text stringlengths 571 40.6k | label int64 0 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 The author derives the Jordan canonical form for the monodromy operator of an isolated singularity of a hypersurface in \({\mathbb{C}}^ n\). He also computes the cohomology with coefficients in \({\mathbb{Z}}\) of the algebraic knot of this singularity. singularities of hypersurfaces; monodromy operator; cohomology of algebraic knots Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Knots and links (in high dimensions) [For the low-dimensional case, see 57M25] A formula for the computation of the integral cohomology for the knot of an isolated singularity of a hypersurface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Denote by \(R(Q,\alpha)\) the affine space of \(\alpha\)-dimensional representations of a quiver \(Q\). The \textit{generic semisimple decomposition of} \(\alpha\) is the sum \(\alpha=\beta_1+\cdots+\beta_k\), where \(R(Q,\alpha)\) contains a dense open subset \(U\) such that for all \(x\in U\), the simple composition factors of the representation \(V_x\) corresponding to \(x\) have dimension vectors \(\beta_1,\dots,\beta_k\). The main results of the paper are a characterisation of the generic semisimple decomposition in terms of so-called orthogonal sequences, and an algorithm to compute it. The author works over an algebraically closed base field of characteristic zero, and uses results of [\textit{L. Le Bruyn} and \textit{C. Procesi}, Trans. Amer. Math. Soc. 317, 585--598 (1990; Zbl 0693.16018)]. We note that the results in loc. cit. were extended to positive characteristic in [\textit{M. Domokos} and \textit{A. N. Zubkov}, Algebr. Represent. Theory 5, 305--317 (2002; Zbl 1061.16023)] and in [\textit{M. Domokos}, NATO Sci. Ser. II, Math. Phys. Chem. 28, 47--61 (2001; Zbl 1004.16013)] (compare Corollary 1.2.5 of the latter paper with Theorem 1.4). So the results of the present work hold in positive characteristic as well. quiver representation; generic decomposition; semisimple representation; orthogonal sequence Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets On generic semi-simple decomposition of dimension vector for an arbitrary quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) parametrizing closed subschemes of \(\mathbb P^n\) with Hilbert polynomial \(p(t)\) has been of great interest every since Grothendieck constructed it in the early 1960s. Early results include the connectedness theorem of \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)] and smoothness of \(\mathrm{Hilb}^{p(t)} (\mathbb P^2)\) due to \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. \textit{A. Reeves} and \textit{M. Stillman} showed that every non-empty Hilbert scheme contains a smooth Borel-fixed point [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)] and \textit{A. P. Staal} classified those with exactly one such fixed point, which are necessarily smooth and irreducible [Math. Z. 296, No. 3--4, 1593--1611 (2020; Zbl 1451.14010)].
The main result classifies Hilbert schemes with two Borel-fixed points over a field \(k\) of characteristic zero. To describe the result, express the Hilbert polynomial \(p(t)\) in the form used by \textit{Gotzmann}, namely
\[
p(t) = \sum_{i=1}^m \binom{t+\lambda_i-i}{\lambda_i-1}
\]
where \(\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_m \geq 1\) [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009)]. Writing \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\), the theorem lists for exactly which \(\mathbf{\lambda}\) the Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) has two Borel-fixed points and further determines when it is (a) smooth, (b) irreducible and singular or (c) a union of two components. In each case the irreducible components are normal and Cohen-Macaulay and the singularities of the Hilbert scheme appear as cones over certain Segre embeddings of \(\mathbb P^a \times \mathbb P^b\). Since the writing of his paper, (a) \textit{A. P. Staal} [``Hilbert schemes with two Borel-fixed points in arbitrary characteristic'', Preprint, \url{arXiv:2107.02204}] has shown that the theorem is valid in all characteristics with a small modification when char \(k=2\) and (b) \textit{R. Skjelnes} and \textit{G. G. Smith} [J. Reine Angew. Math. 794, 281--305 (2023; Zbl 07640144)] have classified the smooth Hilbert schemes are described their geometry.
Despite the difficulty of the content, the paper is readably written. Section 1 gives preliminaries on Borel-fixed (strongly stable) ideals and the resolution of \textit{S. Eliahou} and \textit{M. Kervaire} [J. Algebra 129, No. 1, 1--25 (1990; Zbl 0701.13006)], while Section 2 identifies the tuples \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\) corresponding to Hilbert schemes with two components. Section 3 uses the comparison theorem of \textit{R. Piene} and \textit{M. Schlessinger} [Am. J. Math. 107, 761--774 (1985; Zbl 0589.14009)] to compute the tangent space of the non-lexicographic Borel-fixed ideal \(I(\mathbf{\lambda})\) and give a partial basis for the second cohomology group of \(k[x_0,\dots,x_n]/I(\mathbf{\lambda})\). These are used in Section 4 where the main theorem is proved to describe the universal deformation space of \(I(\mathbf{\lambda})\) and hence the nature of singularities of the Hilbert schemes. Finally in Section 5 the author gives examples of Hilbert schemes with three Borel-fixed points. The last three examples relate to Hilbert schemes studied in the literature [\textit{S. Katz}, in: Zero-dimensional schemes. Proceedings of the international conference held in Ravello, Italy, June 8-13, 1992. Berlin: de Gruyter. 231--242 (1994; Zbl 0839.14001); \textit{D. Chen} and \textit{S. Nollet}, Algebra Number Theory 6, No. 4, 731--756 (2012; Zbl 1250.14004); \textit{D. Chen} et al., Commun. Algebra 39, No. 8, 3021--3043 (2011; Zbl 1238.14012)]. Hilbert scheme; singularities; Borel-fixed points; deformations of ideals Syzygies, resolutions, complexes and commutative rings, Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Singularities of surfaces or higher-dimensional varieties Hilbert schemes with two Borel-fixed 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 study binomial \(D\)-modules, which generalize \(A\)-hypergeometric systems. We determine explicitly their singular loci and provide three characterizations of their holonomicity. The first of these is an equivalence of holonomicity and \(L\)-holonomicity for these systems. The second refines the first by giving more detailed information about the \(L\)-characteristic variety of a non-holonomic binomial \(D\)-module. The final characterization states that a binomial \(D\)-module is holonomic if and only if its corresponding singular locus is proper. binomial \(D\)-modules; \(A\)-hypergeometric systems Berkesch, Christine; Matusevich, Laura Felicia; Walther, Uli, Singularities and holonomicity of binomial \(D\)-modules, J. Algebra, 439, 360-372, (2015) Sheaves of differential operators and their modules, \(D\)-modules, Singularities in algebraic geometry, Other hypergeometric functions and integrals in several variables, Toric varieties, Newton polyhedra, Okounkov bodies Singularities and holonomicity of binomial \(D\)-modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A sandwiched singularity is a local ring \(\mathcal O\) which is a birational extension of a two-dimensional regular local ring \(R\). Let us consider a surface \(S\) which has a point whose local ring is a sandwich singularity. The authors study in this paper the equisingularity classes of all birational projections of \(S\) to a plane. This problem was dealt with in \textit{M. Spivakovky}'s paper [Ann. Math. (2) 131, 411--491 (1990; Zbl 0719.14005)]. Spivakovsky divides this problem into two parts: discrete and continuous. In this paper the authors deal with the discrete part.
Any birational projection from a sandwiched singularity to a plane is obtained by the morphism of blowing up a complete \(\mathfrak m_{ O}\)-primary ideal \(J\) in the local ring of a regular point \(O\) on the plane. The goal of this paper is to give the equisingularity type of these ideals. More precisely: Let \(\mathcal O\) be a birational normal extension of a regular local ring \((R,\mathfrak m_O)\); the authors describe the equisingularity type of any complete \(\mathfrak m_O\)-primary ideal \(J\subset R\) such that its blow-up \(\text{Bl}_J(R)\) has some point \(Q\) whose local ring is analytically isomorphic to \(\mathcal O\). This is done by the Enriques diagram of the cluster of base points of any such ideal. Recall that an Enriques diagram is a tree together with a binary relation---proximity--- representing topological equivalence classes of clusters of points in the plane.
Therefore, in section 1 the authors give an overview concerning the language of infinitely near points, sandwiched surface singularities and graphs. A good source for this is the book of \textit{E. Casas-Alvero} [Singularities of Plane Curves. Cambridge University Press (2000; Zbl 0967.14018)]. In section two the authors introduce a technical device, namely the consept of contraction for a sandwiched singularity \(\mathcal O\). A contraction for a sandwiched singularity is the resolution graph \(\Gamma_{\mathcal O}\) of \(\mathcal O\), enriched by some proximity relations between their vertices. After having fixed a sandwich graph, the problem is to find the whole list of possibilities for such proximities. This is achieved in section 3. In section 4 the authors study the problem of describing the equisingularity classes of the ideals for a given sandwiched surface singularity, i.e., they describe all the possible Enriques diagram for~\(\mathcal O\). normal surface singularity; sandwiched singularity; complete ideal; Enriques diagram; equisingularity Alberich-Carramiñana, M.; Fernández-Sánchez, J.: Equisingularity classes of birational projections of normal singularities to a plane, Adv. math. 216, 753-770 (2007) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Regular local rings Equisingularity classes of birational projections of normal singularities to a 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 The authors explicitly compute the set of limits of tangents of a quasi-ordinary complex hypersurface singularity in terms its special monomials which are determined by the parametrization of the hypersurface near the singularity. As a corollary they show that the limits of the tangents is a topological invariant of the singularity. quasi-ordinary singularity; limits of tangents Singularities in algebraic geometry, Local complex singularities, Complex surface and hypersurface singularities Limits of tangents of quasi-ordinary hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this article, we study the asymptotic cardinality of the set of algebraic points of fixed degree and bounded height of a surface defined over a number field, when the bound on the height tends to infinity. In particular, we show that this can be connected to the Batyrev-Manin-Peyre conjecture, i.e. the case of rational points, on some punctual Hilbert scheme. Our study shows that these associated Hilbert schemes provide, under certain conditions, new counterexamples to the Batyrev-Manin-Peyre conjecture. However, in the cases of \(\mathbb{P}^1 \times \mathbb{P}^1\) and \(\mathbb{P}^2\) detailed in this article, the associated Hilbert schemes satisfy a slightly weaker version of the Batyrev-Manin-Peyre conjecture. number theory; algebraic point; Hilbert scheme; heights Rational points, Varieties over global fields, Parametrization (Chow and Hilbert schemes), Heights Algebraic points of bounded height 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 As a measure of the complexity of the singularities of a variety, one can study the sequence of jumping numbers. One way of defining the jumping numbers is through an embedded resolution of singularities and the definition is independent of the chosen resolution. In this article the authors define what it means for an exceptional divisor on such a resolution to contribute to a specific jumping number. Their main result is that an exceptional divisor \(E\) of a minimal embedded resolution of a curve \(C\) on a smooth surface contributes to the sequence of jumping numbers if and only if \(E\) has non-trivial intersection with at least three of the other components of the full transform of \(C\). multiplier ideal; jumping number; plane curve Karen E. Smith and Howard M. Thompson, Irrelevant exceptional divisors for curves on a smooth surface, Algebra, geometry and their interactions, Contemp. Math., vol. 448, Amer. Math. Soc., Providence, RI, 2007, pp. 245 -- 254. Singularities in algebraic geometry, Singularities of curves, local rings, Local complex singularities Irrelevant exceptional divisors for curves on a smooth 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 In the paper under review, the authors compute two point extremal Gromov-Witten invariants of Hilbert schemes of points on surfaces, and determine the extremal quantum boundary operator. The main ideas are to use the localized virtual fundamental cycle and to apply the results of Okounkov-Pandharipande on the quantum cohomology of Hilbert schemes of points on the affine plane \(\mathbb C^2\).
In Section 2, the authors review standard results about the Hilbert schemes \(X^{[n]}\) of points on smooth algebraic surfaces \(X\), and mentioned Ruan's Cohomological Crepant Resolution Conjecture. In Section 3, based on methods of Kiem-J. Li, some reduction lemmas about the localized virtual fundamental cycle of stable maps to \(X^{[n]}\) were proved. These stable maps are extremal in the sense that their images are contracted to points by the Hilbert-Chow morphisms. In Section 4, a universal formula about two point extremal Gromov-Witten invariants of \(X^{[n]}\) is established. In Section 5 and Section 6, the authors studies the extremal quantum boundary operator, and determines it when \(X\) is the projective plane \(\mathbb P^2\). In Section 7, two point extremal Gromov-Witten invariants and the extremal quantum boundary operator for an arbitrary surface \(X\) are computed. These results are then applied to partially confirm Ruan's conjecture. Hilbert schemes; Gromov-Witten invariants; localization technique Li, J; Li, W-P, Two point extremal Gromov-Witten invariants of Hilbert schemes of points on surfaces, Math. Ann., 349, 839-869, (2011) Parametrization (Chow and Hilbert schemes), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vertex operators; vertex operator algebras and related structures Two point extremal Gromov-Witten invariants 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 For a fixed term ordering one can consider the affine variety parametrizing all ideals with a fixed initial ideal, referred to here as a Gröbner stratum. These stratify the Hilbert scheme \(H\) (of subschemes of projective space with fixed Hilbert polynomial) and therefore their structure may reveal important information about the \(H\).
In this paper the authors prove several theorems involving Gröbner strata, first of which is a very nice intrinsic definition of its defining ideal. Furthermore the authors are able to prove many rationality results for components of the Hilbert scheme. For example they show, using Gr\"boner strata, that any smooth, irreducible component of \(H\), as well as the Reeves and Stillman component (which contains the Lex ideal) is rational.
The paper is a an excellent resource on Gröbner strata and will be useful to anybody interested in the study of the Hilbert scheme. It is dotted with illuminating examples. The authors also present some useful algorithms as well as obtain results which can improve explicit computation of equations defining these subsets. Hilbert schemes; Gröbner bases; initial ideals Lella, P., Roggero, M.: Rational components of Hilbert schemes. Rendiconti del Seminario Matematico dell'Università di Padova \textbf{126}, 11-45 (2011) Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Rational 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 This paper gives reasonable and explicit bounds on the central fiber of the degeneration of surfaces of general type with given \(\chi({\mathcal O}_X)\) and \(K^2_X\).
Section 1 presents reasonable bounds for the number of singularities outside the double curve on the central fiber of a relatively minimal permissible degeneration of surfaces of general type (theorem 10), and the number of components in a relative canonical model (theorem 11).
In section 2, we figure out how to change discrepancies for the iteration of singularity of class \(T\) (theorem 15). This provides several inductive formulas used in section 3 for a singularity of class \(T\). Section 3 clarifies the reason behind the bound of the index of singularity of the central fiber under the simplifying assumption that the central fiber is irreducible, with resolution a surface of general type (theorem 23). Also we present explicit bounds for the index of singularity for this case in theorem 23. degenerations of surfaces of general type; number of singularities; number of components; index of singularity Lee, Y., Numerical bounds for degenerations of surfaces of general type, International Journal of Mathematics, 10, 79-92, (1999) Global theory and resolution of singularities (algebro-geometric aspects), Fibrations, degenerations in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Surfaces of general type, Minimal model program (Mori theory, extremal rays) Numerical bounds for degenerations of surfaces of general type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A total mass is the weighted count of continuous homomorphisms from the absolute Galois group of a local field to a finite group. In the preceding paper [Int. Math. Res. Not. 2015, No. 23, 12590--12619 (2015; Zbl 1387.11088)], the authors observed that in a particular example two total masses coming from two different weightings are dual to each other. We discuss the problem of how generally such a duality holds and relate it to the existence of simultaneous resolution of singularities, using the wild McKay correspondence and the Poincaré duality for stringy invariants. We also exhibit several examples. mass formulas; local Galois representations; quotient singularities; dualities; McKay correspondence; equisingularities; stringy invariants Galois theory, Ramification and extension theory, Varieties over finite and local fields, Global theory and resolution of singularities (algebro-geometric aspects), McKay correspondence Mass formulas for local Galois representations and quotient singularities. II: Dualities and resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In [\textit{J. Briançon}, \textit{Y. Laurent} and \textit{Ph. Maisonobe}, C. R. Acad. Sci., Paris, Sér. I 313, No. 5, 285-288 (1991; Zbl 0743.32011)] the equivalence between the existence of relative Bernstein polynomials associated with a deformation \(F\) of isolated singularities hypersurfaces and equisingularity properties was proved. After having defined a natural equisingularity notion for an analytic map \((F_1,\dots,F_p)\) defined on \(\mathbb{C}^3\), we show that the existence of relative Bernstein polynomials still characterizes the equisingularity and is also equivalent to a ``non-characteristic'' type condition. existence of relative Bernstein polynomials; equisingularity H. Biosca, Caractérisation de l'existence de polynômes de Bernstein relatifs associés à une famille d'applications analytiques, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 4, 395-398. Equisingularity (topological and analytic), Deformations of complex singularities; vanishing cycles, Global theory and resolution of singularities (algebro-geometric aspects) Characterization of the existence of relative Bernstein polynomials associated to a family of analytic maps | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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{G. Ellingsrud} and \textit{Ch. Peskine} [Invent. Math. 95, No, 1, 1-11 (1989; Zbl 0676.14009)] proved a theorem which states that there are only a finite number of components in the Hilbert scheme parameterizing smooth surfaces in \(\mathbb{P}^4\) not of general type. On the other hand, \textit{R. Braun} and \textit{G. Fløystad} [Compos. Math. 93, No. 2, 211-229 (1994; Zbl 0823.14021)] proved that a smooth surface in \(\mathbb{P}^4\) not of general type has degree \(d\leq 105\).
In this note, the author proves that there is only a finite number of components in the Hilbert scheme of surfaces in \(\mathbb{P}^r\) parameterizing integral surfaces of degree \(d\). Then he deduces that a smooth surface in \(\mathbb{P}^4\) not of general type has degree \(d\leq 105\). Thus he obtains a simpler and quicker proof of the above theorem. The proof in this note essentially relies on Castelnuovo's theory and on a lower bound for the genus of the generic hyperplane section of a smooth surface in \(\mathbb{P}^4\). Enriques' classification; Hilbert scheme; Castelnuovo's theory; surfaces in \(\mathbb{P}^4\) Gennaro, V, A note on smooth surfaces in \({\mathbb{P}}^4\), Geometriae Dedicata, 71, 91-96, (1998) Parametrization (Chow and Hilbert schemes), Surfaces of general type, Surfaces and higher-dimensional varieties, Projective techniques in algebraic geometry, Low codimension problems in algebraic geometry A note on smooth surfaces in \(\mathbb{P}^4\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present book is a monograph on the topology of real and complex analytic spaces around an isolated singular point. The first chapters are dedicated mostly to complex singularities; they contain the classical theory of Milnor (chap. 1), the relation between 3-dimensional Lie groups and 2-dimensional complex singularities (chap. 2 and 3), special cases of the general index theorem of Atiyah-Singer (chap. 4), the geometry and topology of quadrics in \(\mathbb{CP}^n\), based on the author's results. The last chapters are about the interplay between complex geometry and real analytic singularities. Chapter 6 contains the necessary material for constructing and studying infinite families of singularities satisfying the strong Milnor condition (chapter 7). Chapter 8 is based on the results of the author on real singularities and open book decomposition of the 3-sphere.
