The Birational Geometry of the Hilbert Scheme of Points on Surfaces

Aaron Bertram () and Izzet Coskun ()
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Aaron Bertram: University of Utah, Department of Mathematics
Izzet Coskun: University of Illinois at Chicago, Department of Mathematics, Statistics and Computer Science

A chapter in Birational Geometry, Rational Curves, and Arithmetic, 2013, pp 15-55 from Springer

Abstract: 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.

Keywords: Modulus Space; Line Bundle; Hilbert Scheme; Ample Line Bundle; Ideal Sheaf (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4614-6482-2_2

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DOI: 10.1007/978-1-4614-6482-2_2

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