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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Hilbert schemes are considered as moduli spaces, and provide a testing ground for conjectures in algebraic geometry and theoretical physics. This book considers the the Hilbert scheme of points and its geometry, and its influence of, and on, infinite dimensional Lie algebras.
For a variety \(X\) of dimension at least three, the corresponding Hilbert scheme is in general singular. When the variety \(X\) is a smooth curve, the corresponding Hilbert scheme of points coincides with the symmetric products of the (intersection cohomology) of the curve. When \(X\) is a smooth algebraic surface, the Hilbert scheme \(X^{[n]}\) parametrizing length \(n\), \(0\)-dimensional closed subschemes of \(X\) is irreducible and smooth, and the Hilbert-Chow morphism \(\rho_n:X^{[n]}\rightarrow X^{(n)}\), where \(X^{(n)}\) denotes the symmetric product, sending an element of \(X^{[n]}\) to its support in \(X^{(n)}\) counted with multiplicity, is a crepant desingularization. This says that \(X^{[n]}\) is an interesting object of study, and the book covers the main results of this, such as its cohomology groups, Chow groups, motive, cobordism class, relation to algebraic combinatorics, the McKay correspondence and integrable systems.
Representation theory of affine Lie algebras on the homology groups of moduli spaces of instantons on ALE spaces, i.e. homology groups of quiver varieties, gives construction of actions of the Heisenberg algebra on the cohomology of the Hilbert schemes \(X^{[n]}\) when \(X\) is a smooth algebraic surface. These geometric constructions of Grojnowski and Nakajima started the field of investigation of the interplay between the Hilbert schemes of points and infinite dimensional Lie algebras, and led to Lehn's geometric construction of the Virasoro algebra and the boundary operator by using the Hilbert schemes \(X^{[n]}\). The book covers the construction of Li, Wang and Qin [\textit{W.-P. Li} et al., Int. Math. Res. Not. 2002, No. 27, 1427--1456 (2002; Zbl 1062.14010)] of the Chern character operators and the \(\mathcal W\) algebras, and \textit{E. Carlsson}'s and \textit{A. Okounkov}'s [Duke Math. J. 161, No. 9, 1797--1815 (2012; Zbl 1256.14010)] construction of the \(\text{Ext}\) vertex operators. It is explained that Lehn's boundary operator is a version of the bosonized Calogero-Sutherland operator, that the Chern character operators are vertex operators of higher conformal weights, and that the \(\mathcal W\) algebras are higher-spin generalizations of Lehn's Virasoro algebras. The \(\text{Ext}\) vertex operators of Carlsson and Okounkov are motivated by the study of Nekrasov partition functions in the setting of Hilbert schemes of points.
The operators and algebras defined above is described as a tool for understanding the finer geometry of the Hilbert schemes \(X^{[n]}\). The Heisenberg algebras give a fundamental language for describing the (co-)homology classes of \(X^{[n]}\) leading to the study of the Gromov-Witten theory of \(X^{[n]}\). The Virasoro operators and the Chern character operators determine the cup-products and the ring structures of the cohomology of \(X^{[n]}\). Then the \(\text{Ext}\) vertex operators is used for the understanding of the intersection theory of \(X^{[n]}\) when its tangent bundle is involved.
This book builds on, and refers to Göttsche's and Nakajima's books on the Hilbert scheme of points. Also Lehn's results are heavily used and referred to. The book gives a detailed survey of the developments on the interplay between the Hilbert scheme of points and the infinite dimensional Lie algebras appearing after the book of Nakajima.
The present book contains five parts and consists of 16 chapters. Part one gives an introduction to Hilbert schemes of points and some of its geometry on the projective plane: The nef cone and flip structure of \((\mathbb P^2)^{[n]}\) is given a thorough study for various \(n\), all necessary results are proved explicitly and several examples are computed in detail.
Part two deals with the construction of various infinite dimensional Lie algebra actions on the cohomology of the Hilbert schemes of points on surfaces and the connections with multiple \(q\)-zeta values. It includes Nakajima's affine Lie algebra actions on the homology of quiver varieties as motivation and background material, the Heisenberg algebras of Grojnowski and Nakajima, the boundary operator and the Virasoro algebras of Lehn, the \(\text{Ext}\) vertex operators of Carlsson and Okounkov, and the Cern character operators and the \(\mathcal W\) algebras of W.-P. Li, W. Wang and Qin. The intersection theory of the Hilbert schemes are by this thoroughly studied, and a lot of results and examples are given. In particular a very deep structure theorem for the higher order derivatives of the Heisenberg operators.
Part three studies the cohomology ring structure of Hilbert schemes of points. The ring generators and the defining ideals are essential results. This part includes the results of Lehn and Sorger when the canonical divisor of the surface is numerically trivial, the results of Costello and Grojnowski in terms of Calogero-Sutherland operators and the Dunkl-Cherednik operators, the integral basis for the cohomology group,and the orbifold cohomology ring of the symmetric product of a surface. It should be pointed out that all results and examples are proved and computed to the full detail. This makes the book a complete reference when working in this area.
Part four considers the equivariant cohomology of Hilbert schemes. Let \(\mathbb T\) be the one-dimensional torus, and let the Jack parameter be interpreted as minus the self-intersection number of the zero section in \(X(\gamma)\). One of the connections between the geometry of the Hilbert schemes \(X^{[n]}\) of points on a (quasi-)projective surface \(X\) and symmetric functions is given by realizing these as certain ordinary cohomology classes of the Hilbert schemes associated to an embedded curve in a surface. Nakajima proved that the Jack polynomials whose Jack parameter is a positive integer \(\gamma\) can be realized as certain \(\mathbb T\)-equivariant cohomology classes of the Hilbert schemes of points on the surface \(X(\gamma)\) which is the total space of the line bundle \(\mathcal O_{\mathbb P^1}(-\gamma)\) over the complex projective line \(\mathbb P^1\). Haiman made connections between the Macdonald polynomials and the geometry of Hilbert schemes, and realized the Macdonald polynomials as certain \(\mathbb T\)-equivariant \(K\)-homology classes of the Hilbert scheme of points on the affine plane \(\mathbb C^2\). Also, this part gives the relation of equivariant cohomology to Toda hierarchies, Hurwitz theory, equivariant local Gromov-Witen theory, and equivariant local Donaldson-Thomas theory. The actions in question are the previously defined actions of the Heisenberg algebras on \(X^{[n]}\) and are interpreted as equivariant cohomology classes of \(X^{[n]}\) in terms of the ring of symmetric functions. These invariant cohomology classes corresponds to the Jack symmetric functions.
The final Part 5 considers the cosection localization technique of Kiem and Li, the gromov-Witten theory of the Hilbert schemes of points, the equivariant quantum corrected boundary operator and the quantum differential equations of Okunkov and Pandharipande, the quantum corrected boundary operator of J. Li and W.-P. Li, and the proof of Ruan's cohomological crepant resolution conjecture.
The book is far from elementary and is suitable for researchers and graduate students with a good knowledge of intersection theory and geometric invariant theory. Starting with this knowledge, it surveys most of the known results and theories of the Hilbert schemes of points, implying infinite dimensional Lie algebras and their actions. Most of the contemporary theory in the field is included in the book, an impressive bibliography is given, and all results are proved in full detail. This can give a feeling of the results drowning in complicated computations, but they can be skipped at a first reading, under the knowledge that they are present in the book, so that the results can be checked on a later occasion. This makes the book complete, and makes it a splendid survey, even textbook, on the subject. ADHM datum; Kac-Moody Lie algebra; Chevalley operator; Crepant resolution conjecture; Heisenberg algebra; Heisenberg operators; Jack symmetric function; quantum corrected boundary operator Research exposition (monographs, survey articles) pertaining to algebraic geometry, 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), Infinite-dimensional Lie (super)algebras, Vertex operators; vertex operator algebras and related structures Hilbert schemes of points and infinite dimensional 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 connection between the singularities of a finite type \(\mathbb Z\)-scheme \(X\) and the asymptotic point count of \(X\) over various finite rings. In particular, if the generic fiber \(X_{\mathbb Q}=X\times_{\mathrm{Spec}\mathbb Z}\mathrm{Spec}\mathbb Q\) is a local complete intersection, we show that the boundedness of \(| X(\mathbb Z/p^n\mathbb Z)|/p^{n\dim X_{\mathbb Q}}\) in \(p\) and \(n\) is in fact equivalent to the condition that \(X_{\mathbb Q}\) is reduced and has rational singularities. This paper completes a recent result of \textit{A. Aizenbud} and \textit{N. Avni} [Duke Math. J. 167, No. 14, 2721--2743 (2018; Zbl 1436.14044)]. rational singularities; complete intersection; analysis on p-adic varieties; asymptotic point count Singularities in algebraic geometry, Rational points On rational singularities and counting points of schemes over finite rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of the article is to describe the classification of simple isolated hypersurface singularities over a field of positive characteristic by certain invariants without computing the normal form. We also give a description of the algorithms to compute the classification which we have implemented in the Singular libraries classifyCeq.lib and classifyReq.lib. simple singularity; normal form; contact equivalent; right equivalent Singularities in algebraic geometry, Singularities of curves, local rings, Singularities of surfaces or higher-dimensional varieties On the classification of simple singularities in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let p be an odd prime, and let F be the p-th Fermat curve, i.e., the complete non-singular curve over \({\mathbb{Q}}\) with affine equation \(x^ p+y^ p=1\). The author constructs the minimal regular model of F over \({\mathbb{Z}}\), by starting with the model \(X=Spec {\mathbb{Z}}[x,y]/(x^ p+y^ p-1))\) and explicitly blowing up its singularities, all of which lie on the fiber above (p), and all of which turn out to be rational double points of type \(A_ n\). The author omits discussion of the singularity at infinity, and so in fact only describes an affine patch of the model, but by considering the action of the symmetric group \(S_ 3\) on the curve one can easily see that this singularity behaves in the same way as the ones labelled 0 and 1 in the diagram on page 255, and so one may readily complete the model. Such a complete model is necessary for the application of Arakelov theory alluded to by the author.
The reviewer carried out the same calculation over \({\mathbb{Z}}_ p[\zeta]\), where \(\zeta\) is a primitive p-th root of unity [`The degenerate fiber of the Fermat curve', in Number theory related to Fermat's last theorem, Proc. Conf., Prog. Math. 26, 57-70 (1982; Zbl 0542.14019)]. minimal regular model of p-th Fermat curve; singularity at infinity; Arakelov theory Chang, H, On the singularities of the Fermat scheme over \({{\mathbb{Z}}}\), Chinese J. Math., 17, 243-257, (1989) Arithmetic ground fields for curves, Singularities in algebraic geometry, Global ground fields in algebraic geometry On the singularities of the Fermat scheme over \({\mathbb{Z}}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this book the author discusses the Hilbert scheme \(X^{[n]}\) of points on a complex surface \(X\). This object is originally studied in algebraic geometry but, as it has been recently realized, it is related to many other branches of mathematics, such as singularities, symplectic geometry, representation theory, and even to theoretical physics. The book reflects this feature on Hilbert schemes and therefore the subjects are analyzed from various points of view. One sees that \(X^{[n]}\) inherits structures of \(X\), e.g., it is a nonsingular complex manifold, it has a holomorphic symplectic form if \(X\) has one, it has a hyper-Kähler metric if \(X= \mathbb{C}^2\), and so on. A new structure is revealed when one studies the homology group of \(X^{[n]}\). The generating function of Poincaré polynomials has a very nice expression. The direct sum \(\bigoplus_n H_* (X^{[n]})\) is a representation of the Heisenberg algebra.
The book, which is nicely written and well-organized, tries to tell the harmony between different fields rather than focusing attention on a particular one. The reader is assumed to have basic knowledge on algebraic geometry and homology groups of manifolds. Some chapters require more background, say spectral sequences, Riemannian geometry, Morse theory, intersection cohomology. symplectic structure; moment map; hyper-Kähler quotients; Dynkin diagrams; vertex algebra; symmetric products; Hilbert scheme of points; Poincaré polynomials; Heisenberg algebra; Morse theory; intersection cohomology H. Nakajima, \textit{Lectures on Hilbert schemes of points on surfaces}, \textit{University Lecture Series}\textbf{18}, American Mathematical Society, Providence RI U.S.A., (1999). Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Special Riemannian manifolds (Einstein, Sasakian, etc.) Lectures on Hilbert schemes of points on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(E\) be the bundle defined by applying a polynomial functor to the tautological bundle on the Hilbert scheme of \(n\) points in the complex plane. By a result of \textit{M. Haiman} [Invent. Math. 149, No. 2, 371--407 (2002; Zbl 1053.14005)], the Čech cohomology groups \(H^i(E)\) vanish for all \(i > 0\). It follows that the equivariant Euler characteristic with respect to the standard two-dimensional torus action has nonnegative integer coefficients in the torus variables \(z_1\), \(z_2\), because they count the dimensions of the weight spaces of \(H^0(E)\). We derive a formula for this Euler characteristic using residue formulas for the Euler characteristic coming from the description of the Hilbert scheme as a quiver variety [\textit{A. Neguţ}, ``Moduli of flags of sheaves and their K-theory'', Preprint, \url{arXiv:1209.4242}; \textit{N. A. Nekrasov}, Adv. Theor. Math. Phys. 7, No. 5, 831--864 (2003; Zbl 1056.81068)]. We evaluate this expression using \textit{N. Jing}'s Hall-Littlewood vertex operator with parameter \(z_1\) [Adv. Math. 87, No. 2, 226--248 (1991; Zbl 0742.16014)], and a new vertex operator formula given in Proposition 1 below. We conjecture that the summand in this formula is a polynomial in \(z_1\) with nonnegative integer coefficients, a special case of which was known to \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. A 286, 323--324 (1978; Zbl 0374.20010)]. representation theory; combinatorics; symmetric functions; Hilbert scheme; Hall-Littlewood polynomials Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations Hall-Littlewood polynomials and vector bundles on the Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This title introduces the theory of arc schemes in algebraic geometry and singularity theory, with special emphasis on recent developments around the Nash problem for surfaces. The main challenges are to understand the global and local structure of arc schemes, and how they relate to the nature of the singularities on the variety. Since the arc scheme is an infinite dimensional object, new tools need to be developed to give a precise meaning to the notion of a singular point of the arc scheme.
Other related topics are also explored, including motivic integration and dual intersection complexes of resolutions of singularities. Written by leading international experts, it offers a broad overview of different applications of arc schemes in algebraic geometry, singularity theory and representation theory.
The articles of this volume will be reviewed individually. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Arcs and motivic integration, Local theory in algebraic geometry, Collections of articles of miscellaneous specific interest Arc schemes and singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article gives a necessary and sufficient condition for the invariant Hilbert scheme studied in [\textit{A. Kubota}, ``Invariant Hilbert scheme resolution of Popov's \(\mathit{SL}(2)\)-varieties'', Preprint, \url{arXiv:1809.01533}] to be the minimal resolution of a 3-dimensional affine normal quasihomogeneous \(\mathrm{SL}(2)\)-variety. invariant Hilbert scheme; spherical variety; minimal resolution Compactifications; symmetric and spherical varieties, McKay correspondence, Singularities in algebraic geometry On minimality of the invariant Hilbert scheme associated to Popov's \(\mathrm{SL}(2)\)-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 The paper under review provides one of the first examples of invariant Hilbert schemes with multiplicities introduced in [\textit{V. Alexeev} and \textit{M. Brion}, J. Algebr. Geom. 14, No. 1, 83--117 (2005; Zbl 1081.14005)]. \(\text{SL}_2\) acts on \((\mathbb{C}^2)^6\) and the moment map \(\mu\) defines the symplectic reduction \(\mu^{-1}(0)// \text{SL}_2\). The paper under review describes explicitly the invariant Hilbert scheme \(\text{Hilb}^{\text{SL}_2}_h(\mu^{-1}(0))\) for the Hilbert function
\[
h:\mathbb{N}_0\to \mathbb{N},\quad d\mapsto d+1,
\]
and proves it is connected and smooth. The Hilbert-Chow morphism
\[
\text{Hilb}^{\text{SL}_2}_h(\mu^{-1}(0))\to \mu^{-1}(0)//\text{SL}_2
\]
factors through the well known symplectic resolutions of \(\mu^{-1}(0)//\text{SL}_2\) given by the cotangent bundles of \(\mathbb{P}^3\) and its dual. The method of the paper provides a general procedure of these calculations that can be applied to similar examples. invariant Hilbert schemes T. Becker, \textit{An example of an} SL\_{}\{2\}-\textit{Hilbert scheme with multiplicities}, Transform. Groups \textbf{16} (2011), no. 4, 915-938. Formal groups, \(p\)-divisible groups, Birational geometry, Special varieties An example of an SL\(_{2}\)-Hilbert scheme with multiplicities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 goal of this paper is to give a proof of a version of inversion of adjunction by means of jet schemes and arc space. The authors present a self contained exposition for this aim. First they introduce the notion of jet schemes and show their existence and the basic properties. They also give the description on the structure of the fibers of the truncation morphisms. They show the formula of minimal log discrepancy in terms of the codimension of cylinders in the arc space. Finally, they prove the inversion of adjunction; i.e., the equality of minimal log discrepancies of a pair and the restriction of the pair onto the normal subvariety. Ein, L., Mustaţă, M.: Jet schemes and singularities. In: Abramovich, D., et al. (eds.) Algebraic Geometry--Seattle 2005, Part 2. Proceedings of Symposia in Pure Mathematics, vol. 80.2, pp. 505-546. American Mathematical Society, Providence (2009) Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Birational geometry, Families, moduli, classification: algebraic theory Jet schemes and 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 his fundamental paper [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV: Les schemes de Hilbert'', Sémin. Bourbaki, exp. 221 (1961; Zbl 0236.14003)], \textit{A. Grothendieck} introduced the so called Hilbert scheme, which is an object which parametrizes all projective subschemes of the projective space with fixed Hilbert polynomial.
In the paper under review, the authors present a more general object, called the multigraded Hilbert scheme, parametrizing all homogeneous ideals with fixed Hilbert function in a graded polynomial ring \(S\).
As in the case of Hilbert schemes, the multigraded Hilbert scheme is a projective scheme (quasi-projective if the grading of \(S\) is not positive), and, when the ground ring is a field, its tangent space at a point corresponding to an ideal \(I\) has a simple description: it is canonically isomorphic to the degree \(0\) piece of \(\Hom(I,S/I)\).
The construction of the multigraded Hilbert scheme is obtained in a great generality, and it enables the authors to prove a conjecture from \textit{D. Bayer}'s thesis [The division algorithm and the Hilbert scheme, Ph.D. thesis, Harvard University (1982)] on equations defining the Hilbert scheme, and to construct a natural morphism from the toric Hilbert scheme to the toric Chow variety, resolving Problem 6.4 appearing in the paper of \textit{B. Sturmfels} [The geometry of A-graded algebras, preprint, \texttt{http://arXiv.org/abs/math.AG/9410032}]. graded polynomial ring; Hilbert function; Chow morphism M. Haiman - B. Sturmfels, Multigraded Hilbert schemes. J. Algebraic Geom., 13 (4) (2004, pp. 725-769. Zbl1072.14007 MR2073194 Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Grassmannians, Schubert varieties, flag manifolds Multigraded Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a simply-connected simple Chevalley group of type \(A\), \(D\) or \(E\) and \(k\) an algebraically closed field whose characteristic is very good for \(G\). According to a conjecture of Grothendieck a semi-universal deformation (also known as miniversal deformation) of the rational double point of the same type can be obtained via a restriction of the categorical quotient morphism \(G\to G//G_{ad}\). More precisely, one has to restrict this morphism to a general slice through a point in the subregular unipotent orbit. In characteristic zero this was proved by \textit{E. Brieskorn} [Actes Congr. internat. Math. 1970, 2, 279-284 (1971; Zbl 0223.22012)] and \textit{P. Slodowy} [Lect. Notes Math. 815 (1980; Zbl 0441.14002)]. Slodowy extended this result to the case where \(\text{char}(k)>4 \operatorname {Cox}(G)-2\). Using work of \textit{V. Hinich} [Isr. J. Math. 76, No. 1/2, 153-160 (1991; Zbl 0810.14001)], this article gives a complete proof of the conjecture, i.e.~there is no assumption on \(\text{char}(k)\) except that it must be very good for \(G\). (\(p\) is very good for \(E_8\) if \(p\neq 2,3,5\); for \(E_6\) and \(E_7\) if \(p\neq 2,3\); for \(D\) if \(p\neq 2\) and for \(A_r\) if \(p\) does not divide \(r+1\).). miniversal deformation; semi-universal deformation; rational double point; simple algebraic group; simple singularity; Chevalley group; quoteint morphism Singularities in algebraic geometry, Linear algebraic groups over arbitrary fields, Local complex singularities On simple groups and 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 \textit{G. Ottaviani} [Ann. Sc. Norm. Pisa, Cl. Sci., IV. Ser. 19, No. 3, 451--471 (1992; Zbl 0786.14026)] proved that, in \({\mathbb{P}}^5\), the only smooth 3-dimensional scrolls over a smooth surface are the Segre scroll, the Bordiga scroll, the Palatini scroll and the \(K3\)-scroll. The first two scrolls are arithmetically Cohen-Macaulay, and so the description of their Hilbert schemes is contained in an article of \textit{G.Ellingsrud} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 8, 423--431 (1975; Zbl 0325.14002)].
