The authors deal with the problem of computing the Hilbert polynomial of the modules of the Kähler differential \(k\)-forms of the coordinate ring of \(W\), \(\Omega^{k}_{R_{W/K}}\). After introducing notation and basic facts in Section 2, they show in Theorem 3.7 that the Hilbert polynomial of \(\Omega^{n+1}_{R_{W/K}}\) is \(\sum_j\binom{m_j+n-2}{n},\) i.e., it is the Hilbert polynomial of the coordinate ring of the fat points scheme \((m_1-1)P_1+\cdots+(m_s-1)P_s\).
This answers positively to a conjecture the authors stated in a previous paper, see Conjecture 5.7 in [\textit{M. Kreuzer} et al., J. Algebra 501, 255--284 (2018; Zbl 1388.13051)].
Moreover, making use of Theorem 3.7, the authors compute the Hilbert polynomial of the modules of the Kähler differential of a scheme of fat points in \(\mathbb P^2\), see Proposition 4.1. Hilbert function; fat point scheme; regularity index; Kähler differential module Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Modules of differentials, Cycles and subschemes Hilbert polynomials of Kähler differential modules for fat point 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 We calculate the cohomology spaces of the Hilbert schemes of points on surfaces with values in local systems. For that purpose, we generalise I. Grojnoswki's and H. Nakajima's description of the ordinary cohomology in terms of a Fock space representation to the twisted case. We make further non-trivial generalisations of M. Lehn's work on the action of the Virasoro algebra to the twisted and the non-projective case. Building on work by M. Lehn and Ch. Sorger, we then give an explicit description of the cup-product in the twisted case whenever the surface has a numerically trivial canonical divisor. We formulate our results in a way that they apply to the projective and non-projective case in equal measure. As an application of our methods, we give explicit models for the cohomology rings of the generalised Kummer varieties and of a series of certain even dimensional Calabi--Yau manifolds. Hilbert schemes of points on surfaces; rational cohomology ring; locally constant systems; generalised Kummer varieties Nieper-Wißkirchen, M.: Twisted cohomology of the Hilbert schemes of points on surfaces, Doc. math. 14, 749-770 (2009) Parametrization (Chow and Hilbert schemes), (Equivariant) Chow groups and rings; motives Twisted cohomology of 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 We prove that for \(m\in\mathbb{N}\), \(m\) large enough, the number of irreducible components of the schemes of \(m\)-jets centered at a point which is a double point singularity is independent of m and is equal to the number of exceptional curves on the minimal resolution of the singularity. We also relate some irreducible components of the jet schemes of an E6 singularity to its ``minimal'' embedded resolutions of singularities. embedded Nash problem; Hilbert-Poincaré series Mourtada, H., Jet schemes of rational double point singularities, \textit{Valuation Theory in Interaction, EMS Ser. Congr. Rep., Eur. Math. Soc.}, (2014) Arcs and motivic integration, Singularities in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Jet schemes of rational double point 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 W be an open irreducible subset of the Hilbert scheme of \({\mathbb{P}}^ r\) parametrizing smooth irreducible curves of degree n and genus g and \(\pi\) : \(W\to {\mathcal M}_ g\) the natural map to the moduli space of curves of genus g. W is said to have the expected number of moduli if \(\dim(\pi (W))=\min \{3g-3,3g-3+\rho \}\), where \(\rho =\rho (g,n,r)\) is the Brill-Noether number.
