On linear stability and syzygy stability for rank 2 linear series

TL;DR

This paper investigates the relationship between linear stability of linear series and slope stability of associated vector bundles, providing a counterexample to a conjecture on smooth plane curves of degree 7.

Contribution

The authors disprove a conjecture linking linear stability and slope stability for linear series on certain curves, specifically providing a counterexample on degree 7 plane curves.

Findings

Counterexample to the conjecture on degree 7 plane curves

Disproof of the equivalence between linear and slope stability in this case

Insights into stability properties of linear series on algebraic curves

Abstract

In previous works, the authors investigated the relationships between linear stability of a generated linear series $|V|$ on a curve $C$, and slope stabillity of the vector bundle $M_{V,L} := \ker (V \otimes \mathcal{O}_C \to L)$. In particular, the second named author and L. Stoppino conjecture that, for a complete linear system $|L|$, linear (semi)stability is equivalent to slope (semi)stability of $M_V$, and the first and third named authors proved that this conjecture holds for hyperelliptic and for generic curves. In this work we provide a counterexample to this conjecture on any smooth plane curve of degree $7$.