Extending methods used by \textit{G. Castelnuovo} [Ven. 1st. Atti (7) II, 855--901 (1891; JFM 23.0865.01)] in the study of Veronese surfaces, in the paper under review the authors examine the Hilbert scheme of Palatini scrolls \(X\). They prove that such a scheme has an irreducible component containing \(X\), which is birational to the Grassmannian \(G(3,\check{\mathbb{P}}^{14})\), and determine the exceptional locus of the birational map. Hilbert scheme; Grassmannian degeneracy locus; webs of linear complexes; linear system; Palatini scroll; secant variety M. L. Fania, E. Mezzetti, On the Hilbert scheme of Palatini threefolds. \textit{Adv. Geom}. 2 (2002), 371-389. MR1940444 Zbl 1054.14052 \(3\)-folds, Parametrization (Chow and Hilbert schemes), Low codimension problems in algebraic geometry, Varieties of low degree, Adjunction problems On the Hilbert scheme of Palatini threefolds. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 punctual Hilbert scheme has been known since the early days of algebraic geometry in EGA style. Indeed it is a very particular case of the Grothendieck's Hilbert scheme which classifies the subschemes of projective space. The general Hilbert scheme is a key object in many geometric constructions, especially in moduli problems. The punctual Hilbert scheme which classifies the 0-dimensional subschemes of fixed degree, roughly finite sets of fat points, while being pathological in most settings, enjoys many interesting properties especially in dimensions at most three. Most interestingly it has been observed in this last decade that the punctual Hilbert scheme, or one of its relatives, the \(G\)-Hilbert scheme of Itô-Nakamura, is a convenient tool in many hot topics, as singularities of algebraic varieties, e.g McKay correspondence, enumerative geometry versus Gromov-Witten invariants, combinatorics and symmetric polynomials as in Haiman's work, geometric representation theory (the subject of this school) and many others topics.
The goal of these lectures is to give a self-contained and elementary study of the foundational aspects around the punctual Hilbert scheme, and then to focus on a selected choice of applications motivated by the subject of the summer school, the punctual Hilbert scheme of the affine plane, and an equivariant version of the punctual Hilbert scheme in connection with the A-D-E singularities. As a consequence of our choice some important aspects are not treated in these notes, mainly the cohomology theory, or Nakajima's theory. for which beautiful surveys are already available in the current literature [\textit{V. Ginzburg}, Lectures on Nakajima's quiver varieties. Paris: Société Mathématique de France. Séminaires et Congrès 24, pt. 1, 145--219 (2012; Zbl 1305.16009); \textit{M. Lehn}, Lectures on Hilbert schemes. Providence, RI: American Mathematical Society (AMS). CRM Proceedings \& Lecture Notes 38, 1--30 (2004; Zbl 1076.14010); \textit{H. Kakajima}, Lectures on Hilbert schemes of points on surfaces. University Lecture Series. 18. Providence, RI: American Mathematical Society (AMS). xi, 132 p. (1999; Zbl 0949.14001)]. scheme; cluster; group; group action; matrix factorization; quotient scheme; singular point Bertin, J., The punctual Hilbert scheme: an introduction, Proceedings of the Summer School 'Geometric Methods in Representation Theory'. I, 1-102, (2012), Soc. Math. France, Paris Actions of groups on commutative rings; invariant theory, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), McKay correspondence, Cluster algebras The punctual Hilbert scheme: an introduction | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 complex algebraic \(K3\) surface with \(\text{Pic} S\) generated by an ample line bundle \(H\) with self-intersection \(H^2 = 2t\) and let \(S^{[n]}\) denote the Hilbert scheme of \(n\) points on \(S\). The group of biregular automorphisms of \(S^{[n]}\) was classified for \(n=2\) by \textit{S. Boissiére} et al. [Prog. Math. 315, 1--15 (2016; Zbl 1375.14015)] and by \textit{A. Cattaneo} [Math. Nachr. 292, No. 10, 2137--2152 (2019; Zbl 1430.14086)], a consequence being that \(\text{Aut} (S^{n]})\) is either trivial or generated by a non-symplectic involution, the answer depending on \(n\) and \(t\).
In the paper under review, the authors classify the group \(\text{Bir} (S^{[n]})\) of birational automorphisms of \(S^{[n]}\). If \(t \geq 2\), there exists a non-trivial birational automorphism \(\sigma \in \text{Aut} (S^{n]}) \iff\) \(t (n-1)\) is not a square and the minimal solution \((X,Y)=(z,w)\) of Pell's equation \(X^2 - t(n-1) Y^2 = 1\) with \(z \equiv \pm 1 \mod (n-1)\) satisfies \(w \equiv 0 \mod 2\) and \((z,z) \equiv (1,1), (1,-1),(-1,-1)\) in \(\mathbb Z_{2(n-1)} \times \mathbb Z_{2t}\), in which case \(\sigma\) is an involution. The authors also determine when \(\sigma\) is symplectic. If \(t=1\) and \((X,Y) = (a,b)\) is the integer solution to \((n-1) X^2 - Y^2 = -1\) with smallest \(a,b > 0\), then \(\text{Bir} (S^{[n]}) = \text{Aut} (S^{n]}) = \mathbb Z_2\) if \(n-1\) is a square or \(b = \pm 1 \mod (n-1)\), otherwise \(n \geq 9\) and \(\text{Bir} (S^{[n]}) \cong \mathbb Z_2 \times \mathbb Z_2\). This extends the result of \textit{O. Debarre} and \textit{E. Macrì} [Int. Math. Res. Not. 2019, No. 22, 6887--6923 (2019; Zbl 1436.14022)] for \(n=2\). Their method combines results of Markman on monodromy operators and the chamber decomposition of the movable cone for hyperkähler manifolds [\textit{E. Markman}, Springer Proc. Math. 8, 257--322 (2011; Zbl 1229.14009)] with the explicit extremal ray computations of \textit{A. Bayer} and \textit{E. Macrì} [Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011)]. irreducible holomorphic symplectic manifolds; Hilbert schemes of points on surfaces; birational equivalence; automorphisms; cones of divisors Automorphisms of surfaces and higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Torelli problem, Complex-analytic moduli problems On birational transformations of Hilbert schemes of points on \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author studies the ramification sets of finite analytic mappings and applications of his results and methods to punctual Hilbert schemes and to finite analytic maps. He uses essentially the technique of ``chaining'' which consists in associating to a finite map a sequence (or chain) of sets, which are components of ramification loci of increasing multiplicity, and then in controlling their dimensions. For that purpose he uses a theorem of Grothendieck about the order of connexity of subgerms of an irreducible analytic germ, and also in the projective case the Fulton-Hansen theorem and a theorem of Deligne.
In {\S} 1, the author introduces three different notions of multiplicity. The topological multiplicity, the stable multiplicity and the algebraic one. In {\S} 2, he gives a fairly general lower bound to the dimension of the ramification set \(T^{d+1}(f)\), the set of points at which the multiplicity is at least \(d+1\). For the topological case he needs a hypothesis about f, called weak multitransversality which guarantees the additivity of multiplicity under deformation. This theorem is proved by a complicated induction involving multiproducts of ramification sets and the theorem of Grothendieck.
In {\S} 3, the author gives applications of {\S} 2, and of the chaining technique to the punctual Hilbert scheme \(Hilb'{\mathcal O}_{X,x}\) which parametrizes in \(Hilb'(X)\) the punctual schemes concentrated at \(x\in X\). The idea consists in identifying the germ of \(Hilb'({\mathcal O}_{X,x})\) at a smoothable element z with the ramification loci an appropriate map obtained by unfolding the equation of z. He thus obtains a lower bound for the local dimension at z of the open set U of smoothable points in X. This bound is (n-1)(\(\ell -1)\) with \(n=\dim (X)\) in the easiest case (X everywhere irreducible). Various, and more complicated results are obtained when we drop the irreducibility hypothesis or consider instead of U the open set of weakly smoothable (i.e. smoothable in a smooth ambient space) element.
Finally in {\S} 4, the author proves similar results for a finite projective morphism \(f:\quad X^ n\to P^ p.\) He generalizes a previous joint result of himself with Lazarsfeld (case \(n=p)\). This consists again in giving cases of non-emptiness for \(T^{d+1}(f)\) under some complicated numerical conditions. ramification sets of finite analytic mappings; punctual Hilbert schemes; ramification loci of increasing multiplicity T. Gaffney, ''Multiple points, chaining and Hilbert schemes,'' Amer. J. Math., vol. 110, iss. 4, pp. 595-628, 1988. Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Multiple points, chaining 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 An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams \(E_6, E_7\), \(E_8\). These curves are non-hyperelliptic of genus 3 or 4. We prove that a positive proportion of each family consists of curves with integral points everywhere locally but no integral points globally. Rational points, Singularities in algebraic geometry, Global ground fields in algebraic geometry, Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus On the arithmetic of simple singularities of type \(E\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(m,n\) be integers and \(\mathcal{Z}_{2}^{m,n}\) be the determinantal variety given by the vanishing of all the \(2\times2\) minors of a generic \(m\times n\) matrix. Let \(\mathcal{Z}_{2,2}^{m,n}\) be the affine variety of first-order jets over \(\mathcal{Z}_{2}^{m,n}\). If \(2<m\leq n\), then \(\mathcal{Z}_{2,2}^{m,n}\) has two irreducible components. One of them is isomorphic to \(\mathbb{A}^{mn}\). The other one is called the principal component, and is denoted by \(Z_0\). Let \(R=\mathbb{F}[x_{i,j},y_{i,j}:1\leq i\leq m, 1\leq j\leq n]\) be the polynomial ring over an algebraically closed field \(\mathbb{F}\) and \(\mathcal{I}=\mathcal{I}_{2,2}^{m,n}\) be the ideal which corresponds to \(\mathcal{Z}_{2,2}^{m,n}\). Let \(\mathcal{I}_0\) be the ideal which corresponds to \(Z_0\). By using combinatorial techniques, the authors compute the Hilbert series of \(R/\mathcal{I}_0\) and prove that this is the square of the Hilbert series of the base variety \(\mathcal{Z}_{2}^{m,n}\). As a consequence, they determine the \(a\)-invariant of \(R/\mathcal{I}_0\) and the Hilbert series of its graded canonical module. They characterize also the property of \(Z_0\) of being Gorenstein. jet schemes; Hilbert series; determinantal varieties Determinantal varieties, Combinatorial aspects of commutative algebra, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Hilbert series of certain jet schemes of determinantal 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 k be an algebraically closed field of characteristic 0, \(d\in {\mathbb{N}}\), and H d the Hilbert scheme parametrizing ideals \({\mathcal I}\subset {\mathcal O}_{{\mathbb{P}}^ 2_ k}\) of \(colength\quad d.\) For \(p\in H\) d, let h(p) denote the Hilbert function of the ideal of \({\mathbb{P}}\) \(2\otimes k(p)\), which corresponds to the point p. If \(\phi: {\mathbb{N}}\to {\mathbb{N}}\) is any function, the subset \(H_{\phi}=\{p\in H\quad d| \quad h(p)=\phi \}\) is locally closed in H d (and possibly empty).
The following result is proved: If \(H_{\phi}\) is provided with the reduced induced scheme structure, then \(H_{\phi}\) is connected and smooth over k.
Moreover, in the appendix a formula is described, by means of which the dimension of \(H_{\phi}\) can be computed in terms of \(\phi\). Hilbert function; stratification of Hilbert scheme of the projective plane Gotzmann, G.: A stratification of the Hilbert scheme of points in the projective plane. Math. Z.199, 539--547 (1988) Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Projective techniques in algebraic geometry A stratification of the Hilbert scheme of points in the projective 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 A particular case in the superstring theory where a finite group \(G\) acts upon the target Calabi-Yau manifold \(M\) in the theory seems to attract both physicists' and mathematician's attention. Define the ``orbifold Euler characteristic'': \(\chi (M,G)= {1\over |G |} \sum_{gh=hg} \chi (M^{\langle g, h\rangle})\), where the summation runs over all the pairs \(g,h\) of commuting elements of \(G\), and \(M^{\langle g,h \rangle}\) denotes the subset of \(M\) of all the points fixed by both of \(g\) and \(h\).
Vafa's formula-conjecture. If a complex manifold \(M\) has trivial canonical bundle and if \(M/G\) has a (nonsingular) resolution of singularities \(\widetilde {M/G}\) with trivial canonical bundle, then we have \(\chi (\widetilde {M/G} ) = \chi (M,G)\).
In the special case where \(M= \mathbb{A}^n\) an \(n\)-dimensional affine space, \(\chi (M,G)\) turns out to be the number of conjugacy classes, or equivalently the number of equivalence classes of irreducible \(G\)-modules. If \(n=2\), then the formula is therefore a corollary to the classical McKay correspondence.
Let \(G\) be a finite subgroup of \(SL(2, \mathbb{C})\) and \(\text{Irr} (G)\) the set of all equivalence classes of nontrivial irreducible \(G\)-modules. Let \(X=X_G: =\text{Hilb}^G (\mathbb{A}^2)\), \(S=S_G: =\mathbb{A}^2/G\), \({\mathfrak m}\) (resp. \({\mathfrak m}_S)\) the maximal ideal of \(X\) (resp. \(S)\) at the origin and \({\mathfrak n}: ={\mathfrak m}_S {\mathcal O}_{\mathbb{A}^2}\). Let \(\pi: X\to S\) be the natural morphism and \(E\) the exceptional set of \(\pi\). Let \(\text{Irr} (E)\) be the set of irreducible components of \(E\). Any \(I\in X\) contained in \(E\) is a \(G\)-invariant ideal of \({\mathcal O}_{\mathbb{A}^2}\) which contains \({\mathfrak n}\).
Definition: \(V(I): =I/({\mathfrak m} I+{\mathfrak n})\).
For any \(\rho\), \(\rho'\), and \(\rho''\in \text{Irr} (G)\) define \(E(\rho): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho)\}\)
\(P(\rho, \rho'): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho) \oplus V(\rho')\}\)
\(Q(\rho, \rho', \rho''): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho) \oplus V(\rho') \oplus V(\rho'')\}\).
Main theorem: (1) The map \(\rho \mapsto E(\rho)\) is a bijective correspondence between \(\text{Irr} (G)\) and \(\text{Irr} (E)\).
(2) \(E(\rho)\) is a smooth rational curve for any \(\rho\in \text{Irr} (G)\).
(3) \(P(\rho, \rho)= Q(\rho, \rho',\rho'') = \emptyset\) for any \(\rho,\rho', \rho''\in \text{Irr} (G)\). Hilbert schemes; orbifold Euler characteristics; irreducible components of exceptional set; superstring theory; McKay correspondence Ito, Y., Nakamura, I.: McKay correspondence and Hilbert schemes. Proc. Japan Acad. Ser. A Math. Sci., 72, 135--138 (1996) Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Global theory and resolution of singularities (algebro-geometric aspects) McKay correspondence and Hilbert schemes | 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 We offer short and conceptual re-proofs of some conjectures of Voevodsky's on the slice filtration. The original proofs were due to Marc Levine using the homotopy coniveau tower. Our new proofs use very different methods, namely, recent development in motivic infinite loop space theory together with the birational geometry of Hilbert schemes. algebraic \(K\)-theory; motivic cohomology; motivic spectral sequence; framed correspondences; Hilbert scheme Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Motivic cohomology; motivic homotopy theory Voevodsky's slice conjectures 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 The authors prove folklore conjectures about rigidity in families of curves in \(\mathbb P^3_{\mathbb C}\). Let \(\mathbf{Hilb}^{dz+1-g} (\mathbb P^r)\) denote the Hilbert scheme of one dimensional closed subschemes in \(\mathbb P^r\) of degree \(d\) and arithmetic genus \(g\) and let \(H_{d,g,r} \subset \mathbf{Hilb}^{dz+g-1} (\mathbb P^r)\) denote the union of irreducible components whose general member is smooth, connected and non-degenerate. An irreducible component \(Z \subset H_{d,g,r}\) is \textit{rigid in moduli} if the image of the natural rational map \(Z \to {\mathcal M}_g\) consists of a single point. It is expected that \(H_{d,g,r}\) has no component rigid in moduli unless \(g=0\). Using varieties of linear series [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume I. Berlim: Springer (1985; Zbl 0559.14017)] and classic tools such as the Accola-Griffiths-Harris bound [\textit{J. Harris}, Curves in projective space. With the collaboration of David Eisenbud. Seminaire de Mathematiques Superieures, 85. Seminaire Scientifique OTAN (NATO Advanced Study Institute), Department de Mathematiques et de Statistique - Universite de Montreal. Montreal, Quebec, Canada: Les Presses de l'Universite de Montreal. (1982; Zbl 0511.14014)], the authors prove this expectation when \(r=3\). They deduce a related conjecture [\textit{J. Harris} and \textit{I. Morrison}, Moduli of curves. New York, NY: Springer (1998; Zbl 0913.14005)], which says that the only family of curves whose deformations are all induced from automorphisms of \(\mathbb P^3\) is the family of rational normal curves. They give partial results for restricted ranges on the triples \((d,g,r)\) with \(r > 3\). Hilbert schemes; algebraic curves; linear series; gonality Parametrization (Chow and Hilbert schemes), Divisors, linear systems, invertible sheaves, Special divisors on curves (gonality, Brill-Noether theory) Irreducibility and components rigid in moduli of the Hilbert scheme 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 The main result is that the Hilbert scheme \(H (d,g)\) of smooth space curves of given genus \(g\) and degree \(d\) may have many irreducible components. Recall that \(H (d,g)\) is irreducible for \(d \geq g + 3\) [\textit{L. Ein}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 469-478 (1986; Zbl 0606.14003)]. On the other hand, for small degree with respect to the genus the scheme \(H (d,g)\) might be highly irreducible. In fact, it is shown that there is no polynomial function \(P (d,g)\) such that the number of components of the Hilbert scheme of space curves of genus \(g\) and degree \(d\), whose generic point corresponds to a smooth connected, projectively normal curve, \(N_{CM} (d,g)\), satisfies \(N_{CM} (d,g) \leq P (d,g)\) for all \(d\) and \(g\). The proof follows from counting the numerical characters \(\chi (C) = (n_ 0, \dots, n_{s - 1})\) associated to a space curve by \textit{L. Gruson} and \textit{C. Peskine} [in: Algebr. Geom., Proc., Tromsø Symp. 1977, Lect. Notes Math. 687, 31-59 (1978; Zbl 0412.14011)] and the theorem that the numerical characters correspond to components of the Hilbert scheme (for projectively normal curves) [\textit{G. Ellingsrud}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 8(1975), 423-431 (1976; Zbl 0325.14002) and \textit{L. Gruson} and \textit{C. Peskine} (loc. cit.)]. This gives the result \(N_{CM} (e) = \sum^ x_{k = 0} {e + 3 - k \choose k + 1}\), where \(e = \max \{t \in \mathbb{Z}, h^ 1 ({\mathcal O}_ C(t)) \neq 0\}\) and \(x = (e+2)/2\) for the number \(N_{CM} (e)\) of curves \(C\), projectively normal, with given speciality index \(e\). Using the fact \(\deg (C) = d = \sum^{s - 1}_{i = 0} (n_ i - i)\) and \(n_ i \leq e + 3\) for the numerical character it follows \(d \leq {e + 4 \choose 2}\) and the Castelnuovo-Halphen bound for the genus (in terms of the degree) gives the result.
Using linkage on proving the proposition: ``If \(X\) is a ribbon (doubling a smooth curve inside a smooth surface) and \(I_ X (k - 1)\) is globally generated, the generic curve \(L_{a,b} (X)\) linked to \(X\) in a complete intersection of type \((a,b)\) \((a \geq b \geq k)\) is smooth'', the authors also get the following result. The number \(N_ L (d,g)\) of linkage classes of generic curves of degree \(d\) and genus \(g\) cannot be bounded by a polynomial function of \(d\) and \(g\). This is obtained by using ribbons on generic union of lines and generic curves linked to it. number of linkage classes; Hilbert scheme; space curves Philippe Ellia, André Hirschowitz, and Emilia Mezzetti, On the number of irreducible components of the Hilbert scheme of smooth space curves, Internat. J. Math. 3 (1992), no. 6, 799 -- 807. Plane and space curves, Parametrization (Chow and Hilbert schemes), Linkage On the number of irreducible components of the 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 concept of primitive form was introduced by \textit{K. Saito} to construct the period mapping for the universal deformation of a holomorphic function with isolated singularities. Since the primitive form is defined by a certain system of bilinear differential equations, the global existence of the primitive form is unknown except the cases of simple singularities und simple elliptic singularities.
