Stability of Tschirnhausen Bundles

TL;DR

This paper proves the semistability and stability of Tschirnhausen bundles associated with general primitive maps of algebraic curves, depending on the genus of the target curve.

Contribution

It establishes the stability properties of Tschirnhausen bundles for general primitive maps of algebraic curves, extending understanding in algebraic geometry.

Findings

Tschirnhausen bundle is semistable if genus of Y is at least 1.

Tschirnhausen bundle is stable if genus of Y is at least 2.

Results hold over fields of characteristic zero or larger than r.

Abstract

Let $\alpha : X \to Y$ be a general degree $r$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $r$. We prove that the Tschirnhausen bundle of $\alpha$ is semistable if $g(Y) \geq 1$ and stable if $g(Y) \geq 2$.