Birational geometry of moduli spaces of sheaves and Bridgeland stability

TL;DR

This paper surveys the birational geometry of moduli spaces of sheaves, highlighting recent advances using Bridgeland stability conditions and derived category techniques to analyze their ample cones and classical results.

Contribution

It connects classical results on moduli spaces with modern Bridgeland stability methods, providing new tools for understanding their birational geometry.

Findings

Classical results on moduli space birational geometry summarized

Bridgeland stability techniques linked to classical questions

Methods for computing cones of ample divisors explained

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 Macr\`i. 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.