The purpose of this paper is to construct the primitive forms associated with simple singularities as reduced symplectic 2-form on simultaneous resolution space. To this end, in \S 1, we give the symplectic geometric construction of Grothendieck's simultaneous resolutions of the adjoint quotients. In \S 2, we give the symplectic geometric construction of simultaneous resolutions of simple singularities (see theorem 2.6). Then we explain that the reduced symplectic 2-form on the simultaneous resolution space is just the primitive form associated with simple singularity (see corollary 2.7). This construction of the primitive forms is closely related to \textit{P. J. Slodowy}'s Lie-theoretic construction of the period mapping. symplectic quotients; simultaneous resolutions of singularities Yamada, H., Symplectic reduction and simultaneous resolution of simple singularities, submitted. Global theory and resolution of singularities (algebro-geometric aspects), Homogeneous spaces and generalizations, Simple, semisimple, reductive (super)algebras, Modifications; resolution of singularities (complex-analytic aspects) Symplectic quotients and simultaneous resolutions of simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the ideals of the rational cohomology ring of the Hilbert scheme \(X^{[n]}\) of \(n\) points on a smooth projective surface \(X\). As an application, for a large class of smooth quasi-projective surfaces \(X\), we show that every cup product structure constant of \(H^*(X^{[n]})\) is independent of \(n\); moreover, we obtain two sets of ring generators for the cohomology ring \(H^*(X^{[n]})\).
Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between \(H^*(X^{[n]}; \mathbb{C})\) and \(H^*_{\text{orb}}(X^n/S_n; \mathbb{C})\) for a large class of smooth quasi-projective surfaces with numerically trivial canonical class. Heisenberg algebra; Hilbert scheme; rational cohomology ring W. Li, Z. Qin, and W. Wang. Ideals of the cohomology rings of Hilbert schemes and their applications. \textit{Trans. Amer. Math. Soc.}, 356(1):245--265 (electronic), 2004. Parametrization (Chow and Hilbert schemes), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Vertex operators; vertex operator algebras and related structures Ideals of the cohomology rings of Hilbert schemes and their 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 F. Klein studied simple singularities, classifying them as quotients of $\mathbb{C}^2$ by the action of a finite subgroup $\Gamma \subseteq \mathrm{SU}_2$. P. Du Val showed that the exceptional divisors of the minimal resolution of the isolated singularity of such a quotient form an arrangement of projective lines whose dual graph is a simply-laced Dynkin diagram $\Delta(\Gamma)$; thus the quotient $\mathbb{C}^2/\Gamma$ is called a simple singularity of type $\Delta(\Gamma)$. P. Slodowy then extended the definition of a simple singularity to the non simply-laced types by adding a second finite subgroup $\Gamma' \subseteq \mathrm{SU}_2$ such that $\Gamma'\supseteq \Gamma$ as a normal subgroup; $\Gamma'/\Gamma =\Omega$ acts on $\mathbb{C}^2/\Gamma$ and this action can be lifted to the minimal resolution of the singularity, inducing an action on the exceptional divisors, which corresponds to a group of automorphisms of the Dynkin diagram of $\mathbb{C}^2/\Gamma$. \par A deformation of a simple singularity $(X_0,\Omega)$ is an $\Omega$-equivariant deformation of the singularity $X_0$ with a trivial action of the automorphism group $\Omega$ on the base space. Setting $\pi : X \rightarrow Y$ as a deformation of $X_0$, a deformation $\psi : X' \rightarrow Y'$ of $X_0$ is induced from $\pi$ by a morphism $\varphi : Y' \rightarrow Y$ if there exist a morphism $\Phi :X' \rightarrow Y'$ such that $\pi\circ \Phi= \varphi\circ\psi$, and given $X_0 \stackrel{i}{\hookrightarrow} X$ and $X_0\stackrel{j}{\hookrightarrow} X'$, $\Phi\circ j=i$. \par A semiuniversal deformation $\pi_0 : X \rightarrow Y$ of a simple singularity $(X_0,\Omega)$ is a deformation of $(X,\Omega)$ such that any other deformation $\psi : X' \rightarrow Y'$ of $(X,\Omega)$ is induced from $\pi_0$ by an $\Omega$-equivariant morphism $\varphi : Y' \rightarrow Y$ with a uniquely determined differential $d_{y'} \varphi :T_{y'} Y' \rightarrow T_y Y$. The quotient of a semiuniversal deformation of a simple singularity of inhomogeneous type $B_r$ ($r \geq 2$), $C_r$ ($r \geq 3$), $F_4$ or $G_2$ by the natural symmetry of the associated Dynkin diagram is a deformation of a simple singularity of homogeneous type $X=D_s$, $E_6$ or $E_7$. \par Letting $\alpha : X_\Gamma \rightarrow \mathfrak{h}/W$ to be the semiuniversal deformation of a simple singularity of type $\Delta(\Gamma)=A_{2r-1}$ ($r \geq 2$), $D_{r}$ ($r \geq 4$) or $E_6$ obtained by the construction of H. Cassens and P. Slodowy, and $\mathfrak{h}$ and $W$ being the Cartan subalgebra and the associated Weyl group of the simple Lie algebra $\mathfrak{g}$ of the same type, respectively, they showed that $\Omega$ of the Dynkin diagram of $\mathfrak{g}$ acts on $X_\Gamma$ and $\mathfrak{h}/W$ such that $\alpha$ is $\Omega$-equivariant. Slodowy then showed that taking the restriction $\alpha^\Omega$ of $\alpha$ over the $\Omega$-fixed points of $\mathfrak{h}/W$ leads to a semiuniversal deformation of a simple singularity, which is inhomogeneous. As $\alpha$ is $\Omega$-equivariant, there is an action of $\Omega$ on every fiber of $\alpha^\Omega$, and the quotient leads to a new morphism $\overline{\alpha^\Omega}$, which is a non-semiuniversal deformation of a simple singularity of homogeneous type $\Delta(\Gamma')$. \par Let $\Gamma$ be a finite subgroup of $\mathrm{SU}_2$, $R$ its regular representation, $N$ its natural representation as a subgroup of $\mathrm{SU}_2$, and $\Delta(\Gamma)$ the associated Dynkin diagram. If $\Omega$ acting on $M(\Gamma)=(\mathrm{End}(R) \otimes N)^\Gamma$ is symplectic, then $\widetilde{\alpha}:X_{\Gamma}\times_{\mathfrak{h}/W}\mathfrak{h}\rightarrow \mathfrak{h}$ and $\alpha:X_{\Gamma}\rightarrow \mathfrak{h}/W$ can be made into $\Omega$-equivariant maps (Theorem 1.4, page 388): letting $M(\Gamma)$ to be the representation space of a McKay quiver built on a Dynkin diagram of type $A_{2r-1}$, $D_{r}$ or $E_6$, there exists a symplectic action of $\Omega=\Gamma'/\Gamma$ on $M(\Gamma)$, inducing the natural action on the singularity $\mathbb{C}^2/\Gamma$; this action then turns $\alpha$ into an $\Omega$-equivariant morphism. \par After A. Caradot shows that the morphism $\alpha^{\Omega} : X_{\Gamma, \Omega} \rightarrow (\mathfrak{h}/W)^{\Omega}$ is $\Omega$-invariant, it follows that $\Omega$ acts on each fiber of $\alpha^{\Omega}$, and the fibers can be quotiented. Furthermore it is known that $(\alpha^{\Omega})^{-1}(\overline{0}) = X_{\Gamma,0}=\mathbb{C}^2/\Gamma$, and thus $(\alpha^{\Omega})^{-1}(\overline{0})/\Omega=X_{\Gamma,0}/\Omega = (\mathbb{C}^2/\Gamma)/(\Gamma'/\Gamma) \cong \mathbb{C}^2/\Gamma'$, which is a simple singularity since $\Gamma'\subseteq \mathrm{SU}_2$ is finite. Thus, $\overline{\alpha^{\Omega}} : X_{\Gamma, \Omega} /\Omega \rightarrow (\mathfrak{h}/W)^{\Omega}$ is a deformation of the simple singularity $\mathbb{C}^2/\Gamma'$ of type $\Delta(\Gamma')$, where the deformation $\overline{\alpha^{\Omega}}$ is obtained through $\Delta(\Gamma)-\Delta(\Gamma,\Gamma')-\Delta(\Gamma')$-procedure. \par Caradot also studies the regularity of the fibers of $\overline{\alpha^{ \Omega}}$ (Theorem 2.3, page 390): assuming $\alpha^\Omega$ is the semiuniversal deformation of a simple singularity of inhomogeneous type $B_r$ ($r \geq 2$), $C_r$ ($r \geq 3$), $F_4$ or $G_2$, every fiber of the quotient $\overline{\alpha^\Omega}$ is singular. \par Finally, after stating a conjecture (Conjecture 3.1, page 396) that there exists a subset $\Theta$ of simple roots of the root system of type $\Delta(\Gamma')$ such that the Dynkin diagram associated to the singular configuration of any fiber of $\overline{\alpha^\Omega}$ is a subdiagram of the Dynkin diagram of type $\Delta(\Gamma')$ containing the vertices associated to $\Theta$, the author proves the conjecture for the types $A_3-B_2-D_4$, $A_5-B_3-D_5$, $D_4-C_3-D_6$, $D_4-G_2-E_6$, $D_4-G_2-E_7$, and $E_6-F_4-E_7$. (Theorem 3.2, page 397). deformations of simple singularities; simple root systems; simple singularities of inhomogeneous types; singular configurations Deformations of singularities, Root systems, Representation theory for linear algebraic groups Root systems and quotients of deformations of simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is known that there exist obstructed smooth curves in the projective space \(\mathbb{P}^r\), \(r\geq 3\), i.e. smooth curves whose corresponding points in the Hilbert scheme \(\text{Hilb} (\mathbb{P}^r)\) are singular. \textit{D. Mumford} [Am. J. Math. 84, 642-648 (1962; Zbl 0114.13106)] gave an example of an everywhere singular irreducible component of \(\text{Hilb} (\mathbb{P}^r)\), parametrizing smooth curves. Recently, \textit{P. Ellia}, \textit{A. Hirschowitz} and \textit{E. Mezzetti} [Int. J. Math. 3, No. 6, 799-807 (1992; Zbl 0824.14024)] showed that the open subset of \(\text{Hilb} (\mathbb{P}^3)\) parametrizing smooth curves with given genus and degree, can have arbitrarily many components when the genus and the degree grow.
In the present paper, the author proves that, for any \(n\geq 3\), there exist infinitely many \(r\) and for each of them a smooth, connected curve \(C_r\) in \(\mathbb{P}^r\) such that \(C_r\) lies on exactly \(n\) irreducible components of \(\text{Hilb} (\mathbb{P}^r)\). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type. projective space curves; Hilbert scheme; moduli of surfaces of general type Fantechi B, Pardini R. On the Hilbert scheme of curves in higher-dimensional projective space. Manuscripta Math, 1996, 90: 1--15 Parametrization (Chow and Hilbert schemes), Plane and space curves, Algebraic moduli problems, moduli of vector bundles, Surfaces of general type On the Hilbert scheme of curves in higher-dimensional 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 The main work of this paper is to determine the full cone \(\text{Eff}\mathbb{P}^{2[n]}\) of effective divisors on the Hilbert scheme \(\mathbb{P}^{2[n]}\) of \(n\) points over \(\mathbb{P}^2\).
The Picard group of \(\mathbb{P}^{2[n]}\) has rank 2 and is generated by \(H\) and \(\Delta/2\), where \(H\) is given by the \(\mathscr{S}_n\)-linearized line bundle \(\mathcal{O}_{\mathbb{P}^2}(1)^{\boxtimes n}\) over \((\mathbb{P}^2)^n\), and \(\Delta\) is the locus of nonreduced schemes, also \(\Delta/2=c_1(\det(R^{\bullet}p(q^*\mathcal{O}_{\mathbb{P}^2}\otimes I_{\mathcal{Z}}))^{\vee})\), with \(q:\mathbb{P}^2\times\mathbb{P}^{2[n]}\rightarrow \mathbb{P}^2\), \(p:\mathbb{P}^2\times\mathbb{P}^{2[n]}\rightarrow \mathbb{P}^{2[n]}\) two projections, and \(I_{\mathcal{Z}}\) the ideal sheaf of the universal family \(\mathcal{Z}\subset \mathbb{P}^2\times\mathbb{P}^{2[n]}\).
It is well-known that \(\Delta\) always spans on edge of the cone \(\text{Eff }\mathbb{P}^{2[n]}\). Also if \(n=\binom{r+2}{2}\), using Gaeta's resolution one can show easily that the other edge of \(\text{Eff }\mathbb{P}^{2[n]}\) is spanned by \(rH-\Delta/2=c_1(\det(R^{\bullet}p(q^*\mathcal{O}_{\mathbb{P}^2}(r)\otimes I_{\mathcal{Z}}))^{\vee})\). For a general \(n\), the author generalized Gaeta's resolution by replacing \(\mathcal{O}_{\mathbb{P}^2}(-r)^{\oplus kn}\) by some semiexceptional bundles. After a number of calculations and analysis on the properties of exceptional bundles, the author finally obtained that for every \(n\in\mathbb{Z}_{+}\), there is a stable bundle \(V\) such that \(\chi(V)/r(V)=n\) and its slop \(\mu(V)>0\) is minimal among all stable bundles \(V'\) with property \(\chi(V')/r(V')\geq n\) and \(\mu(V')>0\). So \(\chi(V\otimes I_{Z})=0\) for every \(Z\in\mathbb{P}^{2[n]}\). Applying some theories of Kronecker modules, the author proved that \(H^i(V\otimes I_Z)=0,~i=0,1,2\) for a general \(Z\). Hence the set \(D_V(n):=\{Z\in\mathbb{P}^{2[n]}\big| H^1(V\otimes I_Z)\neq 0.\}\) is a divisor of class \(\mu(V)H-\Delta/2\) on \(\mathbb{P}^{2[n]}\). The last thing is to check \(D_V(n)\) is extremal. It is enough to construct a complete curve \(C\subset \mathbb{P}^{2[n]}\) such that \(C\cap D_V(n)=\emptyset\). The existence of such curve \(C\) follows from the fact that general \(I_Z\) can be resolved by two fixed bundles \(E_1,E_2\) as follows
\[
0\rightarrow E_1\rightarrow{\phi} E_2\rightarrow I_Z\rightarrow 0,
\]
and the maps failing to give torsion-free cokernels form a subset of codimension \(\geq 2\) inside \(\text{Hom}(E_1,E_2)\).
At last, the author discussed the Bridgeland stability of \(I_Z\) and showed that the collapsing wall for \(\mathbb{P}^{2[n]}\) corresponds exactly to the nontrivial edge \(\mu H-\Delta/2\), which provides an important evidence to the conjecture asserting the correspondence between Bridgeland and Mori walls for moduli spaces of semistable sheaves over \(\mathbb{P}^2\). effective cones; Hilbert schemes of points; exceptional bundles; projective plan Huizenga, J.: Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles (2012). arXiv:1210.6576 Parametrization (Chow and Hilbert schemes), Vector bundles on surfaces and higher-dimensional varieties, and their moduli Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article considers four classes of 3-folds in \(\mathbb P^n\) with \(n \geq 6\): scrolls over \(\mathbb P^2\) or over a smooth quadric surface \(Q\), and quadric or cubic fibrations over \(\mathbb P^1\), with an eye toward describing the Hilbert scheme parameterizing such 3-folds. In particular, the authors show that when these 3-folds have degree \(7 \leq d \leq 11\) then they are unobstructed, i.e. they correspond to smooth points of the Hilbert scheme, and the authors find the dimension of the component of the Hilbert scheme at such points (even giving explicit formulas for this dimension). In some cases, a dense open subset of such an irreducible component is shown to be the locus of good determinantal schemes with the given Hilbert polynomial. To be ``good determinantal'' means that the scheme is defined by the ideal of maximal minors of a homogeneous matrix of appropriate size, and furthermore that the removal of a ``generalized row'' has maximal minors which again define a scheme of the appropriate codimension -- see \textit{M. Kreuzer, U. Nagel, C. Peterson} and the reviewer [J.\ Pure Appl. Algebra 150, No. 2, 155--174 (2000; Zbl 0999.14014)], for the definitions and basic results on good determinantal schemes. The results on good determinantal schemes complement results in [\textit{J. O. Kleppe, R. Miró-Roig, U. Nagel, C. Peterson} and the reviewer, Gorenstein liaison, complete intersection liaison invariants and unobstructedness. Mem. Am. Math. Soc. 154, 732 (2001; Zbl 1006.14018)] on unobstructedness, and the last section of the paper under review addresses the relationship between the two works. scroll; quadric fibration; cubic fibration; determinantal; unobstructed; Cohen-Macaulay G. M. Besana, M. L. Fania, The dimension of the Hilbert scheme of special threefolds. \textit{Comm. Algebra} 33 (2005), 3811-3829. MR2175469 Zbl 1093.14058 \(3\)-folds, Low codimension problems in algebraic geometry, Varieties of low degree, Adjunction problems, Parametrization (Chow and Hilbert schemes) The dimension of the Hilbert scheme of special threefolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 multiview moduli problems that arise in computer vision. We show that these moduli spaces are always smooth and irreducible, in both the calibrated and uncalibrated cases, for any number of views. We also show that these moduli spaces always admit open immersions into Hilbert schemes for more than two views, extending and refining work of \textit{C. Aholt} et al. [Can. J. Math. 65, No. 5, 961--988 (2013; Zbl 1284.13035)]. We use these moduli spaces to study and extend the classical twisted pair covering of the essential variety. multiview geometry; Hilbert scheme; computer vision; moduli theory; deformation theory Stacks and moduli problems, Computational aspects of higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Infinitesimal methods in algebraic geometry, Machine vision and scene understanding Two Hilbert schemes in computer vision | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme \(X^{[a]}\) of points on a complex manifold \(X\) is a compactification of the configuration space of \(a\)-element subsets of \(X\). The integral cohomology of \(X^{[a]}\) is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of \(X^{[2]}\) for any complex manifold \(X\), and the integral cohomology of \(X^{[2]}\) when \(X\) has torsion-free cohomology. Totaro, B., The integral cohomology of the Hilbert scheme of two points, Forum Math. Sigma, 4, (2016) Parametrization (Chow and Hilbert schemes), Discriminantal varieties and configuration spaces in algebraic topology, Singular homology and cohomology theory The integral cohomology of the Hilbert scheme of two 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 Given two integers d, r, with \(d\geq r\geq 3\), there is an upper bound, discovered by Castelnuovo in 1883, for the arithmetic genus of an irreducible, nondegenerate curve of degree d in \(P^ r\). Curves for which the bound is attained are called Castelnuovo curves. This paper deals with the study of some local properties of the Hilbert scheme at points corresponding to smooth Castelnuovo curves. The main result concerns a geometrical characterization of obstructed Castelnuovo curves, namely smooth Castelnuovo curves for which the corresponding point in the Hilbert scheme is singular. These turn out to be curves in \(P^ r\), \(r\geq 4\), lying on rational normal cones, and, if \(r=4\), they are in fact complete intersection curves on a rational normal cubic cone. The proof consists in explicitly computing the dimension of each component of the Hilbert scheme of such curves, and in comparing it with the dimension of the Zariski tangent space.
The first may be done by means of geometrical arguments. In order to know the latter one has, of course, to compute the cohomology of the normal bundle to smooth Castelnuovo curves, and this is worked out with techniques which may be of some use in other similar questions. arithmetic genus; Hilbert scheme at points corresponding to smooth Castelnuovo curves; Zariski tangent space [C2] Ciliberto C.,On the Hilbert scheme of curves of maximal genus in a projective space, Math. Z. 194 (1987) no. 3, 351--363 Parametrization (Chow and Hilbert schemes), Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry On the Hilbert scheme of curves of maximal genus in a 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 ''With any simple curve singularity (plane, complex, affine-algebraic) of Dynkin-type \(\Delta\) we associate the category of all finitely generated torsionfree modules over its complete local ring. For each of these module categories we calculate the Auslander-Reiten quiver. We suggest the construction of the ''twisted quiver'' of a quiver with involution and valuation of arrows which gives rise to a (purely combinatorial) one-to-one correspondence between the Auslander-Reiten quiver and the Dynkin diagram \(\Delta\).''
Let us briefly indicate the idea of the proof: It is shown in {\S}2 that for each Dynkin-diagram \(\Delta\), \(\Lambda_{\Delta}=K[[X,Y]]/(f_{\Delta}(X,Y))\), \(f_{\Delta}\) the polynomial of the simple plane curve singularity of type \(\Delta\), is a Gorenstein-order and the category of finitely generated torsion free \(\Lambda_{\Delta}\)-modules coincides with the category of \(\Lambda\)- lattices. - Therefore for each \(\Delta\) it is possible to construct the Auslander-Reiten quiver by using Auslander-Reiten sequences ({\S}3). Auslander-Reiten quiver; twisted quiver; Dynkin diagram; simple plane curve singularity Dieterich E., Wiedemann A.: The Auslander Reiten quiver of a simple curve singularity. Trans. Am. Math. Soc. 294, 455--475 (1986) Singularities of curves, local rings, Representation theory of associative rings and algebras, Singularities in algebraic geometry The Auslander-Reiten quiver of a simple curve singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the paper under review, the author studies interactions among vertex operators, Grassmannians, and Hilbert schemes. The infinite Grassmannian is approximated by finite-dimensional cutoffs, and a family of fermionic vertex operators is introduced as the limit of geometric correspondences on the equivariant cohomology groups with respect to a one-dimensional torus actions on the cutoffs. The author proves that in the localization basis, these operators are the fermionic vertex operators on the infinite wedge representation. Moreover, the boson-fermion correspondence, locality and intertwining properties with the Virasoro algebra are the limits of relations on the cutoffs. The author further shows that these operators coincide with the vertex operators defined by A. Okounkov and the author in an earlier work on the equivariant cohomology groups of the Hilbert schemes of points on the affine plane with respect to a special torus action. Vertex operators; Grassmannians; Hilbert schemes Carlsson, E.: Vertex operators, grassmannians, and Hilbert schemes, Comm. math. Phys. 300, No. 3, 599-613 (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vertex operators; vertex operator algebras and related structures, Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds Vertex operators, Grassmannians, 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 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 [For the entire collection see Zbl 0695.00010.]