The purpose of the paper is the construction of such families of curves for negative \(\rho\) 's, extending the previous existence range shown by \textit{E. Sernesi} [Invent. Math. 75, 25-57 (1984; Zbl 0541.14024)]. For \(r\geq 12\) or \(r=10\), it is shown the existence of families having the expected number of moduli in the range \(-(5r+\epsilon)g/(4r+9- \epsilon)+f(r)\leq \rho \leq 0\) where \(\epsilon\) is defined by \((r- \epsilon)/3=\lceil (r-1)/3\rceil\) and f(r) is a rational function of r asymptotically like \(5/4r^ 2\) (for the explicit expression see section 1 of the paper). - Note that the range where the existence problem makes sense is \(\rho \geq -3g+3\) and that the above result gives an affirmative answer roughly for \(\rho \geq -5(g-r^ 2)/4.\)
The proof has two parts:
(a) A careful study of the normal bundle of general nonspecial curves in \({\mathbb{P}}^ r\). This is accomplished by studying the special case of nonspecial curves lying in suitable rational normal surface scrolls;
(b) an inductive proof starting from curves \(C\subset {\mathbb{P}}^ r\) whose existence was proved by E. Sernesi [loc. cit.] and attaching general nonspecial curves \(\Gamma\) at \(r+4\) general points. The results in (a) are then used to show that this construction is possible and that the curves \(C\cup \Gamma\) are flatly smoothable (the latter by standard deformation theory techniques developed by E. Sernesi [loc. cit.]). Hilbert scheme; expected number of moduli; Brill-Noether number Lopez, AF, On the existence of components of the Hilbert scheme with the expected number of moduli, Math. Ann., 289, 517-528, (1991) Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) On the existence of components of the Hilbert scheme with the expected number of moduli | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors construct an exact fundamental solution to the Fuchsian system \(\Phi_{\lambda}=\sum_{j=1}^N\frac{A_j}{\lambda-\lambda_j}\Phi\), where \(A_j\) are \(N\times N\) matrices of the special structure corresponding to the so-called Hurwitz Frobenius manifold of dimension \(N\). In this case, \(A_j=-E_j(V+qI)\) where \(E_j=\text{diag}(0,\dots,1,\dots,0)\), \(V=[\Gamma, U]\) with \(\Gamma\) the off-diagonal matrix of the so-called rotation coefficients, \(U=\text{diag}(\lambda_1,\dots,\lambda_N)\) and \(q=-1/2\). Entries of the vector solutions \(\Phi^{(\mathbf{s})}(\lambda)\) to the above Fuchsian equation are given by the integrals over the contours \(\mathbf{s}_k\), \(k=1,\dots,2g+d+m-2=N\), of the particular meromorphic differentials \(W(P,P_j)\), \(j=1,\dots,N\), and the meromorphic function \(f(P)\) of degree \(d\) both defined on a Riemann surface of genus \(g\). The set of the contours of integration consists of \(2g\) contours of the homology basis to the Riemann surface, \(m-1\) closed contours encircling the punctured pre-images \(f^{-1}(\infty)\), and \(d-1\) open contours connecting \(\lambda^{(n)}\) and \(\lambda^{(n+1)}\) from the pre-image \(f^{-1}(\lambda)\). The proof is based on the direct verification of the authors' formula using transformation properties and the Rauch variational formulas for the canonical normalized bidifferential \(W(P,Q)\). The authors also construct the fundamental solutions to the Fuchsian system for any negative half-integer \(q\) and some vector solutions for positive half-integer \(q\). They also study the monodromy group of such \(\Phi(\lambda)\), the action of the braid group and compute the monodromy matrices explicitly in various special cases. Frobenius manifolds; Fuchsian system; monodromy group; braid group; Riemann surface; canonical bidifferential Korotkin, D., Shramchenko, V.: Riemann--Hilbert problem for Hurwitz Frobenius manifolds: regular singularities. J. Reine und Angew. Math. (to be published) Isomonodromic deformations for ordinary differential equations in the complex domain, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Riemann-Hilbert problem for Hurwitz Frobenius manifolds: regular 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 a classification of simple hypersurface singularities by the classification of irreducible Weyl groups not using the normal forms. He proves: For any singularity the following conditions are equivalent:
(1) the singularity is simple;
(2) the singularity is elliptic;
(3) the monodromy group of the singularity is finite;
(4) the monodromy group of the singularity is isomorphic to a Weyl group of type \(A_K\), \(D_K\), \(E_6\), \(E_7\), \(E_8\);
(5) the mixed Hodge structure in the vanishing cohomologies of the singularity is trivial;
(6) the length of the spectrum of the singularity is less than one.
