The effective cone of the moduli space of sheaves on the plane

Coskun, Izzet; Huizenga, Jack; Woolf, Matthew

doi:10.4171/jems/696

Cited by 49 publications

(70 citation statements)

References 4 publications

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“…We do need to recall the resolution by exceptional sheaves from [CHW14]. A stable vector bundle E on P 2 is an exceptional bundle if Ext 1 (E, E) = 0.…”

Section: Resolutions By Exceptional Bundlesmentioning

confidence: 99%

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Universal Series for Hilbert Schemes and Strange Duality

Johnson

2018

International Mathematics Research Notices

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We examine a sequence of examples of pairs of moduli spaces of sheaves on P 2 where Le Potier's strange duality is expected to hold. One of the moduli spaces in these pairs is the Hilbert scheme of two points. We compute the sections of the relevant theta bundle as a representation of SL(X), where P 2 = P(X). For the higher rank space, we construct a moduli space using the resolution of exceptional bundles from Coskun, Huizenga, and Woolf [CHW14]. We compute a subspace of the sections of the theta bundle which is dual to the sections on the Hilbert scheme.In the second part, we use a result from Goller and Lin [GL19] to rigorously apply the "finte Quot scheme method", introduced in Marian and Oprea [MO07] and Bertram, Goller, and Johnson [BGJ16]. This requires us to prove that the kernels in appearing in the Quot scheme have the appropriate resolution by exceptional bundles.

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“…We do need to recall the resolution by exceptional sheaves from [CHW14]. A stable vector bundle E on P 2 is an exceptional bundle if Ext 1 (E, E) = 0.…”

Section: Resolutions By Exceptional Bundlesmentioning

confidence: 99%

“…In Section 5, we will address M (e). In [CHW14], Coskun, Huizenga, and Woolf show how to construct nice resolution for a general semistable sheaf of a fixed Chern character. We recall this construction in Section 4.…”

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confidence: 99%

Universal Series for Hilbert Schemes and Strange Duality

Johnson

2018

International Mathematics Research Notices

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“…In particular cases, Theorem 8.6 can be be made more precise and general. In the case of the projective plane [CHW14,CH15,LZ16] and of K3 surfaces [BM14a,BM14b], given any primitive vector, varying stability conditions corresponds to a directed Minimal Model Program for the corresponding moduli space. This allows to completely describe the nef cone, the movable cone, and the pseudo-effective cone for them.…”

Section: Nef Divisors On Moduli Spaces Of Bridgeland Stable Objectsmentioning

confidence: 99%

“…Typical question are what their nef and effective cones are and what the stable base locus decomposition of the effective cone is. The case of P 2 was studied in many articles such as [ABCH13,CHW14,LZ13,LZ16,Woo13]. The study of the abelian surfaces case started in [MM13] and was completed in [MYY14,YY14].…”

Section: Introductionmentioning

confidence: 99%

Lectures on Bridgeland Stability

Macrì

Schmidt

2017

Lecture Notes of the Unione Matematica Italiana

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Abstract. In these lecture notes we give an introduction to Bridgeland stability conditions on smooth complex projective varieties with a particular focus on the case of surfaces. This includes basic definitions of stability conditions on derived categories, basics on moduli spaces of stable objects and variation of stability. These notes originated from lecture series by the first author at the summer school Recent advances in algebraic and arithmetic geometry, Siena, Italy, August 24-28, 2015 and at the school Moduli of Curves, CIMAT, Guanajuato, Mexico, February 22 -March 4, 2016.

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“…Using these, we pull the bundle back to the universal family, push it down onto the Hilbert scheme, and take the determinant (c.f [LP97]. &[CHW16]). …”

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confidence: 99%

Nef Cones of Nested Hilbert Schemes of Points on Surfaces

Ryan

Yang

2018

International Mathematics Research Notices

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Let X be the projective plane, a Hirzebruch surface, or a general K3 surface. In this paper, we study the birational geometry of various nested Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes on X. We calculate the nef cone for two types of nested Hilbert schemes. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces. 1 NEF CONES OF NESTED HILBERT SCHEMES OF POINTS ON SURFACES 2 product of projective lines, on Hirzebruch surfaces, and on del Pezzo surfaces of degree at least two [BC13]. Recently, there has been tremendous progress using Bridgeland stability to compute the nef cones of Hilbert schemes of points on K3 surfaces [BM14b], Abelian surfaces ([MM13], [YY14]), Enriques surfaces [Nue16], and all surfaces with Picard number one and irregularity zero [BHL + 16].To calculate the nef cone for nested Hilbert schemes, we must first understand the Picard group and the Néron-Severi group.Proposition A. Let X be a smooth projective surface of irregularity zero and fix n ≥ 2. ThenIn particular, the Néron-Severi group N 1 (X [n+1,n] ) has rank 2(ρ(X) + 1), where ρ(X) is the picard number of X.Knowing the Picard groups, we can describe the nef cones. To easily state our theorem, let us first recall the nef cone of the Hilbert schemes of points on P 2 . The nef cone of P 2[n] is spanned by the two divisorswhere H[n] is the class of the pull-back of the ample generator via the Hilbert-Chow morphism and B[n] is the exceptional locus.Theorem B. The nef cone of P 2[n+1,n] , n > 1, is spanned by the four divisors, and pr * a (D n [n + 1]). Similar results hold for the Hirzebruch surfaces F i , i ≥ 0, and general K3 surfaces S as well as for the universal families on all of these surfaces.Knowing the nef cone allows us to recover a theorem of Butler about projective normality of line bundles on Hirzebruch surfaces [But94].Proposition C. Let X be a Hirzebruch surface, and A be an ample line bundles on X, then L = K X + nA is projectively normal for n ≥ 4.

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The effective cone of the moduli space of sheaves on the plane

Cited by 49 publications

References 4 publications

Universal Series for Hilbert Schemes and Strange Duality

Universal Series for Hilbert Schemes and Strange Duality

Lectures on Bridgeland Stability

Nef Cones of Nested Hilbert Schemes of Points on Surfaces

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