The book is written in a clear and accessible style, for graduates, postgraduates and researchers in complex analysis, geometry and topology. Seade, José, On the topology of isolated singularities in analytic spaces, Progress in Mathematics 241, xiv+238 pp., (2006), Birkhäuser Verlag, Basel Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Global theory of complex singularities; cohomological properties, Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry On the topology of isolated singularities in analytic 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 Some technical results on the deformations of varieties of general type and on permanence of semi-log-canonical singularities are proved. These results are applied to show that the connected component of the moduli space of stable surfaces containing the moduli point of a product of stable curves is the product of the moduli spaces of the curves, assuming the curves have different genera. An application of this result shows that even after compactifying the moduli space and fixing numerical invariants, the moduli spaces are still very disconnected. semi-log-canonical singularities; components of moduli schemes Michael A. van Opstall, Moduli of products of curves, Arch. Math. (Basel) 84 (2005), no. 2, 148 -- 154. Families, moduli, classification: algebraic theory, Families, moduli of curves (algebraic), Formal methods and deformations in algebraic geometry, Singularities in algebraic geometry Moduli of products of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Editor's comment: This paper is a direct copy of the paper by \textit{M. L. Spreafico}, ``Local Bertini theorems for geometric properties over a nonperfect field'', Rend. Mat. Appl. (7) 13, No. 3, 561-572 (1993; Zbl 0820.14003). Local ground fields in algebraic geometry, Singularities in algebraic geometry Application to global Bertini theorems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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, X be irreducible compact complex spaces, let \(S\subset C(S)\), the space of cycles of X, and let \(Z_ s\) be the cycle of X parametrized by \(s\in S\). Assume that S covers X (i.e.: \(\forall x\in X\), \(\exists s\in S\) s.t.: \(x\in Z_ s)\). The so-called Kodaira map \(K_ s: X\to C(S)\) was constructed by the author in Invent. Math. 63, 187-223 (1981; Zbl 0436.32024), p. 202 in this situation, generalizing the classical map \(\Phi_ L: X\to {\mathbb{P}}(H^ 0(X,L)')\) associated to a linear system of divisors.
In this paper, a Kodaira map is more generally constructed when S, not necessarily compact, contains a compact subspace R which covers X. The natural factorization \(\alpha\) : \(K_ S(X)\to K_ R(X)\) of \(K_ R\) through \(K_ S\) is then algebraic. From this, the rigidity of the algebraic reduction \(e: X\to A\) follows (i.e.: its fibers form an irreducible component of C(S)). parametrization; equivalent meromorphic mappings; modification; space of cycles; Kodaira map; rigidity; algebraic reduction Algebraic dependence theorems, Compact analytic spaces, Parametrization (Chow and Hilbert schemes), (Equivariant) Chow groups and rings; motives Rigidité des fibres des réductions algébriques. (Rigidity of the fibers of algebraic reductions) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 presentation is to outline a new canonical algorithm of resolution of singularities over fields of characteristic zero. The new algorithm we present here improves that of the previous one [\textit{O. Villamayor}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 1-32 (1989; Zbl 0675.14003)] because the number of steps required to transform an arbitrary subvariety \(X\) into a regular one is in general strikingly smaller. We illustrate and compare with examples the efficiency of our new algorithm over our old one in section 4. algorithm; resolution of singularities Computational aspects of higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) On good points and a new canonical algorithm of resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this extremely brief note, the author deals with homological mirror symmetry for curves of higher genus. Namely if \(M\) is a compact complex curve of genus \(g\geq 2\) (viewed as a real surface with a symplectic form), its mirror dual is a pair \((X,W)\), where \(X\) is a \(3-\)dimensional smooth complex algebraic variety and \(W:X\longrightarrow\mathbb{C}\) is a regular function. The author presents a brief outline of his proof of homological mirror symmetry for \(M\), namely of the existence of an exact equivalence between the category \(D^{\pi}(\mathcal{F}(M))\) of perfect complexes over the Fukaya category of \(M\), and the Karoubian completion \(\overline{D}_{sg}(H)\) of the triangulated category \(D_{sg}(H)=D^{b}(H)/Perf(H)\) of the singularities of the unique singular fibre \(H\) of \(W\). The proof of this result can be found in [Homological mirror symmetry for curves of higher genus, arxiv/0907.3903], which generalizes the proof given by \textit{P. Seidel} [Homological mirror symmetry for genus 2 curves, \url{arxiv:0812.1171}] for curves of genus \(g=2\). Calabi-Yau manifolds (algebro-geometric aspects), Derived categories, triangulated categories, Singularities in algebraic geometry, Symplectic aspects of Floer homology and cohomology A note on mirror symmetry for curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(d\) and \(g\) be integers, and consider the Hilbert scheme \(H(d,g)\) parametrizing smooth, irreducible projective space curves of degree \(d\) and genus \(g\). The author proves that if \(H(d,g)\) is nonempty, then it contains a generically smooth component of the ``expected'' dimension. Moreover, the cohomological properties of a general curve in this component is studied. Hilbert scheme of smooth connected space curves; degree; genus Kleppe, J.O.: On the existence of nice components in the Hilbert Scheme \(\text{H}(d,g)\) of Smooth Connected Space Curves. Boll. U.M.I (7) 8-B, 305-326 (1994) Families, moduli of curves (algebraic), Plane and space curves, Parametrization (Chow and Hilbert schemes), Vector bundles on curves and their moduli On the existence of nice components in the Hilbert scheme \(H(d,g)\) of smooth connected space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of \textit{H. Clemens} and \textit{Z. Ran} [Am. J. Math. 126, No. 1, 89--120 (2004; Zbl 1050.14035)] to prove that a very general hypersurface of degree \(\frac{3n+1}{2} \leq d \leq 2n - 3\) in \(\mathbb{P}^n\) contain lines but no other rational curves. Hypersurfaces and algebraic geometry, Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Grassmannians, Schubert varieties, flag manifolds Rational curves on general type hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This work introduces a notion of complexes of maximal depth, and maximal Cohen-Macaulay complexes, over a commutative noetherian local ring. The existence of such complexes is closely tied to the Hochster's ``homological conjectures'', most of which were recently settled by André. Various constructions of maximal Cohen-Macaulay complexes are described, and their existence is applied to give new proofs of some of the homological conjectures, and also of certain results in birational geometry. homological conjectures; maximal Cohen-Macaulay complex; multiplier ideal; resolution of singularities Syzygies, resolutions, complexes and commutative rings, Homological conjectures (intersection theorems) in commutative ring theory, Local cohomology and commutative rings, Global theory and resolution of singularities (algebro-geometric aspects), Multiplier ideals Maximal Cohen-Macaulay complexes and their uses: a partial survey | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 polynomial map \(f:{\mathbb C}^n\rightarrow {\mathbb C}\) there exists a finite set \(B\subset {\mathbb C}\) such that the restriction \({\mathbb C}^n\backslash f^{-1}(B)\rightarrow {\mathbb C}\backslash B\) is a locally trivial fibration. The elements of the minimal possible such set \(B\) are called bifurcation points. By using sheaf-theoretical methods such as constructible sheaves, the authors generalize the formula of Libgober-Sperber concerning the zeta functions of monodromy at infinity of polynomial maps into various directions. In particular, they obtain some formulas for the zeta functions of global monodromy along the fibers of bifurcation points of polynomial maps. Milnor fibers; monodromy; Newton polyhedra; constructible sheaves; bifurcation point [19] Yutaka Matsui &aKiyoshi Takeuchi, &Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves&#xMath. Z.268 (2011) no. 1-2, p.~409Article | &MR~28 | &Zbl~1264. Singularities in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Projective and enumerative algebraic geometry, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this article, it is important that \(k\) is a fixed algebraically closed field of characteristic \(0\). The authors prove that the center of a homologically homogeneous, finitely generated \(k\)-algebra has rational singularities. Assume \(X=\text{Spec} R\) for an affine, Gorenstein \(k\)-algebra \(R\). In this article, a commutative resolution of singularities is a crepant homomorphism \(f:Y\rightarrow X,\) i.e. \(f^\ast\omega_Y=\omega_X.\) Bondal and Orlov conjectured that two such resolutions are derived equivalent, and this was later proved by Bridgeland. The authors generalize this to a \textit{third} noncommutative crepant resolution explaining Bridgeland's proof. This observation leads to different approaches to the Bondal-Orlov conjecture and related topics. The question now is how the existence of a noncommutative crepant resolution affects the original commutative singularity. It is known that if a Gorenstein singularity has a crepant resolution then it has rational singularities. The authors asks wether this is true for a noncommutative crepant resolution. The article answers this affirmatively.
Let \(\Delta\) be a prime affine \(k\)-algebra that is finitely generated as a module over its center \(Z(\Delta).\) \(\Delta\) is called homologically homogeneous of dimension \(d\) if all simple \(\Delta\)-modules have the same projective dimension \(d.\) The properties of homologically homogeneous rings are close to commutative regular rings, and the idea is to use such a ring \(\Delta\) as a noncommutative analogue of a crepant resolution. Formally, a noncommutative crepant resolution of \(R\) is any homologically homogeneous ring of the form \(\Delta=\text{End}_R(M)\) where \(M\) is a reflexive and finitely generated \(R\)-module. The main result of the article is the following:
Theorem. Let \(\Delta\) be a homologically homogeneous \(k\)-algebra. Then the center \(Z(\Delta)\) has rational singularities. In particular, if a normal affine \(k\)-domain \(R\) has a noncommutative crepant resolution, then it has rational singularities.
Also, examples are given proving that this theorem may fail in positive characteristic.
The article starts with the properties of homologically homogeneous rings, based on tame orders: If \(\Delta\) is a prime ring with simple Artinian ring of fractions \(A\) (i.e. \(\Delta\) is a prime order in \(A\)), \(\Delta\) is called a \textit{tame \(R\)-order} if it is a finitely generated and reflexive \(R\)-module such that \(\Delta_{\mathfrak p}\) is hereditary for all prime ideals \(\mathfrak p\) in \(R\) of height \(1\).
A homologically homogeneous ring \(\Delta\) of dimension \(d\) is Cohen Macaulay (CM) over its center \(Z(\Delta)\), both GK\(\dim\Delta\) and the global homological dimension gl\(\dim\Delta\) of \(\Delta\) equal \(d\), the center \(Z=Z(\Delta)\) is an affine CM normal domain, and finally, \(\Delta\) is a tame \(Z\)-order.
The rest of the article is then used to prove the main theorem. This involves reduction to the Calabi-Yau case for proving that \(Z\) has rational singularities by a generalization of the commutative method where one constructs a Gorenstein cover of a \(\mathbb Q\)-Gorenstein singularity.
The article ends with examples proving, among other things, that the main theorem may fail in the case where \(k\) has positive characteristic.
The article is precise, and illustrates noncommutative algebraic geometry in a concrete way. It also give useful criterions and ideas to be followed in other settings in noncommutative geometry. noncommutative crepant resolution; Rees ring; homologically homogeneous rings; tame orders J. T. Stafford and M. Van den Bergh, Noncommutative resolutions and rational singularities, Michigan Math. J. 57 (2008), 659-674. Special volume in honor of Melvin Hochster. Noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Rings arising from noncommutative algebraic geometry, Homological dimension (category-theoretic aspects) Noncommutative resolutions and 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 Let \(\mathbb{Z} \times BU\) be the space representing complex \(K\)-theory, and let \(K(\mathbb{Z},\text{even}) = \prod^ \infty_{j = 0} K(\mathbb{Z},2j)\) be the product of Eilenberg-MacLane spaces representing the functor \(\prod^ \infty_{j = 0} H^{2j}(-)\). The authors construct an infinite loop space structure on \(K(\mathbb{Z},\text{even})\) such that the total Chern class \(c: \mathbb{Z} \times BU \to K(\mathbb{Z},\text{even})\) is an infinite loop map, and they prove a similar result for the total Stiefel-Whitney class. The constructions use algebraic cycles [see \textit{H. B. Lawson jun.}, Ann. Math., II. Ser. 129, No. 2, 253-291 (1989; Zbl 0688.14006)].
The \(H\)-space structure induced on \(K(\mathbb{Z},\text{even})\) by the infinite loop space structure represents the cup product. There is another infinite loop space structure on \(K(\mathbb{Z},\text{even})\) representing the cup product due to \textit{G. Segal} [Quart. J. Math., Oxford II. Ser. 26, 289-293 (1975; Zbl 0321.55017)]. But the infinite loop space structure used here is different from Segal's; indeed, with Segal's structure, \(c\) is not an infinite loop map. complex \(K\)-theory; infinite loop space; total Chern class; infinite loop map; total Stiefel-Whitney class; algebraic cycles Boyer, CP; Lawson, HB; Lima-Filho, P; Mann, B; Michelson, M-L, Algebraic cycles and infinite loop spaces, Invent. Math., 113, 373-388, (1993) Infinite loop spaces, Topological \(K\)-theory, Algebraic cycles, Parametrization (Chow and Hilbert schemes) Algebraic cycles and infinite loop spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The articles of this volume will be reviewed individually. For the preceding Abel symposium (7; 2010) see Zbl 1231.35003. Proceedings, conferences, collections, etc. pertaining to associative rings and algebras, Proceedings of conferences of miscellaneous specific interest, Representations of quivers and partially ordered sets, Cluster algebras, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Derived categories and associative algebras, Derived categories, triangulated categories Algebras, quivers and representations. The Abel symposium 2011. Selected papers of the 8th Abel symposium, Balestrand, Norway, June 20--23, 2011. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 noncommutative differential calculus, Jacobi algebra (or potential algebra) plays the role of Milnor algebra in the commutative case. The study of Jacobi algebras is of broad interest to researchers in cluster algebra, representation theory and singularity theory. In this article, we study the quasi-homogeneity of a potential in a complete free algebra over an algebraic closed field of characteristic zero. We prove that a potential with finite dimensional Jacobi algebra is right equivalent to a weighted homogeneous potential if and only if the corresponding class in the \(0\)th Hochschlid homology group of the Jacobi algebra is zero. This result can be viewed as a noncommutative version of the famous theorem of Kyoji Saito on isolated hypersurface singularities. Jacobi algebra; quasi-homogeneous potential; Jordan-Chevalley decomposition; noncommutative differential calculus Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Singularities in algebraic geometry Quasi-homogeneity of potentials | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((X,0)\) be a germ of an analytic space and \(f\) a germ of an analytic function on \(X\). The Milnor fibre \(F\) of \(f\) at zero is an analytic invariant of \(f\). Its topology can be reconstructed with the help of a desingularization \(P:Z\rightarrow X\) (in which the special fibre of \(f\) is a normal crossings divisor \(D\)) and a stratified projection of \(F\) over \(D\). These constructions give rise to Hironaka's filtration which is a valuative filtration on \(F\). Another technique due to the third author [in: Real and compl. singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 397--403 (1977; Zbl 0428.32008)] consists in considering a projection \(g\) of \(X\) on a disc and in using a polar curve (relative to \(f\) and associated to \(g\)) and a polar discriminant (or Cerf's diagram) to obtain a polar filtration on \(F\). In the present paper the authors show that these filtrations are homeomorphic and piecewise-diffeomorphic. This links invariants associated to the singularities of the projection to those associated to a desingularization. local Milnor fibre; polar construction; resolution of singularities Singularities in algebraic geometry, Local complex singularities, Milnor fibration; relations with knot theory Invariants of a desingularization and singularities of morphisms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let V be a complex algebraic variety of dimension \(\leq 2\). If V is nonsingular it is straight forward to define the Dolbeault cohomology groups of V. If V is singular, then there is a variety of approaches one can use to define \(L^ 2\)-Dolbeault cohomology groups on the incomplete Kähler manifold V-Sing(V) [see \textit{W. L. Pardon}, Topology 28, No.2, 171-195 (1989; Zbl 0682.32024)], \textit{P. Haskell}, Proc. Am. Math. Soc. 107, No.2, 517-526 (1989; Zbl 0684.58040)] or the author, Publ. Res. Inst. Math. Sci., 24, No.6, 1005-1023 (1988; Zbl 0711.14003)].
In each case the boundary operator is applied to smooth forms on V- Sing(V), but the domains of the forms are restricted in one way or another that takes into account the metric. The author explores the differences in the cohomology defined in these cases. singular algebraic surfaces; Dolbeault cohomology; incomplete Kähler manifold Nagase, M.: Remarks on the L 2-Dolbeault cohomology groups of singular algebraic surfaces and curves. Publ. Res. Inst. Math. Sci. 26(5), 867--883 (1990) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), de Rham cohomology and algebraic geometry, Singularities in algebraic geometry Remarks on the \(L^ 2\)-Dolbeault cohomology groups of singular algebraic surfaces and curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers the structure and uniqueness of \(\mathbb Q\)-factorial terminalizations of nilpotent orbits of classical simple groups. It follows from the existence of minimal models for varieties of general type, see \textit{C. Birkar, P. Cascini, C. Hacon, J. McKernan} [math.AG/0610203], that every complex algebraic variety \(X\) admits a \(\mathbb Q\)-factorial terminalization, i.e. a birational projective morphism \(f:Y\rightarrow X\) with \(K_Y=f^*K_X\), where \(Y\) has only \(\mathbb Q\)-factorial terminal singularities.
Now let \(G\) be a classical simple complex Lie group and \(\mathcal O\) the orbit of a nilpotent element of \(\mathfrak g\) with respect to the adjoint action of \(G\) on \(\mathfrak g\). Let \(\widetilde{\mathcal O}\) denote the normalization of the Zariski closure \(\overline{\mathcal O}\) of \(\mathcal O\) in \(\mathfrak g\). Then \(\widetilde{\mathcal O}\) has only rational Gorenstein singularities [cf. \textit{D. I. Panyushev}, Funct. Anal. Appl. 25, No. 3, 225--226 (1991; Zbl 0749.14030)], and the author gives an explicit construction of a \(\mathbb Q\)-factorial terminalization of \(\widetilde{\mathcal O}\) in terms of the Levi decomposition \(\mathfrak q=\mathfrak l\oplus\mathfrak n\) of a suitable parabolic subalgebra \(\mathfrak q\) of \(\mathfrak g\): There is a nilpotent \(L\)-orbit \(\mathcal O'\) in \(\mathfrak l\) such that \(\mathcal O\) is the induced orbit of \(\mathcal O'\) (the uniquely determined nilpotent orbit in \(\mathfrak g\) which contains a dense open subset of \(\mathfrak n +\mathcal O'\)) and the normalization of \(G\times^Q(\mathfrak n(\mathfrak q)+\mathcal O')\) is a \(\mathbb Q\)-factorial terminalization of \(\widetilde{\mathcal O}\) via the generalized Springer map \([g,z] \mapsto\) Ad\(_gz\). Any two such constructions are connected by a sequence of Mukai flops [cf. \textit{Y. Namikawa}, Advanced Studies in Pure Mathematics 45, 75--116 (2006; Zbl 1117.14018)], and if \(\widetilde{\mathcal O}\) does not have \(\mathbb Q\)-factorial terminal singularities, then every \(\mathbb Q\)-factorial terminalization of \(\widetilde{\mathcal O}\) can be constructed in this way. The proof is based on a detailed description of the set of nilpotent orbits in \(\mathfrak g\) and the Jacobson-Morozov resolution of their Zariski closure.
