The Kodaira dimension of even spin strata of Abelian differentials

TL;DR

This paper proves that for large genus, the minimal even spin components of Abelian differential strata are of general type, advancing understanding of their birational geometry.

Contribution

It establishes the general type of minimal even spin components for large genus, using effective divisor class computations.

Findings

Minimal even spin components are of general type for large genus.

Effective divisor classes are computed and estimated on these components.

Complements previous work on odd spin and general strata.

Abstract

The even spin components of the strata of Abelian differentials are difficult to handle from a birational geometry perspective due to the fact that their spin line bundles have more sections than expected. Nevertheless, in this paper, we prove that for large genus, the minimal even spin components are of general type. This result complements the previous work by the second and third authors, together with Costantini, on the Kodaira dimension of general strata and the minimal odd spin components of Abelian differentials. Our main technical tool is the computation and estimation of a series of effective divisor classes on the even spin components.