The Deligne-Mumford and the Incidence Variety Compactifications of the Strata of $\Omega\mathcal{M}_{g}$

TL;DR

This paper constructs and analyzes compactifications of strata in the moduli space of Abelian differentials, enabling the computation of their Kodaira dimensions and extending existing geometric structures.

Contribution

It introduces a new compactification approach for strata of Abelian differentials, extending parity concepts and applying plumbing constructions to study their geometry.

Findings

Computed Kodaira dimensions for certain strata.

Detailed analysis of hyperelliptic and odd minimal strata in genus three.

Extended parity and plumbing methods to compactifications.

Abstract

The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of the odd minimal stratum in genus three.