Furthermore, if two simple singularities have isomorphic monodromy groups then they are stably equivalent. simple hypersurface singularities; classification of irreducible Weyl groups; monodromy group Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Singularities in algebraic geometry On the A-D-E classification of the simple singularities of functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective variety over the complex numbers \(\mathbb{C}\) and \(X^{[3]}\) the Hilbert scheme of subschemes of length 3 of \(X\), which is known to be smooth. In this work the ring structure of the cohomology ring \(H^*(X^{[3]},\mathbb{Q})\) is determined, for an arbitrary smooth projective variety \(X\), in terms of the structure of the cohomology ring and the Chern classes of \(X\). This is done by first describing the cohomology of \(X^{[2,3]}\), a variety parametrizing couples of subschemes \(Z_ 2\subset Z_ 3\) of \(X\) of lengths 2 and 3 respectively introduced by Elencwaig and Le Barz. Then \(H^*(X^{[3]},\mathbb{Q})\) is described as a subring of \(H^*(X^{[2,3]},\mathbb{Q})\) and the Chern classes of universal bundles on \(X^{[3]}\) are computed. Hilbert scheme of subschemes of length 3; parametrizing couples of subschemes; Chern classes of universal bundles DOI: 10.1515/crll.1993.439.147 Étale and other Grothendieck topologies and (co)homologies, Parametrization (Chow and Hilbert schemes), Characteristic classes and numbers in differential topology The cohomology ring of the Hilbert scheme of 3 points on a smooth projective variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We show that Hilbert schemes of points on supersingular Enriques surface in characteristic 2, \(\mathrm{Hilb}^n(X)\), for \(n \geq 2\) are simply connected, symplectic varieties but are not irreducible symplectic as the Hodge number \(h^{2 , 0} > 1\), even though a supersingular Enriques surface is an irreducible symplectic variety. These are the classes of varieties which appear only in characteristic 2 and they show that the Hodge number formula for \textit{L. Göttsche} and \textit{W. Soergel} [Math. Ann. 296, No. 2, 235--245 (1993; Zbl 0789.14002)] does not hold over characteristic 2. It also gives examples of varieties with trivial canonical class which are neither irreducible symplectic nor Calabi-Yau, thereby showing that there are strictly more classes of simply connected varieties with trivial canonical class in characteristic 2 than over \(\mathbb{C}\) as given by Beauville-Bogolomov decomposition theorem. Hilbert scheme of points; Enriques surface; positive characteristic; lifting to characteristic zero; irreducible symplectic varieties Positive characteristic ground fields in algebraic geometry, Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory, \(K3\) surfaces and Enriques surfaces, Holomorphic symplectic varieties, hyper-Kähler varieties Pathologies of the Hilbert scheme of points of a supersingular Enriques surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathcal H_{d,g,r}\) be the union of the Hilbert scheme components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree d and genus g in \(\mathbb P^r\). In this paper the authors study \(\mathcal H_{d,g,r}\) for the case \(r =4\) and \(d = g+2\). The main result shows that any non-empty \(\mathcal H_{g+2,g,4}\) is irreducible without any restriction on \(g\). The same result was previously established by \textit{H. Iliev} [Proc. Am. Math. Soc. 134, No. 10, 2823--2832 (2006; Zbl 1097.14022)] with several low genus cases left untreated. Hilbert scheme; algebraic curves; linear series Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) Irreducibility of the Hilbert scheme of smooth curves in \(\mathbb {P}^4\) of degree \(g+2\) and genus \(g\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 explicitly constructs \(\Theta\)-invariants in the case \(\tilde E_6\) using Jacobi forms recently introduced by \textit{K. Wirthmüller} [Compos. Math. 82, 293-354 (1992; Zbl 0780.17006)] and the structure theorem for the graded ring of all these forms. See also Proc. Japan Acad., Ser A 69, 247-251 (1993; Zbl 0805.32018). surface singularity; Jacobi forms; theta-invariants; graded ring Jacobi forms, Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Simple, semisimple, reductive (super)algebras, Complex surface and hypersurface singularities, Period matrices, variation of Hodge structure; degenerations, Local structure of morphisms in algebraic geometry: étale, flat, etc. Flat structure for the simple elliptic singularity of type \(\tilde E_6\) and Jacobi forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider a blow-up square of schemes of the form
\[
\begin{tikzcd} Y \arrow[r] \arrow[d] & X\arrow[d, "f"]\\ V \arrow[r] & \mathrm{Spec}(A)\end{tikzcd}
\]
Fix a non-negative integer \(n\). Motivated by analogies with Gersten's conjecture, the authors seek conditions under which the map
\[
K_n(A,I)\rightarrow K_n(A/I^r,I/I^r)\oplus K_n(X,Y_{\text{red}})\eqno(1)
\]
is injective for all sufficiently large \(r\). When \(A\) is local, noetherian, quasi-excellent and contains a field of characteristic \(0\), the authors establish injectivity if \(Y_{\text{red}}\) is regular. (For \(n=2\), this last condition can be dropped.)
