The Fermat cubic and special Hurwitz loci in M_g

TL;DR

This paper computes a new divisor class in the moduli space of curves related to Fermat cubics and special ramification, revealing novel geometric properties and enumerative applications in algebraic geometry.

Contribution

It introduces the first geometric divisor on M_g not derived from a pseudo-stable moduli space, linked to Fermat cubic tails and ramification loci.

Findings

Computed the class of a locus with two triple ramification points in M_g

Identified the divisor as the first non-pullback divisor on the pseudo-stable moduli space

Provided enumerative results on coverings with special ramification behavior

Abstract

We compute the class of the locus in M_g of curves having a pencil with two unspecified triple ramification points. This is the first example of a geometric divisor on M_g which is not the pull-back of a divisor on the moduli space of pseudo-stable curves. This space, in which elliptic tails are replaced by cusps, appears as a result of the first divisorial contraction in the minimal model program for M_g. In particular, we show that our divisor picks-up the locus of Fermat cubic tails when restricted to the boundary divisor of elliptic tails. We also give various enumerative applications concerning coverings of the generic curve having special ramification behaviour.