Stability and applications

We give a brief overview of Bridgeland’s theory of stability conditions, focusing on applications to algebraic geometry. We sketch the basic ideas in Bayer’s proof of the Brill–Noether Theorem and in the authors’ proof of a theorem by Gruson–Peskine and Harris on the genus of space curves.

This note originated from the lecture of the first author at the conference From Algebraic Geometry to Vision and AI: A Symposium Celebrating the Mathematical Work of David Mumford, held at the Center of Mathematical Sciences and Applications, Harvard University, August 18–20, 2018.

Keywords

Brill–Noether theorem, derived categories, space curves, stability conditions

2010 Mathematics Subject Classification

Primary 14J60. Secondary 14D20, 14F05.

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The first-named author was partially supported by the NSF FRG-grant DMS-1664215 and, during the writing of this paper, by the Institut des Hautes Études Scientifiques (IHÉS), and by a Poste Rouge CNRS at Université Paris-Sud.

The second-named author was partially supported by an AMS-Simons Travel Grant.

Received 27 September 2019

Accepted 22 February 2020

Published 12 May 2021