\textit{B. Fu} [arXiv:0809.5109] has shown that the above result also holds for the exceptional simple complex Lie groups. nilpotent orbit; induced orbit; Mukai flop; Springer map; terminal singularity; Jacobson-Morozov resolution Namikawa Y. Induced nilpotent orbits and birational geometry. Adv Math, 2009, 222: 547--564 Semisimple Lie groups and their representations, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Deformations of complex singularities; vanishing cycles, Rational and birational maps, General properties and structure of complex Lie groups Induced nilpotent orbits and birational geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We define elliptic generalization of W-algebras associated with arbitrary quiver using our construction [Lett. Math. Phys. 108, No. 6, 1351--1381 (2018; Zbl 1388.81850)] with six-dimensional gauge theory. supersymmetric gauge theories; conformal field theories; W-algebras; quantum groups; quiver; instanton T. Kimura and V. Pestun, \textit{Quiver elliptic W-algebras}, arXiv:1608.04651 [INSPIRE]. Supersymmetric field theories in quantum mechanics, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Quantum groups and related algebraic methods applied to problems in quantum theory, Representations of quivers and partially ordered sets Quiver elliptic W-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 Using tools from algebraic and symplectic geometry, we study some properties of orbit spaces associated to partial representations of quivers over \(\mathbf C\). Special attention is paid to the compactness and to the construction of suitable compactifications. We also analyse whether analogous ideas could be applied in the study of moduli spaces of quiver bundles. sympletic geometry; quivers; moduli spaces of quiver bundles Representations of quivers and partially ordered sets, Algebraic moduli problems, moduli of vector bundles, Momentum maps; symplectic reduction, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) Representations of quivers and associated gauge-theoretical problems. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((V,0)\) be an isolated hypersurface singularity defined by the holomorphic function \(f:(\mathbb{C}^n,0)\rightarrow(\mathbb{C},0)\). The \(k\)-th Yau algebra \(L^k(V)\) is defined to be the Lie algebra of derivations of the \(k\)-th moduli algebra \(A^k(V):=\mathcal{O}_n/(f, m^kJ(f))\), where \(k\geq 0\), \(m\) is the maximal ideal of \(\mathcal{O}_n\). I.e., \(L^k(V):=\mathrm{Der}(A^k(V),A^k(V))\). These new series of derivation Lie algebras are quite subtle invariants since they capture enough information about the complexity of singularities. In this paper we formulate a conjecture for the complete characterization of ADE singularities by using generalized Cartan matrix \(C^k(V)\) associated to \(k\)-th Yau algebras \(L^k(V)\), \(k\geq 1\). In this paper, we provide evidence for the conjecture and give a new complete characterization for ADE singularities. Furthermore, we compute their other various invariants that arising from the 1-st Yau algebra \(L^1(V)\). isolated singularity; Lie algebra; generalized Cartan matrix Singularities in algebraic geometry, Local complex singularities On the generalized Cartan matrices arising from \(k\)-th Yau algebras of isolated hypersurface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we introduce new invariants to a singularity \((V, 0)\), i.e., the derivation Lie algebras \(\mathcal{L}_k(V)\) of the higher Nash blow-up local algebra \(\mathcal{M}_k(V)\). A new conjecture about the non-existence of negative weighted derivations of \(\mathcal{L}_k(V)\) for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture partially. Moreover, we compute the Lie algebra \(\mathcal{L}_2(V)\) for binomial isolated singularities. We also formulate a sharp upper estimate conjecture for the dimension of \(\mathcal{L}_k(V)\) for weighted homogeneous isolated hypersurface singularities and verify this conjecture for a large class of singularities. derivations; Hessian algebra; weighted homogeneous isolated hypersurface singularity Singularities in algebraic geometry, Local complex singularities Higher Nash blow-up local algebras of singularities and its derivation Lie 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 We study the complexity of an algorithm we gave in a former paper to compute an embedded resolution of an irreducible singular algebraic curve. This is more complex than just finding the Puiseux expansion associated to the curve, but this computation is also more interesting because it gives not only the Puiseux pairs, and the Puiseux series but also a way to work on some other invariants of the curve. For example it will allow the mathematician to work on the mixed Hodge structure of the curve. This complexity is shown to be polynomial in terms of the degree \(d\) of the polynomial of the curve. We will try to make a study of the complexity strongly related to the real algorithm we are using; in fact this study comes after, and is motivated by, two implementations we made of resolutions of irreducible curves. Henry, J. P. G., & Merle, M. (1987). Complexity of computation of embedded resolution of algebraic curves. In J. H. Davenport (Ed.),\textit{Lecture notes in computer science 378}, \textit{EUROCAL '87} (pp. 381-390). New York: Springer. Software, source code, etc. for problems pertaining to algebraic geometry, Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Symbolic computation and algebraic computation Complexity of computation of embedded resolution of algebraic curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective variety of dimension \(n\) defined over the complex number field, and let \(L\) be an ample line bundle on \(X\). In this paper, the author mainly investigates the following conjecture, which is called the Fujita conjecture:
Let \(X\), \(n\) and \(L\) be as above. Then \(K_{X}+mL\) is base point free for any integer \(m\geq n+1\).
If \(\dim X=1\), then this is verified by means of the Riemann-Roch theorem. If \(\dim X=2\), then \textit{I. Reider} [Ann. of Math. 127 (2), 309-316 (1988; Zbl 0663.14010)] proved that the above conjecture is true. If \(\dim X=3\) (resp. \(\dim X=4\)), then this was proved by \textit{L. Ein} and \textit{R. Lazarsfeld} [J. Am. Math. Soc. 6 (4), 875-903 (1993; Zbl 0803.14004)] (resp. by \textit{Y. Kawamata} [Math. Ann. 308 (3), 491-505 (1997; Zbl 0909.14001)]). In arbitrary dimension \(n\), several authors investigated this conjecture. (For example \textit{U. Angehrn} and \textit{Y.-T. Siu} [Invent. Math. 122 (2), 291-308 (1995; Zbl 0847.32035)], \textit{S. Helmke} [Duke Math. J. 88 (2), 201-216 (1997; Zbl 0876.14004)], [Math. Ann. 313 (4), 635-652 (1999; Zbl 0935.14003)], and \textit{H. Tsuji} [Nagoya Math. J. 142, 5-16 (1996; Zbl 0861.32018 )].) In particular it was proved that \(K_{X}+mL\) is base point free for any integer \(m\) that is no less than a number roughly of order \(n^{2}\).
In this paper the author improves the bound and proves the following: \(K_{X}+mL\) is base point free for any integer \(m\) with \(m\geq (e+1/2)n^{4/3}+(1/2)n^{2/3}+1\), where \(e\) is Euler's number. adjoint linear system; vanishing theorem; Fujita conjecture Heier G.: Effective freeness of adjoint line bundles. Doc. Math. 7, 31--42 (2002) Divisors, linear systems, invertible sheaves, Vanishing theorems in algebraic geometry, Singularities in algebraic geometry Effective freeness of adjoint line 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 manuscript under review is an expanded version of a course of lectures given by B. Teissier in the framework of the ``2ndo Congreso Latinoamericano de Matemáticos'' held in Cancun, Mexico, on July 20--26, 2004. The aim of this work, which contains a large amount of various information, is to acquaint people in the mathematical community with the ideas and methods of the theory of polar varieties developed in the past 5-6 decades. Above all, in the special ``dedicatory'' the authors also address their article ``to those administrators of research who realize how much damage is done by the evaluation of mathematical research solely by the rankings of the journals in which it is published, or more generally by bibliometric indices''.
The study of polar varieties has its origin in a quite general observation, appeared in the late 1960s, that basic properties of complex spaces or varieties with singularities can be properly explained with the use of suitable partitions of singular varieties into finitely many nonsingular complex analytic manifolds. These manifolds, or strata, have the property that the local geometry of the space under consideration is constant on each stratum. Thus, in his earliest works \textit{H. Whitney} (see, e.g., [Ann. Math. 81, No. 3, 496--549 (1965; Zbl 0152.27701)]) initiated the study in complex analytic geometry of limits of tangent spaces at nonsingular points, the theory of stratifications and some related topological and differential-geometric constructions (see also [\textit{R. Thom}, Bull. Am. Math. Soc. 75, 240--284 (1969; Zbl 0197.20502)], [\textit{J. N. Mather}, Proc. Sympos. Univ. 1971, 195--232 (1973; Zbl 0286.58003)]).
The authors emphasize that one of the main points of the present article is to describe such partitions algebraically using polar varieties. Of course, for evident reasons, this approach can be applied in the case of reduced spaces only. As usually, the term ``local'' means that ``sufficiently small'' representatives of germs of the corresponding space are considered. In addition, the authors also assume that the spaces under consideration are equidimensional, i.e., all their irreducible components have the same dimension.
At the beginning of the manuscript, in the introduction, a very interesting and detailed historical background is given; the authors' attention focusses on those ideas and results which are closely related to the birth of the concept of polar varieties and its further development (see, e.g., [\textit{B. Teissier}, Apparent contours from Monge to Todd, 1830--1930: A century of geometry, Lect. Notes Phys. 402, 55--62 (1992)], [\textit{R. Piene}, Lect. Notes Comput. Sci. 8942, 139--150 (2015; Zbl 1439.14141)]). Then they discuss the basic properties of limits of tangent spaces, the notions of conormal spaces, projective duality, tangent cones, multiplicity, and describe many other standard and useful constructions in a quite elementary manner. The most important properties of normal cones and polar varieties, the basic relations between the conormal space and its Semple-Nash modification, are discussed in the third section. Then the authors prove that for a given complex analytic space, there is a unique minimal (coarsest) Whitney stratification. Any other Whitney stratification of the space is obtained by adding strata inside the strata of the minimal one (see [\textit{B. Teissier}, Lect. Notes Math. 961, 314--491 (1982; Zbl 0585.14008)]). Among other things, it is explained in an explicit form and formulas how the multiplicities of local polar varieties are related to local topological invariants, local vanishing Euler-Poincaré characteristics, the degree of the dual of a projective variety, etc.
It should be noted that the authors' expressive exposition of highly nontrivial materials contains many nice pictures, examples, comments and remarks, as well as the list of bibliographies, including 127 items. Without any doubt, all this inspires enthusiastic readers to begin studying of the subject or to continue their own research at a higher level than before. polar varieties; equidimensional varieties; singularities; stratifications; Whitney stratifications; Whitney conditions; tubular neighborhoods; tangent cones; limits of tangent spaces; conormal spaces; projective duality, multiplicity; Nash modifications; Plücker-type formulas; Todd's formulas; characteristic classes; Chern classes Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, History of several complex variables and analytic spaces, History of mathematics in the 20th century, History of mathematics in the 21st century, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Stratified sets, Germs of analytic sets, local parametrization, Characteristic classes and numbers in differential topology, Families, moduli of curves (analytic), Local complex singularities, Singularities in algebraic geometry, History of algebraic geometry Local polar varieties in the geometric study of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Singularities of formal power series \(f\in K[[x_1,\ldots,x_n]]\) with \(K\) an algebraically closed field of positive characteristic were studied by Boubakri, Greuel and Markwig [\textit{Y. Boubakri} et al., Rev. Mat. Complut. 25, No. 1, 61--85 (2012; Zbl 1279.14004)]. In the same spirit the authors extend finite determinacy results for two other problems to arbitrary characteristic. Their results concern the case of isolated line singularities, where the automorphisms used preserve a smooth line, and the case of functions in the presence of a hypersurface, where the automorphisms used preserve this hypersurface. isolated line singularities; finite detrminacy Complex surface and hypersurface singularities, Singularities in algebraic geometry The finite \(\mathcal{S}\)-determinacy of singularities in positive characteristic \(\mathcal{S}=\mathcal{R}_{\mathcal{G}},\mathcal{R}_{\mathcal{A}}, \mathcal{K}_{\mathcal{G}},\mathcal{K}_{\mathcal{A}}\) | 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 \(C\) be a space curve of degree \(d\), not lying on a surface of degree \(s-1\), and of maximal genus \(G(d,s)\). For \(d\geq(s-1)^ 2+1\), \(G(d,s)\) has been determined by \textit{L. Gruson} and \textit{C. Peskine} [in Algebraic Geometry, Proc., Tromsø Symp. 1977, Lect. Notes Math. 687, 31-59 (1978; Zbl 0412.14011)]. They also describe the curves \(C\).
In this paper, the author describes the curves \(C\) for \(d=(s-1)^ 2-r\), \(0\leq r\leq s-4\), and the family of these curves in the Hilbert scheme. The value of \(G(d,s)\) in this range has been computed by \textit{Ph. Ellia} [J. Reine Angew. Math. 413, 78-87 (1991; Zbl 0711.14015)]. linkage; deficiency module; space curve; maximal genus; Hilbert scheme Plane and space curves, Linkage, Parametrization (Chow and Hilbert schemes) On certain curves of maximal genus 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 Let \(M_c=M(2,0,c)\) be the moduli space of \({\mathcal O}(1)-\)semistable rank 2 torsion-free sheaves with Chern classes \(c_1=0\) and \(c_2=c\) on a \(K3\) surface \(X,\) where \({\mathcal O}(1)\) is a generic ample line bundle on \(X.\) When \(c=2n\geq 4\) is even, \(M_c\) is a singular projective variety equipped with a holomorphic symplectic structure on the smooth locus. In particular \(M_c\) has trivial canonical divisor.
Following \textit{K. G. O'Grady} [J. Reine Angew. Math. 512, 47--117 (1999; Zbl 0928.14029)] who asks if there is any symplectic desingularization of \(M_{2n}\) for \(n\geq 3\), the author shows that there is no crepant resolution of \(M_{2n}\) for \(n\geq3.\) And consequently that there is no symplectic desingularization. crepant resolution; moduli spaces; projective variety DOI: 10.4134/JKMS.2007.44.1.035 Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Algebraic moduli problems, moduli of vector bundles, Symplectic structures of moduli spaces, \(K3\) surfaces and Enriques surfaces, Global theory and resolution of singularities (algebro-geometric aspects) Nonexistence of a crepant resolution of some moduli spaces of sheaves on a \(K3\) surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{F. Severi} claimed in the 1920s that the Hilbert scheme \({\mathcal H}_{d,g,r}\) of smooth irreducible non-degenerate curves \(C \subset \mathbb P^r\) of degree \(d\) and genus \(g\) is irreducible for \(d \geq g+r\) [Vorlesungen über algebraische Geometrie. Übersetzung von E. Löffler. Leipzig u. Berlin: B. G. Teubner (1921; JFM 48.0687.01)]. \textit{L. Ein} proved Severi's claim for \(r=3\) and \(r=4\) [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 469--478 (1986; Zbl 0606.14003); Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], but there are counterexamples of various authors for \(r \geq 6\).
It has been suggested by \textit{C. Ciliberto} and \textit{E. Sernesi} [in: Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co. 428--499 (1989; Zbl 0800.14002)] that Severi intended irreducibiity of the Hilbert scheme \({\mathcal H}^{\mathcal L}_{d,g,r} \subset {\mathcal H}_{d,g,r}\) of curves whose general member is linearly normal: indeed, the counterexamples above arise from families whose general member is not linearly normal.
Here the authors prove irreducibility for \(g+r-2 \leq d \leq g+r\) (the Hilbert scheme is empty for \(d > g+r\) by Riemann-Roch) and for \(d=g+r-3\) under the additional assumption that \(g \geq 2r+3\). This extends work of \textit{C. Keem} and \textit{Y.-H. Kim} [Arch. Math. 113, No. 4, 373--384 (2019; Zbl 1423.14028)]. Hilbert scheme; linear series; linearly normal curves Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) On the Hilbert scheme of linearly normal curves in \(\mathbb{P}^r\) of relatively high degree | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This work introduces a new framework for understanding uniform behavior of singularity measures such as Hilbert-Kunz multiplicity, Hilbert-Samuel multiplicity, and F-rational signature, for ideals varying in families of rings. Namely, the author calls the combination of a ring map $R \rightarrow A$ with an ideal $I$ of $A$ an \textit{affine $I$-family} if $A/I$ is module-finite over $R$, $I \cap R = 0$, and certain dimension formulas hold. (This is a restricted version of Lipman's notion of an $I$-family from [\textit{J. Lipman}, Lect. Notes Pure Appl. Math. 68, 111--147 (1982; Zbl 0508.13013)]). Then one analyzes the ideals $I(\mathfrak p)$, $I(\mathfrak p)^{[p^e]}$ (when char $k(\mathfrak p) = p>0$) and $I(\mathfrak p)^n$ for $\mathfrak p \in $Spec$(R)$ and various $e, n \in \mathbb N$ in the rings $R(\mathfrak p)$, where the notation $(\mathfrak p)$ means to tensor over $R$ with the residue field $k(\mathfrak p)$ of $\mathfrak p$.
This framework allows the author to recover Lipman's result [loc. cit.] on upper semicontinuity of Hilbert-Samuel multiplicity on the prime spectrum, by showing that the \textit{terms} defining Hilbert-Samuel multiplicity as a limit are also upper semicontinuous in the family.
He also recovers some of his own results (see [Compos. Math. 152, No. 3, 477--488 (2016; Zbl 1370.13006)]) on semicontinuity of Hilbert-Kunz multiplicity on the prime spectrum, again by analyzing the terms, where in this case one has an affine $I$-family $R \rightarrow S$ with reduced fibers of dimension $=$ height$(I)$, where $R$ is F-finite. He further obtains upper semicontinuity of Hilbert-Kunz multiplicity in an affine family where char $R=0$ and the characteristics of the fibers can vary but all residue fields of $R$ are F-finite when they are positive characteristic. In particular, when $R = \mathbb Z$, this answers a question of Claudia Miller from [\textit{H. Brenner} et al., J. Algebra 372, 488--504 (2012; Zbl 1435.13014)] in pursuance of obtaining a sensible notion of Hilbert-Kunz multiplicity in equal characteristic zero.
The author also parlays his methods to show that for local algebras essentially of finite type over a prime characteristic field, the infimum in Hochster and Yao's definition of F-rational signature is actually achieved. He thus recovers a special case of the result in [\textit{M. Hochster} and \textit{Y. Yao}, ``F-rational signature and drops in the Hilbert-Kunz multiplicity'', Preprint] that the F-rational signature of the ring is positive if and only if the ring is F-rational. multiplicity; Hilbert-Samuel polynomial; Hilbert-Kunz multiplicity; semicontinuity; families Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Singularities in algebraic geometry, Deformations of singularities, Fibrations, degenerations in algebraic geometry On semicontinuity of multiplicities in 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 It seems natural to relate the study of (moduli of) vector bundles over smooth complex, projective varieties with the study of the corresponding (families of) embedded projective bundles, when the vector bundle is very ample. Inspired by the case of rank-two vector bundles over curves (see the Introduction of the paper under review and references therein) this paper takes part of a series of papers which deals with the case of rank-two vector bundles over Hirzebruch surfaces \(\mathbb{F}_e\). As a natural continuation of previous works, the authors consider the case in which \(\mathcal{E}\) is a vector bundle over \(\mathbb{F}_e\) with first Chern class \(c_1(\mathcal{E})=4C_0+\lambda f\) (\(C_0\) stands for the minimal section and \(f\) for the fiber of \(\mathbb{F}_e\)) which is known to be very ample.
