Totally Geodesic Subvarieties of the Moduli Space of Curves and Linear Systems

Abstract

We construct a linear system on a general curve in a totally geodesic subvariety of the moduli space of curves. As a consequence, one obtains rank bounds for totally geodesic subvarieties of dimension at least two. Furthermore, this leads to a classification of totally geodesic subvarieties of dimension at least two in strata with at most two zeros.