Moduli of sheaves supported on curves of genus two in a quadric surface

Abstract

We study the moduli space of stable sheaves of Euler characteristic 2, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers and we give a classification of the stable sheaves involving locally free resolutions.

Birational geometry of the moduli space of pure sheaves on quadric surface

“…As applications, we compute the Poincaré polynomial of M and show the rationality of M (Corollary 3.8) which were obtained by Maican by different methods ( [Mai16]). Since each step of the birational transform is described in terms of blow-ups/downs along explicit subvarieties, in principle the cohomology ring and the Chow ring of M can be obtained from that of G. Also one may aim for the completion of Mori's program for M. We will carry on these projects in forthcoming papers.…”

Bridgeland wall-crossing for some Simpson moduli spaces on $\mathbb{P}^1 \times \mathbb{P}^1$

“…Let M = M(0, (2, 3), 5m + 2); first of all, by [Sim94] and [LP93] we have that M is a projective variety of dimension 13. Then we have the following Theorem from [Mai17] describing the strata of M in terms of minimal resolutions:…”

“…When crossing on the other side of W 1 , M 1 is replaced by the space of extensions in the other direction; since Ext 1 (O(0, 1), Q) C (see Appendix), we have that the wall crossing just contracts the P 10 -bundle onto its base. The space M obtained this way therefore has an open subset isomorphic to M 0 , and the complement is given by two disjoint components isomorphic to P 1 and P 1 × P 1 respectively: by Proposition 4.1 and 4.2 in [Mai17], M is isomorphic to a GIT quotient W/G of a certain subset…”

“…Let N = M(0, (2, 3), 5m + 1); by [Sim94] and [LP93] we have that N is a projective variety of dimension 13. As before, we have a structure Theorem from [Mai18] describing the strata of N in terms of minimal resolutions: Theorem 3.6. There exists a decomposition of N into four strata N 0 , N 1 , N 2 and N 3 .…”

“…In the present paper we address the case of certain Simpson moduli spaces of Giesekersemistable torsion sheaves on P 1 × P 1 , for which a lot of birational geometry is known from the work in [Mai17], [CM17] or [CM14]; we proceed as follows: in Section 2, we recall the basic results from the theory of Bridgeland stability conditions on surfaces and we present some preliminary definitions and computations for the case in exam; in Section 3 we go over the wall-crossing behavior for three different moduli spaces of torsion sheaves, recovering most of the birational results from [Mai17], [CM17] and [CM14]; finally, in the Appendix we carry out some calculations needed in the Theorems from Section 3.…”