The authors prove, see Theorem 4.2, that when \(e \leq 2\) the corresponding ruled 3-folds are smooth points of the proper component of the Hilbert scheme, which in fact has the expected dimension. Moreover, see Theorem 4.6, these scrolls fill up either their whole component (\(e=0,1\)) or a codimension one subvariety of their component (\(e=2\)). Some interesting questions -- general elements of the components of the Hilbert schemes, degenerations in terms of vector bundles -- beyond the generalizations to \(e\geq 3\), are also provided (see Section 5). vector bundles; rational ruled surfaces; ruled threefolds; Hilbert schemes; moduli spaces \(3\)-folds, Rational and ruled surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Varieties of low degree, Adjunction problems On families of rank-\(2\) uniform bundles on Hirzebruch surfaces and Hilbert schemes of their scrolls | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get started, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes.
For the second edition, several mistakes and many smaller errors and misprints have been corrected.
See the review of the first edition in [Zbl 1213.14001]. textbook (algebraic geometry); schemes and morphisms; prevarieties; quasi-coherent sheaves; vector bundles; divisors; algebraic curves; determinantal varieties; singularities Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Schemes and morphisms, Divisors, linear systems, invertible sheaves, Sheaves in algebraic geometry, Group schemes, Determinantal varieties, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry Algebraic geometry I. Schemes. With examples and exercises | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider an irreducible projective plane curve, with finitely many singular points \(p_1, \ldots, p_\nu\). For \(i=1, \ldots, \nu\) let \(\Delta_i(t)\) be the characteristic polynomial associated to the germ \((C,p_i)\), and let \(\Delta(t)\) be the product of the \(\Delta_i(T)\). Let \(Q(t)\) be defined by the equation \(\Delta(t) = 1 + (t-1)\delta + (t-1)^2 Q(t)\). For \(\ell = 0, \ldots, d-3\), let \(c_\ell\) be the coefficient of \(t^{(d-3-\ell)d}\) in \(Q(t)\). The authors formulate the astonishing conjecture, that if \(C\) is rational, then for each \(\ell\) the inequality \(c_\ell \leq (\ell +1)(\ell+2)/2\) holds. They support this conjecture by proving it for the case that the logarithmic Kodaira dimension of the complement of \(C\) is smaller than two. In case it is two, they prove it in some particular cases. Fernández de Bobadilla, J.; Luengo, I.; Melle-Hernández, A.; Némethi, A., On rational cuspidal projective plane curves, Proc. Lond. Math. Soc., 92, 1, 99-138, (2006) Singularities of curves, local rings, Plane and space curves, Singularities in algebraic geometry On rational cuspidal projective plane curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Chow variety is a parameter space for effective algebraic cycles on \(\mathbb{P}^n\) (or \(\mathbb{A}^n\)) of given dimension and degree. We construct its analog for differential algebraic cycles on \(\mathbb{A}^n\), answering a question of \textit{X.-S. Gao} et al. [Trans. Am. Math. Soc. 365, No. 9, 4575--4632 (2013; Zbl 1328.14010)]. The proof uses the construction of classical algebro-geometric Chow varieties, the theory of characteristic sets of differential varieties and algebraic varieties, the theory of prolongation spaces and the theory of differential Chow forms. In the course of the proof, several definability results from the theory of algebraically closed fields are required. Elementary proofs of these results are given in the appendix. Parametrization (Chow and Hilbert schemes), Differential algebra, Varieties and morphisms, Model-theoretic algebra Differential Chow varieties exist | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 obtains upper bounds for the absolute values of exponential sums associated to polynomials over finite fields. If \(f\in k[x_1,\dots,x_n]\) and \(\Psi\) is a non-trivial additive character, he puts \(S(\Psi,f) = \sum_{x\in k^n} \Psi(\text{Tr}_{k/{\mathbb F}_p}(f(x)))\). Supposing that \(k\) has \(q\) elements, it is proved that
\[
|S(\Psi,\overline{f})|\leq\dim_{\mathbb C} { { {\mathbb C}[x_1,\dots,x_n] } \over { \bigl(\delta f/\delta x_1, \dots, \delta f/\delta x_n\bigr)} } q^{n/2}.
\]
The proof is based on the reduction to the case \(f=f_d + x_n^{d-1}\), on techniques of \textit{N. Katz} on exponential sums [Sommes exponentielles, Astérisque 79 (1980; Zbl 0469.12007)] and of \textit{M. Raynaud} on cohomology of abelian varieties [in Sémin. Bourbaki 17, Années 1964/65, Exp. No. 286 (1966; Zbl 0204.54301)]. exponential sums; singular hypersurfaces R. Livne, \textit{Cubic exponential sums and Galois representations}, in \textit{Current Trends in Arithmetical Algebraic Geometry, Arcata 1985}, Contemporary Mathematics \textbf{67} (1987), 247-261. Estimates on exponential sums, Exponential sums, Singularities in algebraic geometry, Trigonometric and exponential sums (general theory) Exponential sums and singular hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let k be a field and \(A=k[M]\) the monoid algebra where M is a cancellative, abelian monoid with unit element 0. The Poincaré series for A is \(P_ A(z)=\sum_{m\geq 0}\#(\dim_ kTor^ A_ m(k,k))z^ m.\quad Suppose\) M carries a simple, connected, standard, monoidal gradation, i.e. a monoid homomorphism \(d:\quad M\to N,\) such that \(d^{- 1}(0)=0\), \(d^{-1}(v)\) finite for all v and M generated by \(d^{- 1}(1)\). Then the Hilbert series for A with respect to d is \(H_ A(z)=\sum_{m\geq 0}(d^{-1}(m))z^ m.\quad In\) this article monoids M for which \(H_ A(z)P_ A(-z)=1\) are studied. The main means for the study is the characterization of monoidal homology due to Laudal, the monoidal version of the Möbius inversion formula due to Lallement and the study of Cohen-Macaulay finite posets due to Stanley, Baclawski etc. A corollary is a former result by the author and N. Manolache that Segre products of Veronese subrings of a polynomial ring satisfy the formula \(H_ A(z)P_ A(-z)=1\). algebraic singularity with linear resolution; Fröberg rings; monoid algebra; Poincaré series; Hilbert series; monoidal homology Homological methods in commutative ring theory, Polynomial rings and ideals; rings of integer-valued polynomials, Singularities in algebraic geometry Monoidal algebraic singularities with linear resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 observes that although singularities occur in series and the simplest one have been given names by Arnol'd, it is not clear how to define what is meant by a series. A topological definition of series, for plane curve singularities, is proposed in this work: Let f be an element of the ring of convergent power series \(C\{\) x,y\(\}\) with a non-isolated singularity. The topological series belonging to f consists of all topological types of isolated singularities whose links arise as the splice of the link of f with some other link. (The splice operation is due to Siebenmann and in this case implies that ``the Milnor fibration of an element of the series differs from that of f only in small neighborhoods of the components with higher multiplicities''.) The motivation for the definition, examples, and calculations of some topological invariants are also given, in a very readable and accessible style. plane curve singularities; non-isolated singularity; topological series; topological types of isolated singularities; links; Milnor fibration Schrauwen ( R. ). - Topological series of isolated plane curves singularities , l'Enseignement Mathématique ( 36 ) ( 1990 ), 115 - 141 . MR 1071417 | Zbl 0708.57011 Singularities of differentiable mappings in differential topology, Milnor fibration; relations with knot theory, Differential topology, Singularities of curves, local rings, Singularities in algebraic geometry Topological series of isolated 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 Let \(R\) be a Noetherian local ring. A finitely generated \(R\)-module \(M\) is said to be maximal Cohen-Macaulay if \(\text{depth}\,M=\dim R\). We say that \(R\) has finite Cohen-Macaulay type if there are only finitely many indecomposable maximal Cohen-Macaulay modules up to isomorphism. In the paper under review, the author constructs examples of two dimensional mixed characteristic rings of finite Cohen-Macaulay type and for a large subclass of these examples gives a complete description of its indecomposable maximal Cohen-Macaulay modules. He also computes its AR-quiver. maximal Cohen-Macaulay; finite Cohen-Macaulay representation type; AR-sequences; AR-quiver Puthenpurakal, T.J., On two dimensional mixed characteristic rings of finite Cohen Macaulay type, J. pure appl. algebra, 220, 1, 319-334, (2016) Cohen-Macaulay modules, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities in algebraic geometry On two dimensional mixed characteristic rings of finite Cohen Macaulay type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In a recent paper [the first and the second author, in: Symmetries, integrable systems and representations. Proceedings of the conference on infinite analysis: frontier of integrability, Tokyo, Japan, July 25--29, 2011 and the conference on symmetries, integrable systems and representations, Lyon, France, December 13--16, 2011. London: Springer. 15--33 (2013; Zbl 1307.14007)], the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper, we give a very short geometrical proof of that formula. Qian, C J, A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems, 4708-4715, (2005) Parametrization (Chow and Hilbert schemes), (Co)homology theory in algebraic geometry A simple proof of the formula for the Betti numbers of the quasihomogeneous 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 \(X\) be a smooth projective variety over an algebraically closed field \(k\) of characteristic zero, and let \(\mathcal{F}\) be a foliation of codimension \(q\) on \(X\), with normal sheaf \(N_\mathcal{F}\). If \(S\) is a scheme over \(k\), and \(0 \in S\) a closed point, then following [\textit{T. Suwa}, in: Singularity theory. Proceedings of the symposium, Trieste, Italy, August 19-- September 6, 1991. Singapore: World Scientific. 817--865 (1995; Zbl 0944.32022)] the authors define an unfolding of \(\mathcal{F}\) over \(S\) as a foliation \(\tilde{\mathcal{F}}\) on \(X\times S\) such that \(\tilde{\mathcal{F}}_{|X \times \{0\}} = \mathcal{F}\). When \(S = \text{Spec}(k[x]/(x^2))\), any unfolding of \(\mathcal{F}\) is called an infinitesimal first order unfolding. A singularity \(p \in X\) of \(\mathcal{F}\) is called persistent if, for every infinitesimal first order unfolding \(\tilde{\mathcal{F}}\) of \(\mathcal{F}\), the point \((p,0)\) is a singularity of \(\tilde{\mathcal{F}}\). In this paper, the authors define a scheme structure for the persistent singularities of \(\mathcal{F}\), in such a way that they are defined by an ideal sheaf \(\mathcal{I}\).
On the other hand, a singularity \(p \in \mathcal{F}\) is called a Kupka singularity, if \(d\omega(p) \neq 0\), where \(\omega \in H^0(X,\Omega^q_X \otimes \det(N_\mathcal{F}))\) is a twisted \(q\)-form defining \(\mathcal{F}\). In this paper, the authors give a more refined definition for the Kupka scheme, which is defined by an ideal sheaf \(\mathcal{K}\). They show that in case \(q = 1\), if the ideal sheaf \(\mathcal{J}\) associated to the singular locus of \(\mathcal{F}\) is a sheaf of radical ideals, \(c_1(N_\mathcal{F}^*) \neq 0\) and \(H^1(X,N_\mathcal{F}^*) = 0\), then \(\mathcal{F}\) has Kupka singularities. Furthermore, they relate Kupka singularities and persistent singularities in the following way: when \(q = 1\), they prove that \(\mathcal{J} \subset \mathcal{I} \subset \mathcal{K}\). This result generalizes the one obtained in [\textit{C. Massri} et al., Asian J. Math. 22, No. 6, 1025--1046 (2018; Zbl 1408.32032)] for foliations on \(\mathbb{P}^n\). Moreover, these inclusions remain true for foliations of higher dimension if the normal bundle of \(\mathcal{F}\) is locally free; however, in general this is not known to be true.
Finally, the authors show that the inexistence of persistent singularities for a foliation \(\mathcal{F}\), satisfying \(\text{Ext}_{\mathcal{O}_X}^1(N_\mathcal{F}^*,\text{Sym}^2(N_\mathcal{F}^*)) = 0\), implies the existence of a connection on \(N_\mathcal{F}^*\). In particular, in this case, all Chern classes of \(N_\mathcal{F}^*\) vanish. foliations; unfoldings of foliations; Kupka singularities; persistent singularities Singularities in algebraic geometry, Singularities of holomorphic vector fields and foliations, Dynamical aspects of holomorphic foliations and vector fields, Complex-analytic moduli problems Foliations with persistent 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 \(G\) be a finite subgroup of SL\((2,C)\) of type A or D. The paper under review studies the geometry and topology of the Hilbert scheme of points on \(C^2/G\), in two different flavors. First, the classical coarse Hilbert scheme Hilb\((C^2/G)\) of the singular quotient space; second, the orbifold Hilbert scheme Hilb\(([C^2/G])\), the moduli space of \(G\)-invariant finite colength subschemes of \(C^2\). The first one decomposes into components according to the number of points, and the second one decomposes according to finite dimensional representations of \(G\). Using these decompositions one defines the generating series of Euler characteristics in the two cases. The coarse generating series hence is a series in one variable, and the orbifold gerenating series is a series in \(n+1\) variables corresponding to the irreducible representations of \(G\).
The goal of the paper is to give explicit combinatorial formulas for these two generating series. The first main result is Theorem 1.4 which proves a decomposition of the orbifold Hilbert scheme to affine spaces (locally closed strata), parametrized by combinatorial objects called Young walls. Using some combinatorics and representation theory the authors show how this theorem implies Nakajima's formula for the Euler characteristic generating series of the orbifold Hilbert scheme. The second main result of the paper, Theorem 1.7, proves that the coarse generating series is a particular specialization of the orbifold one, essentially at roots of unities. Although this result was known in type A (by Dijkgraaf-Sulkowski, Toda) the direct combinatorial proof of this paper is new, as well as the result in type D. The authors formulate an analogous conjecture in type E. Hilbert scheme; singularities; Euler characteristic; generating series; Young wall Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial aspects of representation theory Euler characteristics of Hilbert schemes of points on simple surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface \(\mathbb C^2/G\) with \(G<\mathrm{SL}(2,\mathbb C)\) a finite subgroup, we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type \(A\) singularities. We announce a proof of our conjecture for singularities of type \(D\). The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding (extended) affine Lie algebra; we discuss possible representation-theoretic consequences of this fact. Our results, respectively conjectures, imply the modularity of the generating function for surfaces with type \(A\) and type \(D\), respectively arbitrary, simple singularities, confirming predictions of \(S\)-duality. Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry, Group actions on varieties or schemes (quotients) Euler characteristics of Hilbert schemes of points on surfaces with simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite cyclic subgroup of \(GL(2, \mathbb{C})\) of order \(n\) which contains no reflections. Let \(\mathbb{A}^2\) be the complex affine plane. We consider a certain subscheme \(\text{Hilb}^G (\mathbb{A}^2)\) of \(\text{Hilb}^n(\mathbb{A}^2)\) consisting of \(G\)-invariant zero-dimensional subschemes of length \(n\). We describe the structure of \(\text{Hilb}^G (\mathbb{A}^2)\) and prove this is the minimal resolution of the quotient surface singularity \(\mathbb{A}^2/G\). Hilbert scheme; resolution; quotient surface singularity Kidoh, R., Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), 91--103. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Hilbert schemes and cyclic quotient surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth connected complex surface.
Haiman defined the isospectral Hilbert scheme of \(n\) points on \(X\) to be the blow up \(B^n \to X^n\) of the product variety
\(X^n\) along the big diagonal \(\Delta_n\) (in [\textit{M. Haiman}, Math. Sci. Res. Inst. Publ. 38, 207--254 (1999; Zbl 0952.05074); J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)] and
proved that \(B^n\) is normal, Cohen-Macaulay and Gorenstein.
Here the author further investigates singularities of \(B^n\), that is, of the pair \((B^n,\emptyset)\), which has log-canonical singularities if and only if \((X^n, {\mathcal I}_{\Delta_n})\) does.
Using Haiman's description of generators of the ideal \({\mathcal I}_{\Delta_n}\) for the local model \(\mathbb C^2\), the
author proves the upper bound on the log-canonical threshold
\[
\text{lct} (X^n,{\mathcal I}_{\Delta_n}) \leq \frac{2n-2}{d_n}
\]
where \(d_n\) is the minimal degree of a generator of \({\mathcal I}_{\Delta_n}\).
This bound leads to the main theorem, which states that the singularities of \(B^n\) are canonical if \(n \leq 5\)
and log-canonical for \(n \leq 7\), but for \(n \geq 9\) they are not log-canonical.
The author conjectures that the log-canonical threshold bound is sharp, which would imply that \(B^n\) has
canonical singularities if and only if \(n \leq 7\) and log-canonical singularities if and only if \(n \leq 8\).
The author also provides two log-resolutions of \(B^3\), one crepant and the other \(S_3\)-equivariant. isospectral Hilbert scheme of points; canonical singularities; log-canonical thresholds; crepant resolutions Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Singularities of the isospectral Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a smooth complex algebraic surface. Given positive integers \(n_1<n_2<\cdots <n_k\), let \(S^{[n_1, n_2, \dots, n_k]}\) denote the nested Hilbert scheme parameterizing nested 0-dimensional sub-schemes of \(S\): \(\xi_{n_1}\subseteq \xi_{n_2}\subseteq \cdots \xi_{n_k}\) of length \(n_i\). The nested Hilbert schemes are natural analogues for the Hilbert schemes \(S^{[n]}\) of points, and some of them have played an important role in the study of syzygies. The present well-written paper gives a quite comprehensive study of \(S^{[n, n+1, n+2]}\).
The first main result is a new proof of the irreducibility of \(S^{[n, n+1, n+2]}\) due to \textit{N. Addington} [Algebr. Geom. 3, No. 2, 223--260 (2016; Zbl 1372.14009)]. The idea here is to realize \(S^{[n, n+1, n+2]}\) as \(\mathbb{P}(\mathscr{I}_{Z_{[n, n+1]}})\), where \(Z_{[n, n+1]}\) is the subscheme of \(S\times S^{[n, n+1]}\) parameterizing triples \((p, \xi_{n}, \xi_{n+1})\) with \(p\in \text{Supp}(\xi_{n+1})\), and use a criterion of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [Trans. Amer. Math. Soc. 350, No. 6, 2547--2552 (1998; Zbl 0893.14001)]. Along the way, with more care the authors establish an estimate on codimension of certain strata of \(Z_{[n, n+1]}\) in \(S\times S^{[n, n+1]}\), which is quadratic in the minimal number \(i\) of generators for the localized ideal. While a linear estimate is sufficient in the criterion mentioned above, the quadratic one is of great interest on its own. Via forgetful and residual point maps, the irreducibility of \(S^{[n, n+2]}, S^{[1, n, n+1, n+2]}, S^{[1, n+1, n+2]}\) and \(S^{[1, n, n+2]}\) are deduced from that of \(S^{[n, n+1, n+2]}\).
The second one is an explicit construction of a family of nested subschemes, indicating that \(S^{[1, 2, \dots, 23]}\) is reducible. As a corollary, \(S^{[n_1, n_2, \dots, n_k]}\) is reducible whenever \(k\ge 23\).
