Moduli Spaces of $α$-stable Pairs and Wall-Crossing on $\mathbb{P}^2$

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.2010 Mathematics Subject Classification. 14D20. Key words and phrases. Semistable pairs, Wall-crossing formulae, Blow-up/down, and Betti numbers.

On the homology of the moduli space of plane sheaves with Hilbert polynomial5m+3

“…Our calculation of the Poincaré polynomial agrees thus with [12]. Intriguingly, this is the same as the Poincaré polynomial of M P 2 (5,1), computed in [12] and [4]. This raises the question whether M P 2 (5,1) and M P 2 (5,3) are (canonically) isomorphic.…”

“…Appendix A. Singular programs ring r=0,(x,y,z),dp; int i,j; poly P, q, q1, q2, l1, l2, l3; P=0; int points, lines; points = 0; lines = 0; list s1, s2, s2_0, s2_1, s2_2, s3, s4, s5, d; proc point_3_1(list l) {points=points+3; return(x^ (2*positive_part(values(l, list(0,1,7)))) +x^ (2*positive_part(values(l, list(1,0,7)))) +x^ (2*positive_part(values(l, list(7,1,0)))));}; proc point_6(list l) {points=points+6; return(x^ (2*positive_part(values(l, list(0,1,7)))) +x^ (2*positive_part(values(l, list(1,0,7)))) +x^ (2*positive_part(values(l, list(7,1,0)))) +x^ (2*positive_part(values(l, list(1,7,0)))) +x^ (2*positive_part(values(l, list(0,7,1)))) +x^ (2*positive_part(values(l, list(7, proc w1(list u, list v) {return(add (-v[1]-u [1],s1)+add (-v[1]-u [2],s1)+list (-v[1]-u [3],-v[1]-u [4]) +add (-v[2]-u [1],s2)+add (-v[2]-u [2],s2)+add (-v[2]-u [3],s1)+add (-v[2]-u [4],s1) +add (-v[3]-u [1],s2)+add (-v[3]-u [2],s2)+add (-v[3]-u [3],s1)+add (-v[3]-u [4],s1) +add (-v[4]-u [1],s2)+add (-v[4]-u [2],s2)+add (-v[4]-u [3],s1)+add (-v[4]-u[4],s1));}; proc g1(list u, list v) {return(add (-v[2]+v [1], s1) + list (-v[2]+v [2], -v [2]+v…”

“…We refer to [5,Section 2] for a brief outline of the Bia lynicki-Birula method. In [5,Sections 5,6] this method was used to study the homology of M P 2 (4,1), relying on a stratification by strata that are easily understood as geometric quotients. We will apply the same technique to M P 2 (5,3).…”

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

“…The geometry of the moduli space M has been studied by many authors [7,10,20,21]. In [12], it was conjectured that genus zero Gopakumar-Vafa (or BPS) invariant defined in M-theory is equal to the Euler characteristic of the moduli space M P 2 (r, 1) up to sign.…”

“…When r = 4, the conjecture was first checked in [20] where the author uses a stratification of the moduli space with respect to the global section spaces. The Poincaré polynomials of the moduli spaces when r = 4 and 5 have been computed in [7] by a wall-crossing technique in the moduli spaces of α-stable pairs, and also in [21] by the classification of the semi-stable sheaves carried out in [10] and [17]. Recently, a B-model calculation in physics computes the Poincaré polynomial up to r = 7 [11, Table 2] in terms of the refined BPS indices.…”