This paper is a sequel to a previous paper by the author [Acta Math. 147, 13-49 (1981; Zbl 0479.14004)], where the method of iteration was used to derive r-fold-point formulas for a proper map \(f:X\to Y\) [see \textit{S. Katz} in Algebraic geometry, Proc. Conf., Sundance/Utah 1986, Lect. Notes Math. 1311, 147-155 (1988; Zbl 0657.14035)]. - In the present paper a defect of the method of iteration is pointed out in the case of torsion and new formulas are derived using the Hilbert scheme. More precisely the new formulas are expressed in terms of effectively computable polynomials in the Chern classes of f with integer coefficients and in terms of three multiple-point cycles \(t_ r, u_ r\) and \(v_ r\) where \(t_ r\) enumerates the points of Y whose fibers contain length r subschemes, \(u_ r\) enumerates the points of X which are parts of length r subschemes of the fibers and \(v_ r\) enumerates the points x in X such that there exists a length r\(+1\) subscheme of the fiber \(f^{-1}fx\) that is an extension of some length r subscheme by x. Finally, the author raises several unanswered questions that could be investigated in the future. iteration; r-fold-point; Hilbert scheme; multiple-point cycles Steven L. Kleiman, Multiple-point formulas. II. The Hilbert scheme, Enumerative geometry (Sitges, 1987) Lecture Notes in Math., vol. 1436, Springer, Berlin, 1990, pp. 101 -- 138. Enumerative problems (combinatorial problems) in algebraic geometry, Parametrization (Chow and Hilbert schemes), Algebraic cycles Multiple-point formulas. II: 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 purpose of this paper is to settle a conjecture raised by \textit{Z. Qin} and \textit{W. Wang} in their paper ``Integral operators and integral cohomology classes of Hilbert schemes'' [Math. Ann. 331, No. 3, 669--692 (2005; Zbl 1081.14006)]. The conjecture states that, under a particular hypothesis, certain homological operators are integral. To this end the authors use topological \(K\)-theory and the algebra of rings of symmetric functions. The result is then used to obtain explicit generators for the integral cohomology of the Hilbert scheme of \(n\) points on a smooth projective surface which has no odd cohomology.
The paper is well written, thorough and contains the necessary information on Hilbert schemes of points on surfaces. Several computations are worked out explicitly. Hilbert schemes; integral cohomology (Co)homology theory in algebraic geometry, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Integral cohomology 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 We provide here an infinite family of finite subgroups \(\{G_n\subset\text{SL}(\mathbb{C}\}_{n\geq 2}\) for which the \(G\)-Hilbert scheme \(G_n\)-Hilb \(\mathbb{A}^n\) is a crepant resolution of \(\mathbb{A}^n/G_n\), via the Hilbert-Chow morphism. The proof is based on an explicit description of the toric structure of \(G_n\)-Hilb \(\mathbb{A}^n\), \(n\geq 2\), in terms of Nakamura's \(G_n\)-graphs. DOI: 10.1016/j.crma.2006.11.033 Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Smooth toric \(G\)-Hilbert schemes via \(G\)-graphs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 describes, in detail, a process for constructing Kummer \(K3\) surfaces, and other ``generalized'' Kummer \(K3\) surfaces. In particular, we look at how some well-known geometrical objects such as the platonic solids and regular polygons can inspire the creation of singular surfaces, and we investigate the resolution of those surfaces. Furthermore, we will extend this methodology to examine the singularities of some complex two-dimensional quotient spaces and resolve these singularities to construct a Kummer \(K3\) and other generalized Kummer \(K3\) surfaces. \(K3\) surfaces and Enriques surfaces, Singularities of surfaces or higher-dimensional varieties Simple surface singularities, their resolutions, and construction of \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a quasiprojective smooth surface defined over an algebraically closed field of positive characteristic. In this note we show that if \(X\) is Frobenius split then so is the Hilbert scheme \(\text{Hilb}^n(X)\) of \(n\) points in \(X\). In particular, we get the higher cohomology vanishing for ample line bundles on \(\text{Hilb}^n(X)\) when \(X\) is projective and Frobenius split. Shrawan Kumar and Jesper Funch Thomsen, Frobenius splitting of Hilbert schemes of points on surfaces, Math. Ann. 319 (2001), no. 4, 797 -- 808. Parametrization (Chow and Hilbert schemes), Vanishing theorems in algebraic geometry, Finite ground fields in algebraic geometry Frobenius splitting 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 smooth complex surface \(S\), the punctual Hilbert scheme \(\text{Hilb}^k(S)\) parametrizing zero-dimensional subschemes of \(S\) of length \(k\) is a smooth complex variety [cf.: \textit{J. Fogarty}, Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. Moreover, the Hilbert-Chow morphism \(c:\text{Hilb}^k(S)\to S^{(k)}\), which to such a subscheme associates the cycle defined by its support and by its local multiplicities, establishes \(\text{Hilb}^k(S)\) as a smooth partial compactification of the Zariski open subset \(S_0^{(k)}\) parametrizing \(k\)-uples of distinct points, and \(\text{Hilb}^k(S)\) is compact when \(S\) is compact. In the case of a smooth almost complex compact surface, i.e., a smooth compact real 4-fold with a distinguished almost complex structure, an analogue of the punctual Hilbert scheme was proposed by the author of the paper under review [Ann. Inst. Fourier, 50, No. 2, 689--722 (2000; Zbl 0954.14002)]. Actually, C. Voisin proposed two different constructions of a punctual Hilbert scheme \(\text{Hilb}^k(X)\) for a smooth almost complex compact fourfold \((X,J)\), without any integrability assumption on the almost complex structure \(J\).
Now, in the subsequent paper under review, the author continues her studies in the particular case of symplectic fourfolds. Given such a symplectic fourfold \((X,\omega)\), a theorem of \textit{M. Gromov} [Invent. Math. 82, 307--347 (1985; Zbl 0592.53025)] asserts that the set of almost complex structures compatible with the symplectic form \(\omega\) on \(X\) is a contractible space. Applying her previous constructions to any such almost complex structure \(J\) on \(X\), the author obtains a differentiable \(2k\)-fold \(\text{Hilb}^k(X)\) which is uniquely determined by \((X,\omega)\), via the choice of such an almost complex structure \(J\), up to diffeomorphisms isotopic to the identity. As to the precise structure of such a punctual Hilbert scheme \(\text{Hilb}^k(X)\), the author provides three main theorems.
The first main theorem concerns the special case where \(\omega\) is a Kähler form and \(J\) can be chosen to be integrable, that is when \((X,J)\) is underlying a complex Kähler surface. Under these assumptions, \(\text{Hilb}^k(X)\) is shown to be a Kähler variety.
The second main theorem states that if \(\omega\) is a Kähler form on a compact surface \(S\), and if \(\lambda>0\) is a suffiently small number, then there is a Kähler metric of Kähler class \(c^* ([\omega]_k)-\lambda\cdot\delta\) on \(\text{Hilb}^k(S)\), where \(\delta \in H^2(\text{Hilb}^k (S),\mathbb{Z})\) satisfies the equality \(2\delta=c_1 (c^{-1}(\Delta))\) for the generalized diagonal \(\Delta\) in the symmetric product \(S^{(k)}\). This theorem gives a refined description of the cohomology in small degree of the punctual Hilbert scheme, thereby providing a more precise version of the author's first main theorem. As to the symplectic, almost complex situation of a compact fourfold, the author's third main theorem is certainly the most important result of the present paper. Its precise statement is as follows:
Let \((X,\omega)\) be a compact symplectic fourfold. Then there is a positive real number \(\lambda_0\) such that for any \(0<\lambda<\lambda_0\), there exists a symplectic form on \(\text{Hilb}^k(X)\) with cohomology class \(c^* ([\omega]_k)-\lambda\cdot \delta\). Furthermore, all these symplectic forms belong to a well-defined deformation class of symplectic forms on the punctual Hilbert scheme \(\text{Hilb}^k(X)\). In the last section of the present paper, possible applications of this third main theorem to the overall study of symplectic fourfolds are discussed, together with some concrete open problems and speculations, mainly with a view toward the construction of new invariants for symplectic fourfolds. compact complex surfaces; Kähler manifolds; almost complex structure Voisin C.: On the punctual Hilbert scheme of a symplectic fourfold. Contemp. Math. 312, 265--289 (2002) Parametrization (Chow and Hilbert schemes), Almost complex manifolds, Compact complex surfaces, Symplectic manifolds (general theory), Compact Kähler manifolds: generalizations, classification, Global differential geometry of Hermitian and Kählerian manifolds On the punctual Hilbert scheme of a symplectic fourfold | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 first part of the paper, we present two criteria to characterise lexicographic sets among Borel sets: one criterion by two combinatorial invariants of a Borel set, the other by an extremal property of a packing problem. In the second part, we apply these results to prove the simple-connectedness of certain Hilbert schemes by Gröbner basis theory. lexicographic Borel set; Hilbert schemes; Gröbner basis [Mal97] D. Mall, Characterizations of lexicographic sets and simply-connected Hilbert schemes, In: Proceedings of AAECC-12, Lecture Notes in Computer Science, Vol. 1255, Springer Verlag, 1997, pp. 221--136. Parametrization (Chow and Hilbert schemes), Combinatorics of partially ordered sets, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Characterisations of lexicographic sets and simply-connected 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 an algebraically closed field of characteristic \(0,\) \(S=K[x_0,\dots,x_n]\) and \({\mathbb P}^n_K=\mathrm{Proj} S.\) Let us consider on the set of the terms of \(S\) a degreverse order. The authors prove that in the Hilbert scheme of points in \({\mathbb P}^n_K,\) the point corresponding to a segment ideal, with respect to a degreverse order, is singular. Unfortunately this result cannot be generalized to Hilbert schemes with a Hilbert polynomial of a positive degree.
Moreover they provide an algorithm for computing all the saturated Borel ideals with a given Hilbert polynomial. Hilbert scheme of points; Borel ideal; segment ideal; Gröbner stratum; degrevlex term order; Gotzmann number Cioffi, F.; Lella, P.; Marinari, M. G.; Roggero, M., Segments and Hilbert schemes of points, Discrete Math., 311, 20, 2238-2252, (2011) Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Segments and Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We present several conjectures on multiple \(q\)-zeta values and on the role which they play in certain problems of enumerative geometry. multiple \(q\)-zeta value; \(q\)-deformation; Hilbert scheme; CW/DT correspondence Okounkov, A, Hilbert schemes and multiple \(q\)-zeta values, Funct. Anal. Appl., 48, 138-144, (2014) Parametrization (Chow and Hilbert schemes), Binomial coefficients; factorials; \(q\)-identities, Enumerative problems (combinatorial problems) in algebraic geometry, Multiple Dirichlet series and zeta functions and multizeta values Hilbert schemes and multiple \(q\)-zeta values | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 first-order deformations of the tangent sheaf of resolutions of Calabi-Yau threefolds that are of the form \(\mathbb{C}^3/\mathbb{Z}_r\), focusing on the cases where the orbifold has an isolated singularity. We prove a lower bound on the number of deformations for any crepant resolution of this orbifold. We show that this lower bound is achieved when the resolution used is the \(G\)-Hilbert scheme, and note that this lower bound can be found using methods from string theory. These methods lead us to a new way to construct the \(G\)-Hilbert scheme using the singlet count. Gaines, B.: (0,2)-deformations and the Hilbert scheme Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Applications of Lie groups to the sciences; explicit representations, Quantum field theory on curved space or space-time backgrounds, Calabi-Yau manifolds (algebro-geometric aspects), Topology and geometry of orbifolds \((0,2)\)-deformations and the \(G\)-Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A simple \(K3\)-singularity is a three-dimensional normal isolated singularity with a certain condition on the mixed Hodge structure on a good resolution. We prove here that a three-dimensional normal isolated singularity is a simple \(K3\)-singularity if and only if the exceptional divisor of a \(\mathbb{Q}\)-factorial terminal modification is an irreducible normal \(K3\)-surface. simple \(K3\)-singularity; mixed Hodge structure; exceptional divisor S. Ishii and K. Watanabe, A geometric characterisation of a simple \(K3\) singularity, Tôhoku Math. J. 44 (1992), 19--24. Singularities in algebraic geometry, \(K3\) surfaces and Enriques surfaces, Mixed Hodge theory of singular varieties (complex-analytic aspects), Modifications; resolution of singularities (complex-analytic aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), \(3\)-folds A geometric characterization of a simple \(K3\)-singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Usually, moduli spaces in geometry are singular varieties. In order to avoid the difficulties related to their singular nature, the recently invented ``Derived Deformation Theory Program (DDT)'' aims at developing appropriate versions of the (non-abelian) derived functor of the respective moduli functor. Rather than ordinary varieties or schemes, the resulting geometric objects are sought to be ``dg-schemes'', i.e., geometric objects whose algebras of functions are commutative differential graded algebras, and which are considered up to quasi-isomorphisms. While the DDT program appears to be well-established in the formal case, mainly in view of the recent fundamental work of M. Kontsevich, S. Barannikov, V. Hinich, M. Manetti, and others, the structure of global derived moduli spaces is much less understood.
In this vein, the aim of the paper under review is to provide a comprehensive DDT-type construction of the derived Hilbert scheme. In a foregoing paper [cf.: \textit{I. Ciocan-Fontanine} and \textit{M. Kapranov}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 34, 403--440 (2001; Zbl 1050.14042)], the authors have already constructed a derived version of a first global algebro-geometric moduli space, namely of Grothendieck's wellknown ``Quot scheme''. Using a somewhat similar but nevertheless different approach, the authors are now investigating another important global moduli space in the context of the DDT program. While in the usual algebraic geometry, the Hilbert scheme is a particular case of the Quot scheme, the two constructions turn out to diverge considerably when passing to the framework of derived categories.
More precisely, let \(k\) be a field of characteristic zero, \(X\) a smooth projective variety over \(k\), and \({\mathcal O}_X(1)\) a very ample line bundle on \(X\) defining a projective embedding. For a given polynomial \(h\), the authors construct a dg-manifold \(\text{RHilb}^{\text{LCI}}_h(X)\) as the derived version of the classical geometric Hilbert scheme \(\text{Hilb}_h(X)\) of closed subschemes of \(X\) with Hilbert polynomial \(h\) relative to the polarization \({\mathcal O}_X(1)\). However, when the polynomial \(h\) is identically 1, then the derived Hilbert scheme turns out to coincide with the variety \(X\) whereas the derived Quot scheme \(\text{RQuot}({\mathcal O}_X)\) is known to be different from \(X\).
As for applications of these DDT-type constructions, the earlier constructed dg-manifolds RQuot are suitable for describing the derived moduli spaces of vector bundles on a fixed variety \(X\). In contrast, the dg-schemes \(\text{RHilb\,}h(X)\) established here are expected to play a similar rôle with regard to the derived moduli spaces of projective varieties themselves, which the authors corroborate by two striking examples. Namely, they use the explicit structure of the dg-schemes \(\text{RHilb\,}h(X)\) to construct two types of geometric derived moduli spaces:
(1) the derived space of maps \(\text{RMap}(C,Y)\) from a fixed projective scheme \(C\) to a fixed smooth projective variety \(Y\) and
(2) the derived stack of stable degree-\(d\) maps \(R\overline M_{g,n}(Y,d)\) from \(n\)-pointed nodal curves of genus \(g\) to a given smooth projective variety \(Y\). The latter example completes some earlier work of \textit{M. Kontsevich} [in: The moduli spaces of curves, Prog. Math. 129, 335--368 (1995; Zbl 0885.14028)] and others (Behrend-Manin, Fulton-Pandharipande), thereby contributing to the mathematical theory of Gromov-Witten invariants. All in all, this is a very comprehensive paper of fundamental importance in derived moduli theory. The conceptual ingredients and refined techniques for the construction of derived Hilbert schemes include cotangent complexes, Harrison homology, derived moduli of operad algebras, derived schemes of ideals in finite-dimensional commutative algebras, and the theory of algebraic stacks. In spite of its highly advanced character, the exposition is very detailed, systematic and clear. moduli spaces; derived categories; operad algebras; Gromov-Witten invariants Ciocan-Fontanine, Ionuţ; Kapranov, Mikhail M., Derived Hilbert schemes, J. Amer. Math. Soc., 15, 4, 787-815, (2002) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Supervarieties, Derived categories, triangulated categories, Nonabelian homological algebra (category-theoretic aspects) Derived 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 \(X \subset \mathbb{C}^3\) be a rational double point singularity, and let \(\Gamma\) be its Newton dual fan in \(\mathbb{R}^3\). Because \(X\) is Newton non-degenerate, any regular subdivision of \(\Gamma\) induces a toric embedded resolution of \(X\). Inspired by the Nash problem, the authors define weights in \(\mathbb{R}^3\) for certain components of the jet schemes of \(X\), and they construct a regular subdivision of \(\Gamma\) whose rays are generated by these weights. In other words, they construct a map from a certain set of jet scheme components to the set of boundary components of a certain toric embedded resolution of \(X \subset \mathbb{C}^3\). In a case by case analysis for each singularity type, the authors show that this map is a bijection, and they show that in all cases except the \(E_8\) singularity, the constructed toric embedded resolution is minimal in the sense that each boundary divisor's valuation is centered at some boundary divisor in any other toric embedded resolution of \(X \subset \mathbb{C}^3\). embedded Nash problem; resolution of singularities; toric geometry Global theory and resolution of singularities (algebro-geometric aspects), Arcs and motivic integration, Toric varieties, Newton polyhedra, Okounkov bodies Jet schemes and minimal toric embedded resolutions of rational double point singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of this note is to announce some results on the GIT problem for the Hilbert and Chow scheme of curves of degree \(d\) and genus \(g\) in the projective space of dimension \(d-g\), whose full details will appear in [the authors, ``GIT for polarized curves'', preprint, \url{arxiv:1109.6908}]. In particular, we extend the previous results of L. Caporaso up to \(d>4(2g-2)\) and we observe that this is sharp. In the range \(2(2g-2) < d < \frac{7}{2} (2g-2)\), we get a complete new description of the GIT quotient. As a corollary, we get a new compactification of the universal Jacobian over the moduli space of pseudo-stable curves. GIT; Hilbert scheme; Chow scheme; stable curves; pseudo-stable curves; compactified universal Jacobian Geometric invariant theory, Parametrization (Chow and Hilbert schemes) On GIT quotients of Hilbert and Chow schemes 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 Let \(X\) be a smooth projective algebraic curve of genus \(g \geq 4\). Denote by \(M(n,\xi)\) the moduli space of stable vector bundles of rank \(n\) and determinant \(\xi\). The objects studied in this paper are the Hilbert scheme \(\text{Hilb}^P_{M(n,\xi)}\), for \(P\) a polynomial of degree \(\geq 2\), the space \(\text{Mor}_P({\mathbb G},M(n,\xi))\) of morphisms to \(M(n,\xi)\) from the Grassmannian \({\mathbb G}= \text{Grass}(n-r,{\mathbb C}^n)\), with \(r\) an integer with \(0 <r <n \), ``where \(P\) is the Hilbert polynomial associated to the Grassmannian \(\mathbb G\)'', and ``the moduli space of stable bundles over \(X\times {\mathbb G}\)''. Namely, the authors describe an open subscheme of \({\mathcal HG}\), ``the irreducible component of the Hilbert scheme \(\text{Hilb}^P_{M(n,\xi)}\) containing \(r\)-Hecke cycles'', and an open subset of the ``irreducible component of the scheme \(\text{Mor}_P({\mathbb G},M(n,\xi))\) that contains the \(r\)-Hecke morphisms'', denoted \(\text{Mor}_P^{{\mathcal H},r}({\mathbb G},M(n,\xi))\).
As a consequence they show the following: ``If \(\text{Mor}_s({\mathbb P}^1,{\mathbb G})\not =\emptyset\) then
\[
\text{Mor}_{2ns}({\mathbb P}^1,M(n,\xi))\not =\emptyset.
\]
Moreover \(\text{dim } \text{Mor}_{2ns}({\mathbb P}^1,M(n,\xi)) \geq (n^2-1)(g-1)+1\)''.
