Birational Properties of Some Moduli Spaces Related to Tetragonal Curves of Genus 7

Let $\mathcal M_{7,n}$ be the (coarse) moduli space of smooth curves of genus 7 with n≥0 marked points defined over the complex field $\mathbb C$. We denote by $\mathcal M^1_{7,n;4}$ the locus of points inside $\mathcal M_{7,n}$ representing curves carrying a $g^1_4$. It is classically known that $\mathcal M^1_{7,n;4}$ is irreducible of dimension 17+n. We prove in this paper that $\mathcal M^1_{7,n;4}$ is rational for 0≤n≤11.