An Effective Divisor in $\overline{M_{g}}$ Defined by Ramification Conditions

TL;DR

This paper introduces a new effective divisor on the moduli space of stable curves, analyzes its class in the Picard group, and explores relations among coefficients similar to Brill-Noether divisors, providing insights into the structure of $ar{M}_g$.

Contribution

It defines a new divisor $ar{S^2 W}$ on $ar{M}_g$, computes its class, and reveals relations among coefficients akin to those of Brill-Noether divisors, advancing understanding of divisor classes.

Findings

Computed the class of $ar{S^2 W}$ in the Picard group.

Found relations among coefficients similar to Brill-Noether divisors.

Calculated the coefficient of $ extlambda$ in the divisor class.

Abstract

We define an effective divisor of the moduli space of stable curves $\overline{M_g}$, which is denoted $\overline{S^{2}W}$. Writing the class of $\overline{S^{2}W}$ in the Picard group of the moduli functor Pic$_{\text{fun}}(\overline{M_{g}})\otimes \mathbb{Q}$ in terms of the so-called Harer basis $\lambda,\delta_0,\ldots,\delta_{[g/2]}$, we prove that the relations among the coefficients of $\delta_1,\ldots,\delta_{[g/2]}$ are the same relations on coefficients as the Brill-Noether divisors. We present a result on effective divisors of $\overline{M_g}$ which could be useful to get the same relations on coefficients for other divisors. We also compute the coefficient of $\lambda$.