The third one is that \(S^{[n, n+1, n+2]}\) is a local complete intersection and has klt singularities. The proof involves showing that a two-step blowup gives a (small) resolution of singularities of \(S^{[n, n+1, n+2]}\).
In the end, the Picard group and the canonical divisor of \(S^{[n, n+1, n+2]}\) are computed in case \(S\) is regular. nested Hilbert schemes; irreducibility; singularities; Picard group Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Irreducibility and singularities of some 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 Let \(k\) be a field and let \(A\) be a local integral noetherian \(k\)-algebra of dimension one such that its normalization \(\bar A\) is a discrete valuation ring with residue field \(k\). The authors study the Hilbert scheme of zero-dimensional subschemes of Spec\((A)\). They prove that its connected components \({\mathcal M}_ \tau\) parametrize the ideals of \(A\) of colength \(\tau\). Furthermore \({\mathcal M}_ \tau\) are embedded in a linear subspace \({\mathcal M}\) of a certain Grassmannian. The authors study the partition of \({\mathcal M}\) by its intersection with the Schubert cells. They end the paper investigating the structure of \({\mathcal M}\) in the case of rings \(A\) with monomial semigroups. curve singularities; Hilbert scheme; Spec; Schubert cells Pfister, G., Steenbrink, J.H.M.: Reduced Hilbert schemes for irreducible curve singularities. J. Pure and Appl. Alg.77, 103--116 (1992) Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Grassmannians, Schubert varieties, flag manifolds Reduced Hilbert schemes for irreducible 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 \textit{G. Pfister} and \textit{J. H. M. Steenbrink} [J. Pure Appl. Algebra 77, No. 1, 103--116 (1992; Zbl 0752.14007)] studied the structure of punctual Hilbert schemes of certain degree for irreducible curve singularities by their intersections with Schubert cells. Using their method, the author and his students proved that punctual Hilbert schemes of degree two for monomial plane curve singularities are isomorphic to a projective line. In this paper, we generalize this result for general monomial curve singularities. punctual Hilbert scheme; monomial curve singularity Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Enumerative problems (combinatorial problems) in algebraic geometry The punctual Hilbert schemes of degree two for monomial 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 establishes an example where the principal component of the Hilbert scheme of \(n\) points in affine \(d\)-space is not Cohen-Macaulay, for any \(n \geq 9, d \geq 8\). This then provides a counter-example to a conjecture of Haiman.
The author gives an explicit description of the counter-example and details its construction and computes local coordinates on the component. The paper is well organized and will be of interest to anyone who studies the Hilbert scheme of points. Hilbert schemes; Cohen-Macaulay rings; ideal projectors; Littlewood-Richardson rule Parametrization (Chow and Hilbert schemes), Cohen-Macaulay modules The singularities of the principal component of 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 ``If \(f : X^n \to Y^p\) is a morphism of smooth complex analytic varieties with \(n < p\), then the multiple points of order \(k\) of \(f\) in the target are those \(y \in Y^p\) with \(k\) preimages, each preimage counted with multiplicity. ... In this paper we give a new construction, using punctual Hilbert schemes, which we offer as an alternative to multi-jets in the study of multiple point singularities. As an illustration of its usefulness, we use it to find a resolution of the closure of the triple point set of any ``good'' map \(f\), and of the multiple point set of a ``good'' \(f\) of any order, provided the map \(f\) has kernel rank at most 2. ... This construction also provides a useful starting point for finding resolutions of multiple point sets for general \(f\), by reducing the problem of resolving the singularities of \(f\) to the problem of resolving the singularities of the corresponding Hilbert schemes''. punctual Hilbert schemes; multiple point singularities; triple point Gaffney, T.; Counting Double Point Singularities, preprint. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Punctual Hilbert schemes and resolutions of multiple 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 The authors prove that the Hilbert functor \(\text{Hilb}_{X/S}\) of flat families of closed subschemes of a projective scheme \(X \to S\) over a locally noetherian base is representable by a scheme that is a disjoint union of locally projective schemes. The existence of the Hilbert scheme has been established by several others [\textit{A. Grothendieck}, ``Techniques de construction et théorèmes d'existence en géométrie algébique. IV: Les schéma de Hilbert'', Sém. Bourbaki 13 (1960/61), No. 221 (1961; Zbl 0236.14002); \textit{D. Mumford}, ``Lectures on curves on an algebraic surface.'' (1966; Zbl 0187.42701); \textit{E. Sernesi}, ``Topics on families of projective schemes'', Queen's Pap. Pure Appl. Math. 73 (1986); \textit{E. A. Strømme}, in: Parameter spaces: enumerative geometry, algebra and combinatorics. Proc. Banach Center conf., Warsaw, Poland, February 1994. Banach Cent. Publ. 36, 179--198 (1996; Zbl 0877.14002); \textit{A. B. Altman} and \textit{S. L. Kleiman}, Adv. Math. 35, 50--112 (1980; Zbl 0427.14015); \textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)]).
The approach in the paper under review is the following. For a finitely generated \(\mathbb{N}\)-graded \(A\)-algebra \(R\), the authors show that the functor \(\text{HomHilb}_R\) of homogeneous ideals in \(R\) having \(A\)-flat quotients is representable by a scheme. To prove this representability result it suffices to consider the functor \(\text{HomHilb}^\varphi_R\) of flat and homogeneous quotients of the polynomial ring \(R=A[x_0, \ldots, x_m]\) having a fixed Samuel function \(\varphi\).
The Homogeneous Hilbert functor is a special case of the Multigraded Hilbert functor introduced by Haiman and Sturmfels [loc. cit.], and the approach given in the paper under review is comparable with the one of Haiman and Sturmfels. However the proof of the representability of the Homogeneous Hilbert functor is appealingly simple:
Let \(G(B^n)\) be the closed subscheme of the Grassmannian of the truncated polynomial ring \(B^n=A[x_0, \ldots, x_m]/(x_0, \ldots, x_m)^n\) that represents \(\text{HomHilb}^{\varphi}_{B^n}\). The inverse limit \(G(B^{n+1})\to G(B^n)\) is the functor \(\text{HomHilb}^{\varphi}_R\). Let \(C_n\subseteq G(B^n)\) denote the intersection of the schematic images of \(G(B^{n+i})\to G(B^n)\). It follows by noetherian induction that \(C_n\subseteq G(B^n)\) is a closed subscheme. There exists, furthermore, an integer \(r\) such that any ideal \(I\) corresponding to a point of \(\text{HomHilb}^{\varphi}_R\) is generated in degrees \(\leq r\), and consequently the maps \(C_{n+r+1}\to C_{n+r}\) are injective. By Zariski's Main Theorem one obtains that \(C_{r+n+1}=C_{r+n}\), which proves representability, and projectivity, of \(\text{HomHilb}^{\varphi}_{R}\). The authors also supply a second proof that uses the theory of Fitting ideals to realize the representing object as a locally closed subscheme inside a product of suitable Grassmannians.
Representability of the Hilbert functor is then obtained in the following way. A flat family \(\text{Proj}(R/I)\to \text{Spec}(A)\) with Hilbert polynomial \(p(n)\) on each fiber is equivalent with the existence of some integer \(r\) such that the \(A\)-module of degree \(n\) elements \((R/I)_n\) is locally free of rank \(p(n)\), for \(n \geq r\). In particular, by truncating the first \(r\) degrees and forming the ideal \(I'=\bigoplus_{n\geq r} I\) one gets that the homogeneous quotient \(R/I'\) is \(A\)-flat. Furthermore, as the integer \(r\) uniformly depends on the Hilbert polynomial only, there is a map \(\text{Hilb}_{\mathbb{P}}^m\to\text{HomHilb}_R\) which is a section of the natural map \(\text{HomHilb}_R \to \text{Hilb}_{\mathbb{P}}^m\) -- giving representability of the Hilbert functor. homogeneous ideals; representable functors; Hilbert scheme A. Álvarez, F. Sancho, P. Sancho, Homogeneous Hilbert scheme. \textit{Proc. Amer. Math. Soc}. \textbf{136} (2008), 781-790. MR2361849 Zbl 1131.14008 Parametrization (Chow and Hilbert schemes) Homogeneous 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 More than 50 years ago \textit{D. Mumford} gave an example of a generically non-reduced irreducible component of the Hilbert scheme of smooth irreducible space curves [Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)]. Expanding on Mumford's example, \textit{J. O. Kleppe} used Hilbert flag schemes to show that if a general curve \(C\) in a Hilbert scheme component \(V\) lies on a smooth cubic surface \(X\) and \(H^1({\mathcal O}_X (-C)(3)) \neq 0\), then \(V\) is generically non-reduced [Lect. Notes Math. 1266, 181--207 (1985; Zbl 0631.14022)]. Recently \textit{J. O. Kleppe} and \textit{J. C. Ottem} extended these ideas to produce examples in which the general curve \(C\) lies on a quartic surface [Int. J. Math. 26, Article ID 1550017, 30 p. (2015; Zbl 1323.14005)], but their method fails for curves lying on smooth surfaces \(X\) of degree \(d \geq 5\) or having Picard number \(\rho (X) > 2\).
Here the author combines Hilbert flag scheme methods and the theory of Hodge loci to construct more examples of non-reduced Hilbert schemes whose general curve \(C\) lies on a smooth surface \(X\) of degree \(d \geq 5\) (and on no quartic) satisfying \(\rho (X) > 2\). He starts with a smooth surface \(X\) of degree \(d \geq 5\) containing two coplanar lines \(L_1\) and \(L_2\), noting that \(\rho (X) > 2\). The Hilbert scheme of extremal curves of the form \(D=2L_1 + L_2 \subset X\) is generically non-reduced by work of \textit{M. Martin-Deschamps} and \textit{D. Perrin} [Ann. Scient. École Norm. Sup. 29, 757--785 (1996; Zbl 0892.14005)]. Letting \(\gamma \in H^{1,1}(X, \mathbb Z)\) denote the cohomology class of of \(D\), the Zariski closure of the associated Hodge locus \(\text{NL}(\gamma)\) in the open set \(U \subset |{\mathcal O}_{\mathbb P^3} (d)|\) of smooth surfaces is irreducible and generically non-reduced. With arguments involving exact sequences and Mumford regularity, he shows that the general member \(C \in |{\mathcal O}_X (D) (m)|\) is a smooth connected curve for \(m \geq 2d-2\). Letting \(X\) and \(C\) vary and taking the closure in the Hilbert scheme gives an irreducible component which is generically non-reduced. The arguments are clearly written and easy to follow. Hilbert scheme of space curves; Hilbert flag scheme; Hodge locus; non-reduced components Parametrization (Chow and Hilbert schemes), Transcendental methods of algebraic geometry (complex-analytic aspects), Variation of Hodge structures (algebro-geometric aspects) On generically non-reduced components of Hilbert schemes of smooth 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 \(X\) be a complex quasi-projective surface and let \(n\) be a positive integer. The Hilbert scheme \(\hbox{Hilb}^n(X)\) parametrizes the zero-dimensional subschemes of \(X\) of length \(n\). The author studies the topological Euler characteristics of these Hilbert schemes, which he collects in a generating series
\[
Z_X(q) = \sum_{n \geq 0} q^n \chi(\hbox{Hilb}^n(X)).
\]
When \(X\) is a smooth surface, these have been carefully studied by Fogarty, and Göttsche described the generating series of the Poincaré polynomial of these Hilbert schemes in terms of the Betti numbers of \(X\). To extend this, the author considers a particular action of \(\mathbb Z_p\) (\(p\) a positive integer) on \(\mathbb C^2\) involving another integer \(q\) coprime to \(p\), and denotes by \(X(p,q)\) the quotient variety. The main result of this paper is a representation of \(Z_{X(p,1)}(q)\) as a coefficient of a two-variable generating function, obtained by studying a torus action on \(X(p,1)\). The author links the combinatorics of this problem to \(p\)-fountains, a generalization of the notion of a fountain of coins. Hilbert scheme; cyclic quotient singularity; \(p\)-fountain Gyenge, Á, Hilbert scheme of points on cyclic quotient singularities of type \((p, 1)\), Period. Math. Hungar., 73, 93-99, (2016) Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Hilbert scheme of points on cyclic quotient singularities of type {\((p,1)\)} | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author shows that the Hilbert scheme \({\mathcal I}'_{d,g,3}\) of smooth, complex, curves of degree \(d\) and genus \(g\) in \(\mathbb{P}^ 3\) is irreducible when \(d=11\), \(g=10\) and when \(d=10\), \(g=9\). Then, by using results of a previous paper in cooperation with \textit{Seon Ja Kim} [J. Algebra 145, 240-248 (1992)], he proves that \({\mathcal I}'_{d,g,3}\) is irreducible for all \(d\) and \(g\) such that \(d>g\) and positive \(\rho(d,g,3)\) (the Brill-Noether number). Some results by \textit{M. Coppens} [Ann. Mat. Pura Appl., IV. Ser. 157, 183-197 (1990; Zbl 0742.14025)] are also used. smooth complex space curves; Hilbert scheme; Brill-Noether number Keem C., Manuscr, Math. 71 (3) pp 307-- (1991) Parametrization (Chow and Hilbert schemes), Plane and space curves A remark on the Hilbert scheme of smooth complex space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove a recent conjecture by \textit{ Gyenge} et al. [Int. Math. Res. Not. 2017, No. 13, 4152--4159 (2017; Zbl 1405.14010); Eur. J. Math. 4, No. 2, 439--524 (2018; Zbl 1445.14007)] giving a formula of the generating function of Euler numbers of Hilbert schemes of points \(\operatorname{Hilb}^N(\mathbb{C}^2/ \Gamma )\) on a simple singularity \({\mathbb{C}^2}/ \Gamma \), where \(\Gamma\) is a finite subgroup of \(\operatorname{SL}(2)\). We deduce it from the claim that quantum dimensions of standard modules for the quantum affine algebra associated with \(\Gamma\) at \(\zeta =\exp (\frac{2\pi \sqrt{-1}}{2({h^{\vee }}+1)})\) are always 1, which is a special case of an earlier conjecture by Kuniba. Here \({h^{\vee }}\) is the dual Coxeter number. We also prove the claim, which was not known for \({E_7}\), \({E_8}\) before. Hilbert schemes of points; quantum affine algebras; quantum dimensions; simple surface singularities Parametrization (Chow and Hilbert schemes), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Quantum groups (quantized enveloping algebras) and related deformations Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine 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 We give a method of counting the number of curves with a given type of singularity in a suitably ample linear series on a smooth surface using punctual Hilbert schemes. The types of singularities for which our results suffice include the topological type with local equation \(x^{a}+y^{b}\) with \({\leqslant}a{\leqslant}3b\). We work out the example of curves with the analytic type of singularity with local equation \(x^{2}+y^{n}\) for \(1\leq n\leq 9\). punctual Hilbert scheme; type of singularity Russell, H.: Counting singular plane curves via Hilbert schemes. Adv. math. 179, No. 1, 38-58 (2003) Enumerative problems (combinatorial problems) in algebraic geometry, Singularities of curves, local rings, Parametrization (Chow and Hilbert schemes) Counting singular plane curves via 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 We discuss two conjectures by Francesco Severi and Joe Harris about the irreducibility and the dimension of the Hilbert scheme parameterizing smooth projective curves of given degree and genus. dimension; Hilbert scheme; irreducibility; smooth curve Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory) A few remarks about the Hilbert scheme of smooth projective 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 For a finite subgroup \(\Gamma\subset \mathrm{SL}(2,\mathbb{C})\) and \(n\geqslant 1\), we construct the (reduced scheme underlying the) Hilbert scheme of \(n\) points on the Kleinian singularity \(\mathbb{C}^2/\Gamma\) as a Nakajima quiver variety for the framed McKay quiver of \(\Gamma\), taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed
McKay quiver by removing an arrow and then `cornering', and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of the stability parameter. Hilbert scheme of points; quiver variety; Kleinian singularity; preprojective algebra; cornered algebra Representations of quivers and partially ordered sets, Parametrization (Chow and Hilbert schemes), McKay correspondence, Singularities in algebraic geometry Punctual Hilbert schemes for Kleinian singularities as quiver varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an integral quasi-projective surface over the complex numbers with at worst Kleinian singularities. The latter condition means that the singular points are 2-dimensional quotient singularities. Let \(d\) be a positive integer and denote by \(\mathrm{Hilb}^d(X)\) the Hilbert scheme of length \(d\) subschemes of \(X\). The main result of this paper is that \(\mathrm{Hilb}^d(X)\) is irreducible of dimension \(2d\). The author points out that this result has also recently been proved by
\textit{A. Craw} et al. [Algebr. Geom. 8, No. 6, 680--704 (2021; Zbl 1494.16013)].
This generalizes work of
\textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)] which showed that when \(X\) is nonsingular then \(\mathrm{Hilb}^d(X)\) is nonsingular and connected, so in particular irreducible. On the other hand,
\textit{R. M. Miró-Roig} and \textit{J. Pons-Llopis} [Commun. Algebra 41, No. 5, 1776--1780 (2013; Zbl 1277.14005)] showed that \(\mathrm{Hilb}^d (X)\) is reducible for sufficiently large \(d\) if \(X\) has an isolated cone singularity over a nonsingular curve of degree at least 5. The proof uses the fact that \(\mathrm{Hilb}^d(X)\) is irreducible if and only if any length \(d\) subscheme of \(X\) is smoothable (i.e. a flat deformation of \(d\) distinct points), then by induction showing that it suffices to show that any length \(d\) subscheme of \(X\) supported at a single singular point is smoothable. Hilbert scheme of points; Kleinian singularities; maximal Cohen-Macaulay modules; matrix factorizations; smoothability Cohen-Macaulay modules, Syzygies, resolutions, complexes and commutative rings, Deformations of singularities, Parametrization (Chow and Hilbert schemes), Singularities of surfaces or higher-dimensional varieties Irreducibility of the Hilbert schemes of points on surfaces with Kleinian singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathcal O\) be the complete local ring of an irreducible curve singularity over an algebraically closed field of characteristic zero. The authors describe the Hilbert schemes of length \(r\) closed subschemes of \(\text{Spec } {\mathcal O}\) as constructed by \textit{G. Pfister} and \textit{J. H. M. Steenbrink} [J. Pure Appl. Algebra 77, No. 1, 103--116 (1992; Zbl 0752.14007)]. Recalling their work, let \(\overline {\mathcal O}\) be the normalization and set \(\delta = \dim_k ({\overline {\mathcal O}}/{\mathcal O})\). Letting \(I(2 \delta) \subset {\overline {\mathcal O}}\) denote the ideal of elements of order at least \(2 \delta\), the subset \(M \subset \mathbb G = \text{Gr} (\delta, {\overline {\mathcal O}}/I(2 \delta))\) consisting of subspaces \(V\) which are \({\mathcal O}\)-submodules maps onto a variety \(\mathcal M = \psi(M)\) via the Plücker embedding \(\psi: \mathbb G \hookrightarrow \mathbb P^N\). Further, if \({\mathcal I}_r\) is the set of ideals \(I \subset {\mathcal O}\) with \(\dim_k {\mathcal O}/I = r\), the map \(\varphi_r: {\mathcal I}_r \to \mathbb G\) given by \(I \mapsto t^{-r}I / I(2 \delta)\) is injective for all \(r\) with Zariski closed image \({\mathcal M}_r \subset \mathcal M\), defining the Hilbert schemes. Moreover \({\mathcal M}_r = \mathcal M\) for \(r \geq 2 \delta\).
