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

摘要

We show that the Hurwitz scheme $mathcal{H}_{g,d}$ parametrizing $d$-sheetedsimply branched covers of the projective line by smooth curves of genus $g$, upto isomorphism, is unirational for $(g,d)=(10,8)$ and $(13,7)$. Theunirationality is settled by using liaison constructions in $mathbb{P}^1imes mathbb{P}^2$ and $mathbb{P}^6$ respectively, and through the explicitcomputation of single examples over a finite field.