The connection to the existing literature on the subject is clearly described in the paper, making it easier for the reader to see the main ideas. moduli of vector bundles; Hilbert scheme; Hecke correspondence; gonality; moduli of stable maps Brambila-Paz, L.; Mata-Gutiérrez, O.: On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve. Manuscripta math. 142, 525-544 (2013) Vector bundles on curves and their moduli, Parametrization (Chow and Hilbert schemes) On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Motivated by the AGT conjecture [\textit{L. F. Alday} et al., Lett. Math. Phys. 91, No. 2, 167--197 (2010; Zbl 1185.81111)], the present article provides a geometric realization of the vacuum representation of \(W\)-algebra \(W_{\kappa}(\mathfrak{gl}_r)\). More precisely, it is shown that \(W_{\kappa}(\mathfrak{gl}_r)\) acts on the (equivariant, localized) Borel-Moore homology of Hilbert schemes of points \(\mathrm{Hilb}(r,n)\) on the \(r\)-th order infinitesimal neighborhood of \(\mathbb{C}^2\subset \mathbb{C}^3\), and this module is precisely the vacuum representation.
The proof of this result relies on the identification of a Verma module of \(W_{\kappa}(\mathfrak{gl}_r)\) with the Borel-Moore homology of the moduli spaces \(\mathcal{M}(r,n)\) of framed torsion free sheaves on \(\mathbb{P}^2\), see [\textit{O. Schiffmann} and \textit{E. Vasserot}, Publ. Math., Inst. Hautes Étud. Sci. 118, 213--342 (2013; Zbl 1284.14008)]. Namely, the authors show that \(\mathrm{Hilb}(r,n)\) is a closed subscheme of \(\mathcal{M}(r,n)\), working with quiver realizations of ADHM type. Next, a careful analysis of the torus fixed loci allows to identify homology of \(\mathrm{Hilb}(r,n)\) with a direct summand of homology of \(\mathcal{M}(r,n)\). Finally, the authors check that this summand is preserved by the Hecke operators of Schiffmann-Vasserot, and thus deduce the action of the \(W\)-algebra.
It is also suggested (see Conjecture 1.3) that it would be more natural to replace the Borel-Moore homology of \(\mathrm{Hilb}(r,n)\) by the vanishing cycle cohomology of the moduli space \(\mathcal{Q}(r,n)\) of Higgs sheaves on \(\mathbb{P}^2\) framed at infinity. Hilbert schemes; quiver representations; \(W\)-algebras Calabi-Yau manifolds (algebro-geometric aspects), Virasoro and related algebras, Relationships between surfaces, higher-dimensional varieties, and physics Hilbert schemes of nonreduced divisors in Calabi-Yau threefolds and \(W\)-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 Let \(D\) be a smooth Cartier divisor on a smooth quasi-projective surface \(S\). The Hilbert scheme \((S \setminus D)^{[n]}\) of \(n\) points on \(S \setminus D\) is not proper, but \textit{J. Li} and \textit{B. Wu} have constructed a compactification relative to \(D\) [``Good degeneration of Quot-schemes and coherent systems'', Preprint, \url{arXiv:1110.0390}], called the relative Hilbert scheme. The author uses the moduli stack of stable ideal sheaves and the stack of expanded degenerations of \textit{J. Li} [J. Differ. Geom. 57, 509--578 (2001; Zbl 1076.14540)] to produce the generating function for the normalized Poincaré polynomial of the relative Hilbert scheme of points analogous to the generating function for the Hilbert scheme of \(n\) points given by \textit{L. Göttsche} and \textit{W. Soergel} [Math. Ann. 296, No. 2, 235--245 (1993; Zbl 0789.14002)]. When \(S = \mathbb P^2\) and \(D \subset \mathbb P^2\) is a line, the cohomology groups of the relative Hilbert scheme are computed and it is shown that the natural map from the Chow group to the Borel-Moore homology is an isomorphism. Hilbert scheme of points; relative Hilbert scheme; Poincaré polynomial Iman Setayesh, Relative Hilbert scheme of points, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.) -- Princeton University. Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory Relative 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 Simple space curve singularities were classified by the first author [Commun. Algebra 27, No.8, 3993--4013 (1999; Zbl 0963.14011)]. The same methods are used to determine the simple isolated determinantal codimension two singularities in other dimensions. By a counting argument simple higher dimensional ones can only exist for dimension at most 4, and only \(2\times 3\) matrices occur. In high dimensions even rigid determinantal singularities exist, but they are not isolated. For the case of fat points in the plane the counting argument does not work, but it turns out that there is only one family, again given by a \(2\times3\) matrix.
In the surface case the simple codimension two determinantal singularities are exactly the rational triple points. All singularities on the lists are quasi-homogeneous, but in dimension four there are examples where one variable has a negative weight.
For simple fat points and space curves the complete list of adjacencies is computed, with computer assistance. simple singularities; space curves; fat points; Hilbert-Burch theorem; classification of singularities; adjacencies Frühbis-Krüger, A.; Neumer, A., Simple Cohen-Macaulay codimension 2 singularities, Commun. Algebra, 38, 454-495, (2010) Local complex singularities, Complex surface and hypersurface singularities, Deformations of singularities, Deformations of complex singularities; vanishing cycles, Singularities of surfaces or higher-dimensional varieties Simple Cohen-Macaulay codimension 2 singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider the construction of refined Chern-Simons torus knot invariants by \textit{M. Aganagic} and \textit{S. Shakirov} [Commun. Math. Phys. 333, No. 1, 187--228 (2015; Zbl 1322.81069)] from the DAHA viewpoint of \textit{I. Cherednik} [Int. Math. Res. Not. 2013, No. 23, 5366--5425 (2013; Zbl 1329.57019)]. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of \textit{O. Schiffmann} and \textit{E. Vasserot} [Compos. Math. 147, No. 1, 188--234 (2011; Zbl 1234.20005); Duke Math. J. 162, No. 2, 279--366 (2013; Zbl 1290.19001)] to relate knot invariants to the Hilbert scheme of points on \(\mathbb{C}^2\). Then, we use the methods of the second author to compute these invariants explicitly in the uncolored case. We also propose a conjecture relating these constructions to the rational Cherednik algebra, as in the work of the first author, A. Oblomkov, J. Rasmussen and V. Shende [\textit{E. Gorsky} et al., Duke Math. J. 163, No. 14, 2709--2794 (2014; Zbl 1318.57010)]. Among the combinatorial consequences of this work is a statement of the \(\frac{m}{n}\) shuffle conjecture. torus knots; Hilbert scheme; double affine Hecke algebra Gorsky, E; Neguţ, A, Refined knot invariants and Hilbert schemes, J. Math. Pures Appl, 9, 104, 403-435, (2015) Parametrization (Chow and Hilbert schemes), Hecke algebras and their representations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Knots and links in the 3-sphere Refined knot invariants 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 The purpose of this paper is to define and prove the existence of the Hilbert scheme. This was originally done by \textit{A. Grothendieck} in Sém. Bourbaki 13 (1960/61), No. 221 (1961; Zbl 0236.14003). A simplified proof was given by \textit{D. Mumford} [``Lectures on curves on an algebraic surface'' (Princeton 1966; Zbl 0187.42701)], and we basically follow that proof, with small modifications. existence of Hilbert scheme Strømme, S. -A.: Elementary introduction to representable functors and Hilbert schemes. Banach center publ. 36, 179-198 (1996) Parametrization (Chow and Hilbert schemes) Elementary introduction to representable functors 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 In an earlier paper [J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)], the author showed that the Hilbert scheme \(H_n= \text{Hilb}^n(\mathbb{C}^2)\) of points in the plane can be identified with the Hilbert scheme of regular orbits \(\mathbb{C}^{2n}//S_n\), as defined by \textit{Y. Itô} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)]. Using this identification and a generalization of the McKay correspondence shown recently by \textit{T. Bridgeland}, \textit{A. D. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], he proves vanishing theorems for tensor powers of tautological bundles on \(H_n\). Then he applies these vanishing theorems to establish several further theorems.
1) The author proves a series of character formulas for spaces of global sections of these vector bundles. In particular, he obtains the character formula for diagonal harmonics conjectured by \textit{A. M. Garsia} and him [J. Algebr. Comb. 5, No. 3, 191--244 (1996; Zbl 0853.05008)]. This result implies that the dimension of the space of diagonal harmonics is \((n+ 1)^{n-1}\) and that the Hilbert series of the bigraded subspace of skew elements is given by the \(q, t\)-Catalan polynomial.
2) The author shows the ``operator conjecture'' stated in a previous paper [J. Algebr. Comb. 3, No. 1, 17--76 (1994; Zbl 0803.13010)] which says that the space of diagonal harmonics is generated by certain \(S_n\)-invariant polarization operators applied to the space of classical harmonics.
One of the main technical tools used in the paper is the polygraph, a subspace arrangement in \(\mathbb{C}^{2n+2l}\) consisting of the tuples \((P_1,\dots, P_n,Q_1,\dots, Q_l)\) such that \(Q_i\in \{P_1,\dots, P_n\}\) for all \(i\). Its homogeneous coordinate ring \(R(n, l)\) is the space of global sections of the vector bundles in question and carries geometric information about the Hilbert scheme.
The author also formulates several conjectures concerning further results on \(\text{Hilb}^n(\mathbb{C}^2)\) and the Hilbert schemes \(\text{Hilb}^n(\mathbb{C}^d)\) with \(d\geq 3\). diagonal harmonics; Catalan polynomial; operator conjecture Blandin, H.: La Conjecture de Polarisation Généralisée. Doctorat en Mathématiques, Thèse. Montréal, (Québec, Canada). Université du Québec a Montréal, QC (2015) Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations, Vanishing theorems in algebraic geometry, Group actions on affine varieties Vanishing theorems and character formulas for the Hilbert scheme of points in the plane | 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 authors calculate the Betti numbers of the Hilbert scheme of points in the plane. Observe that the maximal torus of SL(3) acts on \(Hilb^ d({\mathbb{P}}^ 2)\) with isolated fixed points. It follows from a result of Birula-Białynicki that \(Hilb^ d({\mathbb{P}}^ 2)\) has a cellular decomposition. Then the calculation of the Betti numbers reduces to a careful study of the representation of the torus at the tangent spaces of the fixed points. As a by-product to their method, the authors also obtain similar results about the punctual Hilbert scheme and the Hilbert scheme of points in the affine plane. Betti numbers of the Hilbert scheme of points in the plane; punctual Hilbert scheme Ellingsrud, Geir; Strømme, Stein Arild, On the homology of the Hilbert scheme of points in the plane, Invent. Math., 87, 343-352, (1987) Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry On the homology of the Hilbert scheme of points in the plane | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Hilbert schemes of suitable smooth, projective threefold scrolls over the Hirzebruch surface \(\mathbb{F}_{e}\), \(e\geq 2\), are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension, and the general point of such a component is described. ruled varieties; vector bundles; rational surfaces; Hilbert scheme Fania, L; Flamini, F, Hilbert schemes of some threefold scrolls over \(\mathbb{F}_{e}\), Adv. Geom., 16, 413-436, (2016) Parametrization (Chow and Hilbert schemes), Elliptic surfaces, elliptic or Calabi-Yau fibrations, \(3\)-folds, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Adjunction problems, Low codimension problems in algebraic geometry, Varieties of low degree Hilbert schemes of some threefold scrolls over \(\mathbb{F}_{e}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If \(X\) is a smooth, \(d\)-dimensional, projective variety defined over an algebraically closed field \(k\), a 0-dimensional subscheme \(Z\subseteq X\) is called fat of type \(n_ 1\geq\cdots\geq n_ t\geq 1\), if Supp \(Z\) consists of \(t\) points \(P_ 1,\dots,P_ t\) and \({\mathcal O}_{Z,P_ i}={\mathcal O}_{X,P_ i}/{\mathfrak m}^{n_ i}_{X,P_ i}\) for \(i=1,\dots,t\). The author shows that the locus corresponding to those schemes is a locally closed subset of the set of \(k\)-rational points of the Hilbert scheme \(\text{Hilb}^ n(X)\), where \(n=\sum^ t_{i=1}{n_ i+d-1\choose d}\). He thus provides a clean scheme-theoretic derivation of a statement which had been mentioned already by \textit{G. Paxia} as a well- known, but not yet proved result in Proc. Am. Math. Soc. 112, No. 1, 19- 23 (1991; Zbl 0733.14001)]where it was demonstrated that the sets under consideration are irreducible and constructible. flat points; variety flat; Hilbert scheme Projective techniques in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Parametrization (Chow and Hilbert schemes) The fat locus 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 \(Y \subset \mathbb P^n\) be a smooth hypersurface of degree \(d\). For a point \(y \not \in Y\) the projection \(\pi: Y \to \mathbb P^{n-1}\) gives a finite morphism of degree \(d\) and one can consider the relative Hilbert scheme \(\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1})\) parametrizing zero-dimensional subschemes of length \(m\) contained in the fibers of the projection for each \(1 \leq m \leq d\). It is easy to see that \(\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1})\) is isomorphic to \(\mathbb P^{n-1}\) for \(m=d\) and to \(Y\) for \(m=1\) or \(m=d\), so the author focuses on the intermediate values \(1 < m < d\). Here it is known from work of \textit{L. Gruson} and \textit{C. Peskine} that \(\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1})\) is a smooth connected projective variety of dimension \(n-1\) for general \(y \not \in Y\) [Duke Math. J. 162, No. 3, 553--578 (2013; Zbl 1262.14058)]. In the paper under review the author gives an explicit embedding \(j:\text{Hilb}_y^{[m]} (Y/\mathbb P^{n-1}) \hookrightarrow \mathbb P\) into a weighted projective space for \(n > 1\) and gives gives equations defining the image \(X \subset \mathbb P\). The equations show that \(X \subset \mathbb P\) is a smooth weighted complete intersection, and hence results of \textit{I. Dolgachev} [Lect. Notes Math. 956, 34--71 (1982; Zbl 0516.14014)] yield the generating function for the dimensions of the graded pieces of \(\oplus_{k=0}^\infty H^0(X, {\mathcal O}_X (k))\), the dualizing sheaf \(\omega_X\), and the Picard group for \(n \geq 4\). relative Hilbert scheme; weighted projective space Parametrization (Chow and Hilbert schemes) The relative Hilbert scheme of projection 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 \(e_1, \ldots, e_c\) be positive integers and let \(Y \subseteq \mathbb{P}^n\) be the monomial complete intersection defined by the vanishing of \(x_1^{e_1}, \ldots, x_c^{e_c}\). In this paper, we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points \(\mathrm{Hilb}^d (Y)\), and discuss applications to the Hilbert scheme of points \(\mathrm{Hilb}^d (X)\) of arbitrary complete intersections \(X \subseteq \mathbb{P}^n\). Clements-Lindström ring; Betti numbers; infinite free resolutions; finite subscheme; strongly stable ideal; Eisenbud-Green-Harris conjecture; Lex-plus-powers conjecture Syzygies, resolutions, complexes and commutative rings, Linkage, complete intersections and determinantal ideals, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Parametrization (Chow and Hilbert schemes) Syzygies in Hilbert schemes of complete intersections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 summarizes recent results concerning singularities with respect to the Mather-Jacobian log discrepancies over an algebraically closed field of arbitrary characteristic. The basic point is that the inversion of adjunction with respect to Mather-Jacobian discrepancies holds under arbitrary characteristic. Using this fact, we will reduce several geometric properties of the singularities to jet scheme problems and try to avoid discussions that are distinctive to characteristic \(0\). Singularities of surfaces or higher-dimensional varieties, Arcs and motivic integration, Jets in global analysis Singularities in arbitrary characteristic via jet schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this article, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on \(K3\) surfaces. algebraic geometry; cubic fourfold; \(K3\) surface; rationality problem Rationality questions in algebraic geometry, Parametrization (Chow and Hilbert schemes), \(K3\) surfaces and Enriques surfaces, Hypersurfaces and algebraic geometry Hilbert schemes of two points on \(K3\) surfaces and certain rational cubic fourfolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 a deformation of the triply graded Khovanov-Rozansky homology of a link \(L\) depending on a choice of parameters \(y_c\) for each component of \(L\), which satisfies link-splitting properties similar to the Batson-Seed invariant. Keeping the \(y_c\) as formal variables yields a link homology valued in triply graded modules over \(\mathbb{Q}[x_c,y_c]_{c\in \pi_{0}(L)}\). We conjecture that this invariant restores the missing \(Q\leftrightarrow T\,Q^{-1}\) symmetry of the triply graded Khovanov-Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points in the plane. We compute this invariant for all positive powers of the full twist and match it to the family of ideals appearing in Haiman's description of the isospectral Hilbert scheme. Khovanov-Rozansky homology; Soergel bimodules; Hilbert schemes Parametrization (Chow and Hilbert schemes), Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) Hilbert schemes and \(y\)-ification of Khovanov-Rozansky homology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(SL(3,\mathbb C)\). The Hilbert scheme \(\text{Hilb}^G(\mathbb C^3)\) is by definition the subscheme of \(\text{Hilb}^{\mid G\mid}(\mathbb C^3)\) parametrizing \(G\)-invariant subschemes. It has been proved that \(\text{Hilb}^G(\mathbb C^3)\) is irreducible, smooth and it is a crepant resolution of \(\mathbb C^3/G\). In this paper, the authors study this Hilbert scheme when \(G\) is a non-abelian simple subgroup of \(SL(3,\mathbb C)\). There are two such subgroups, \(G_{60}\) and \(G_{168}\), of order 60 and 168 respectively. \(G_{60}\) is isomorphic to the alternating group of degree 5 and \(G_{168}\) is isomorphic to \(PSL(2,7)\). The authors are particularly interested in giving a precise description of the fibre over the origin of \(\mathbb C^3/G\). It turns out that in the first case this fibre is a connected union of four smooth rational curves and in the second one it is a union of a smooth rational curve and a doubly blown-up projective plane, with infinitely near centres. \(G\)-invariant subschemes; crepant resolution; Hilbert scheme Gomi, Y., Nakamura, I., Shinoda, K.: Hilbert schemes of G-orbits in dimension three. Asian J. Math. 4(1), 51--70 (2000; Kodaira's issue) Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Group actions on varieties or schemes (quotients), Linear algebraic groups over the reals, the complexes, the quaternions Hilbert schemes of \(G\)-orbits in dimension three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an arbitrary algebraically closed field. For any finite subgroup \(G\) of \(\text{SL}(3,k)\) of order prime to the characteristic of the ground field \(k\) the quotient space \(\mathbb{A}^3_k/G\) is then a normal Gorenstein variety with trivial canonical sheaf. In birational geometry, a natural choice of a crepant smooth resolution of the singular variety \(\mathbb{A}^3_k/G\) would be very convenient, but it seems that the general theories in this framework, such as the theory of minimal models and the theory of torus embeddings, do not canonically provide a resolution of this type.
In the paper under review, the author approaches this problem by studying a particular Hilbert scheme \(\text{Hilb}^G(\mathbb{A}^3_k)\), which finally turns out to be a canonical crepant resolution of the quotient variety \(\mathbb{A}^3_k/G\). The so-called \(G\)-orbit Hilbert scheme \(\text{Hilb}^G:= \text{Hilb}^G(\mathbb{A}^3_k)\) is, by definition, the scheme parametrizing all \(G\)-invariant smoothable \(0\)-dimensional subschemes of \(\mathbb{A}^3_k\) of length \(n:=|G|\). This object, which may be regarded as a certain substitute for the quotient \(\mathbb{A}^3_k/G\) was introduced by the author and \textit{Y. Itô} in [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)] as a suitable tool in the study of resolutions of quotient singularities via the famous McKay correspondence.
In the present paper, the \(G\)-orbit Hilbert scheme \(\text{Hilb}^G\) is described for a finite abelian subgroup \(G\) of \(\text{SL}(3,k)\) resulting in the fact that \(\text{Hilb}^G\) appears then as a smooth torus embedding associated to a crepant fan in \(\mathbb{R}^3\) with apices constructed from the group \(G\). Furthermore, it is shown that the commutativity of \(G\) implies the nonsingularity of that associated fan.
This finally establishes the author's main theorem (Theorem 0.1.) stating the following: For any abelian subgroup \(G\) of \(\text{SL}(3,k)\) of order prime to the characteristic of the ground field \(k\) the \(G\)-orbit Hilbert scheme \(\text{Hilb}^G(\mathbb{A}^3_k)\) is a crepant resolution of the quotient space \(\mathbb{A}^3_k/G\).
The first half of the present article is devoted to describing a \(G\)-orbit Hilbert scheme as a toric variety in arbitrary dimension. This part is based on a fine analysis of the corresponding lattices and \(G\)-graphes, which is highly interesting and important for its own sake. In the second half of the paper, the particular case of dimension three and an abelian subgroup \(G\) of \(\text{SL}(3,k)\) is inspected more closely by means of the special appearing \(G\)-graphs, culminating in the author's main theorem mentioned above. At the end of the article, the author discusses some interesting examples in dimension three and four, thereby illustrating the variety of possibilities that can occur already in those low-dimensional cases.
