Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces

Friedman, Robert; Qin, Zhenbo

doi:10.4310/cag.1995.v3.n1.a2

Cited by 49 publications

(63 citation statements)

References 22 publications

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“…This has been studied by a number of authors (e.g. [49], [23], [18]). Assume for simplicity that S is simply connected.…”

Section: Moduli Of Vector Bundlesmentioning

confidence: 99%

“…[n] × S [m] followed by blow downs in another direction [23], [18]. The change of the Betti and Hodge numbers under wallcrossing can be explicitly determined and this can be used e.g.…”

Section: Moduli Of Vector Bundlesmentioning

confidence: 99%

“…[n] × S [m] of Hilbert schemes of points [18], [23]. The leading terms of these expressions can be explicitly evaluated.…”

Section: Moduli Of Vector Bundlesmentioning

confidence: 99%

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On the Motive of the Hilbert scheme of points on a surface

Göttsche

2001

Mathematical Research Letters

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The Hilbert scheme S[n] of points on an algebraic surface S is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power S (n) . For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert schemes of points have connections to several fields of mathematics, including moduli spaces of sheaves, Donaldson invariants, enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras and also to theoretical physics. This talk will try to give an overview over these connections. IntroductionThe Hilbert scheme S[n] of points on a complex projective algebraic surface S is a a parameter variety for finite subschemes of length n on S. It is a nice (crepant) resolution of singularities of the n-fold symmetric power S (n) of S. If S is a K3 surface or an abelian surface, then S[n] is a compact, holomorphic symplectic (thus hyperkähler) manifold. Thus S [n] is at the same time a basic example of a moduli space and an example of a nice resolution of singularities of a singular variety. There are a number of conjectures and general phenomena, many of which originating from theoretical physics, both about moduli spaces for objects on surfaces and about nice resolutions of singularities. In all of these the Hilbert scheme of points can be viewed as a model case and sometimes as the main motivating example. Hilbert schemes of points on a surface have connections to many topics in mathematics, including moduli spaces of sheaves and vector bundles, Donaldson invariants, Gromov-Witten invariants and enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras, noncommutative geometry and also theoretical physics.

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“…This has been studied by a number of authors (e.g. [49], [23], [18]). Assume for simplicity that S is simply connected.…”

Section: Moduli Of Vector Bundlesmentioning

confidence: 99%

“…[n] × S [m] followed by blow downs in another direction [23], [18]. The change of the Betti and Hodge numbers under wallcrossing can be explicitly determined and this can be used e.g.…”

Section: Moduli Of Vector Bundlesmentioning

confidence: 99%

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On the Motive of the Hilbert scheme of points on a surface

Göttsche

2001

Mathematical Research Letters

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“…On the other hand, elementary transforms, refer to [10] and [2,Appendix] about general information, has the following advantages: (i) birational transforms obtained there are blowing-ups whose centers are derived by a canonical, moduli-theoretic way; (ii) when two parameters α and α ′ defining stability conditions of objects are given, one not only connects the moduli scheme of α-semistable objects with that of α ′ -semistable ones, but also relates their universal families, if exist. Ellingsrud-Göttsche [1] and Friedman-Qin [3] proposed to apply elementary transform to the case where r = 2 and the wall of type c separating H − and H + is good, so the natural subset…”

Section: Introductionmentioning

confidence: 99%

Blowing-Ups Describing the Polarization Change of Moduli Schemes of Semistable Sheaves of General Rank

Yamada

2010

Communications in Algebra

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Abstract. Let H and H′ be two ample line bundles over a smooth projective surface X, and M (H) (resp. M (H ′ )) the coarse moduli scheme of H-semistable (resp. H ′ -semistable) sheaves of fixed type (r, c 1 , c 2 ). We construct a sequence of blowing-ups which describes how M (H) differs from M (H ′ ) not only when r = 2 but also when r is arbitrary. Means we here utilize are elementary transforms and the notion of a sheaf with flag.

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“…The geometry and topology of these moduli spaces have been extensively studied. General results include the nonemptiness (see 21,18,13]), generic smoothness and irreducibility (see 3,8,27,10,24,11,25]), and variations of these moduli spaces (see 12,14,6,9,23]). There are also some speci c results for these moduli spaces over particular surfaces (see 4,5,16,18,19,26,28] for some of the papers).…”

Section: Introductionmentioning

confidence: 99%