Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces

Li, Wei Ping; Qin, Zhenbo; Wang, Weiqiang

doi:10.1007/s002080200330

Cited by 63 publications

(126 citation statements)

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“…A set of ring generators for H * (X [n] ) when X is P 2 (which is easily shown to be equivalent to the set given in the above Theorem) or C 2 was first found by Ellingsrud and Strømme [ES2]. Lehn's new proof [Lehn] using the Heisenberg algebra for this result when X = C 2 has been very inspiring for our approach in [LQW1]. On the other hand, the approach of Ellingsrud and Strømme has been extended to other rational surfaces and ruled surfaces in [Bea], and to K3 surfaces by Markman [Mar].…”

Section: Heisenberg Algebra and Hilbert Schemes Vafa And Wittenmentioning

confidence: 99%

“…Lehn [Lehn] realized geometrically the Virasoro algebra and applied it to study connections between Heisenberg generators and cup products on the cohomology groups of Hilbert schemes. In our joint work with Li and Qin [LQW1,LQW2], we have further developed the connection between vertex algebras and Hilbert schemes and used it to obtain new results on the cohomology ring structure of X [n] associated to an arbitrary projective surface X (which in general has been unaccessible by classical methods in algebraic geometry). On the other hand, in the framework of wreath products, there has been a group theoretic construction of the Virasoro algebra given by I. Frenkel and the author [FW] acting on R Γ which uses the construction of Heisenberg algebra in [Wa1].…”

Section: Introductionmentioning

confidence: 99%

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Algebraic structures behind Hilbert schemes and wreath products

Wang¹

2002

Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory

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Abstract. In this paper we review various strikingly parallel algebraic structures behind Hilbert schemes of points on surfaces and certain finite groups called the wreath products. We explain connections among Hilbert schemes, wreath products, infinite-dimensional Lie algebras, and vertex algebras. As an application we further describe the cohomology ring structure of the Hilbert schemes. We organize this paper around several general principles which have been supported by various works on these subjects.

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Section: Heisenberg Algebra and Hilbert Schemes Vafa And Wittenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Algebraic structures behind Hilbert schemes and wreath products

Wang¹

2002

Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory

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“…This is a sequel to [LQW1]- [LQW4] and [QW]. We continue the study of the cohomology rings of the Hilbert schemes X [n] of n points on a smooth surface X and the Chen-Ruan orbifold cohomology rings of the symmetric products X n /S n .…”

Section: Introductionmentioning

confidence: 99%

“…In [Lehn], [LQW1]- [LQW4], [LS2], which were in turn built on the earlier works [Got], [VW], [Na1], [Na2], [Gro] and others, the connections between vertex operators and the multiplicative structure of the rational cohomology group H * (X [n] ) when X is projective have been developed. These connections have been successfully applied to unravel various structures on the cohomology ring of X [n] .…”

Section: Introductionmentioning

confidence: 99%

“…It allows us to introduce a universal ring G X for a smooth quasi-projective surface X with the S-property, which governs the cohomology rings H * (X [n] ) for a fixed X and all n. This ring G X will be called the FH ring associated to X, as it is analogous to the one arising in the framework of symmetric groups and wreath products [FH], [Wa]. We further determine the ring structure of G X , and obtain two sets of ring generators of H * (X [n] ) which are the quasi-projective counterparts of the main results in [LQW1], [LQW2]. We remark that a universal ring termed as the Hilbert ring was introduced in [LQW3] which governs the cohomology rings H * (X [n] ) for a fixed projective X and all n. These two universal rings reflect distinct structures of the corresponding cohomology rings.…”

Section: Introductionmentioning

confidence: 99%

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Ideals of the cohomology rings of Hilbert schemes and their applications

Qin

Wang

2003

Trans. Amer. Math. Soc.

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Abstract. 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] ; C) and H * orb (X n /Sn; C) for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.

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Hilbert Schemes of Points on Surfaces

Göttsche¹

2005

Frobenius Splitting Methods in Geometry and Representation Theory

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n distinct points of X, if d ≥ 2 (Lemma 7.1.6). On the other hand, every X (n) is nonsingular if d = 1 (Exercise 7.1.E.5).Section 7.2 presents some general results on Hilbert schemes of points on quasiprojective schemes: their existence (Theorem 7.2.3) and the description of their Zariski tangent spaces (Lemma 7.2.5). Also, the punctual Hilbert scheme X [n] x (parametrizing length-n subschemes supported at a given point x) is introduced, and it is shown that X [n] x is projective and connected (Proposition 7.2.9). The Hilbert-Chow morphism is introduced and studied in Section 7.3, again in the setting of quasi-projective schemes. The existence of a projective morphism of schemes γ : X [n] −→ X (n) which yields the cycle map on closed points is deduced from work of Iversen (Theorem 7.3.1). The fibers of γ are products of punctual Hilbert schemes; their connectedness implies that X [n] is connected if X is (Corollary 7.3.4). Then, the loci X [n] * * , resp. X [n] * , consisting of subschemes supported at n distinct points, resp. at least n − 1 distinct points, are considered. In particular, it is shown that X [n] * is a nonsingular variety of dimension nd, and the complement X [n] * \ X [n] * * is a nonsingular 208 Chapter 7. Hilbert Schemes of Points prime divisor, if X is a nonsingular variety of dimension d (Lemma 7.3.5). Section 7.4 is devoted to Hilbert schemes of points on a nonsingular surface X. Each X [n] is shown to be a nonsingular variety of dimension 2n (Theorem 7.4.1) and the Hilbert-Chow morphism is shown to be birational, with exceptional set being a prime divisor (Proposition 7.4.5). Finally, the Hilbert-Chow morphism is shown to be crepant (Theorem 7.4.6), a result due to Beauville in characteristic 0 and to Kumar-Thomsen in characteristic p ≥ 3.Section 7.5 begins with the observation that any symmetric product of a split quasiprojective scheme is split as well (Lemma 7.5.1). Together with the crepantness of the Hilbert-Chow morphism, this implies the splitting of X [n] , where X is a nonsingular split surface (Theorem 7.5.2). In turn, this yields the vanishing of higher cohomology groups of any ample invertible sheaf on X [n] , if X is split and proper over an affine variety (Corollary 7.5.4). This applies, in particular, to the nonsingular projective split surfaces and also to the nonsingular affine surfaces (since these are split by Proposition 1.1.6). Further, we obtain a relative vanishing result for the Hilbert-Chow morphism (Corollary 7.5.5).Notation. Throughout this chapter, X denotes a quasi-projective scheme over an algebraically closed field k of characteristic p ≥ 0, and n denotes a positive integer. By schemes, as earlier in the book, we mean Noetherian separated schemes over k; their closed points will just be called points.

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Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces

Abstract: Abstract. Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers.

Cited by 63 publications

References 21 publications

Algebraic structures behind Hilbert schemes and wreath products

Algebraic structures behind Hilbert schemes and wreath products

Ideals of the cohomology rings of Hilbert schemes and their applications

Hilbert Schemes of Points on Surfaces

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