Moduli of sheaves, Fourier–Mukai transform, and partial desingularization

Abstract

Abstract. We study birational maps among 1) the moduli space of semistable sheaves of Hilbert polynomial 4m + 2 on a smooth quadric surface, 2) the moduli space of semistable sheaves of Hilbert polynomial m 2 + 3m + 2 on P 3 , 3) Kontsevich's moduli space of genus-zero stable maps of degree 2 to the Grassmannian Gr(2, 4). A regular birational morphism from 1) to 2) is described in terms of Fourier-Mukai transforms. The map from 3) to 2) is Kirwan's partial desingularization. We also investigate several geometric properties of 1) by using the variation of moduli spaces of stable pairs.

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

“…This example is more straightforward than the previous two and it was already present in [CM14] in some form; we report it here for the sake of completeness.. Let S = M(0, (2, 2), 4m + 2); by [Sim94] and [LP93] we have that S is a projective variety of dimension 9. As before, we have a structure Theorem from [CM14] describing the strata of S in terms of minimal resolutions: Theorem 3.10. There exists a decomposition of S into three disjoint strata S 0 , S 1 and S 2 .…”

“…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.…”

Geometry of moduli spaces of rational curves in linear sections of Grassmannian Gr(2,5)

“…Let N(Gr(2, 5)) be the relative Kronecker quiver space N(U; 2, 2) with the fiber N(4; 2, 2) (for the precise definition of quiver space, see [6]). Then there exists a divisorial contraction M 2 (Gr(2, U)) → N(Gr(2, 5)) contracting the stable maps whose images are planar ( [6]). In summary, we obtain (4,5).…”

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

“…In this case, because of the self-duality of Gr (2,4), the complete description is particularly clear. Essentially all of birational models in this case have been described in [2,9]. For the reader's convenience, we leave the statement and references.…”

“…For the reader's convenience, we leave the statement and references. (8), (9) are obtained by the duality map Φ in Lemma 6.1. Note that for any divisor D on the boundary of Eff(M), M(D) is a contraction with positive dimensional fibers of one of normal varieties K, K * , and Gr(3, ∧ 2 V ).…”