The birational geometry of the hilbert scheme of points on surfaces

Abstract

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 K22. 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 Arcara et al. (The birational geometry of the Hilbert scheme of points on P2 and Bridgeland stability, arxiv:1203.0316, 2012) to these surfaces.