The integral Picard groups of low-degree Hurwitz spaces

TL;DR

This paper computes the integral Picard groups of low-degree Hurwitz stacks, specifically for degree 4 and 5 covers, and explores their properties for simply branched covers, revealing finite groups with genus-dependent order.

Contribution

It provides the first explicit calculations of the integral Picard groups for these specific Hurwitz stacks, advancing understanding of their geometric and arithmetic structure.

Findings

Picard groups for degree 4 and 5 covers computed

Integral Picard groups for simply branched covers determined

Picard groups are finite with order depending on genus

Abstract

We compute the Picard groups with integral coefficients of the Hurwitz stacks parametrizing degree $4$ and $5$ covers of $\mathbb{P}^1$. As a consequence, we also determine the integral Picard groups of the Hurwitz stacks parametrizing simply branched covers. For simple branching, the Picard groups are finite, with order depending on the genus.