The Birational Geometry of the Hilbert Scheme of Points on Surfaces

In this paper, we study the birational geometry of the Hilbert scheme of points on a smooth, projective surface, with special emphasis on rational surfaces such as P2,P1 × P1 and F1. We discuss constructions of ample divisors and determine the ample cone for Hirzebruch surfaces and del Pezzo surfaces with K2 ≥ 2. As a corollary, we show that the Hilbert scheme of points on a Fano surface is a Mori dream space. We then discuss effective divisors on Hilbert schemes of points on surfaces and determine the stable base locus decomposition completely in a number of examples. Finally, we interpret certain birational models as moduli spaces of Bridgeland stable objects. When the surface is P1 × P1 or F1, we find a precise correspondence between the Mori walls and the Bridgeland walls, extending the results of [ABCH] to these surfaces.