Maximal Rank Divisors on $\overline{\mathcal{M}}_{g,n}$

TL;DR

This paper calculates specific divisor classes on moduli spaces of curves and uses these to prove that certain moduli spaces are of general type, advancing understanding of their geometric properties.

Contribution

It explicitly computes divisor classes related to quadrics on curves and applies these results to determine the general type of specific moduli spaces.

Findings

The divisor class for loci where curves lie on quadrics is computed.

The moduli spaces 16,8 and 17,8 are shown to be of general type.

Existing divisor classes imply 12,10 is of general type.

Abstract

We compute the class of the effective divisors on $\overline{\mathcal{M}}_{g,n}$, which are set theoretically equal to the locus of moduli points $[C,p_1,\dots ,p_n]$ where $C$ lies on a quadric under the map given by the linear series $|K_C-p_1-\dots -p_n|$. Using this divisor class we show that the moduli spaces $\overline{\mathcal{M}}_{16,8}$ and $\overline{\mathcal{M}}_{17,8}$ are of general type. We also note that the divisor classes computed in the papers \cite{FV1} and \cite{FV2} can be used to show that $\overline{\mathcal{M}}_{12,10}$ is of general type as well.