Birational geometry of moduli spaces of sheaves and Bridgeland stability

Abstract

Moduli spaces of sheaves and Hilbert schemes of points have experienced a recent resurgence in interest in the past several years, due largely to new techniques arising from Bridgeland stability conditions and derived category methods. In particular, classical questions about the birational geometry of these spaces can be answered by using new tools such as the positivity lemma of Bayer and Macrì. In this article we first survey classical results on moduli spaces of sheaves and their birational geometry. We then discuss the relationship between these classical results and the new techniques coming from Bridgeland stability, and discuss how cones of ample divisors on these spaces can be computed with these new methods. This survey expands upon the author's talk at the 2015 Bootcamp in Algebraic Geometry preceding the 2015 AMS Summer Research Institute on Algebraic Geometry at the University of Utah. Contents 1. Introduction 1 2. Moduli spaces of sheaves 2 3. Properties of moduli spaces 9 4. Divisors and classical birational geometry 15 5. Bridgeland stability 24 6. Examples on P 2 31 7. The positivity lemma and nef cones 35 References 40

Lectures on Bridgeland Stability

“…G k the k-vector space G ⊗ k for a field k and abelian group G X a smooth projective variety over C I Z the ideal sheaf of a closed subscheme Z ⊂ X D b (X) the bounded derived category of coherent sheaves on X ch(E) the Chern character of an object E ∈ D b (X) K 0 (X) the Grothendieck group of X K num (X) the numerical Grothendieck group of X NS(X) the Néron-Severi group of X N 1 (X) NS(X) R Amp(X) the ample cone inside N 1 (X) Pic d (C) the Picard variety of lines bundles of degree d on a smooth curve C many explanations on the topics of these notes. We are also grateful to Jack Huizenga for sharing a preliminary version of his survey article [Hui16] with us and to the referee for very useful suggestions which improved the readability of these notes. The first author would also like to thank very much the organizers of the two schools for the kind invitation and the excellent atmosphere, and the audience for many comments, critiques, and suggestions for improvement.…”

“…There is also a recent survey [Hui16] focusing more on both the classical theory of semistable sheaves and concrete examples still involving Bridgeland stability. The note [Bay16a] focuses instead on deep geometric applications of the theory (the classical Brill-Noether theorem for curves).…”