Semistable Sheaves With Symmetric  on a Quadric Surface

Abe, Takeshi

doi:10.1017/nmj.2016.50

Cited by 7 publications

(3 citation statements)

References 22 publications

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“…See [CHW17, §4]. On F 0 , Abe [Abe17] shows that there are balanced slopes µF 1 + µF 2 with the same property.…”

Section: Proofmentioning

confidence: 99%

Existence of semistable sheaves on Hirzebruch surfaces

Coskun¹,

Huizenga²

2019

Preprint

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Let Fe denote the Hirzebruch surface P(O P 1 ⊕O P 1 (e)), and let H be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves M Fe,H (r, c1, c2) is nonempty. Our algorithm relies on certain stacks of prioritary sheaves. We first solve the existence problem for these stacks and then algorithmically determine the Harder-Narasimhan filtration of the general sheaf in the stack. In particular, semistable sheaves exist if and only if the Harder-Narasimhan filtration has length one.We then study sharp Bogomolov inequalities ∆ ≥ δH (c1/r) for the discriminants of stable sheaves which take the polarization and slope into account; these inequalities essentially completely describe the characters of stable sheaves. The function δH (c1/r) can be computed to arbitrary precision by a limiting procedure. In the case of an anticanonically polarized del Pezzo surface, exceptional bundles are always stable and δH (c1/r) is computed by exceptional bundles. More generally, we show that for an arbitrary polarization there are further necessary conditions for the existence of stable sheaves beyond those provided by stable exceptional bundles. We compute δH (c1/r) exactly in some of these cases. Finally, solutions to the existence problem have immediate applications to the birational geometry of M Fe,H (v).

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“…See [CHW17, §4]. On F 0 , Abe [Abe17] shows that there are balanced slopes µF 1 + µF 2 with the same property.…”

Section: Proofmentioning

confidence: 99%

Existence of semistable sheaves on Hirzebruch surfaces

Coskun¹,

Huizenga²

2019

Preprint

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“…Similar results for the the Hilbert scheme on other rational surfaces were obtained in [BC13], for example including nef cones of all Hilbert schemes points on Hirzebruch surfaces. In the case of P 1 × P 1 , the effective cone of many moduli spaces of sheaves have been determined in [Rya16], and in all cases where c 1 is symmetric in [Abe16].…”

Section: Birational Geometry Of Moduli Spaces Of Sheaves: a Quick Surveymentioning

confidence: 99%

Wall-crossing implies Brill-Noether: Applications of stability conditions on surfaces

Bayer¹

2018

Algebraic Geometry: Salt Lake City 2015

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Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces.We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces.The intended reader is an algebraic geometer with a limited working knowledge of derived categories.

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“…More generally, there has been a lot of work on (e.g., [ABCH], [BMW], [CH4], [CH1], [CH2], [DLP], [Hui2], [Hui1], [CZ2], [CZ1], [Woo]). Although much of this work has been extended to more rational and ruled surfaces (e.g., [Abe], [Bal], [Göt], [Kar], [Moz], [Qin]), no general method to compute the entire effective cone of a moduli space of sheaves on has been given. This is because the proof in [CHW] relies heavily on properties that are unique to .…”

Section: Introductionmentioning

confidence: 99%

The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface

Ryan¹

2017

Nagoya Math. J.

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Abstract. Let ξ be a stable Chern character on P 1 × P 1 , and let M (ξ) be the moduli space of Gieseker semistable sheaves on P 1 × P 1 with Chern character ξ. In this paper, we provide an approach to computing the effective cone of M (ξ). We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of M (ξ) which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on P 1 × P 1 .

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Semistable Sheaves With Symmetric on a Quadric Surface

Cited by 7 publications

References 22 publications

Existence of semistable sheaves on Hirzebruch surfaces

Existence of semistable sheaves on Hirzebruch surfaces

Wall-crossing implies Brill-Noether: Applications of stability conditions on surfaces

The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface

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