The unirationality of the Hurwitz schemes $\mathcal{H}_{10,8}$ and $\mathcal{H}_{13,7}$

TL;DR

This paper proves that certain Hurwitz schemes parametrizing branched covers of the projective line are unirational for specific genus and degree pairs, using liaison constructions and explicit computations over finite fields.

Contribution

It establishes the unirationality of $\

Findings

Unirationality of $\\mathcal{H}_{10,8}$ and $\\mathcal{H}_{13,7}$ proven.

Liaison constructions in projective spaces used for proofs.

Explicit finite field computations support the theoretical results.

Abstract

We show that the Hurwitz scheme $\mathcal{H}_{g,d}$ parametrizing $d$-sheeted simply branched covers of the projective line by smooth curves of genus $g$, up to isomorphism, is unirational for $(g,d)=(10,8)$ and $(13,7)$. The unirationality is settled by using liaison constructions in $\mathbb{P}^1 \times \mathbb{P}^2$ and $\mathbb{P}^6$ respectively, and through the explicit computation of single examples over a finite field.