Weak stability conditions on coherent systems of genus four curves

TL;DR

This paper investigates the stability conditions on the derived category of coherent systems for genus four curves, showing how these conditions degenerate to those on a related threefold, linking curve theory with threefold geometry.

Contribution

It extends the understanding of stability conditions on coherent systems for genus four curves and connects them to stability conditions on Kuznetsov components of cubic threefolds.

Findings

Stability conditions degenerate to those on Kuznetsov components of cubic threefolds.

The study applies Feyzbakhsh--Novik stability conditions to genus four curves.

Establishes a link between curve stability and threefold geometry.

Abstract

The derived category of coherent systems is an interesting triangulated category associated with a smooth, projective curve $C$. These categories admit Bridgeland stability conditions, as recently shown by Feyzbakhsh and Novik. Their construction depends explicitly on the higher rank Brill-Noether theory of $C$. In this short note, we study the Feyzbakhsh--Novik stability conditions for a general curve of genus four. We show that these stability conditions degenerate to a stability condition on the Kuznetsov component of the corresponding nodal cubic threefold, using a result of Alexeev-Kuznetsov.