Unirationality of the Hurwitz space \(\mathcal {H}_{9,8}\) (Q1682402)

The authors study the Hurwitz space \(\mathcal H_{9,8}\) which parameterizes \(8\)-sheeted covers of \(\mathbb P^1\) by curves of genus \(9\). The main result shows that \(\mathcal H_{9,8}\) is unirational. The idea of proof is to deduce the unirationality from the corresponding Brill-Noether locus. As a consequence, the authors develop an explicit computer code which can randomly produce a nodal curve of genus \(9\) with \(12\) nodes and a pencil of degree \(8\).