Uniruledness of some moduli spaces of pointed spin curves

Abstract

The moduli space ${\mathcal{S}}_{g,2n}$ parametrizes pointed curves with spin structure. These are tuples $[C,p_1,\dots ,p_{2n},\eta ]$ where $\eta \in \text{Pic}(C)$ such that ${\eta }^{\otimes 2}\cong {\omega }_C(-p_1-\cdots -p_{2n})$. We prove that ${\mathcal{S}}_{2,4}$, ${\mathcal{S}}_{2,6}$, ${\mathcal{S}}_{3,2}$, ${\mathcal{S}}_{3,4}$, ${\mathcal{S}}_{3,6}$, ${\mathcal{S}}_{4,2}$, ${\mathcal{S}}_{4,4}$, ${\mathcal{S}}_{5,2}$ and ${\mathcal{S}}_{5,4}$ are uniruled.

v3: Minor changes