Generic Beauville's Conjecture

TL;DR

This paper proves Beauville's conjecture that the pushforward of a general vector bundle under a finite cover of smooth curves is semistable or stable depending on the genus of the base curve, for general covers.

Contribution

It establishes Beauville's conjecture for general covers in any component of the Hurwitz space of smooth curves.

Findings

Proves semistability of pushforward bundles when genus of Y is at least 1.

Proves stability of pushforward bundles when genus of Y is at least 2.

Validates Beauville's conjecture for a broad class of covers in the Hurwitz space.

Abstract

Let $\alpha: X \to Y$ be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under $\alpha$ is semistable if the genus of $Y$ is at least $1$ and stable if the genus of $Y$ is at least $2$. We prove this conjecture if the map $\alpha$ is general in any component of the Hurwitz space of covers of an arbitrary smooth curve $Y$.