Derived categories and the genus of space curves

TL;DR

This paper extends classical results on the genus of space curves to abelian threefolds using derived category wall-crossing techniques, providing new bounds for stable objects and insights into longstanding conjectures.

Contribution

It generalizes genus bounds to abelian threefolds and introduces wall-crossing methods for ideal sheaves, also offering new approaches to classical conjectures.

Findings

Bounds for Chern characters of stable sheaves

Extension of genus bounds to abelian threefolds

Initial progress on Hartshorne-Hirschowitz conjecture

Abstract

We generalize a classical result about the genus of curves in projective space by Gruson and Peskine to principally polarized abelian threefolds of Picard rank one. The proof is based on wall-crossing techniques for ideal sheaves of curves in the derived category. In the process, we obtain bounds for Chern characters of other stable objects such as rank two sheaves. The argument gives a proof for projective space as well. In this case these techniques also indicate an approach for a conjecture by Hartshorne and Hirschowitz and we prove first steps towards it.