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