It follows from this and some additional argument that if \(k\) is a field of characteristic \(0\) and \(C\hookrightarrow {\mathbb A}^{N+1}_k\) is the cone over a smooth projective variety with \((A,{ m})\) the local ring at the singular point, then we have injectivity of
\[
K_n(A)\rightarrow K_n(A/m^r)\oplus K_n(Spec(A)-\{m\})\eqno(2)
\]
again for sufficiently large \(r\).
The authors show by counterexample that (2) need not be injective for general isolated singularities, even in dimension one. They conjecture, however, that (1) holds in much greater generality.
The proofs rely heavily on computations in cyclic and Hochschild homology. Gersten's conjecture; algebraic \(K\)-theory; singular schemes; Hochschild homology; cyclic homology \(K\)-theory of schemes, Singularities in algebraic geometry Analogues of Gersten's conjecture 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 Let \(G\subset\mathrm{GL}(V)\) be a reductive algebraic subgroup acting on the symplectic vector space \(W=(V\oplus V^\ast)^{\oplus m}\), and let \(\mu:W\to\mathrm{Lie}(G)^\ast\) be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction \(\mu^{-1}(0)//G\) for classes of examples where \(G=\mathrm{GL}(V)\), \(O(V)\), or \(\mathrm{Sp}(V)\). For these classes of examples, \(\mu^{-1}(0)//G\) is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert-Chow morphism with the (well-known) symplectic desingularizations of \(\mu^{-1}(0)//G\). invariant Hilbert scheme;a canonical desingularization; symplectic reduction R. Terpereau, \textit{Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups}, Math. Z. \textbf{277} (2014), no. 1-2, 339-359. Parametrization (Chow and Hilbert schemes), Group actions on affine varieties, Linear algebraic groups over arbitrary fields, Momentum maps; symplectic reduction, Group actions on varieties or schemes (quotients) Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We provide a criterion for when Hilbert schemes of points on \(K3\) surfaces are birational. In particular, this allows us to generate a plethora of examples of non-birational Hilbert schemes which are derived equivalent. Parametrization (Chow and Hilbert schemes), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, \(K3\) surfaces and Enriques surfaces Derived equivalent Hilbert schemes of points on \(K3\) surfaces which are not birational | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 submodules of analytic Hilbert modules defined over certain algebraic varieties in bounded symmetric domains, the so-called Jordan-Kepler varieties \(V_\ell\) of arbitrary rank \(\ell \). For \(\ell> 1\), the singular set of \(V_\ell\) is not a complete intersection. Hence the usual monoidal transformations do not suffice for the resolution of the singularities. Instead, we describe a new higher rank version of the blow-up process, defined in terms of Jordan algebraic determinants, and apply this resolution to obtain the rigidity of the submodules vanishing on the singular set. analytic Hilbert module; algebraic variety; symmetric domain; reproducing kernel; curvature; rigidity Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), Determinantal varieties, Associated manifolds of Jordan algebras, Toeplitz operators, Hankel operators, Wiener-Hopf operators Singular Hilbert modules on Jordan-Kepler varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a smooth projective algebraic surface over an algebraically closed field \(\kappa\) and let \(H_d= \text{Hilb}_bS\) be the Hilbert scheme of 0-dimensional subschemes of length \(d\) in \(S\). Taking a point \(x\in S\), we shall study the set \(H_d[x]= \{\xi\in H_d |\text{Supp} \xi =x\}\). The set \(H_d[x]\) is obviously endowed with a variety structure, i.e., it is a reduced irreducible scheme of dimension \(d-1\) over \(\kappa\). Moreover, the set \(H_d[x]\) is a subscheme in \(H_d\). It is referred to as the punctual Hilbert scheme of the surface. For \(d=1\) and 2, its description is trivial: \(H_1[x]= \{\text{point}\}\) and \(H_2[x]= P(T_xS) \simeq \mathbb{P}^1\). Even for \(d=3\), the set \(H_d[x]\) acquires singularities: The set \(H_3[x]\) is a surface isomorphic to a cone over the space cubic curve in \(\mathbb{P}^3\). For higher dimensions \(d\), the singularities of the set \(H_d[x]\) are quite complicate. The description can, possibly, be made in terms of Iarrobino's stratification.
