The uniruledness of the Prym moduli space of genus 9 (Q6547687)

The Prym moduli space \(\mathcal{R}_g\) is known to be uniruled for \(g\le 8\), and of general type for \(g\ge 13\) except possibly for \(g=16\); moreover \(\mathcal{R}_{12}\) has non-negative Kodaira dimension. This paper shows that also \(\mathcal{R}_9\) is uniruled, leaving only \(g=10,\,11,\,16\) as completely open cases.\N\NThe proof involves a study of the Brill-Noether divisor \(\mathcal{D}_9\) that parametrises pairs \((C,\eta\in \operatorname{Pic}^0(C)[2])\in\mathcal{R}_9\) for which there is a line bundle \(L\) of degree \(8\) on \(C\) with \(h^0(C,L)\ge 3\) and \(H^0(C,L\otimes\eta)\neq 0\). It is fairly easy to check that this is a divisor. Significantly harder is the calculation of the class of \(\overline{\mathcal{D}}_9\) in \(\overline{\mathcal{R}}_9\): it uses the fact that the pencil of Prym curves on a Nikulin surface (a \(K3\) surface with a symplectic involution), which is proved here following an idea due to Lelli-Chiesa.\N\NThe components of \(\overline{\mathcal{D}}_9\) are uniruled and in fact covered by rational curves \(R\) satisfying \(R.K_{\overline{\mathcal{R}}_9}<0\) and \(R.{\overline{\mathcal{D}}_9}\ge 0\). If one knew that \({\overline{\mathcal{D}}_9}\) were irreducible, the result would follow easily from a pseudo-effectivity argument, but instead it is necessary to pick out a component whose intersection with \(R\) is non-negative. That is in fact true for all components, and is sufficient for the pseudo-effectivity argument, but there is a transversality condition that must be checked on one component to complete the proof.