Tschirnhausen Bundles of Quintic Covers of $\mathbb{P}^1$

TL;DR

This paper classifies Tschirnhausen bundles arising from degree 5 covers of the projective line, proving the existence of smooth covers with specific pushforward properties and analyzing the moduli space of such covers.

Contribution

It provides a classification of Tschirnhausen bundles for quintic covers and demonstrates the existence of smooth covers with these properties using a novel dimension counting approach.

Findings

Classified Tschirnhausen bundles for degree 5 covers.

Proved the existence of smooth covers with desired pushforward.

Developed a minimization theorem analogous to Bhargava's sieving argument.

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 $d = 5$. Equivalently, we classify which $\mathbb{P}^3$-bundles over $\mathbb{P}^1$ contain smooth irreducible degree $5$ covers of $\mathbb{P}^1$. Our main contribution is proving the existence of \emph{smooth} covers whose structure sheaf has the desired pushforward. We do this by showing that the substack of singular curves has positive codimension in the moduli stack of finite flat covers with desired pushforward. To compute the dimension of the space of singular curves, we prove a (relative) ``minimization theorem'', which is the geometric analogue of Bhargava's sieving argument when computing the densities of discriminants of quintic number fields.