Wall-Crossing implies Brill-Noether. Applications of stability conditions on surfaces

TL;DR

This paper explores how wall-crossing techniques for Bridgeland stability conditions on surfaces can be used to reprove classical results like Lazarsfeld's Brill-Noether theorem and discusses recent applications in algebraic geometry.

Contribution

It demonstrates the use of wall-crossing methods to reprove classical theorems and surveys recent developments in stability conditions on surfaces.

Findings

Reproves Lazarsfeld's Brill-Noether theorem using wall-crossing.

Highlights applications of stability conditions in birational geometry.

Provides a survey of recent advances in the field.

Abstract

Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces. We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces. The intended reader is an algebraic geometer with a limited working knowledge of derived categories. This article is based on the author's talk at the AMS Summer Institute on Algebraic Geometry in Utah, July 2015.