The Fermat cubic and special Hurwitz loci in $\overline{\mathcal{M}}_g$

Abstract

We compute the class of the compactified Hurwitz divisor $\overline{\mathfrak{TR}}_d$ in $\overline{\mathcal{M}}_{2d-3}$ consisting of curves of genus $g=2d-3$ having a pencil $\mathfrak g^1_d$ with two unspecified triple ramification points. This is the first explicit example of a geometric divisor on $\overline{\mathcal{M}}_g$ which is not pulled-back form the moduli space of pseudo-stable curves. We show that the intersection of $\overline{\mathfrak{TR}}_d$ with the boundary divisor $\Delta_1$ in $\overline{\mathcal{M}}_g$ picks-up the locus of Fermat cubic tails.