Totally geodesic subvarieties of the moduli space of curves and linear systems

TL;DR

This paper constructs linear systems on general curves within totally geodesic subvarieties of the moduli space of curves, deriving rank bounds and classifying certain subvarieties based on their dimension and zeros.

Contribution

It introduces a method to analyze linear systems on curves in totally geodesic subvarieties, providing new bounds and classifications in the moduli space context.

Findings

Constructed linear systems on general curves in totally geodesic subvarieties.

Established rank bounds for these subvarieties of dimension at least two.

Classified totally geodesic subvarieties with at most two zeros in certain strata.

Abstract

We construct a linear system on a general curve in a totally geodesic subvariety of the moduli space of curves. As a consequence, we obtain rank bounds for totally geodesic subvarieties of dimension at least two. Furthermore, we classify totally geodesic subvarieties of dimension at least two in strata with at most two zeros.