# Cubic curves and totally geodesic subvarieties of moduli space

### Abstract

In this paper we present the first example of a primitive, totally geodesic subvariety $F \subset \mathcal{M}_{g,n}$ with $\dim(F)>1$. The variety we consider is a surface $F \subset \mathcal{M}_{1,3}$, defined using the projective geometry of plane cubic curves. We also obtain a new series of Teichmüller curves in $\mathcal{M}_4$, and new $\mathrm{SL}_2(\mathbb{R})$–invariant varieties in the moduli spaces of quadratic differentials and holomorphic $1$-forms.

## Authors

Curtis T. McMullen

Mathematics Department, Harvard University, Cambridge, MA 02138-2901

Ronen E. Mukamel

Mathematics Department, Rice University, Houston, TX 77005

Alex Wright

Mathematics Department, Stanford University, Stanford, CA 94305