Complete curves in the strata of differentials

TL;DR

This paper offers an alternative proof that the strata of holomorphic differentials with fixed zero orders do not contain complete algebraic curves, using positivity of divisor classes on moduli spaces of curves.

Contribution

It presents a new proof method for a known result, replacing the maximum modulus principle with divisor class positivity techniques.

Findings

Strata of holomorphic differentials lack complete algebraic curves.

Positivity of divisor classes can be used to prove non-completeness.

Provides an alternative proof to Gendron's result.

Abstract

Gendron proved that the strata of holomorphic differentials with prescribed orders of zeros do not contain complete algebraic curves by applying the maximum modulus principle to saddle connections. Here we provide an alternative proof for this result by using positivity of divisor classes on moduli spaces of curves.