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

In this paper, we provide an approach to computing the effective cone of moduli spaces of sheaves on a smooth quadric surface. We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of the moduli space 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 a smooth quadric surface.