Torus action on the moduli spaces of torsion plane sheaves of multiplicity four

Choi, Jinwon; Maican, Mario

doi:10.1016/j.geomphys.2014.05.005

Cited by 15 publications

(45 citation statements)

References 19 publications

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“…Also, these are consistent with the results in [26,2,31,4,22]. (2) It has been conjectured that the topological Euler characteristics of M(d, 1) are equal to the genus zero Gopakumar-Vafa invariants up to sign [14].…”

Section: Remark 54supporting

confidence: 87%

Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^2$

Choi

Chung

2016

J. Math. Soc. Japan

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Abstract. We study the wall-crossing of the moduli spaces M α (d, 1) of α-stable pairs with linear Hilbert polynomial dm+ 1 on the projective plane P 2 as we alter the parameter α. When d is 4 or 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincaré polynomials of the moduli spaces M(d, 1) of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan-Hölder filtrations is three.1. Introduction 1.1. Motivation and Results. In moduli theory, for a given quasi-projective moduli space M 0 , various compactifications stem from the different view points for the moduli points of M 0 . After we obtain various compactified moduli spaces of M 0 , it is quite natural to ask the geometric relationship among them. Sometimes, this question is answered by birational morphisms between them, which enables us to obtain some geometric information (for example, the cohomology groups) of one space from that of the other [28,5].In this paper, we study the moduli space of semistable sheaves of dimension one on smooth projective surfaces [27], which recently gains interests in both mathematics and physics. This is an example of compactifications of the relative Jacobian variety, where we regard its general point as a sheafon a smooth curve C with pole along points p i of general position. In general, the moduli space of semistable sheaves is hard to study due to the lack of geometry of its boundary points. However, if n is equal to the genus of C, the sheaf F has a unique section up to scalar. So, we may alternatively consider the general point as a sheaf with a section, which in turn leads to another compactification, so called the moduli space of α-semistable pairs (more generally, the coherent systems [18]). When α is large, it can be shown that the moduli spaces of α-stable pairs are nothing but the relative 2010 Mathematics Subject Classification. 14D20.

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Section: Remark 54supporting

confidence: 87%

Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^2$

Choi

Chung

2016

J. Math. Soc. Japan

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“…They studied in [2,7] and [8] the (C * ) 2 -fixed locus of M(4, 1), M(5, 1) and M(5, 2) respectively, using a different description of the moduli spaces. By their results, the fixed locus of M(4, 1) consists of 180 isolated points and 6 projective lines P 1 (Theorem 1.1 in [2]), the fixed locus of M(5, 1) consists of 1,407 isolated points, 132 projective lines and 6 irreducible components of dimension 2 isomorphic to P 1 × P 1 (Theorem 2 in [7]), and the fixed locus of M(5, 2) consists of 1,329 isolated points, 174 projective lines and three isomorphic surfaces obtained by blowing up P 1 × P 1 at three points on the diagonal, then blowing down the strict transform of the diagonal (Theorem in [8]). Our result is compatible with theirs, because what we have obtained are just the cellular decompositions of their fixed loci.…”

Section: Remark 13mentioning

confidence: 99%

“…In Sect.3, we study the fixed locus of M(4, 1). The last two moduli spaces, M(5, 1) and M (5,2), are dealt in the last section, Sect. 4.…”

Section: Remark 13mentioning

confidence: 99%

“…There are many interesting results on moduli spaces of 1-dimensional semistable sheaves on surfaces, mainly with surfaces K3, abelian or rational, for instance [2,3,[5][6][7][8][11][12][13] and [14]. Recently because of its close relation to PT-invariants (defined in [9]) on local Calabi-Yau 3-fold, this kind of moduli spaces over Fano surfaces become even more worthy of study.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.1 implies that the formulas given in [14] not only provide motive decompositions but also cellular decompositions of the moduli spaces. Remark 1.2 Theorem 6.1 in [14] shows that [M(5, 1)] = [M (5,2)]. But these two moduli spaces don't have the same (C * ) 2 -fixed locus, hence we know there is no (C * ) 2 -equivariant isomorphisms between them.…”

Section: Introductionmentioning

confidence: 99%

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Affine pavings for moduli spaces of pure sheaves on $$\mathbb {P}^2$$ P 2 with degree $$\le 5$$ ≤ 5

Yao

2014

Geom Dedicata

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Let M(d, r ) be the moduli space of semistable sheaves of rank 0, Euler characteristic r and first Chern class d H (d > 0), with H the hyperplane class in P 2 . In [14] we gave an explicit description of the class [M(d, r )] of M(d, r ) in the Grothendieck ring of varieties for d ≤ 5 and g.c.d(d, r ) = 1. In this paper we compute the fixed locus of M(d, r ) under some (C * ) 2 -action and show that M(d, r ) admits an affine paving for d ≤ 5 and g.c.d(d, r ) = 1. We also pose a conjecture that for any d and r coprime to d, M(d, r ) would admit an affine paving.Keywords Moduli spaces of 1-dimensional semistable sheaves on the projective plan ·

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The geometry of the moduli space of one-dimensional sheaves

Choi

Chung

2014

Sci. China Math.

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Abstract. Let M d be the moduli space of stable sheaves on P 2 with Hilbert polynomial dm + 1. In this paper, we determine the effective and the nef cone of the space M d by natural geometric divisors. Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the GrothendieckRiemann-Roch theorem. We also present the stable base locus decomposition of the space M 6 . As a byproduct, we obtain the Betti numbers of the moduli spaces, which confirm the prediction in physics.

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Torus action on the moduli spaces of torsion plane sheaves of multiplicity four

Cited by 15 publications

References 19 publications

Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^2$

Moduli spaces of $\alpha$-stable pairs and wall-crossing on $\mathbb{P}^2$

Affine pavings for moduli spaces of pure sheaves on $$\mathbb {P}^2$$ P 2 with degree $$\le 5$$ ≤ 5

The geometry of the moduli space of one-dimensional sheaves

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