The authors consider the case \(\mathcal O = k[[t^2, t^{2d+1}]] \subset {\overline {\mathcal O}} = k[[t]]\) for fixed \(d\): denote the Hilbert schemes above by \({\mathcal A}_d = {\mathcal M}\) and \({\mathcal A}_{d,r} = {\mathcal M}_r\). Pfister and Steenbrink [loc. cit.] had already shown that \({\mathcal A}_{d,r} = {\mathcal A}_d\) is a rational variety of dimension \(d\) for \(r \geq 2d\), so they focus on the case \(1 \leq r \leq 2d\). Using their previous work [\textit{Y. Sōma} and \textit{M. Watari}, J. Singul. 8, 135--145 (2014; Zbl 1312.14016)] and semi-groups generated by orders of elements to study the form of the ideals, the authors prove that \({\mathcal A}_{d,2s} \cong {\mathcal A}_{d,2s+1}\) if \(0 < s < d\) and \({\mathcal A}_{d,r} \cong {\mathcal A}_{e,r}\) for \(0 < r < \min\{2d,2e\}\). Applying these results, they show that for \(1 \leq r \leq 2d\) and \(s = [ r/2 ]\), \({\mathcal A}_{d,r}\) is a rational projective variety of dimension \(s\) which is isomorphic to \({\mathcal A}_s\) if \(r \geq 2\). Moreover \({\mathcal A}_{d,r}\) consists of a single point for \(r=1\), is isomorphic to \(\mathbb P^1\) for \(r=2,3\) and is a singular rational projective variety for \(4 \leq r \leq 2d\) whose singular locus is described. punctual Hilbert schemes Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings The punctual Hilbert schemes for the curve singularities of type \(A_{2d}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main result of this paper is that \(H_{d,g,n}\), the Hilbert scheme of smooth irreducible curves of degree \(d\) and genus \(g\) in \({\mathbb{P}}^ n\), is irreducible provided that \(d>((2n-3)g+n+3)/n\). A particular case is \(n=3\), when the condition reduces to \(d\geq g+3\). - It is also shown that any integral space curve satisfying \(d\geq p_ a+3\) is smoothable in \({\mathbb{P}}^ 3\). The results on space curves were asserted by Severi but without complete proofs. special curves; Hilbert scheme of smooth irreducible curves; space curves L. Ein, Hilbert scheme of smooth space curves, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 469--478. Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) Hilbert scheme of smooth space curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of the present paper is to study the structure of the punctual Hilbert schemes for the curve singularities of types \(E_6\) and \(E_8\). Our analysis uses computational methods to decompose a punctual Hilbert scheme into affine cells. We also use known results about the compactified Jacobians of singular curves. punctual Hilbert schemes; curve singularities of types \(E_6\) and \(E_8\); compactified Jacobians of singular curves Y. S\={}oma, M. Watari: Punctual Hilbert schemes for irreducible curve singularities of types E6and E8. J. Singularites. 8, 135-145, (2014). Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings The punctual Hilbert schemes for the curve singularities of types \(E_6\) and \(E_8\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers a germ of a nodal curve \(X\), and studies the Hilbert scheme of length \(m\) subschemes of \(X\). The knowledge of this Hilbert scheme Hilb\(_m(X)\) is an important step for attaching some enumerative problems via the degeneration of curves to nodal curves.
The author proves that Hilb\(^0_m(X)\), the Hilbert scheme of length \(m\) subschemes of the node, consists in a union of \(m-1\) rational curves, meeting transversally. Then the author describes Hilb\(_m(X)\) as a formal scheme defined along Hilb\(^0_m\). Let \(\tilde X\to B\) denote the deformation of some smooth germ of curve to a nodal one (formally defined by \(xy-t\)). The author uses the previous result to show that, when the total space of the family is smooth, then the relative Hilbert scheme Hilb\(_m(\tilde X/B)\) is a smooth, \((m+1)\)-dimensional formal variety. Similarly, some relative flag Hilbert schemes for \(\tilde X/B\) are proven to be normal and complete intersection. nodal curves Ran, Z., A note on Hilbert schemes of nodal curves, J. Algebra, 292, 2, 429-446, (2005) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) A note on Hilbert schemes of nodal 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 \(\mathcal {M}_g\) be the moduli space of smooth curves of genus \(g\). Set \(\rho (g,r,d):= g-(r+1)(g-d+r)\) (the Brill-Noether number). Set \(\mathcal {M}^{r}_{g,d}:= \{C\in \mathcal {M}_g:\) \(C\) has a \(g^r_k\}\). If \(\rho (g,r,d)=-1\), then \(\mathcal {M}^{r}_{g,d}\) is a divisor of \(\mathcal {M}_g\) [\textit{D. Eisenbud} and \textit{J. Harris}, Ann. Sci. Éc. Norm. Supér. (4) 22, No. 1, 33--53 (1989; Zbl 0691.14006); \textit{F. Steffen}, Invent. Math. 132, No. 1, 73--89 (1998; Zbl 0935.14018)].
Here the authors prove that if \(\rho (g,r,d) = \rho (g,s,e)=-1\) and \(r\neq s\), then \(\mathcal {M}^{r}_{g,d} \neq \mathcal {M}^{s}_{g,e}\), unless \(e=2g-2-d\) (Serre duality gives that the two divisors are the same if \(e =2g-2-d\)). This result is very useful: previously just proving that \(\mathcal {M}^{1}_{23,12}\), \(\mathcal {M}^{2}_{23,17}\) and \(\mathcal {M}^{3}_{23,20}\) are mutually distinct was a key step to prove that \(\mathcal {M}_{23}\) has Kodaira dimension \(\geq 2\) [\textit{G. Farkas}, J. Reine Angew. Math. 539, 185--200 (2001; Zbl 0994.14022)].
In the paper under review the authors also give other applications (the existence of a unique irreducible component of the Hilbert scheme of smooth curves in \(\mathbb {P}^r\) with fixed degree and genus and the non-esistence of \((d-e)\)-secant \((r-s-1)\)-plane on embeddings by the general \(g^r_d\) on a general \(C\in \mathcal {M}^{r}_{g,d}\) if \(e\neq 2g-2-d\)). Brill-Noether theory; moduli space of curves, divisors on the moduli space of curves; linear series; special divisors Choi, Y.; Kim, S.; Kim, Y.: Remarks on brill-Noether divisors and Hilbert schemes, J. pure appl. Algebra 216, 377-384 (2012) Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory, Parametrization (Chow and Hilbert schemes) Remarks on Brill-Noether divisors 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 Here the author gives several uper bounds --- both new and classical ones (but with a new proof) for the dimension of the tangent space of the Hilbert scheme of a subvariety \(X\) of a fixed variety \(V\subset\mathbb{P}^ n\); usually \(X\) is a linear space or at least a complete intersection in \(\mathbb{P}^ n\). Sample result:
Theorem: Set \(\delta(r,m):=(m+1)(r-m)\); let \(X\) be an integral complete intersection in \(\mathbb{P}^ n\) with \(m=\dim(X)\); let \(V\) be an integral variety with \(X\subset V\subset\mathbb{P}^ n\), \(X\cap V_{reg}\neq\emptyset\), and \(r=\dim(V)\); let \(t_{X,V}\) the dimension of the tangent space at \(X\) of the Hilbert scheme of subvarieties of \(V\). Let \(L\hat{}\) be the intersection of all tangent spaces to \(V\) (seen as subspaces of \(\mathbb{P}^ n)\) at the points of \(X\cap V_{reg}\). Then \(t_{X,V}\leq\delta(r,m)\) and equality holds if and only if \(\dim(L\hat {})=r\) (i.e. the tangent spaces are the same at all such points). Assume \(X\) linear; if \(t_{X,V}=\delta(r,m)-m\), then \(\dim(L\hat {})=r-1\); if \(\dim(L\hat {})=r-1\), then \(\delta(r,m)-1\leq t_{X,V}\leq \delta(r,m)\).
The proofs use the comparison between the deformation functor of \(X\) in \(V\) and of their affine cones and explicit computations (using the very simple cohomology of \(X\)) of the tangent space to the second functor. dimension of the tangent space of the Hilbert scheme; subvariety; deformation functor; affine cones Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves Some observations on the singularities of the Hilbert scheme which parametrizes the linear varieties contained in a 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 A curve X in \({\mathbb{P}}^ 3\) is said to be \textit{obstructed} if the corresponding point of the Hilbert scheme is singular. A long-standing problem has been to give a geometrical characterization of non- obstructedness. Let \({\mathcal I}_ X\) be the ideal sheaf of X in \({\mathbb{P}}^ 3\). A curve is said to be of \textit{maximal rank} if for each n, at most one of \(h^ 0({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) and \(h^ 1({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) is non-zero. A curve is said to have \textit{natural cohomology} if at most one of \(h^ 0({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\), \(h^ 1({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) and \(h^ 2({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) is non-zero for each n. Prior to this paper, several examples were known of smooth obstructed space curves, but it was an open question (due to E. Sernesi) whether there exists one which furthermore has maximal rank.
For much of the paper X is assumed to have the property that \(h^ 1({\mathbb{P}}^ 3,{\mathcal I}_ X(n))\) is nonzero for exactly one n. The first set of results go to show that the ``generic'' such X (in a precise sense) is non-obstructed. The second part of the paper gives some useful techniques for constructing obstructed curves, using liaison. The authors apply these techniques in the third part to produce a concrete example of a smooth, obstructed curve with maximal rank.
The authors point out that their examples of a smooth obstructed maximal rank curve has also been constructed independently (and with different techniques) by \textit{C. Walter} (``Some examples of obstructed curves in \({\mathbb{P}}^ 3\)'', Proc., Trieste 1989). They also point out that their example does not have natural cohomology, and indeed Walter asked whether all curves with this property may be non-obstructed. However, recently \textit{M. Martin-Deschamps} and \textit{D. Perrin} have produced a curve (but not a smooth one) which has natural cohomology but is obstructed (``Courbes gauches et modules de Rao''). linkage; Hilbert scheme; maximal rank; obstructed curves; liaison; natural cohomology Bolondi, G.; Kleppe, J. O.; Miro-Roig, R. M.: Maximal rank curves and singular points of the Hilbert scheme. Compos. math. 77, 269-291 (1991) Plane and space curves, Linkage, Parametrization (Chow and Hilbert schemes) Maximal rank curves and singular points 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 Fix an algebraically closed field \(k\) of characteristic zero and let \(H^d (\mathbb A^n_k)\) denote the Hilbert scheme parametrizing zero dimensional closed subschemes of length \(d\). The \textit{principal component} of \(H^d (\mathbb A^n_k)\) consists of flat limits of \(d\) distinct points, but sometimes there are other components, the first examples being due to \textit{A. Iarrobino} [Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)]. An \textit{elementary component} of \(H^d (\mathbb A^n_k)\) is one whose general member corresponds to a closed subscheme supported at a single point and is generically smooth. \textit{A. Iarrobino} and \textit{J. Emsalem} constructed an explicit example of an elementary component some 40 years ago [Compos. Math. 36, 145--188 (1978; Zbl 0393.14002)]. After describing their example carefully, the author uses the theory of border basis schemes due to \textit{M. Kreuzer} and \textit{L. Robbiano} [Collect. Math. 59, No. 3, 275--297 (2008; Zbl 1190.13022)]; J. Pure Appl. Algebra 215, No. 8, 2005--2018 (2011; Zbl 1216.13018)] to extend the Iarrobino-Ensalem construction, obtaining more examples of elementary components. Hilbert scheme of points; elementary component Parametrization (Chow and Hilbert schemes) Some elementary components of 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 We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme \(\text{Hilb}^2 \mathbb{P}^2\) of two points on \(\mathbb{P}^2\) and the dense open set parametrizing non-planar clusters in the punctual Hilbert scheme \(\text{Hilb}^4_0(\mathbb{A}^3)\) of clusters of length four on \(\mathbb{A}^3\) with support at the origin. We find explicit equations in projective, respectively affine, embeddings for these spaces. In particular, we answer a question of Bernd Sturmfels who asked for a description of the latter space that is amenable to further computations. While the explicit equations we find are controlled in a precise way by the representation theory of \(\operatorname{SL}_3\), our arguments also rely on computer algebra. binomial ideals; sparse polynomials; numerical algebraic geometry; witness sets; Macaulay dual spaces Geometric aspects of numerical algebraic geometry, Parametrization (Chow and Hilbert schemes), Numerical algebraic geometry, Symbolic computation and algebraic computation, Commutative rings defined by binomial ideals, toric rings, etc. On the equations defining some 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 Main topic: Let \(H^ d\) be the Hilbert scheme of \(P^ 2_{{\mathbb{C}}}\) of length \(d\) and \(H_{\geq \phi}\) its subscheme of \(p\in H^ d\) with the Hilbert function \(h(p)\geq \phi\), \(\phi\) any function \(\phi: N\to N\). Then \(H_{\geq \phi}\) is simple connected. The proofs needs punctual Hilbert schemes. - A more or less direct corollary of this theorem is: The Hilbert scheme of curves of degree \(d\) and genus \(g\) in the projective space \(P^ 3_{{\mathbb{C}}}\) is simple connected. connectivity of Hilbert scheme of curves; punctual Hilbert schemes Parametrization (Chow and Hilbert schemes), Topological properties in algebraic geometry, Families, moduli of curves (algebraic) Einfacher Zusammenhang der Hilbertschemata von Kurven im komplex-projektiven Raum. (Simple connectedness of the Hilbert schemes of curves in complex projective space) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Hilbert schemes \(\mathrm{Hilb}^{p(t)} (\mathbb P^m)\) parametrizing closed subschemes \(X \subset \mathbb P^m\) with Hilbert polynomial \(p(t)\) have received much attention from algebraic geometers since their construction in the early 1960s. Although connected [\textit{R. Hartshorne}, Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)], \(\mathrm{Hilb}^{p(t)} (\mathbb P^m)\) exhibits many bad behaviors, for example it can have non-reduced components [\textit{D. Mumford}, Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)] and every singularity type appears in some Hilbert scheme [\textit{R. Vakil}, Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)]. In the paper under review, the authors classify all smooth Hilbert schemes.
Recall that a Hilbert polynomial of a closed subscheme \(X \subset \mathbb P^m\) can be uniquely written in the form \(p(t) = \sum_{i=1}^r \binom{t+\lambda_i-i}{\lambda_i-1}\) where \(\lambda = (\lambda_1, \dots, \lambda_r)\) is a partition of integers satisfying \(\lambda_1 \geq \dots \geq \lambda_r \geq 1\) [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009)]. The authors prove that the partitions corresponding to a smooth Hilbert scheme are precisely those in the following list:
(1) \(m = 2 \geq \lambda_1\).
(2) \(m \geq \lambda_1\) and \(\lambda_r \geq 2\).
(3) \(\lambda = (1)\) or \(\lambda = (m^{r-2}, \Lambda_{r-1},1)\), where \(r \geq 2\) and \(m \geq \lambda_{r-1} \geq 1\).
(4) \(\lambda = (m^{r-s-3}, \lambda_{r-s-s}^{s+2},1)\), where \(r-3 \geq s \geq 0\) and \(m-1 \geq \lambda_{r-s-2} \geq 3\).
(5) \(\lambda = (m^{r-s-5}, 2^{s+4},1)\), where \(r-5 \geq s \geq 0\).
(6) \(\lambda = (m^{r-3},1^3)\), where \(r \geq 3\).
(7) \(\lambda = (m+1)\) or \(r=0\).
Moreover, the authors describe the schemes parametrized by each family listed. For example, the general member of family (3) is a union of a hypersurface of degree \(r-2\), a linear subspace of dimension \(\lambda_{r-1}\) and a point while the general member of family (5) is a union of a hypersurface of degree \(r-s-3\), a hypersurface of degree \(s+2\) of a linear subspace of dimension \(\lambda_{r-s-2}\) and a point. The families in (7) correspond to the one-point Hilbert scheme parametrizing \(\mathbb P^m\) and the empty scheme. All families in the theorem were previously known to be smooth: smoothness for families (1) and (6) follows from work of \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)], families (2) and (3) were shown smooth by
\textit{A. P. Staal} [Math. Z. 296, No. 3--4, 1593--1611 (2020; Zbl 1451.14010)] and families (4) and (5) were shown smooth by \textit{R. Ramkumar} [J. Algebra 617, 17--47 (2023; Zbl 1503.13007)] in his work on Hilbert schemes with at most two Borel-fixed ideals. The main contribution here is that this list is complete.
As to the strategy of the proof, it is known from work of \textit{A. Reeves} and \textit{M. Stillman} that the lexicographical point is always smooth on the Hilbert scheme and determines a unique irreducible component of \(\mathrm{Hilb}^{p(t)} (\mathbb P^m)\) of computable dimension [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)]. In theory one could attempt to show smoothness by computing the dimension of the Zariski tangent space at the other Borel-fixed points, but this is unwieldy. Instead, the authors construct families of subschemes corresponding to points on Hilbert schemes that necessarily singular. To describe these points, define a \textit{residual inclusion} \(X \subset Y \subset \mathbb P^m\) to be a closed immersion such that there is a linear subspace \(\Lambda \subset \mathbb P^m\) containing \(X\) and a hypersurface \(D \subset \Lambda\) with \(Y\) the residual scheme of \(D \subset X\) in \(\Lambda\). A \textit{residual flag} is a flag \(\emptyset \subset X_e \subset X_{e-1} \subset \dots \subset X_1\) of residual inclusions. For most Hilbert polynomials not in the list, the authors produce such an \(X_1\) near the lexicographic point which corresponds to a singular point on the Hilbert scheme. The others not on the list are handled with three other singular families. The proof is valid over \(\mathrm{Spec}\,\mathbb Z\). The authors credit \texttt{Macaulay2} [\url{http://www.math.uiuc.edu/Macaulay2/}] for many experimental computations that were indispensable in discovering their results. Hilbert schemes; Borel fixed ideals; partial flag varieties Parametrization (Chow and Hilbert schemes), Fibrations, degenerations in algebraic geometry, Geometric invariant theory, Actions of groups on commutative rings; invariant theory Smooth Hilbert schemes: their classification and geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Following the approach in the book [Commutative algebra. With a view toward algebraic geometry. Berlin: Springer-Verlag (1995; Zbl 0819.13001)], by \textit{D. Eisenbud}, where the author describes the generic initial ideal by means of a suitable total order on the terms of an exterior power, we introduce first the \textit{generic initial extensor} of a subset of a Grassmannian and then the \textit{double-generic initial ideal} of a so-called \textit{GL-stable subset} of a Hilbert scheme. We discuss the features of these new notions and introduce also a partial order which gives another useful description of them. The double-generic initial ideals turn out to be the appropriate points to understand some geometric properties of a Hilbert scheme: they provide a necessary condition for a Borel ideal to correspond to a point of a given irreducible component, lower bounds for the number of irreducible components in a Hilbert scheme and the maximal Hilbert function in every irreducible component. Moreover, we prove that every isolated component having a smooth double-generic initial ideal is rational. As a by-product, we prove that the Cohen-Macaulay locus of the Hilbert scheme parameterizing subschemes of codimension 2 is the union of open subsets isomorphic to affine spaces. This improves results by \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)] and \textit{R. Treger} [J. Algebra 125, No. 1, 58--65 (1989; Zbl 0705.14047)]. generic initial ideal; extensor; Hilbert scheme; irreducible component; maximal Hilbert function; rationality Parametrization (Chow and Hilbert schemes), Computational aspects of higher-dimensional varieties, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Rationality questions in algebraic geometry, Exterior algebra, Grassmann algebras Double-generic initial ideal and 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 Many authors have studied the Hilbert scheme \(\text{Hilb}^d (X)\) parametrizing closed subschemes \(Z \subset X\) of length \(d\) on a variety \(X\). \textit{J. Fogarty} showed that \(\text{Hilb}^d (X)\) is smooth and irreducible when \(X\) is a smooth connected surface [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)], but for \(n>2\) and \(d \gg 0\) \textit{A. Iarrobino} [Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)] proved that \(\text{Hilb}^d (\mathbb A^n)\) is reducible by constructing \textit{elementary} components \(Z \subset \text{Hilb}^d (\mathbb A^n)\), those parametrizing subschemes supported at a single point.
