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

Abstract

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.

Local BPS Invariants: Enumerative Aspects and Wall-Crossing

“…To compute the Poincaré polynomial, we relate M β birationally with the moduli spaces of δ-stable pairs by wall-crossing. This approach is taken in [6] to compute the Betti numbers for M β when S = P 2 and β = 4 and 5. See also [9].…”

“…This wall-crossing phenomenon can be explained by elementary modification of pairs. See [35, §3], [15,Lemma.4.24] and [6]. Now each Ext group can be computed using the following proposition.…”

“…We use the wall-crossing in the moduli space of δ-stable pairs. The same strategy is used in [6] to study M β when S = P 2 . See § 3.3 for a review of δ-stable pair theory.…”

Mori's Program for the Moduli Space of Conics in Grassmannian

“…In the study of Fano manifolds, the space of lines in Gr(2, n) has been one of the main tools to study the geometry of the linear or quadratic sections of Gr(2, n) ( [31]). In fact, the codimension two linear section of Gr (2,5) is the answer for Hirzebruch's question in dimension 4: Classify all minimal compactifications of C 4 .…”

“…In [15], the authors provided new such a pair by using the double cover (the so-called double symmetroid) of the determinantal symmetroid in the space of quadrics P(Sym 2 C 5 * ). One of the main steps of the construction is to find an explicit birational relation between the double symmetroid and the Hilbert scheme of conics in the Grassmannian Gr(3, 5) ∼ = Gr (2,5) of planes. This relation has been established in [16] in a broader setting, namely, for the space of quadrics in P(Sym 2 C n+1 * ) and the Hilbert scheme of conics in Gr(n − 1, n + 1) ∼ = Gr(2, n + 1).…”