The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface

Abstract

Let $\xi$ be a stable Chern character on ${\mathbb{P}}^1\times {\mathbb{P}}^1$, and let $M(\xi )$ be the moduli space of Gieseker semistable sheaves on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ with Chern character $\xi$. In this paper, we provide an approach to computing the effective cone of $M(\xi )$ after showing that it is a Mori dream space for all $\xi$. We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(\xi )$ which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on ${\mathbb{P}}^1\times {\mathbb{P}}^1$.

42 pages

References in corpus (7)

• Birational models of moduli spaces of coherent sheaves on the projective plane
• Projectivity and Birational Geometry of Bridgeland Moduli spaces on an Enriques Surface
• The Space of Stability Conditions on Abelian Threefolds, and on some Calabi-Yau Threefolds
• Nef cones of Hilbert schemes of points on surfaces
• Derived category of coherent sheaves and counting invariants
• Wall crossing of the moduli spaces of perverse coherent sheaves on a blow-up
• Nef divisors for moduli spaces of complexes with compact support

Cited by in corpus (2)

• Nef cones of nested Hilbert schemes of points on surfaces
• Wall-Crossing implies Brill-Noether. Applications of stability conditions on surfaces