On the rationality of some moduli spaces related to pointed trigonal curves

Abstract

For each $g\ge1$, let $\mathcal{T}_{g,n}\subseteq \mathcal{M}_{g,n}$ be the locus of points corresponding to curves carrying a base-point-free $g^1_3$. Here we give a proof that $\mathcal{T}_{g,n}$ is rational if $g\ge4$ and $1\le n\le 2g+7$.