Tschirnhausen bundles of sextic covers of $\mathbb{P}^1$

TL;DR

This paper classifies Tschirnhausen bundles arising from degree 6 covers of the projective line, revealing that all constraints are explained by algebra multiplication and that all such bundles are realized by covers with subcovers.

Contribution

It provides a complete classification of Tschirnhausen bundles for sextic covers and links constraints to algebraic multiplication structures.

Findings

All constraints on the pushforward bundles are explained by algebra multiplication.

Every possible pushforward bundle is realized by a cover with a nontrivial proper subcover.

The classification applies specifically to degree 6 covers of the projective line.

Abstract

A degree $d$ genus $g$ cover of the complex projective line by a smooth irreducible curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. We classify the bundles that arise this way when $d = 6$. Interestingly, our methods show that all constraints on the pushforward are ``explained'' by multiplication in an algebra. Finally, we show that all possible pushforwards are realized by covers with a nontrivial proper subcover.