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 \({\mathbb{P}}^{2}, {\mathbb{P}}^{1} \times {\mathbb{P}}^{1}\) and \(\mathbb{F}_{1}\). We discuss constructions of ample divisors and determine the ample cone for Hirzebruch surfaces and del Pezzo surfaces with K 2≥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 \({\mathbb{P}}^{1} \times {\mathbb{P}}^{1}\) or \(\mathbb{F}_{1}\), 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 \({\mathbb{P}}^{2}\) and Bridgeland stability, arxiv:1203.0316, 2012) to these surfaces.
To Jim Simons, with gratitude
Mathematics Subject Classification codes (2000): 14E30, 14C05, 14D20, 14D23
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Acknowledgements
During the preparation of this paper the first author was partially supported by the NSF grant DMS-0901128 and the second author was partially supported by the NSF CAREER grant DMS-0950951535 and an Alfred P. Sloan Foundation Fellowship. The second author would like to thank the Simons Foundation and the organizers of the Simons Symposium on Rational Points over Non-algebraically Closed Fields, Fyodor Bogomolov, Brendan Hassett, and Yuri Tschinkel, for a very productive and enlightening conference. It is a pleasure to thank Daniele Arcara, Arend Bayer, Jack Huizenga, and Emanuele Macrì for discussions about Bridgeland stability and the birational geometry of Hilbert schemes.
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Bertram, A., Coskun, I. (2013). The Birational Geometry of the Hilbert Scheme of Points on Surfaces. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds) Birational Geometry, Rational Curves, and Arithmetic. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6482-2_2
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