In this paper, we use a different approach based on the natural birational model \(X_d\) of the scheme \(H_d[x]\), which is obtained by ``lifting'' the scheme \(H_d[x]\) to the Hilbert scheme of complete flags \(\Gamma_{12 \dots d} =\{(\xi_1, \dots, \xi_d)\in \prod^d_{k=1} H_k |\xi_1 \subset\xi_2 \subset \cdots \subset \xi_d\}\). The model thus constructed is helpful in giving an algebraic-geometric description of the three-dimensional variety \(H_4[x]\) and its singularities. punctual Hilbert scheme of a surface; complete flags; singularities A. S. Tikhomirov, ''A smooth model of punctual Hilbert schemes of a surface,''Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],208, 318--334 (1995). Parametrization (Chow and Hilbert schemes), Surfaces and higher-dimensional varieties A smooth model for the punctual Hilbert scheme of a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective complex surface. Long ago \textit{J. Fogarty} showed that the Hilbert scheme \(X^{[n]}\) parametrizing schemes of finite length \(n\) is smooth and irreducible of dimension \(2n\) [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. Here the authors describe the nef cone of \(X^{[n]}\) and the dual cone \(\mathrm{NE} (X^{[n]})\) spanned by effective curves when \(n \geq 2\) in terms of the corresponding cones on \(X\) under certain conditions.
To state the main theorem, suppose that \(H^1(X, {\mathcal O}_X)=0\). If \(B_n \subset X^{[n]}\) denotes the boundary divisor consisting of non-reduced subschemes \(Z \subset X\), there is an isomorphism \(\Psi: \mathrm{Pic} (X^{n]}) \cong \mathrm{Pic} (X) \oplus \mathbb Z \cdot (B_n / 2)\), due to \textit{J. Fogarty} [Am. J. Math. 95, 660--687 (1973; Zbl 0299.14020)]. Now assume that (a) the nef cone of \(X\) is spanned by \(F_1, \dots, F_t\) and \(\mathrm{NE} (X)\) is spanned by curves \(C_1, \dots, C_t\) satisfying \(F_i \cdot C_j = \delta_{i,j}\) and (b) the divisor \(L=(n-1) \sum F_i\) is \((n-1)\)-very ample in the sense of \textit{M. Beltrametti} and \textit{A. J. Sommese} [in: Problems in the theory of surfaces and their classification. Papers from the meeting held at the Scuola Normale Superiore, Cortona, Italy, October 10-15, 1988. London: Academic Press; Rome: Istituto Nazionale di Alta Matematica Francesco Severi. 33--48; appendix: 44--48 (1991; Zbl 0827.14029)], meaning that \(H^0(X,L) \to H^0(X, {\mathcal O}_Z \otimes L)\) is surjective for each \(Z \in X^{[n]}\). Then the authors prove that (1) the nef cone of \(X^{[n]}\) is spanned by \(D_{F_1}, \dots, D_{F_t}, (n-1) \sum D_{F_i} - B_n / 2\), where \(D_{F_i} \in \mathrm{Pic} (X^{[n]}\) corresponds to \(F_i\) under the isomorphism \(\psi\) and (2) \(\mathrm{NE} (X^{[n]})\) is spanned by the classes
\[
\beta_{C_1} - (n-1) \beta_n, \dots, \beta_{C_t} - (n-1) \beta_n, \beta_n,
\]
where \(\beta_{C_i} = \{x + x_1 + \dots + x_{n-1}: x \in C_i\}\) for some fixed points \(x_1, \dots, x_{n-1}\) not on the \(C_i\) and \(\beta_n = \{ Z + x_2 + \dots + x_{n-1} \in X^{[n]}: \mathrm{Supp} (Z) = \{x_1\} \}\). This extends earlier work of \textit{W.-P. Li} et al. [Contemp. Math. 322, 89--96 (2003; Zbl 1057.14012)].
They apply the theorem to a Hirzebruch surface \(X\), recovering a result of \textit{A. Bertram} and \textit{I. Coskun} [in: Birational geometry, rational curves, and arithmetic. Based on the symposium ``Geometry over closed fields'', St. John, UK, February 2012. New York, NY: Springer. 15--55 (2013; Zbl 1273.14032)]. They also classify the curves whose homology classes are in the list above which have minimal degree in the sense that their intersection numbers with certain very ample divisors are all equal to one, proving that their moduli spaces are smooth of the expected dimension. nef cones; Hilbert schemes Parametrization (Chow and Hilbert schemes) The nef cones of and minimal-degree curves in the Hilbert schemes of points on certain surfaces | 0 |
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