In a sense, the present work may be regarded as a complement to the related earlier results by \textit{Y. Itô} and \textit{M. Reid} [in: Higher-dimensional complex varieties. Proc. Int. Conf. Trento, Italy, June 15--24, 1994. 221--240 (1996; Zbl 0894.14024)] and by \textit{Y. Itô} and \textit{H. Nakajima} [Topology 39, 1155--1191 (2000; Zbl 0995.14001)]. quotient varieties; quotient singularities; resolution of singularities; toric varieties I. Nakamura, \textit{Hilbert schemes of abelian group orbits}, J. Algebraic Geom. \textbf{10} (2001), no. 4, 757-779. Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Hilbert schemes of abelian group orbits | 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 Denoting by \({\mathcal I}'(d,g,3)\) the subscheme of the Hilbert scheme, whose general point corresponds to smooth irreducible and nondegenerate curves of degree \(d\) and genus \(g\) in \(\mathbb{P}^ 3\), it is proved that \({\mathcal I}'(d,g,3)\) is irreducible in the following cases: (i) \(d \geq g+3\), (ii) \(d=g+2\) and \(g \geq 5\), (iii) \(d=g+1\) and \(g \geq 11\).
It should be pointed out that a similar result in the case of \(d \geq g+3\), \(r=3\) was also obtained by \textit{L. Ein} in Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 469-478 (1986; Zbl 0606.14003) independently (and earlier). complex space curves; Hilbert scheme \textsc{C. Keem and S. Kim}, Irreducibility of a subscheme of the Hilbert scheme of complex space curves, J. Algebra, \textbf{145} (1992), 240-248. Parametrization (Chow and Hilbert schemes), Schemes and morphisms, Families, moduli of curves (algebraic) Irreducibility of a subscheme of the Hilbert scheme of 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 In the paper under review, the authors consider a family of integral plane curves and prove that if the relative Hilbert scheme of points is smooth, then the pushforward of the constant sheaf on the relative Hilbert scheme contains no summands other than the expected ones.
As a corollary, they show that the perverse filtration on the cohomology of the compactified Jacobian of an integral plane curve encodes the cohomology of all Hilbert schemes of points on the curve. An interesting consequence for the enumerative geometry of Calabi-Yau three-folds is also mentioned. locally planar curves; Hilbert scheme; compactified Jacobian; versal deformation; perverse cohomology; decomposition theorem L. Migliorini and V. Shende, \textit{A support theorem for Hilbert schemes of planar curves}, J. Eur. Math. Soc. 15, 2353-2367. Parametrization (Chow and Hilbert schemes), Jacobians, Prym varieties, Formal methods and deformations in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) A support theorem for Hilbert schemes of planar curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let $K$ be an algebraically closed field. A parametrization of a germ of a plane curve singularity is given by a pair $(x(t), y(t))$ of power series, $x(t)$, $y(t)\in K[\![t]\!]$. $\Gamma=\{\mathrm{ord}_t(h)\mid h\in K[\![x(t),y(t)]\!]\}$ is the semigroup of the parametrization. It is assumed that the parametrization is primitive, i.e., $\dim_K(K[\![t]\!]/K[\![x(t),y(t)]\!]) <\infty$. Furthermore, it is assumed that $\mathrm{ord}_t(x(t))=:n<\mathrm{ord}_t(y(t))=:m$ and that $n\nmid m$. \par Two parametrizations $(x(t),y(t))$ and $(\overline x(t),\overline y(t))$ are said to be $\mathcal A$-equivalent if there exist automorphisms $\psi: K[\![t]\!]\to K[\![t]\!]$ and $\varphi=(\varphi_1,\varphi_2): K[\![x,y]\!]\to K[\![x,y]\!]$ such that $(x(\psi(t)),y(\psi(t)))=(\varphi_1(\overline x(t),\overline y(t)),\varphi_2(\overline x(t),\overline y(t))))$. A parametrization $(x(t),y(t))$ is called \textit{simple} if there are only finitely many $\mathcal A$-equivalent classes in a deformation of $(x(t),y(t))$. \par \textit{J. W. Bruce} and \textit{T. J. Gaffney} [J. Lond. Math. Soc., II. Ser. 26, 465--474 (1982; Zbl 0575.58008)] classified the simple parametrized plane curve singularities over the complex numbers $\mathbb C$; \textit{K. Mehmood} and \textit{G. Pfister} [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 60(108), No. 4, 417--424 (2017; Zbl 1399.57008)] gave a different proof for this classification and gave also the classification of unimodal singularities. Furthermore, they extended the results to a classification over the real numbers $\mathbb R$. \par In the paper under review the authors give a similar classification in positive characteristic. A parametrization is not simple if $\Gamma$ has a minimal system of generators consisting of more than three elements. For the cases of simple parametrizations the authors describe $\Gamma$ by its generators and give normal forms for the parametrization. characteristic \(p\); parametrized plane curves; simple singularities Singularities of curves, local rings, Singularities in algebraic geometry, Plane and space curves Simple singularities of parametrized plane curves in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is known by the results of \textit{J. Kollár} [Shafarevich maps and automorphic forms. Princeton, NJ: Princeton University Press (1995; Zbl 0871.14015)], \textit{L. Ein} and \textit{R. Lazarsfeld} [J. Am. Math. Soc. 10, No. 1, 243--258 (1997; Zbl 0901.14028)], \textit{O. Debarre} and \textit{C. D. Hacon} [Manuscr. Math. 122, No. 2, 217--228 (2007; Zbl 1140.14040)] that effective divisors representing principal and other low-degree polarizations on complex abelian varieties have mild singularities. In this note, we extend these results to all polarizations of degree \(<g\) on simple \(g\)-dimensional abelian varieties, settling a conjecture of Debarre and Hacon [loc. cit.]. Divisors, linear systems, invertible sheaves, Minimal model program (Mori theory, extremal rays), Abelian varieties and schemes Singularities of divisors of low degree on simple abelian 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 \(p(t)\) be an admissible Hilbert polynomial in \(\mathbb{P}^{n}\) of degree d. The Hilbert scheme \(\mathcal Hilb_{p(t)}^n\) can be realized as a closed subscheme of a suitable Grassmannian \(\mathbb{G}\), hence it could be globally defined by homogeneous equations in the Plücker coordinates of \(\mathbb{G}\) and covered by open subsets given by the non-vanishing of a Plücker coordinate, each embedded as a closed subscheme of the affine space \(\mathbb{A}^{D}\), \(D=\dim(\mathbb{G})\). However, the number \(E\) of Plücker coordinates is so large that effective computations in this setting are practically impossible. In this paper, taking advantage of the symmetries of \(\mathcal Hilb_{p(t)}^n\) we exhibit a new open cover, consisting of marked schemes over Borel-fixed ideals, whose number is significantly smaller than \(E\). Exploiting the properties of marked schemes, we prove that these open subsets are defined by equations of degree \(\leq d+2\) in their natural embedding in \(A^{D}\). Furthermore we find new embeddings in affine spaces of far lower dimension than D, and characterize those that are still defined by equations of degree \(\leq d+2\). The proofs are constructive and use a polynomial reduction process, similar to the one for Gröbner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases. Hilbert scheme; Borel-fixed ideal; marked scheme Green, M.L.: \textit{Generic initial ideals}, Six lectures on commutative algebra (Bellaterra, 1996), Progr. Math., vol. 166. Birkhäuser, Basel, 1998, pp. 119-186 Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials, Symbolic computation and algebraic computation A Borel open cover 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 Let \(K\) be an algebraically closed field of characteristic \(p>0\). Isolated simple singularities \(f\in K[[x_1, \dots, x_n]]\) are classified with respect to right equivalence. The corresponding classification with respect to contact equivalence was done by \textit{G. M. Greuel} and \textit{H. Kröning} [Math. Z. 203, No. 2, 339--354 (1990; Zbl 0715.14001)]. Here the result was similar to Arnold's classification in characteristic \(0\). In case of right equivalence it turns out that there are only finitely many simple singularities.
The classification is based on the generalization of the notion of modality to the algebraic setting. It is proved that the modality is semicontinuous in any characteristic. simple singularity; classification right equivalence; characteristic \(p\) Greuel, G.-M.; Hong Duc, N., Right simple singularities in positive characteristic, \textit{J. Reine Angew. Math.}, 712, 81-106, (2016) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Right simple singularities in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors generalize linear algebraic and monad descriptions of the Hilbert schemes \(\text{Hilb}^{[c]} (\mathbb C^2)\) of length \(c\) subschemes of \(\mathbb C^2\) given by \textit{H. Nakajima} [Lectures on Hilbert schemes of points on surfaces. Providence, RI: American Mathematical Society (1999; Zbl 0949.14001)]. To state the main result, let \(V\) be a complex \(c\)-dimensional vector space. For commuting operators \(B_1, \dots, B_n\) on \(V\) and a map \(I:\mathbb C \to V\), the tuple \((B_1, \dots, B_n, I)\) is \textit{stable} if there is no proper subspace \(S \subset V\) in the image of \(I\) and invariant under each \(B_i\). Then \(\text{GL} (V)\) acts on the stable tuples and the authors prove a bijection between {\parindent=0.7cm \begin{itemize}\item[1.] Ideals \(J \subset \mathbb C [x_1, \dots, x_n]\) with quotient of dimension \(c\); \item[2.] Stable tuples \((B_1, \dots, B_n,I)\) with \(\dim V = c\) modulo the action of \(\text{GL} (V)\); \item[3.] \textit{perfect extended monads}; these are complexes
\[
0 \to V_{1-n} \otimes {\mathcal O}_{\mathbb P^n} (1-n) \to V_{2-n} \otimes V_{2-n} \otimes {\mathcal O}_{\mathbb P^n} (2-n) \dots \to V_0 \otimes {\mathcal O}_{\mathbb P^n} \to V_1 \otimes {\mathcal O}_{\mathbb P^n} (1) \to 0
\]
with \(V_1=V, V_0 = V^{\oplus n} \oplus \mathbb C\) and \(V_i = V^{\oplus {{n}\choose{1-i}}}\) for \(i<0\) which are exact except possibly at degree \(0\).
\end{itemize}} Consequently \(\text{Hilb}^{[c]} (\mathbb C^n)\) is isomorphic to a GIT quotient of \(\mathcal C (n,c) \times \text{Hom} (\mathbb C, V)\), where \(\mathcal C (n,c)\) is the variety of \(n\) commuting \(c \times c\) matrices. The correspondence between (1) and (2) and the Hilbert scheme description is known (for example, see [\textit{F. Vaccarino}, J. Algebra 317, No. 2, 634--641 (2007; Zbl 1155.13007)]), but the correspondence with perfect extended monads is new. The authors give a similar description of the Hilbert scheme of points on any affine variety \(Y \subset \mathbb C^n\). Using recent results of \textit{K. Šivic} [Linear Algebra Appl. 437, No. 2, 393--460 (2012; Zbl 1323.15011)] the authors show irreducibility of \(\text{Hilb}^{[c]} (\mathbb C^3)\) for \(c \leq 10\), improving on the result for \(c \leq 8\) due to \textit{D. A. Cartwright} et al. [Algebra Number Theory 3, No. 7, 763--795 (2009; Zbl 1187.14005)]. On the other hand, \(\text{Hilb}^{[c]} (\mathbb C^3)\) is reducible for \(c \geq 78\) by work of \textit{A. Iarrobino} [Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)]. Hilbert scheme of points; commuting matrices; monads Parametrization (Chow and Hilbert schemes) Commuting matrices and the Hilbert scheme of points on affine spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In 1960, \textit{A. Grothendieck} [Sém. Bourbaki 13, No. 221 (1961; Zbl 0236.14003)] proved the existence of a projective scheme \(H= \text{Hilb}_{p(t)} \mathbb{P}^n\) parametrizing closed subschemes of the projective space \(\mathbb{P}^n\) with given Hilbert polynomial \(p(t)\). There are few general results about these schemes and they have only been studied in special cases. For instance, in 1975, \textit{G. Ellingsrud} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 8, 423-432 (1975; Zbl 0325.14002)] proved that arithmetically Cohen-Macaulay closed subschemes of \(\mathbb{P}^n\) of codimension 2 are parametrized by smooth points of an open subset \(A\) of \(H\) and computed the dimension of \(A\).
Turning to the codimension 3 case, in the present paper the authors compute the dimension of the open smooth subset of the Hilbert scheme \(H\) parametrizing arithmetically Gorenstein closed subschemes of \(\mathbb{P}^n\) of codimension 3. Hilbert scheme; Hilbert polynomial; codimension 3; arithmetically Gorenstein closed subschemes Kleppe, J.; Miró-Roig, R. M., The dimension of the Hilbert scheme of Gorenstein codimension 3 subschemes, J. Pure Appl. Algebra, 127, 73-82, (1998) Parametrization (Chow and Hilbert schemes), Low codimension problems in algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The dimension of the Hilbert scheme of Gorenstein codimension \(3\) subschemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme \(\text{Hilb}^{p(m)}(\mathbb P ^n)\) parametrizes closed subschemes of \(\mathbb P^n\) with given Hilbert polynomial \(p(m)\). By a result of Groethendieck, it is known that \(\text{Hilb}^{p(m)}(\mathbb P ^n)\) is a projective scheme and by a result of Hartshorne, it is connected. In the paper under review, the authors study the component \(H_n\) of the Hilbert scheme whose general point parametrizes a pair of codimension two linear subspaces of \(\mathbb P ^n\) for \(n\geq 3\). In particular they show that \(H_n\) is smooth and isomorphic to the blow up of the symmetric square of \(\mathbb G (n-2,n)\) along the diagonal, and that \(H_n\) intersects only one other component of \(\text{Hilb}^{p(m)} (\mathbb P ^n)\). Moreover, they show that \(H_n\) is a Mori dream space. Hilbert scheme; Mori dream space; Stable base locus decomposition D. Chen, I. Coskun & S. Nollet, Hilbert scheme of a pair of codimension two linear subspaces. Comm. Algebra 39, no. 8, 3021--3043, 2011.arXiv:0909.5170 Rational and birational maps, Minimal model program (Mori theory, extremal rays), Grassmannians, Schubert varieties, flag manifolds, Fine and coarse moduli spaces, Parametrization (Chow and Hilbert schemes) Hilbert scheme of a pair of codimension two linear subspaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 In this paper, the author studies the integral cohomology of the Hilbert scheme of points on a surface.
Let \(X\) be a smooth complex projective surface. For a positive integer \(n\), let \(X^{[n]}\) be the Hilbert scheme of points parametrizing length-\(n\) \(0\)-dimensional closed subschemes of \(X\). It is known that \(X^{[n]}\) is irreducible and smooth, and has dimension \(2n\).
The author proves that if \(X\) has torsion-free cohomology, then so does the Hilbert scheme \(X^{[n]}\) for every \(n\).
Moreover, if the integral Chow motive of \(X\) is trivial, then so is the integral Chow motive of \(X^{[n]}\) for every \(n\). The main idea is to use E. Markman's approach when \(X\) has a nontrivial Poisson structure and the reduced obstruction theory, due to A. Gholampour and R. P. Thomas, for the nested Hilbert schemes of surfaces. Hilbert scheme of points; integral cohomology; Chow motive Parametrization (Chow and Hilbert schemes), Algebraic cycles, Discriminantal varieties and configuration spaces in algebraic topology The integral cohomology of the Hilbert scheme of points on a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We explain Sklyanin's separation of variables in geometrical terms and construct it for Hitchin and Mukai integrable systems. We construct Hilbert schemes of points on \(T^*\Sigma\) for \(\Sigma= \mathbb{C}, \mathbb{C}^*\) or an elliptic curve, and on \(\mathbb{C}^2/\Gamma\) and show that their complex deformations are integrable systems of Calogero-Sutherland-Moser type. We present the hyper-Kähler quotient constructions for Hilbert schemes of points on cotangent bundles to the higher genus curves, utilizing the results of Hurtubise, Kronheimer and Nakajima. Finally, we discuss the connections to physics of \(D\)-branes and string duality. Sklyanin's separation of variables; Hitchin and Mukai integrable systems; Calogero-Sutherland-Moser type; hyper-Kähler quotient constructions; duality A. Gorsky, N. Nekrasov, and V. Rubtsov, Hilbert schemes, separated variables and \(D\)-branes , Comm. Math. Phys. 222 (2001), 299--318. String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Parametrization (Chow and Hilbert schemes), Relationships between algebraic curves and integrable systems, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests Hilbert schemes, separated variables, and \(D\)-branes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 degeneration $X\rightarrow C$ gives a degeneration of the corresponding Hilbert schemes, and the study of such degenerations is of obvious interest in algebraic geometry. \par New techniques for studying degenerations were introduced by \textit{J. Li} and \textit{B. Wu} [Commun. Anal. Geom. 23, No. 4, 841--921 (2015; Zbl 1349.14014)], with an approach based on the technique of expanded degenerations. The method is very general and can be used to study degenerations of various types of moduli problems such as Hilbert schemes and moduli spaces of sheaves. They also used degenerations of Quot-schemes and coherent systems to obtain degeneration formulae for Donaldson-Thomas invariants and Pandharipande-Thomas stable pairs. \par In the present article, the authors search to understand degenerations of irreducible holomorphic symplectic manifolds. As a start, the authors study degenerations of $K3$ surfaces and their Hilbert schemes, a guiding example being type II degenerations of $K3$ surfaces leading to the investigation of degenerations of the Hilbert scheme of points for simple degenerations $X\rightarrow C$ with no a priori restriction on the type or dimension of the fibre. The assumption of a simple degeneration gives that the total space is smooth, and that the central fibre $X_0$ over the point $0\in C$ of the $1$-dimensional base $C$ has normal crossing along smooth varieties. In this paper, a technique for the construction of degenerations of Hilbert schemes is developed, which allow to control the geometry of the degenerate fibres. \par The authors describe the equalities and differences between their approach and that of Li and Wu [loc. cit.]: First of all, this article considers only Hilbert schemes of points, whereas Li and Wu consider general Hilbert schemes of ideal sheaves with arbitrary Hilbert polynomial, and also Quot schemes. Both approaches apply Li's method of expanded degenerations $X[n]\rightarrow C[n]$ which in the case of constant Hilbert polynomial means the construction of a family whose special fibre over $0$ parametrises length $n$ subschemes of the degenerate fibre $X_0.$ The main problem in this setting is to describe the subschemes whose support meets the singular locus of $X_0,$ and the idea of the construction is that whenever a subscheme approaches a singularity in $X_0,$ a new ruled component is inserted into $X_0$ and it will be sufficient to work with subschemes supported on the smooth loci of the fibres of $X[n]\rightarrow C[n].$ The dimension of the base $C[n]$ is increased at each step of increasing $n,$ and finally one has to take equivalence classes of subschemes supported on the fibres of $X[n]\rightarrow C[n].$ Additionally, the construction of expanded degenerations also includes an action of an $n$-dimensional torus $G[n]\subset\text{SL}(n)$ acting on $X[n]\rightarrow C[n]$ such that $C[n]/\!\!/ G[n]=C.$ \par Li and Wu proceed by constructing the stack $\mathfrak{X}/\mathfrak{C}$ of expanded degenerations associated to $X\rightarrow C,$ giving a notion of equivalence. For fixed Hilbert polynomial $P,$ a definition of stable ideal sheaf is given, and this is used to define a stack $I^P_{\mathfrak{X/C}}$ over $C$ parametrizing those (the Li-Wu stack). In the case where the constant Hilbert polynomial $P=n,$ this gives subschemes of length $n$ supported on the smooth locus of a fibre of an expanded degeneration, having finite automorphism group. In the present article, the method is not to use the Li-Wu stack, but to use GIT with respect to the action of $G.$ \par The body of the article is the construction of a setup allowing to apply GIT-methods. One has to assume that the dual graph $\Gamma(X_0)$ associated to the singular fibre $X_0$ is bipartite, i.e. that it has no cycles of odd length. One can always perform a quadratic base change so that this holds, so the assumption is rather mild. At first, the authors construct a relatively ample line bundle $\mathcal L$ on $X[n]\rightarrow C[n].$ Then the bipartite assumption allows to construct a $G[n]$-linearisation on $\mathcal L$ which is proven to be applicable to Hilbert schemes. The choice of the correct $G[n]$- linearisation is the most important technical tool of the present article. Using $\mathcal L$, the authors construct an ample line bundle $\mathcal M_l$ on the relative Hilbert scheme defined by $\mathbf{H}^n:=\text{Hilb}^n(X[n]/C[n]),$ with a natural $G[n]$-linearisation. In this explicit situation, GIT stability can be analysed using a relative version of the Hilbert-Mumford numerical criterion: It is proved that (semi-)stability of a point $[Z]\subset\mathbf{H}^n$ only depends on the degree $n$-cycle associated to $Z.$ \par Fixing the $G[n]$-linearised sheaf $\mathcal L,$ the construction depends on several choices: The orientation of the dual graph $\Gamma(X_0)$ has two possible choices as it is bipartite, and both give isomorphic GIT quotients. A suitable $l$ is chosen in the construction of $\mathcal M_l,$ but the characterisation of stable $n$-cycles shows that the final result is independent of this. \par Let $Z\subset X[n]_q$ for some point $q\in C[n].$ Using a local étale coordinate $t$ it is obtained coordinates $t_1,\dots,t_{n+1}$ on $C[n]$ and then $\{a_1,\dots,a_r\}$ is defined to be the subset indexing coordinates with $t_i(q)=0.$ Put $a_0=1,\;a_{r+1}=n+1,$ then $\mathbf{a}=(a_0,\dots,a_{r+1})\in\mathbb Z^{r+2}$ determines a vector $\mathbf{v_a}\in\mathbb Z^{r+1}$ with $i$-th component $a_i-a_{i-1}.$ Now, by definition, $Z$ has smooth support if it is supported in the smooth part of the fibre $X[n]_q.$ Then each point $P_i$ in the support of $Z$ is contained in a unique component of $X[n]_q$ with multiplicity say $n_i.$ This leads to the definition of the numerical support $\mathbf{v}(Z)\in\mathbb Z^{r+1}.$ \par The first main result then describes the stable locus in $\mathbf{H}^n$ with respect to $\mathcal M_l$ when $l\gg 2n^2:$ If $[Z]\in\mathbf{H}^n$ has smooth support, then $[Z]\in\mathbf{H}^n(\mathcal M_l)^{\text{ss}}$ if and only if $\mathbf{v}(Z)=\mathbf{v_a}$ (then also $[Z]\in\mathbf{H}^n_{\text{GIT}}:=\mathbf{H}^n(\mathcal M_l)^s$), if $[Z]\in\mathbf{H}^n$ does not have smooth support, then $[Z]\notin\mathbf{H}^n(\mathcal M_l)^{\text{ss}}.$ \par Given this result, the authors define the main object of study in the paper, the GIT quotient $I_{X/C}=\mathbf{H}_{\text{GIT}}^n/G[n].$ The advantage is that the GIT stable points can be controlled very explicitly so that the geometry of the fibres of the degenerate Hilbert-schemes can be controlled in detail. On the other side, the authors define the stack quotient $\mathcal I_{X/C}=[\mathbf{H}^n_{\text{GIT}}/G[n]].$ The next main result gives the relation between the two concepts, saying that the GIT quotient $I^n_{C/X}$ is projective over $C,$ and that the stack $\mathcal I^n_{X/C}$ is a Deligne-Mumford stack, proper and of finite type over $C$ with $I_{X/C}$ as coarse moduli space. In addition, the morphism $f:\mathcal I^n_{X/C}\rightarrow T^n_{\mathfrak{X/C}}$ is an isomorphism of Deligne-Mumford stacks. \par The above results prove that the GIT approach and the Li-Wu construction of degenerations of Hilbert schemes of points are equivalent. One advantage are the tools to explicitly describe the degenerate Hilbert schemes, and this is thoroughly illustrated with an example of degree $n$ Hilbert schemes on two components. \par As mentioned, a main objective of the article is to construct good degenerations of Hilbert schemes of $K3$ surfaces. It turns out that the technique with relative Hilbert schemes cannot be applied directly, but the situation is analysed based on the results in the paper, and a sketch of further work is given. \par This is a very deep and good article, showing the force of GIT theory compared to stack-theory. This proves that techniques of classical algebraic geometry are appropriate for solving algebraic geometric problems, and gives the theory for further studies (in particular of degenerations of Hilbert schemes of $K3$ surfaces). In addition, the article is very well written, and can be used as a guide for authors in the field. GIT; geometric invariant theory; degeneration; Hilbert scheme; expanded degenerations; Donaldson-Thomas invariants; Pandharipande-Thomas stable pairs; $K3$ surfaces; Hilbert scheme of points; Li-Wu stack; bipartite graph; smooth support; numerical support; GIT quotient; stack quotient; Deligne-Mumford stack Fibrations, degenerations in algebraic geometry, Parametrization (Chow and Hilbert schemes), Geometric invariant theory, Stacks and moduli problems A GIT construction of degenerations 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 The author classifies simple hypersurface K3 singularities (X,x) defined by a non-degenerated polynomial \(f(z_ 0,z_ 1,z_ 2,z_ 3)\) and he studies the minimal resolution \(\pi: (\tilde X,E)\to (X,x)\) and the singularities on the exceptional divisor E.