The author proves that if \(R \subset \mathbb A^n\) is supported at the origin, then the corresponding point \([R] \in \text{Hilb}^d (\mathbb A^n)\) lies on an elementary component if \(R\) has \textit{trivial negative tangents}, meaning that the tangent map \(\langle \partial_1, \dots, \partial_n \rangle \to \text{Hom}(I_R,{\mathcal O}_R)_{<0}\) of the orbit of \([R]\) under translation is surjective. Conversely, if \(Z \subset \text{Hilb}^d (\mathbb A^n)\) is a generically reduced elementary component and char \(k =0\), then the general point \([R] \in Z\) has trivial negative tangents (it is unknown whether there exist generically non-reduced components).
The main tools in the proof are obstruction theory and an extension of the decomposition theorem of \textit{A. Bialynicki-Birula} [Ann. Math. (2) 98, 480--497 (1973; Zbl 0275.14007)] from smooth proper varieties to the singular non-proper Hilbert schemes \(\text{Hilb}^d (\mathbb A^n)\).
The author uses his criterion to construct infinitely many smooth points of distinct elementary components of \(\text{Hilb}^d (\mathbb A^4)\); these examples show that the Gröbner fan need not distinguish components of \(\text{Hilb}^d (\mathbb A^n)\). He also reduces the question of whether \(\text{Hilb}^d (\mathbb A^n)\) is reduced to a testable conjecture involving the trivial negative tangents condition. Hilbert scheme of points; elementary components Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Deformations and infinitesimal methods in commutative ring theory Elementary components of Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(<\) be a monomial well-ordering on \(R=K[x_0, \dots,x_n]\), \(K\) an algebraically closed field For a monomial ideal \(I_0\) with Hilbert polynomial \(p(t)\) consider \(M_{I_0}= \{J\subset R\mid J\) homogeneous and saturated ideal and \(\text{in}(J)=I_0\}\). Here \(\text{in}(J)\) denotes the initial ideal of \(J\). It is proved that \({\mathcal M}_{I_0}\) carries the structure of a locally closed subscheme in \(\text{Hilb}^{p(t)}_{\mathbb{P}^n}\). \({\mathcal M}_{I_0}\) is affine if \(I_0\) is saturated. In this case, an explicit construction of the coordinate ring is possible. The set of all \({\mathcal M}_{I_0}\), \(I_0\) monomial with Hilbert polynomial \(p(t)\), leads to a natural stratification of \(\text{Hilb}_{\mathbb{P}^n}^{p(t)}\).
As an application, the singular locus of the component of \(\text{Hilb}^{4t+1}_{\mathbb{P}^4}\) containing the arithmetically Cohen-Macaulay curves of degree 4 is described. Hilbert scheme; stratification; Gröbner basis; Hilbert polynomial R. Notari and M.L. Spreafico, A stratification of Hilbert schemes by initial ideals and applications , Manuscr. Math. 101 (2000), 429-448. MR1759253 %(2001b:14008)
Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Computational aspects of higher-dimensional varieties, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) A stratification of Hilbert schemes by initial ideals 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 By a theorem of F. S. Macaulay, Hilbert polynomials of ideals in \({\mathbb{P}}^ n\) correspond to sequences of integers \(0<a_ 0\leq a_ 1\leq...\leq a_ s\) with \(s\leq r-1\). The author shows that the ideals with \(s\leq r-2\) have a simple canonical form. The irreducibility of the corresponding Hilbert schemes is immediate. The proof uses the technique of comparison with monomial ideals. irreducibility of Hilbert schemes Gotzmann G. (1989). Some irreducible Hilbert schemes. Math. Z. 201: 13--17 Parametrization (Chow and Hilbert schemes) Some irreducible 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 several previous papers [for example \textit{M. Artin} and \textit{J. J. Zhang}, Adv. Math. 109, 228-287 (1994; Zbl 0833.14002)], the authors have developed a theory of noncommutative algebraic geometry. Starting with a not necessarily commutative ring \(A\) (graded by the natural numbers \(\mathbb{N}\)) they do not define the scheme Proj(\(A\)) directly as a physical object, but instead define the category of ``coherent (respectively quasicoherent) sheaves on Proj(\(A\))'' as the quotient of the category of finitely generated (respectively arbitrary) graded \(A\)-modules by the subcategory of torsion modules (torsion means killed by some power of the graded maximal ideal of \(A\)). If \(A\) is commutative then under suitable hypotheses the resulting categories are equivalent to the categories of coherent or quasicoherent sheaves on the usual scheme Proj(\(A\)).
The following is a somewhat simplified description of what the authors do in the present paper. They replace the categories of \(A\) modules involved in the construction of Proj by a \(k\)-linear Grothendieck category \({\mathcal C}\), where \(k\) is a commutative ring. Among other things the category \({\mathcal C}\) is abelian, and for objects \(V\) and \(W\) in \({\mathcal C}\), Hom\(_{\mathcal C}(V,W)\) is a \(k\)-module in a natural way. Typical examples are Proj(\(A\)) (regarded as an abelian category as described above), the category Mod(\(A\)) of right \(A\)-modules, and the category Gr(\(A\)) of graded right \(A\)-modules. If \(R\) is a Noetherian \(k\)-algebra and \(P\) is a Noetherian object in \({\mathcal C}\) then the Hilbert functor Quot\(_P\) associated to \(P\) assigns to \(R\) the set isomorphism classes of quotients of \(P\otimes_k R\) which are flat over \(R\). The authors' goal, following Grothendieck in his construction of the classical Hilbert scheme, is to show that (under suitable hypotheses) the Hilbert functor is representable. The paper is very long, much of it taken up by carefully stating the hypotheses necessary to carry out the authors' program and developing the necessary categorical machinery (tensor products, Hom, Tor, Ext, base change, completion, deformations). The authors' conclusion in section E3 is that under suitable hypotheses the Hilbert functor Quot\(_P\) is represented by a separated algebraic space locally of finite type over \(k\). More specifically (section E4) if \({\mathcal C}=\) Gr then the Hilbert functor of quotients with specified Hilbert function is represented by a projective scheme over \(k\), and (section E5) if \({\mathcal C}=\) Proj(\(A\)) the Hilbert functor is represented by a scheme which is a countable union of projective closed subschemes. The authors conjecture that the Hilbert functor of quotients with specified Hilbert function is represented by a projective scheme. base change; Grothendieck category; Hilbert scheme; noncommutative projective scheme Artin, M., Zhang, J.: Abstract Hilbert schemes. Algebr. Represent. Theory 4, 305--394 (2001) Parametrization (Chow and Hilbert schemes), Category-theoretic methods and results in associative algebras (except as in 16D90), Noncommutative algebraic geometry, Grothendieck categories Abstract 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 [For the entire collection see Zbl 0653.00008.]
Let Q be a smooth complex quadric of dimension n-1 and let \(Z^*(d,g,n,Q)\) be the subset of the Hilbert scheme of Q consisting of the smooth nondegenerated connected curved of degree d and genus g. The author proves that if \(n\geq 7\) and \(g\leq (n/2)-1\) then \(Z^*(d,g,n,Q)\) is smooth and irreducible. If \(d>n\), a general element of the closure of \(Z^*(d,0,n,Q)\) in the Hilbert scheme is shown to have maximal rank. deformation; degeneration; complex quadric; Hilbert scheme Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry On the Hilbert scheme of curves in a smooth quadric | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 defined over an algebraically closed field \(k\). Let \(X^{[n]} := \text{Hilb}(X)\) be the Hilbert scheme of zero-dimensional subschemes of \(X\) of length \(n\). The main theme of the book is to study the Hilbert schemes \(X^{[n]}\), with particular reference to understand the cohomology and the Chow ring of \(X^{[n]}\).
For \(n = 1,2\), the Hilbert scheme \(X^{[n]}\) is very well understood. In fact, \(X^{[1]}\) is just \(X\) itself and \(X^{[2]}\) can be obtained by blowing up \(X \times X\) along the diagonal and taking the quotient by the natural involution that exchanges factors in \(X \times X\). The Hilbert scheme \(X^{[n]}\) is smooth if \(n \leq 3\) or \(\dim X \leq 2\). If \(X\) is a curve, then \(X^{[n]}\) is isomorphic to the \(n\)-th symmetric product, \(X^{(n)}\), of \(X\). In general, in any dimension, the natural set- theoretic map \(X^{[n]} \to X^{(n)}\) sending each 0-scheme to its support (with multiplicities) gives a desingularization of \(X^{(n)}\) if \(X^{[n]}\) is smooth. When \(\dim X = 2\), the canonical bundle of \(X^{[n]}\) is the pullback of the dualizing sheaf of \(X^{(n)}\).
In chapter 1 some fundamental facts and background material are recalled. In \S1.1 the definition and the most important properties of \(X^{[n]}\) are given. In \S1.2 the author discusses the Weil conjectures in the form he needs to compute Betti numbers of Hilbert schemes. In \S1.3 the punctual Hilbert scheme, which parametrizes subschemes concentrated in a point of a smooth variety is studied.
Chapter 2 is devoted to the computation of the Betti numbers of Hilbert schemes. In \S\S2.1, 2.2 the structure of the closed subscheme of \(X^{[n]}\) which parametrizes subschemes of length \(n\) on \(X\) concentrated on a variable point of \(X\), and the punctual Hilbert schemes \(\text{Hilb}(k[[x,y]])\) are studied. \S2.3 contains the explicit computation of the Betti numbers of \(S^{[n]}\) for a smooth surface \(S\), using the Weil conjectures. The Betti numbers of all the \(S^{[n]}\) are computed as simple power series expressions in terms of the Betti numbers of \(S\). Similar results are obtained in \S2.4, where the Betti numbers of higher order Kummer varieties \(KA_ n\) are also computed. These varieties were previously defined by A. Beauville, as new examples of Calabi-Yau manifolds. \S2.5 is devoted to the computation of the Betti numbers for triangle varieties, which parametrize length 3 0-dimensional subschemes. The main powerful tool used throughout this chapter are the Weil conjectures.
The second part of the book, chapters 3 and 4, is devoted to the computation of the cohomology and the Chow ring of Hilbert schemes. In chapter 3, \S\S3.1, 3.2, varieties of second and higher order data are constructed and studied. Such varieties are needed to give precise solutions to classical problems in enumerative algebraic geometry concerning contacts of families of subvarieties of projective space. As an application (see \S\S3.2, 3.3) a formula for the numbers of higher order contacts of a smooth projective variety with linear subvarieties in the ambient space is computed.
The last chapter, chapter 4, is the most elementary and classical of the book. In this chapter the Chow ring of relative Hilbert schemes of projective bundles is studied. In \S4.1 the author constructs the embeddings of relative Hilbert schemes into Grassmannian bundles and studies them. The case of the relative Hilbert scheme of a \(\mathbb{P}^ 1\)- bundle over a smooth variety is studied in more detail. In \S4.2 the Chow ring of the variety \(\widetilde{\text{Hilb}}^ 3(\mathbb{P}^ 2)\), parametrizing triangles in \(\mathbb{P}^ 2\) with a marked side is computed. \S4.3 is devoted to a generalization of this result to a relative situation. In \S4.3 the author studies the relative Hilbert scheme \(\text{Hilb}^ 3(\mathbb{P}(E)/X)\) of subschemes of length 3 in the fibers of the projectivization \(\mathbb{P}(E)\) of a vector bundle \(E\).
The various chapters are rather independent from each other. To read this book the reader only needs to know the basics of algebraic geometry. The book is therefore of interest not only to experts but also to graduate students and researchers in algebraic geometry not familiar with Hilbert schemes of points. -- This book is also very well organized and very nicely written. Finally, it contains a wide up-to-date bibliography on the topic. Bibliography; punctual Hilbert scheme; Betti numbers; higher order Kummer varieties; triangle varieties; Chow ring Göttsche, L. 1994.Hilbert Schemes of Zero-dimensional Subschemes of Smooth Varieties, Lecture Notes in Math. #1572 1--196. Berlin: Springer-Verlag. Parametrization (Chow and Hilbert schemes), Topological properties in algebraic geometry, Algebraic cycles, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Hilbert schemes of zero-dimensional subschemes of smooth varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this dissertation, the author studies invariants of Hilbert schemes of zero-dimensional subschemes of smooth varieties. He simplifies his arguments from Math. Ann. 286, No. 1-3, 193-207 (1990; Zbl 0679.14007) to compute the Betti numbers of the Hilbert scheme \(\text{Hilb}^n (X)\) of zero-dimensional subschemes of length \(n\) of a smooth projective surface. The proof is based on an extension of a cell decomposition of the local Hilbert scheme presented by \textit{G. Ellingsrud} and \textit{S. A. Strømme} [in Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)], and on finding a good reduction mod \(p\) and using the Weil conjectures. The author proceeds to use similar methods to find the Betti numbers of \(\text{Hilb}^n (X)\) for Kummer varieties \(X\) of higher order, and for several kinds of Hilbert schemes of triangles.
The second part of the thesis deals with cases in which the Chow ring of Hilbert schemes can be computed. The author succeeds in the case of varieties of second and higher order data and applies his formulas to give enumerative results about contact varieties of projective varieties with linear spaces in \(\mathbb{P}^N\). He also describes the Chow ring of \(\text{Hilb}^3 (\mathbb{P} ({\mathcal E}),X)\), the relative Hilbert scheme of a projective bundle of a vector bundle \({\mathcal E}\) of rank 3 over a smooth variety \(X\), in analogy with the results of \textit{G. Elencwajg} and \textit{P. Le Barz} [Compos. Math. 71, No. 1, 85-119 (1989; Zbl 0705.14004)]. Also several varieties of triangles are treated. The results get rather messy and were determined using the aid of a computer.
The exposition is clear and on a very high level. The reader is assumed to have a thorough knowledge of a good portion of the work contained in the 108 references. invariants of Hilbert schemes of zero-dimensional subschemes; Betti numbers; Kummer varieties; Chow ring Göttsche, L.: Hilbert schemes of zero-dimensional subschemes of smooth varieties. Lect. Notes Math. vol. 1572, Berlin Heidelberg New York: Springer 1993 Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Algebraic cycles, Algebraic moduli problems, moduli of vector bundles Hilbert schemes of zero-dimensional subschemes of smooth varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is motivated by a construction of \textit{I. Gordon} and \textit{T. Stafford} [Adv. Math. 198, No.1, 222--274 (2005; Zbl 1084.14005)] concerning the representation theory of a symplectic reflection algebra \(H_c\) and the spherical subalgebra \(U_c\) of \(H_c.\) If \(G\) is a finite subgroup of \(\text{SL}_2\mathbb{C}\) and \(\Gamma=G\wr S_m\), the wreath product by the symmetric group, then \(\Gamma\) acts on \(V=(\mathbb{C})^m\) preserving the natural symplectic structure.
Let \(Y_{\Gamma,m}\) be the set of \(\Gamma\)-invariant ideals \(I\) in the Hilbert scheme of \(m| G| \) points in \(\mathbb{C}^2\) such that the quotient \(\mathbb{C}[x,y]/I\) is isomorphic to a direct sum of \(m\) copies of the regular representation of \(G\). Then a crepant resolution \(Y_{\Gamma,m}\to V/\Gamma\) exists. The algebra \(U_c\) has a filtration such that the associated graded algebra is isomorphic to \(\mathcal O(V/\Gamma),\) and Gordon and Stafford suggested that there is a suitable category completing the square
\[
\begin{tikzcd} U_c\text{-mod}\ar[r,"\cong"]\ar[d,"\mathrm{gr}" '] & ???\ar[d,"\mathrm{gr}"]\\ \mathcal O(V/\Gamma)\text{-mod}\ar[r] & \mathcal O_{Y_{\Gamma,m}}\text{-mod}\rlap{\,.}\end{tikzcd}
\]
They proved that this is possible in the case where \(G=1\), implying \(\Gamma=S_m\). When \(m=1\), implying \(G=\Gamma\), \(V/\Gamma\) is a Kleinian singularity and the algebras \(H_c\) is introduced. If \(\Gamma\) is cyclic, \(Y_{\Gamma,m}\) is the \(\Gamma\)-Hilbert scheme \(\text{Hilb}_{\Gamma}\mathbb{C}^2\) that parameterizes \(\Gamma\)-invariant ideals \(I\) of \(\mathbb{C}[u,v]\) such that \(\mathbb{C}[u,v]/I\) is isomorphic to the regular representation of \(\Gamma\). The purpose of the paper is to solve the problem of Gordon and Stafford when \(\Gamma\) is cyclic of order \(n\).
