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

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.