A three-dimensional singularity (X,x) is a simple K3 singularity if (X,x) is quasi-Gorenstein and if the exceptional divisor E of any minimal resolution is a normal K3 surface, where a minimal resolution \(\pi: (\tilde X,E)\to (X,x)\) is a proper morphism with only terminal singularities on \(\tilde X,\) with \(\tilde X\simeq X\setminus \{x\}\) and with \(K_{\tilde X}\) nef with respect to \(\pi\).
- The simple K3 singularities could be regarded as three-dimensional generalizations of simple elliptic singularities.
If the simple K3 singularity (X,x) is defined by a non-degenerated polynomial f(z), then \((1,1,1,1)\in \Gamma(f)\). The weight \(\alpha =\alpha(f)=(\alpha_ 1,\alpha_ 2,\alpha_ 3,\alpha_ 4)\) of the quasi-homogeneous polynomial \(f_{\Delta_ 0}\) associated to the face \(\Delta_ 0\) containing (1,1,1,1) verifies \(\sum^{4}_{i=1}\alpha_ i =1\). - Then to classify the simple K3 singularities we need to study the set \(W_ 4\) of weights: \(W_ 4=\{\alpha \in {\mathbb{Q}}^ 4_+| \quad \sum^{4}_{i=1}\alpha_ i =1,\alpha_ 1\geq...\geq \alpha_ 4\) and \((1,1,1,1)\in Int(C(\alpha))\},\) where \(C(\alpha\)) is the closed cone in \({\mathbb{R}}^ 4\) generated by the set \(T(\alpha)=\{\nu \in {\mathbb{Z}}^ 4_ 0| \alpha.\nu =1\}.\)
The author shows that the cardinality of \(W_ 4\) is 95, and for each weight \(\alpha\) he gives a quasi-homogeneous f of weight \(\alpha\) which defines a simple K3 singularity and such that \(\Delta_ 0=\Gamma (f)\) is the convex hull of \(T(\alpha\)). Then he constructs a minimal resolution \(\pi: \tilde X\to X\) using torus embedding: if the weight \(\alpha(f)=(p_ 1/p,...,p_ 4/p)\), where \(p_ 1,...,p_ 4\) are relatively prime integers, the filtered blow-up with weight \((p_ 1,...,p_ 4)\), \(\Pi: (V,F)\to ({\mathbb{C}}^ 4,0)\), induces a minimal resolution of (X,x). In the last part the author shows that under some conditions on the nondegenerate polynomial f defining the simple K3 singularity (X,x), the type and the number of the singularities on E are determined by the weight \(\alpha(f)\), independently of f. type of singularities; simple hypersurface K3 singularities; minimal resolution; exceptional divisor; number of the singularities; weight Yonemura, T., Hypersurface simple \textit{K}3 singularities, Tohoku Math. J. (2), 42, 3, 351-380, (1990), MR 1066667 Singularities of surfaces or higher-dimensional varieties, \(K3\) surfaces and Enriques surfaces, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Hypersurface simple K3 singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C\) be a smooth projective curve of genus \(g \geq 3\) with Jacobian \(J = \text{Pic}^0 C\) and \(\alpha: C \hookrightarrow J\) an Abel-Jacobi map. The image of \(\alpha\) gives a corresponding point in \(\text{Hilb}_J\) which lies on a unique irreducible component \(\text{Hilb}_{C,J} \subset \text{Hilb}_J\). [\textit{H. Lange} and \textit{E. Sernesi}, Ann. Mat. Pura Appl. (4) 183, No. 3, 375--386 (2004; Zbl 1204.14012)]; see also [\textit{P. A. Griffiths}, J. Math. Mech. 16, 789--802 (1967; Zbl 0188.39302)] proved that if \(C\) is nonhyperelliptic, then \(\text{Hilb}_{C/J}\) is smooth of dimension \(g\),
while if \(C\) is hyperelliptic, then \(\text{Hilb}_{C/J}\) is irreducible of dimension \(g\) and everywhere nonreduced with Zariski tangent space of dimension \(2g-2\): in both cases, the only deformations of \(C\) in \(J\) are by translation. In the hyperelliptic case, the author clarifies the non-reduced scheme structure by proving that \(\text{Hilb}_{C/J} \cong J \times R_g\) where \(R_g = \text{Spec}[s_1,\dots,s_{g-2}]/\mathfrak m^2\) and \(\mathfrak m = (s_1, \dots, s_{g-2})\) is the maximal ideal. He uses this result to describe the scheme theoretic fibers of the Torelli morphism \(\tau_g: \mathcal M_g \to \mathcal A_g\) along the non-hyperelliptic locus, where \(\mathcal M_g\) is the moduli stack of genus \(g\) curves and \(\mathcal A_g\) is the moduli stack of principally polarized abelian varieties of dimension \(g\).
There is an application to the moduli space of Picard sheaves on the Jacobian. Fix a smooth curve \(C\) of genus \(g \geq 2\) with Jacobian \(J\) and dual \(\hat J\). For \(p \in C\) and \(1 \leq d \leq g-1\), view the line bundle \(\xi = {\mathcal O}_C (dp)\) as a sheaf on \(\hat J\) by first pushing it forward along an Abel-Jacobi map \(\alpha: C \hookrightarrow J\) and then using the identification of \(J\) with \(\hat J\): applying his Fourier transform to this sheaf on \(\hat J\), \textit{Mukai} constructs an associated \textit{Picard sheaf} \(F\) on \(J\). Letting \(M(F)\) be the connected component containing \(F\) in the moduli space \(\text{Spl}_J\) of simple coherent sheaves on \(J\), \textit{Mukai} proves that if \(g=2\) or \(C\) is non-hyperelliptic, then the natural morphism \(\hat J \times J \to M(F)\) is an isomorphism [\textit{S. Mukai}, Nagoya Math. J. 81, 153--175 (1981; Zbl 0417.14036)] and when \(C\) is hyperelliptic it is an isomorphism onto \(M(F)_{\text{red}}\) [\textit{S. Mukai}, Adv. Stud. Pure Math. 10, 515--550 (1987; Zbl 0672.14025)]. Analogous to the main theorem, the author describes the non-reduced scheme structure by proving that \(M(F) \cong \hat J \times J \times R_g\). Jacobian; Torelli morphism; Hilbert schemes; Picard sheaves; Fourier-Mukai transform Parametrization (Chow and Hilbert schemes), Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard 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 paper, the authors compute the sum of Betti numbers of smooth Hilbert schemes on the complex projective spaces. For fixed positive integer \(n\) and polynomial \(p(t)\), let \(\mathbb P^{n[p]}\) denote the Hilbert scheme of closed subschemes of the complex projective space \(\mathbb P^{n}\) with Hilbert polynomial \(p(t)\). It is previously known that the Hilbert scheme \(\mathbb P^{n[p]}\) is nonempty if and only if \(p(t)= \sum_{i=1}^r \binom{t+\lambda_i - i}{\lambda_i - 1}\) for some integer partition \(\lambda = (\lambda_1, \ldots, \lambda_r)\) satisfying \(\lambda = (n+1)\), \(r = 0\), or \(n \ge \lambda_1 \ge \ldots \ge \lambda_r \ge 1\). Moreover, \(\mathbb P^{n[p]}\) is smooth if and only if one of the following seven cases is true:
\begin{itemize}
\item[1.] \(n \le 2\);
\item[2.] \(\lambda_r \ge 2\);
\item[3.] \(\lambda = (1)\) or \(\lambda = (n^{r-2}, \lambda_{r-1}, 1)\) where \(r \ge 2\);
\item[4.] \(\lambda = (n^{r-s-3}, \lambda_{r-s-2}^{s+2}, 1)\) where \(r \ge s+3\);
\item[5.] \(\lambda = (n^{r-s-5}, 2^{s+4}, 1)\) where \(r \ge s+5\);
\item[6.] \(\lambda = (n^{r-3}, 1^3)\) where \(r \ge 3\);
\item[7.] \(\lambda = (n+1)\) or \(r = 0\).
\end{itemize}
The main theorem of the paper presents explicit formulas for the sum \(H_{n, \lambda}\) of the Betti numbers of \(\mathbb P^{n[p]}\) in the above seven cases except Case~2. For instance,
\[
H_{n, \lambda} = \binom{n+r-2}{r-2} \binom{n+1}{\lambda_{r-1}} (n + 1 - \lambda_{r-1})(\lambda_{r-1} + 1)
\]
when \(\lambda = (n^{r-2}, \lambda_{r-1}, 1)\) is in Case 3 with \(n > \lambda_{r-1} > 1\).
The main ideas in the proofs are to use the \(\mathrm{PGL}(n + 1)\)-action on \(\mathbb P^{n[p]}\) induced from the \(\mathrm{PGL}(n + 1)\)-action on \(\mathbb P^{n}\) and to apply the classical theorem of A.~Bialynicki-Birula. These ideas enable the authors to translate the computation of the ranks of the homology groups into counting saturated monomial ideals and then to translate that into counting choices of orthants in an \((n + 1)\)-dimensional lattice. Hilbert scheme; Betti number; cohomology; homology; saturated monomial ideal Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings The sum of the Betti numbers 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 We propose some conjectures on the generating series of (equivariant) Euler characteristics of some vector bundles constructed from the tautological bundles on Hilbert schemes of points on affine \(k\)-spaces. We establish the surface case of these conjectures and present some verifications of the higher dimensional cases. Elagin, A.D.: On equivariant triangulated categories. arXiv:1403.7027, (2014) Parametrization (Chow and Hilbert schemes), Vector bundles on surfaces and higher-dimensional varieties, and their moduli Tautological sheaves on Hilbert schemes of points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let $C$ be an integral proper complex curve with compactified Jacobian $J$. Letting $C^{[n]}$ denote the Hilbert scheme of length $n$ subschemes of $C$, the Abel-Jacobi morphism $\varphi: C^{[n]} \to J$ sends a closed subscheme $Z$ to ${\mathcal I}_Z \otimes {\mathcal O}(x)^{\otimes n}$, where $x \in C$ is a nonsingular point. When $C$ has at worst planar singularities, both $C^{[n]}$ and $J$ are integral schemes with local complete intersection singularities according to \textit{A. B. Altman} et al. [in: Real and compl. Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 1--12 (1977; Zbl 0415.14014)] and \textit{J. Briancon} et al. [Ann. Sci. Éc. Norm. Supér. (4) 14, 1--25 (1981; Zbl 0463.14001)]. Furthermore $\varphi$ has the structure of a $\mathbb P^{n-g}$-bundle for $n \geq 2g-1$ by work of \textit{A. B. Altman} and \textit{S. L. Kleiman} [Adv. Math. 35, 50--112 (1980; Zbl 0427.14015)] so that the rational homology group $H_* (C^{[n]})$ is determined by $H_* (J)$. Recent work of \textit{D. Maulik} and \textit{Z. Yun} [J. Reine Angew. Math. 694, 27--48 (2014; Zbl 1304.14036)] and \textit{L. Migliorini} and \textit{V. Shende} [J. Eur. Math. Soc. (JEMS) 15, No. 6, 2353--2367 (2013; Zbl 1303.14019)] endows $H^* (J)$ with a certain perverse filtration $P$ for which $H^* (C^{[n]})$ can be recovered from the $P$-graded space $\text{gr}_*^P H^* (J)$. \par Motivated by these results and a suggestion of Richard Thomas, the author shows how $H_* (C^{[n]})$ can be recovered from a filtration on $H_* (J)$ using a method not reliant on perverse sheaves. Taking an approach inspired by work of \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)], he defines two pairs of creation and annihilation operators acting on $V(C)=\bigoplus_{n \geq 0} H_* (C^{[n]})$. The first pair $\mu_{\pm} [\text{pt}]$ corresponds to adding or removing a nonsingular point in $C$. The second pair $\mu_{\pm}[C]$ come from the respective projections $p,q$ from the flag Hilbert scheme $C^{[n,n+1]}$ to $C^{[n]}$ and $C^{[n+1]}$, namely $q_* p^{!}$ and $p_* q^{!}$ for appropriate Gysin maps $p^!$ and $q^!$. The main theorem states that the subalgebra of $\text{End} (V(C))$ generated by $\mu_{\pm} [\text{pt}], \mu_{pm}[C]$ is isomorphic to the Weyl algebra $\mathbb Q [x_1, x_2, \partial_1, \partial_2]$ and that the natural map $W \otimes \mathbb Q [\mu_+ [\text{pt}], \mu_+ [C]] \to V(C)$ is an isomorphism, where $W$ is the intersection of the kernels of $\mu_- [\text{pt}]$ and $\mu_- [C]$; moreover the Abel-Jacobi pushforward map $\varphi_*: V(C) \to H_* (J)$ induces an isomorphism $W \cong H_* (J)$. Dual variations for cohomology groups recover and strengthen the results of Maulik-Yun [Zbl 1304.14036] and Migliorini-Shende [Zbl 1303.14019]. locally planar curves; Hilbert scheme; compactified Jacobian; Weyl algebra Parametrization (Chow and Hilbert schemes), Jacobians, Prym varieties, Singularities of curves, local rings Homology of Hilbert schemes of points on a locally planar curve | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 theorem of \textit{L. Göttsche} [Hilbert schemes of zero-dimensional subschemes of smooth varieties. Berlin: Springer-Verlag (1994; Zbl 0814.14004), p. 37] establishes a connection between cohomological invariants of a complex projective surface \(S\) and corresponding invariants of the Hilbert scheme of \(n\) points on \(S\). This relationship is encoded in certain infinite product \(q\)-series which are essentially modular forms. Here we make use of the circle method to arrive at exact formulas for certain specializations of these \(q\)-series, yielding convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the \(q\)-series' coefficients. Hilbert schemes; circle method; modular forms; topological invariants Parametrization (Chow and Hilbert schemes), Modular and automorphic functions, \(K3\) surfaces and Enriques surfaces, Surfaces of general type Exact formulas for invariants 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 Let \(\text{Hilb}^{p(t)} (\mathbb P^n)\) be the Grothendieck Hilbert scheme that parametrizes closed subschemes \(X \subset \mathbb P^n\) with Hilbert polynomial \(p(t)\). \textit{A. Reeves} and \textit{M. Stillman} showed that the point in \(\text{Hilb}^{p(t)} (\mathbb P^n)\) corresponding to the lexicographic ideal is smooth [J. Algebr. Geom. 6, 235--246 (1997; Zbl 0924.14004)] and determines a canonical \textit{lexicographic component} of \(\text{Hilb}^{p(t)} (\mathbb P^n)\). \textit{I. Peeva} and \textit{M. Stillman} gave a version of this theorem for toric Hilbert schemes [Duke Math. J. 111, 419--449 (2002; Zbl 1067.14005)]. With \textit{M. Haiman} and \textit{B. Sturmfels} extending the Grothendieck Hilbert schemes to standard graded Hilbert schemes \(\mathcal H^{\mathfrak h} (R)\) parametrizing homogeneous ideals \(I\) with fixed Hilbert function \(\mathfrak h\) in a graded ring \(R\) [J. Algebr. Geom. 13, 725--769 (2004; Zbl 1072.14007)] it becomes natural to ask what the lexicographic points look like in that setting. \textit{D. Maclagan} and \textit{G. G. Smith} gave an analog to the Reeves-Stillman for standard multigraded Hilbert schemes in two variables [Adv. Math. 223, 1608--1631 (2010; Zbl 1191.14007)].
The authors give examples showing that the lexicographic point does not behave so nicely in \(\mathcal H^{\mathfrak h} (R)\) for more variables. They show that for \(S = k[x,y,z]\) and \(\mathfrak h = (1,3,4,4,3,3,3, \dots)\), the Hilbert scheme \(\mathcal H^{\mathfrak h} (S)\) is the union of two irreducible components of dimension \(8\) containing the lexicographic point in their intersection, so the lexicographic point is singular and does not correspond to a canonical component. They also give an example where the lexicographic point is not even Cohen-Macaulay. They also show for the exterior algebra \(E = \bigwedge k^5\) and \(\mathfrak h = (1,5,7,2)\) that the standard graded Hilbert scheme \(\mathcal H^{\mathfrak h} (E)\) is the union of two irreducible components of dimensions \(14\) and \(15\) which contain the lexicographic point in their intersection. standard graded Hilbert scheme; lexicographic component; lexicographic ideal; reducible scheme; exterior algebra Parametrization (Chow and Hilbert schemes), Polynomial rings and ideals; rings of integer-valued polynomials, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Exterior algebra, Grassmann algebras On the smoothness of lexicographic points on Hilbert schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite group \(G \subset \mathrm{GL}(n, \mathbb{C})\), the \(G\)-Hilbert scheme is a fine moduli space of \(G\)-clusters, which are 0-dimensional \(G\)-invariant subschemes \(Z\) with \(H^0(\mathcal{O}_Z)\) isomorphic to \(\mathbb{C}[G]\). In many cases, the \(G\)-Hilbert scheme provides a good resolution of the quotient singularity \(\mathbb{C}^n/G\), but in general it can be very singular. In this note, we prove that for a cyclic group \(A \subset \mathrm{GL}(n, \mathbb{C})\) of type \(\frac{1}{r}(1, \dots, 1, a)\) with \(r\) coprime to \(a\), \(A\)-Hilbert Scheme is smooth and irreducible. \(A\)-Hilbert schemes; cyclic quotient singularities Parametrization (Chow and Hilbert schemes), Vanishing theorems in algebraic geometry, Arcs and motivic integration, Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Variation of Hodge structures (algebro-geometric aspects) A-Hilbert schemes for \(\displaystyle\frac{1}{r}(1^{n-1},a)\) | 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 The present article continues work of the same authors on the cohomology ring of the Hilbert scheme \(X^{[n]}\) of points (i.e. of length \(n\) closed subschemes of \(X\) of dimension \(0\)) on a smooth projective surface \(X\), linking it to Heisenberg - and vertex algebras following Vafa-Witten, Göttsche, Grojnowski, Nakajima and Lehn.
