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

Ryan, Tim

doi:10.1017/nmj.2017.24

Cited by 9 publications

(6 citation statements)

References 73 publications

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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: General Surfacesmentioning

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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Section: General Surfacesmentioning

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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“…The effective and ample cones of the Hilbert scheme of Del Pezzo surfaces are discussed in [BC]. Ryan [Ry] studies the effective cone of the moduli space of sheaves on a quadric surface without the assumption symmetric. The ample and movable cone of the moduli space of sheaves on a surface is also studied by several authors [BM], [CH1], [CH2], [LZ], [Y].…”

Section: Effective Cone Of the Moduli Spacementioning

confidence: 99%

Semistable Sheaves With Symmetric on a Quadric Surface

Abe¹

2016

Nagoya Math. J.

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For moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface, we pursue analogy to some results known for moduli spaces of sheaves on a projective plane. We define an invariant height, introduced by Drezet in the projective plane case, for moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface and describe the structure of moduli spaces of height zero. Then we study rational maps of moduli spaces of positive height to moduli spaces of representation of quivers, effective cones of moduli spaces, and strange duality for height-zero moduli spaces.

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“…On the other hand, it can be quite challenging to verify that H 0 (E ⊗ F ) = 0 for a general E ∈ M H (v). These types of questions have been studied in [CHW16] in the case of P 2 and [Rya16] in the case of P 1 × P 1 . Interesting effective divisors arising in the birational geometry of moduli spaces frequently arise in this way.…”

Section: Theorem 426 ([Li94]mentioning

confidence: 99%

Birational geometry of moduli spaces of sheaves and Bridgeland stability

Huizenga

2016

Preprint

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Moduli spaces of sheaves and Hilbert schemes of points have experienced a recent resurgence in interest in the past several years, due largely to new techniques arising from Bridgeland stability conditions and derived category methods. In particular, classical questions about the birational geometry of these spaces can be answered by using new tools such as the positivity lemma of Bayer and Macrì. In this article we first survey classical results on moduli spaces of sheaves and their birational geometry. We then discuss the relationship between these classical results and the new techniques coming from Bridgeland stability, and discuss how cones of ample divisors on these spaces can be computed with these new methods. This survey expands upon the author's talk at the 2015 Bootcamp in Algebraic Geometry preceding the 2015 AMS Summer Research Institute on Algebraic Geometry at the University of Utah. Contents 1. Introduction 1 2. Moduli spaces of sheaves 2 3. Properties of moduli spaces 9 4. Divisors and classical birational geometry 15 5. Bridgeland stability 24 6. Examples on P 2 31 7. The positivity lemma and nef cones 35 References 40

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

Cited by 9 publications

References 73 publications

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

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

Semistable Sheaves With Symmetric on a Quadric Surface

Birational geometry of moduli spaces of sheaves and Bridgeland stability

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