To state the main result: There is an action of \(\Gamma\) on the first Weyl algebra \(\mathbb{C}[\delta,y]\) and on the localization \(\mathbb{C}[\delta,y^{\pm}]\). For \(\mathbf{k}\in\mathbb{C}^{n-1}\), the author constructs \(U_{\mathbf{k}}\) and \(H_{\mathbf{k}}\) as subalgebras of the crossed product \(\mathbb{C}[\delta,y^{\pm}]\ast\Gamma.\) Then, for suitable \(\mathbf{k},\mathbf{k}^\prime\in\mathbb{C}^{n-1}\), \(U_{\mathbf{k}^\prime}\)-\(U_{\mathbf{k}}\)-bimodules \(B(\mathbf{k}^\prime,\mathbf{k})\) are constructed and a sufficient condition for these to induce a Morita equivalence is given. The bimodules \(B(\mathbf{k}^\prime,\mathbf{k})\) are assembled to form a Morita \(\mathbb{Z}^{n-1}\)-algebra \(R\) which is a \(\mathbb{Z}^{n-1}\times\mathbb{Z}^{n-1}\)-graded algebra without identity. The algebras \(U_{\mathbf{k}}\) and the bimodules \(B(\mathbf{k}^\prime,\mathbf{k})\) are contained in \(\mathbb{C}[\delta,y^{\pm}]\ast\Gamma\) and have a differential operator (order) filtration inherited to \(R\). The associated graded algebra of \(U_{\mathbf{k}}\) is isomorphic to \(\mathcal O(V/\Gamma).\) \(\text{Coh}(\text{Hilb}_{\Gamma}\mathbb{C}^2)\) denotes the category of coherent sheaves on \(\text{Hilb}_{\Gamma}\mathbb{C}^2.\) For a graded algebra \(\mathcal{R},\) \(\mathcal{R}\)-qgr denotes the quotient category of finitely generated graded \(\mathcal{R}\)-modules modulo torsion. The main result is then
If \(\mathbf{k}\) is dominant, then: (1) there is an equivalence of categories \(R-\text{qgr}\cong U_{\mathbf{k}-\text{mod}};\) (2) there is equivalence of categories \(\text{gr} R-\text{qgr}\cong U_{\mathbf{k}-\text{mod}}\cong\text{Coh}(\text{Hilb}_{\Gamma}\mathbb{C}^2).\)
To prove this theorem, \(\mathbb{C}^2/\Gamma\) has to be controlled. When \(\Gamma\) is cyclic of order \(n\), the author computes the Hilbert-Chow morphism (the crepant morphism discussed above) using toric varieties. The insight in global sections of the scheme in question involves computation of graded Poincare series and Picard schemes. To construct \(R\), a discussion of various categories over multi-homogeneous coordinate rings is needed. This is done, and is a beautiful application of quotient categories. Also a definition of ample systems is of importance. Then \(\mathbb{Z}^n\)-algebras and Morita theory for spherical subalgebras are discussed, and adding up, this is sufficient to prove the main theorem above in a very nice and readable matter. The deformation theory however, is present, but somewhat hidden in the general results about the Weyl algebra. Finally, the concluding remarks prove that the theory considered here is consistent with earlier results. Weyl algebra; torsion module; multigraded algebra I. Musson, Hilbert schemes and noncommutative deformations of type A Kleinian singularities, J. Algebra, 293 (2005), 102--129. math.RT/0504543. Deformations of singularities Hilbert schemes and noncommutative deformations of type A Kleinian singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0509.00008.]
Author's abstract: ''I will give a simple description of how to locally represent the punctual Hilbert scheme \(Hilb^ n{\mathbb{C}}^ 2\) as a local flattener of some unfolding of a finite map-germ and how to derive information on the geometry of the Hilbert scheme \(Hilb^ n{\mathbb{C}}\{x,y\}\) from the study of the ramification loci of that unfolding; and conversely I will also describe how to obtain information on the topological classification of finite stable map-germs of type \(\Sigma_ 2\) using properties of \(Hilb^ n{\mathbb{C}}\{x,y\}\), following \textit{M. Granger}, ''Géométrie des schémas de Hilbert ponctuels'' (Thèse, Nice 1980); see also Mém. Soc. Math. Fr., Nouv. Sér. 8 (1983; Zbl 0534.14002), and \textit{J. Briançon, M. Granger}, and \textit{J. P. Speder}, Ann. Sci. Éc. Norm. Super., IV. Sér. 14, 1-25 (1981; Zbl 0463.14001) and \textit{J. Damon} and the author, Invent. Math. 32, 103- 132 (1976; Zbl 0333.57017).''
The paper is short and contains sketches of proofs. punctual Hilbert scheme; local flattener; unfolding of a finite map-germ; topological classification of finite stable map-germs Galligo, A.: Hilbert scheme as flattener. Proc. sympos. Pure math. 40, 449-452 (1983) Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Deformations of submanifolds and subspaces, Germs of analytic sets, local parametrization Hilbert scheme as flattener | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 an algebraic manifold and \(Y \subset X\) a submanifold with normal bundle \(N\).
Grothendieck's estimate states that any component \(H\) of the Hilbert scheme \(\text{Hilb}_ X\) containing \(\{Y\}\) satisfies \(\dim H \geq h^ 0 N - h^ 1N\). In 1972, Bloch improved this result and he proved that if the semiregularity map \(\pi : H^ 1N \to H^{p + 1} (\Omega_ X^{p - 1})\), \(p = \text{codim} (Y,X)\), is injective then \(\dim H = h^ 0N\) and hence \(H\) is smooth at \(\{Y\}\).
In this paper the author improves Bloch's estimate and he shows that
\[
\dim H \geq h^ 0 N - h^ 1N + \dim (\text{im} (\pi)).
\]
He also finds interesting applications of this result. normal bundle; Hilbert scheme Ran, Z.: Hodge theory and the Hilbert scheme. J. Differ. Geom. 37, 191--198 (1993) Parametrization (Chow and Hilbert schemes), Transcendental methods, Hodge theory (algebro-geometric aspects) Hodge theory and the Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper studies n-dimensional varieties in \({\mathbb{P}}^ N\) which are hypersurfaces in some linear subspace. These are called n-hypersurfaces. The first theorem in the paper states that if \(n>0\), then a subscheme of \({\mathbb{P}}^ N\) with the Hilbert polynomial of an n-hypersurface is indeed an n-hypersurface. It follows that such a Hilbert polynomial is minimal among all Hilbert polynomials of subschemes of the same dimension and degree.
The second result is the following: Let X be a projective scheme with Hilbert polynomial \(P(t)=\sum^{n}_{i=0}a_ i\left( \begin{matrix} t+n-i\\ n-i\end{matrix} \right).\) Then \(2a_ 1\geq a_ 0(a_ 0-1)\), and if X is of pure dimension, equality implies that X is an n-hypersurface (of degree \(a_ 0)\). Hilbert scheme; n-hypersurfaces; Hilbert polynomial Ådlandsvik, B.: Hilbert schemes of hypersurfaces and numerical criterions. Math. Scand.56, 163-170 (1985) Parametrization (Chow and Hilbert schemes), \(n\)-folds (\(n>4\)), Fine and coarse moduli spaces, Special surfaces Hilbert schemes of hypersurfaces and numerical criterions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field. Work of \textit{F. S. Macaulay} [Proc. Lond. Math. Soc. (2) 26, 531--555 (1927; JFM 53.0104.01)] shows that the Hilbert scheme \(\text{Hilb}^{p(t)} (\mathbb P_k^n)\) parametrizing proper closed subschemes \(X \subset \mathbb P_k^n\) with Hilbert polynomial \(p(t)\) is non-empty precisely when \(p(t) = \sum_{j=1}^r \binom{t+b_j-j+1}{b_j}\) for uniquely determined integers \(b_1 \geq b_2 \geq \dots \geq b_r \geq 0\). Let \(\text{Hilb}^{p(t)} (\mathbb P_k^n)\) be the Hilbert scheme parametrizing closed subschemes \(X \subset \mathbb P_k^n\) with Hilbert polynomial \(p(t)\). It follows from work of Macaulay [loc. cit.] that the polynomials \(p(t) \in \mathbb Q [t]\) for which \(\text{Hilb}^{p(t)} (\mathbb P_k^n)\) is non-empty have unique expressions \(p(t) = \sum_{j=1}^r \binom{t+b_j-j+1}{b_j}\) for integers \(b_1 \geq b_2 \geq \dots \geq b_r \geq 0\). The author proves here that the lexicographic ideal is the unique saturated strongly stable ideal of codimension \(c\) with Hilbert polynomial \(p\) if and only if at least one of the following holds: (1) \(b_r>0\), (2) \(c \geq 2\) and \(r \leq 2\), (3) \(c=1\) and \(b_1=b_r\) or (4) \(c=1\) and \(r-s \leq 2\), where \(b_1=b_2=\dots=b_s > b_{s+1}\). When \(k\) is algebraically closed, the lex ideal is the unique saturated Borel-fixed ideal of codimension \(c\) with Hilbert polynomial \(p\) if and only if at least one of (1)-(4) hold. Using the result of \textit{A. Reeves} and \textit{M. Stillman} [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)] that the scheme corresponding to the lex ideal is a smooth point of \(\text{Hilb}^{p(t)} (\mathbb P_k^n)\), it follows that \(\text{Hilb}^{p(t)} (\mathbb P_k^n)\) is nonsingular and irreducible if any of (1)-(4) hold, where \(n = c + \deg p(t)\). These include examples of \textit{G. Gotzmann} [Math. Z. 201, No. 1, 13--17 (1989; Zbl 0696.14001)].
The author uses the results above to study the geography of Hilbert schemes. He defines two operators \(A\) and \(\Phi\) on the set of Hilbert polynomials (\(A\) corresponds to adding a point, \(\Phi\) roughly corresponds to taking a projective cone). For each codimension \(c>0\), the graph with vertices \(\text{Hilb}^p (\mathbb P^n)\) such that \(\deg p + c = n\) and edges given by pairs \((\text{Hilb}^p (\mathbb P^n), \text{Hilb}^{A(p)} (\mathbb P^n))\) and \((\text{Hilb}^p (\mathbb P^n), \text{Hilb}^{\Phi (p)} (\mathbb P^{n+1}))\) is an infinite full binary tree with root \(\text{Hilb}^1 (\mathbb P^c)\) called the \textit{Hilbert tree} \(\mathcal H_c\).The \textit{Hilbert forest} is the disjoint union \(\mathcal H\) of Hilbert trees taken over all \(c>0\). Defining a probability distribution on \(\mathcal H\) for which all vertices at a fixed height are equally likely, the author proves that the probability of a vertex in \(\mathcal H\) corresponding to a smooth irreducible Hilbert scheme is greater than \(1/2\). The paper is well-written with helpful examples.
Reviewers remark: Since the dimension $n$ of $\mathbb P^n$ varies in the Hilbert forest, one cannot draw similar conclusions about the geography of Hilbert schemes for subschemes of a fixed $\mathbb P^n$. Indeed, \textit{M. Martin-Deschamps} and \textit{D. Perrin} [Ann. Sci. Éc. Norm. Supér. (4) 29, No. 6, 757--785 (1996; Zbl 0892.14005)] have shown that for $d \geq 6$, there are onlyfinitely many arithmetic genera $g$ for which the open subset $H_{d,g} \subset \text{Hilb}^{dt+1-g}(\mathbb P_k^3)$ corresponding to locally Cohen-Macaulay curves is reduced or irreducible. smooth Hilbert schemes; lexicographic ideal; geography of Hilbert schemes Parametrization (Chow and Hilbert schemes) The ubiquity of smooth 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 authors offer a conjecture on the relation between weighted Euler characteristics of Hilbert schemes of points of integral complex curves with at worst locally planar singularities, and the HOMFLY polynomials of the links of these singularities. The link of a singularity \(C\) is its intersection with a small surrounding \(3\)-sphere. For non-singular curves a formula for the weighted Euler characteristic is well-known, and the authors show that each singularity produces a multiplicative correction to it. Each correction is expressed by an explicit conjectural formula in terms of the singularity's HOMFLY polynomial. Since it is known how to express the weighted Euler characteristic of \(C\) in terms of the Gopakumar-Vafa invariants the conjecture, if true, will provide an indirect evidence in favor of the large \(N\) duality between the Gromov-Witten and the Chern-Simons theories. The latter gives rise to the HOMFLY polynomials along the lines explained by Witten.
Most of the paper is dedicated to proving some corollaries of the conjectured formula, and verifying its special cases. In particular, the limit case when the HOMFLY polynomial reduces to the Alexander polynomial is derived from a theorem of Campillo, Delgado and Gusein-Zade. Symmetries implied by the conjecture are verified for the integrals giving the weighted Euler characteristic. Finally, for \(y^k=x^n\) singularities with gcd\((k,n)=1\), whose links are \((k,n)\)-torus knots, and for a singularity, whose link is the \((2,13)\) cable of the right-handed trefoil, the two sides of the formula are matched by direct computations. Hilbert schemes of points; locally planar singularity; weighted Euler characteristics; link of singularity; Gopakumar-Vafa invariants; HOMFLY polynomial; Alexander polynomial A. Oblomkov, J. Rasmussen, and V. Shende, The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link, \textit{Duke Math. J.}, 161 (2012), no. 7, 1277--1303.Zbl 1256.14025 MR 2922375 Singularities of curves, local rings, Knots and links in the 3-sphere The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 investigate affine varieties that are homogeneous with respect to a nonstandard grading. An important special case of such a variety occurs as the Gröbner strata of a monomial ideal on the Hilbert scheme.
It is shown that a homogeneous variety can be embedded into its tangent space at the origin. Thus it can be smooth if and only if it is isomorphic to affine space. Furthermore the authors show that such varieties are rationally chain connected by examining the action of a suitable torus corresponding to the grading.
The paper is succinct, well-written, and contains many examples. It is important to anyone studying the structure and defining equations of the Hilbert scheme. Homogeneous varieties; Hilbert schemes; G-graded rings; initial ideals G. Ferrarese - M. Roggero, Homogeneous varieties for Hilbert schemes. Int. J. Algebra, 3 (9-12) (2009), pp. 547-557. Zbl1193.13001 MR2545200 Graded rings, Polynomial rings and ideals; rings of integer-valued polynomials, Parametrization (Chow and Hilbert schemes), Group actions on affine varieties Homogeneous varieties for 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 New techniques for dealing with problems of numerical stability in computations involving multivariate polynomials allow a new approach to real world problems. Using a modelling problem for the optimization of oil production as a motivation, we present several recent developments involving border bases of polynomial ideals. After recalling the foundations of border basis theory in the exact case, we present a number of approximate techniques such as the eigenvalue method for polynomial system solving, the AVI algorithm for computing approximate border bases, and the SOI algorithm for computing stable order ideals. To get a deeper understanding for the algebra underlying this approximate world, we present recent advances concerning border basis and Gröbner basis schemes. They are open subschemes of Hilbert schemes and parametrize flat families of border bases and Gröbner bases. For the reader it will be a long, tortuous, sometimes dangerous, and hopefully fascinating journey from oil fields to Hilbert schemes. oil field; polynomial system solving; eigenvalue method; Buchberger-Möller algorithm; border basis; approximate algorithm; border basis scheme; Gröbner basis scheme; Hilbert scheme; \texttt{CoCoA} Kreuzer, M.; Poulisse, H.; Robbiano, L., From oil fields to Hilbert schemes, (Robbiano, L.; Abbott, J., Approximate Commutative Algebra, Texts and Monographs in Symbolic Computation, (2009), Springer-Verlag Wien), 1-54 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Solving polynomial systems; resultants, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Parametrization (Chow and Hilbert schemes) From oil fields to 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 his fundamental paper [``Techniques de construction et théoremes d'existence en géométrie algébrique. IV: Les schéma de Hilbert'', Sém. Bourbaki 13(1960/61), No. 221 (1961; Zbl 0236.14003)], \textit{A. Grothendieck} introduced the so called Hilbert scheme, which parametrizes all projective subschemes of the projective space with fixed Hilbert polynomial.
Next many authors gave other proofs of the existence of the Hilbert scheme, see for instance \textit{D. Mumford} [``Lectures on curves on an algebraic surface'' (1966; Zbl 0187.42701)], \textit{A. B. Altman} and \textit{S. L. Kleiman} [Adv. Math. 35, 50--112 (1980; Zbl 0427.14015)], \textit{E. Sernesi} [``Topics on families of projective schemes'', Queen's Pap. Pure Appl. Math. 73 (1986)], \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)].
In particular, many efforts have been devoted to the study of Hilbert scheme of points (see e.g. \textit{A. Iarrobino} [Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)], \textit{H. Nakajima} [Lectures of Hilbert schemes of points on surfaces, University Lecture Series 18 (1999; Zbl 0949.14001)], \textit{M. E. Huibregtse} [Pac. J. Math. 223, No. 2, 269--315 (2006; Zbl 1113.14007)]).
In the paper under review, the authors present an explicit construction of the Hilbert scheme of \(n\) points of a projective scheme, giving explicit natural equations (of an affine covering) of it. This approach is close to the one by M. E. Huibregtse, but more intrinsic in nature. The authors' construction of the Hilbert scheme \(H\) of \(n\) points of an affine scheme Spec\((R)\) over an affine base scheme Spec\((A)\) relies on two ideas: the first, is to use the description of the \(R\)-module structure on an \(A\)-module \(F\) in terms of \(A\)-algebra morphisms form \(R\) to End\(_A(F)\). The second is the observation that the \(A\)-module morphisms from an \(A\)-module \(M\) to End\(_A(F)\) correspond to \(A\)-algebra morphisms from the symmetric algebra Sym\(_A(M\otimes_A\text{End}_A(F)^{\vee})\) to \(A\). This enables the authors to recognize that the \(A\)-algebra morphisms from Sym\(_A(M)\) to End\(_A(F)\) correspond to the algebra morphisms \(H\to A\), where \(H\) is the residue of the algebra Sym\(_A(M\otimes_A\text{End}_A(F)^{\vee})\) by the ideal corresponding to commuting \(n\times n\) matrices, and with coefficients in Sym\(_A(M\otimes_A\text{End}_A(F)^{\vee})\).
At the end of the paper, the authors give some examples showing how their methods can be used: they describe the Hilbert scheme of \(n\) points in Spec\((S^{-1}A[X])\), where \(S\) is a multiplicative set in the polynomial ring \(A[X]\) in the variable \(X\) over \(A\), and describe an open subset of a component of the Hilbert scheme containing many subschemes of \(n\) points with support at a fixed point (this set can be used to show that the Hilbert scheme of \(n\) points in Spec\((A[Y_1,\dots,Y_m])\) is reducible when \(m\geq 3\) and \(n\gg 0\) (compare with the quoted paper of A. Iarrobino)). Hilbert scheme of points; functor; commuting matrices T. S. Gustavsen, D. Laksov, R. M. Skjelnes, An elementary, explicit, proof of the existence of Hilbert schemes of points. \textit{J. Pure Appl. Algebra}\textbf{210} (2007), 705-720. MR2324602 Zbl 1122.14004 Parametrization (Chow and Hilbert schemes), Commutativity of matrices, Endomorphism rings; matrix rings An elementary, explicit, proof of the existence of Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Luna's slice theorem [\textit{D. Luna}, Bull. Soc. Math. Fr., Suppl., Mém. 33, 81--105 (1973; Zbl 0286.14014)] is a classical result to study the local structure of an affine variety equipped with a reductive group action. In this paper, the author obtains a Luna's slice type theorem for certain moduli spaces called \textit{invariant Hilbert schemes}. These moduli spaces were introduced by Haiman and Sturmfels for diagonalizable groups [\textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] and then by \textit{V. Alexeev} and \textit{M. Brion} for arbitrary reductive algebraic groups [J. Algebr. Geom. 14, No. 1, 83--117 (2005; Zbl 1081.14005); \textit{M. Brion}, in: Handbook of moduli. Volume I. Somerville, MA: International Press; Beijing: Higher Education Press. 64--117 (2015; Zbl 1322.14001)]. They appear in many situations: classification of spherical varieties, invariant deformation theory of affine schemes with reductive group actions, construction of canonical resolutions of quotient singularities, etc. However little is know about the geometric properties of the invariant Hilbert schemes (connectedness, smoothness, reducedness, etc) except in some particular cases.
The author obtains Luna's slice type results to study the local structure of invariant Hilbert schemes (Theorems 2.3 and 2.4). Then, he deduces numerical smoothness criteria for certain class of invariant Hilbert schemes (Corollaries 2.5 and 2.6). Also, a more involved example of invariant Hilbert scheme where these results apply to prove smoothness is provided in the last section of the paper. invariant Hilbert scheme; Luna's étale slice theorem; smoothness criterion Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes) On the invariant Hilbert schemes and Luna's étale slice theorem | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.