Denote by \(K\) the canonical class of \(X\). For an element \(\xi\in X^{[n]}\), let \(I_{\xi}\) be the corresponding sheaf of ideals. Define the universal subscheme
\[
{\mathcal Z}_n\,=\,\{(\xi,x)\in X^{[n]}\times X\,| \,x\in\text{ supp}(\xi)\}\subset X^{[n]}\times X.
\]
Let \(p_1\) and \(p_2\) be the projections of \(X^{[n]}\times X\) to \(X^{[n]}\) and \(X\), respectively. Let
\[
{\mathbb H}_X\,=\,\bigoplus_{n=0}^{\infty}H^*\left( X^{[n]}\right)
\]
be the direct sum of the rational cohomology spaces of the Hilbert schemes \(X^{[n]}\). For \(m\geq 0\) and \(n>0\), let \(Q^{[m,m]}=\emptyset\), and define \(Q^{[m+n,m]}\) to be the closed subset
\[
Q^{[m+n,m]}\,=\,\left\{(\xi,x,\eta)\in X^{[n+m]}\times X\times X^{[m]}\,| \,\xi\subset\eta\,\,\,\,\text{ and}\,\,\,\,\,\text{ supp}(I_{\eta}/I_{\xi})=\{x\}\right\}.
\]
The main point of this approach to the cohomology of Hilbert schemes is that the \textit{Nakajima operators} \({\mathfrak a}_{i}\in \text{ End}({\mathbb H}_X)\) for \(i\in{\mathbb Z}\) defined by
\[
{\mathfrak a}_{-n}(\alpha)(a)\,=\,\tilde{p}_{1*}\left(\left[Q^{[m+n,m]}\right]\cdot\tilde{\rho}^*\alpha\cdot\tilde{p}^*_2a\right),
\]
where \(a\in H^*(X^{[m]})\) and \(\tilde{p}_{1}\), \(\tilde{\rho}\), and \(\tilde{p}_2\) are the projections from \(X^{[n+m]}\times X\times X^{[m]}\) to \(X^{[n+m]}\), \(X\), and \(X^{[m]}\), respectively, and \({\mathfrak a}_{n}\) defined by \((-1)^n\) times the operator obtained from switching the roles of \(\tilde{p}_{1}\) and \(\tilde{p}_2\) in the definition of \({\mathfrak a}_{-n}\), render \({\mathbb H}_X\) an irreducible module over the Heisenberg algebra. \textit{M. Lehn} [Invent. Math. 136, 157--207 (1999; Zbl 0919.14001)] then identified a Virasoro algebra generated by operators \({\mathfrak L}_m(\alpha)\) by bosonization, the central term involving the Euler class \(e\) of \(X\).
The main emphasis here is on the operators \({\mathfrak G}_i(\gamma)\): let \(G(\gamma,n)\) be defined by
\[
G(\gamma,n)\,=\,p_{1*}\left(\text{ ch}({\mathcal O}_{{\mathcal Z}_n})\cdot p_2^*\,\,\text{ td}(X)\cdot p_2^*\gamma\right)\,\,\in\,\,H^*\left( X^{[n]}\right),
\]
where \(\gamma\in H^*(X)\), and whose homogeneous component in \(H^{| \gamma| +2i}\) for homogeneous \(\gamma\) is denoted by \(G_i(\gamma,n)\). The \textit{Chern character operator} \({\mathfrak G}_i(\gamma)\in\text{ End}({\mathbb H}_X)\) is defined to be the operator on \(H^*(X^{[n]})\) arising from the cup product with \(G_i(\gamma,n)\). The interest in these operators stems from the fact that the cohomology ring of \(X^{[n]}\) is generated by the classes \(G_i(\gamma,n)\) for \(0\leq i<n\) and \(\gamma\) running over a linear basis of \(H^*(X)\).
The first part of the article under review computes an explicit formula (theorem 4.7) for \({\mathfrak G}_i(\alpha)\) with three terms in which Heisenberg operators occur which are indexed by generalized partitions (i.e. partitions \(\lambda\) into positive and negative integers) and which are applied to \(\alpha\), to \(e\alpha\), and to \(\epsilon\alpha\) for \(\epsilon=K,K^2\), respectively. The last term contains numbers \(g_{\epsilon}(\lambda)\) of which one only knows that they do not depend on \(X\) or \(\alpha\); it disappears for \(K=0\). Therefore one arrives at a description of the ring structure of \(H^*(X^{[n]})\) for numerically trivial canonical bundle, the description remaining only partial in the general case.
In the second part, the authors investigate a \({\mathcal W}\)-algebra of operators on \({\mathbb H}_X\): the tensor product of graded commutative algebra like a subalgebra \({\mathcal A}\subset H^*(X)\) and a Lie algebra like the Lie algebra of differential operators on the circle is canonically a (current super-) Lie algebra. It admits a central extension denoted \(\widehat{\mathcal W}({\mathcal A})\). On the other hand, define \({\mathcal W}_X\) to be the vector spaces generated for \(p\geq 0\), \(n\in{\mathbb Z}\), and \(\alpha\in H^*(X)\) by \(\text{ id}_{{\mathbb H}_X}\) and the operators \({\mathfrak J}_n^p(\alpha)\in\text{ End}({\mathbb H}_X)\). \({\mathcal W}_X\) contains as special cases the operators \({\mathfrak a}_{n}(\alpha)\), \({\mathfrak L}_m(\alpha)\), and \({\mathfrak G}_i(\alpha)\). The main theorem (theorem 5.5) computes the commutation relations of the \({\mathfrak J}_n^p(\alpha)\) via operator product expansions. It turns out that the leading term in the commutation relations gives precisely the \({\mathcal W}\)-algebra \(\widehat{\mathcal W}(H^*(X))\). In this sense, \({\mathcal W}_X\) is a (global) deformation of \(\widehat{\mathcal W}(H^*(X))\) in \(K\) and \(e\), because \(\widehat{\mathcal W}({\mathcal B}_X)\) is isomorphic to \({\mathcal W}_X^{\mathcal B}\), where \({\mathcal B}_X\subset H^*(X)\) is the ideal consisting of classes \(\alpha\) with \(e\alpha=K\alpha=0\), and where \({\mathcal W}_X^{\mathcal B}\) is the linear span of the \({\mathfrak J}_n^p(\alpha)\) with \(\alpha\in{\mathcal B}_X\). Hilbert scheme of points on a surface; Nakajima operators; Chern character operator; cohomology ring Wei-Ping Li, Zhenbo Qin, and Weiqiang Wang, Hilbert schemes and \? algebras, Int. Math. Res. Not. 27 (2002), 1427 -- 1456. Parametrization (Chow and Hilbert schemes), Relationships between surfaces, higher-dimensional varieties, and physics, Classical real and complex (co)homology in algebraic geometry, Vertex operators; vertex operator algebras and related structures Hilbert schemes and \(\mathcal W\) 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 Let \( X \) be the Segre-Veronese embedding of \(\mathbb{P}^a\times\mathbb{P}^b\times\mathbb{P}^c\times \cdots\) with degree \( L=(l_1,l_2,l_3,\dots).\) Corresponding to it we have the multigraded ring \[ S[X]=\mathbb{C}[\alpha_0,\dots,\alpha_a,\beta_0,\dots,\beta_b,\gamma_0,\dots,\gamma_c,\dots] .\]
The ring \( S[X] \) has two dual interpretations. The first, more geometric, as ``functions'' on \(X\) and the second, more algebraic, in terms of derivations. We have then the well known apolarity lemma: \[F\in \langle \{ p_1,\dots,p_r \} \rangle \Leftrightarrow I(\{ p_1,\dots,p_r \}) \subset \mbox{Ann}(F)\] where \(F\in S[X], \ p_1,\dots,p_1\in X\) and \(\mbox{Ann}(F)\subset S[X]\) is the annihilator of \(F\) in the space of derivations.
This lemma gives a characterization of rank \(r\) tensors. In the present paper the authors introduce a result for border rank similar to the apolarity lemma. The main result goes as follows: Suppose a tensor or polynomial \(F\) has border rank at most \(r.\) Then there exists a multihomogeneous ideal \(I\subset S[X]\) such that \(I\subset \mbox{Ann}(F)\) and for each multidegree \(D\) the \(D\)th graded piece \(I_D\) of \(I\) has codimension equal to \(\min(r,\dim S[X]_D).\)
In fact, the result stated above is a consequence of more general results that they show for a smooth toric variety. They also have if and only if kind of results using more technical hypotheses.
Moreover, the results are applied to the cases \(\mathbb{P}^2\times \mathbb{P}^1 \times \cdots \times \mathbb{P}^1\) with arbitrary degree and \(\mathbb{P}^a\times \mathbb{P}^b\times \mathbb{P}^c\) with degree \((1,1,1)\) obtaining new bounds for the border rank of such Segre-Veronese varieties. apolarity; border rank; Cox ring; invariant ideals; multigraded Hilbert scheme; variety of sums of powers Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies, Multilinear algebra, tensor calculus, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) Apolarity, border rank, and multigraded Hilbert scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper the author studies the configuration of simple singularities in a reduced sixtic curve in \(\mathbb{P}^2\). Recall that singularities are described by their dual graph in an embedded resolution of singularities and simple singularities are in correspondence with simple Dynkin diagrams, of type \(A_i\) \((i\geq 1)\), \(D_j\) \((j\geq 4)\) and \(E_k\) \((k=6,7,8)\). The main theorem in this article is:
There exists a reduced sixtic curve in \(\mathbb{P}^2\) with only simple singularities given by a finite Dynkin graph \(G\) if and only if \(G\) is a subgraph of a graph listed by the author.
For the proof the author uses full version of Nikulin's embedding theorem of even lattices. resolution of singularities; Dynkin diagrams; sixtic curve Jin-Gen Yang, Sextic curves with simple singularities, Tôhoku Math. J. 48 (1996), 203--227. Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Computational aspects of algebraic curves Sextic curves 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 We observe that there exists a Białynicki-Birula decomposition of the Hilbert scheme \(\text{Hilb}^P_n\) such that the cells are homeomorphic to Gröbner strata of homogeneous ideals with fixed initial ideal. Using such a decomposition, we show that \(\text{Hilb}^P_n\) is singular at a monomial scheme if the corresponding Gröbner stratum is singular at \(J\). Hilbert scheme; Białynicki-Birula decomposition; Gröbner bases Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials Computable Białynicki-Birula decomposition 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 Authors' abstract: ``Let \(k\) be an algebraically closed field of characteristic \(p>3\). Let \(X\) be an irreducible smooth projective surface over \(k\). Fix an integer \(n\ge 2\) and let \(\mathcal{Hilb}^{n}_X\) be the Hilbert scheme parameterizing effective \(0\)-cycles of length \(n\) on \(X\). The aim of the present article is to find the \(S\)S-fundamental group scheme and Nori's fundamental group scheme of the Hilbert scheme \(\mathcal{Hilb}^{n}_X\).''
The aim is achieved and the reader may get from the nice first part a full description of the main definitions of the different fundamental groups and related papers. it is also pointed out where the assumption \(p>3\) is needed. finite vector bundle; \(S\)-fundamental group-scheme; Hilbert scheme; semistable bundle; Tannakian category; Hilbert-Chow morphism Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Homotopy theory and fundamental groups in algebraic geometry, Group schemes, Parametrization (Chow and Hilbert schemes) Fundamental group schemes of Hilbert scheme of \(n\) points on a smooth projective surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(S\) be a smooth projective surface over the complex numbers; let \(S^{(r)}\) be its \(r\)-fold symmetric product and \(S^{[r]}\) the Hilbert scheme of 0-dimensional subschemes of length \(r\).
In case \(K_S\) is trivial, the deformation theory of \(S^{[r]}\) has been studied by Beauville and Fujiki in order to construct examples of higher-dimensional symplectic manifolds. In that case \(S^{[r]}\) has deformations which are not Hilbert schemes of points on a surface. -- We prove that under suitable hypotheses (e.g. if \(S\) is of general type) this cannot happen; every (small) deformation of \(S^{(r)}\) and \(S^{[r]}\) is induced naturally by a deformation of \(S\) (in particular, all deformations of \(S^{(r)}\) are locally trivial). Hilbert schemes of points; \(r\)-fold symmetric product of projective surface; deformation theory Fantechi, B., Deformation of Hilbert schemes of points on a surface, Compos. Math., 98, 2, 205-217, (1995), MR 1354269 Formal methods and deformations in algebraic geometry, Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Deformation of Hilbert schemes of points on a surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a nontrivial finite subgroup of \(SL_n (\mathbb{C})\). Suppose that the quotient singularity \(\mathbb{C}^n/G\) has a crepant resolution \(\pi:X\to \mathbb{C}^n/G\) (i.e. \(K_X= {\mathcal C}_X)\). There is a slightly imprecise conjecture, called the McKay correspondence, stating that there is a relation between the Grothendieck group (or (co)homology group) of \(X\) and the representations (or conjugacy classes) of \(G\) with a ``certain compatibility'' between the intersection product and the tensor product. The purpose of this paper is to give more precise formulation of the conjecture when \(X\) can be given as a certain variety associated with the Hilbert scheme of points in \(\mathbb{C}^n\). We give the proof of this new conjecture for an abelian subgroup \(G\) of \(SL_3(\mathbb{C})\). group action; homology group; quotient singularity; crepant resolution; McKay correspondence; Grothendieck group; intersection product; Hilbert scheme of points Ito, Y., Nakajima, H.: McKay correspondence and Hilbert schemes in dimension three. Topology, 39, 1155--1191 (2000) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, \(K\)-theory of schemes McKay correspondence and Hilbert schemes in dimension three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0575.00008.]
The paper reports on the classification of reduced curve singularities which are of finite Cohen-Macaulay representation type, which was obtained by the author and \textit{H. Knörrer} in Math. Ann. 270, 417-425 (1985; Zbl 0553.14011). For new developments see the forthcoming Proceedings of the Symposium on Representation of Algebras, Singularities and Vector Bundles (Lambrecht 1985, to appear in Lect. Notes Ser.). classification of reduced curve singularities; Cohen-Macaulay representation Singularities of curves, local rings, Singularities in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Torsion free modules and simple curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities According to [\textit{P. Dunin-Barkowski} et al., Commun. Math. Phys. 328, No. 2, 669--700 (2014; Zbl 1293.53090)] and [the author, Duke Math. J. 163, No. 9, 1795--1824 (2014; Zbl 1327.14051)], the ancestor correlators of any semisimple cohomological field theory satisfy \textit{local} Eynard-Orantin recursion. In this paper, we prove that for simple singularities, the local recursion can be extended to a global one. The spectral curve of the global recursion is an interesting family of Riemann surfaces defined by the invariant polynomials of the corresponding Weyl group. We also prove that for genus \(0\) and \(1\), the free energies introduced in [\textit{B. Eynard} and \textit{N. Orantin}, Commun. Number Theory Phys. 1, No. 2, 347--452 (2007; Zbl 1161.14026)] coincide up to some constant factors with respectively the genus \(0\) and \(1\) primary potentials of the simple singularity. Milanov, T, The eynard-orantin recursion for simple singularities, Commun. Number Theory Phys., 9, 707-739, (2015) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Relationships between surfaces, higher-dimensional varieties, and physics The Eynard-Orantin recursion for 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 Hausel and Rodriguez-Villegas (2015, \textit{Astérisque} 370, 113-156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes \((\mathbb{C}^2)^{[n]}\) on \(n\) points, as \(n\rightarrow +\infty\), is a \textit{Gumbel distribution}. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes \(((\mathbb{C}^2)^{[n]})^{T_{\alpha ,\beta}}\) that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer \(A\geq 2\). Furthermore, if \(p_k(A;n)\) denotes the number of partitions of \(n\) with exactly \(k\) parts that are multiples of \(A\), then we obtain the asymptotic
\[
p_k(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^k}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}},
\]
a result which is of independent interest. Betti numbers; Hilbert schemes; partitions Parametrization (Chow and Hilbert schemes), (Co)homology theory in algebraic geometry, Analytic theory of partitions Limiting Betti distributions of Hilbert schemes on \(n\) 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 propose a variation of the classical Hilbert scheme of points, the \textit{double nested Hilbert scheme of points}, which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov-Witten invariants of local curves of Bryan-Pandharipande, as predicted by the Maulik-Nekrasov-Okounkov-Pandharipande (MNOP) correspondence. Finally, we discuss \(K\)-theoretic refinements à la Nekrasov-Okounkov. double nested Hilbert schemes; local curves; stable pair invariants Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Double nested Hilbert schemes and the local stable pairs theory 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 Fix integers \(g\geq 4\) and \(r\geq 2\). Let \(H'[d,g,r]\) denote the open subset of \(\text{Hilb} (\mathbb P^{d+r-rg-1})_{\text{red}}\) parametrizing the smooth \(r\)-dimensional and degree \(d\) non-degenerate scrolls over a general smooth genus \(g\) curve. Here we prove that if \(d\geq r^2g+rg\) the algebraic set \(H'[d,g,r]\) has at least \(r\) irreducible components. Parametrization (Chow and Hilbert schemes), Vector bundles on curves and their moduli, Rational and ruled surfaces, Projective techniques in algebraic geometry Dominant irreducible components of the Hilbert schemes of 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 Let \(G\) be a finite group acting on a compact differentiable manifold \(X\). Then an orbifold Euler number of \(X\) is defined by \(e(X,G): ={1\over |G|} \sum_{gh =hg} e(X^g \cap X^h)\), where the sum runs over all commuting pairs in \(G\) and \(X^g\) denotes the set of fixed points of the action of \(g\). Let \(X\) be a compact Kähler manifold or a Moishezon manifold of complex dimension \(d\). The Hodge polynomial \(h(X,x,y)\) is defined by \(h(X,x,y): =\sum_{p,q} h^{p,q} (X)x^py^q\), where the \(h^{p,q} (X)\) are the Hodge numbers of \(X\).
Orbifold Hodge numbers \(h^{p,q} (X,G)\) are as follows: For any point \(x\in X^g\), the eigenvalues of \(g\) on the tangent space \(T_{X,x}\) are roots of unity \(e^{2\pi i \alpha_1}, \dots, e^{2\pi i \alpha_d}\), where \(0\leq \alpha_j <1\) and the \(\alpha_j\) are locally constant functions on \(X^g\). Let \(X_1(g), \dots, X_r(g)\) be the connected components of \(X^g\). For \(i=1, \dots, r\) we put \(F_i(g)\) equal to the value of \(\sum^d_{j=1} \alpha_j\) on \(X_i(g)\) and set
\[
h_g^{p,q} (X,G): =\sum^r_{i=1} \dim\biggl(H^{p-F_i(g), q-F_i(g)} \bigl(X_i(g) \bigr) ^{C(g)} \biggr).
\]
Here \(C(g)\) is the centralizer of \(g\) in \(G\) and \((\cdot)^{C(g)}\) denotes the \(C(g)\)-invariant part. Then the orbifold Hodge numbers are defined by \(h^{p,q} (X,G): =\sum_{[g]} h_g^{p,q} (X,G)\), where \(g\) runs again over a system of representatives for the conjugacy classes of \(G\). The orbifold Hodge polynomial of \(X\) is \(h(X,G,x,y): =\sum_{p,q} h^{p,q} (X,G)x^p y^q\).
Theorem 1. \(h(S^n,G(n),x,y) =h(\text{Hilb}^n (S),x,y)\).
Theorem 2. \(h(A^{n-1}, G(n),x,y) =h(K_{n-1}, x,y)\), where \(K_{n-1}\) denotes a higher order Kummer variety. action of finite group; orbifold Euler number; Kähler manifold; Moishezon manifold; orbifold Hodge polynomial Lothar Göttsche, Orbifold-Hodge numbers of Hilbert schemes, Parameter spaces (Warsaw, 1994) Banach Center Publ., vol. 36, Polish Acad. Sci. Inst. Math., Warsaw, 1996, pp. 83 -- 87. Parametrization (Chow and Hilbert schemes), Transcendental methods, Hodge theory (algebro-geometric aspects), Group actions on varieties or schemes (quotients) Orbifold-Hodge numbers of Hilbert schemes | 0 |
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