The Hurwitz space Picard rank conjecture for $d>g-1$

TL;DR

This paper proves that for degrees greater than g-1, the simple Hurwitz space has a trivial rational Picard group and is uniruled when the degree exceeds g+1, advancing understanding of its geometric properties.

Contribution

It establishes the triviality of the rational Picard group for certain Hurwitz spaces and demonstrates their uniruledness in specific degree ranges.

Findings

Trivial rational Picard group for $ ext{d} > ext{g}-1$

Uniruledness of the space for $ ext{d} > ext{g}+1$

Advances understanding of Hurwitz space geometry

Abstract

We show the simple Hurwitz space $\mathcal{H}_{g,d}$ has trivial rational Picard group for $d>g-1$ and is uniruled for $d>g+1$.