Strata of $k$-differentials

TL;DR

This paper provides a comprehensive description of the compactification of strata of $k$-differentials on Riemann surfaces, including residue conditions and geometric properties, for all values of $k$.

Contribution

It introduces a complete compactification framework for $k$-differential strata, incorporating global residue conditions and admissible covers, extending previous work to all $k$.

Findings

Complete description of compactified strata of $k$-differentials.

Global $k$-residue conditions reformulated via admissible covers.

Analysis of deformation, residues, and flat structures of $k$-differentials.

Abstract

A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical line bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification of the moduli space of $k$-differentials. In this paper we give a complete description for the compactification of the strata of $k$-differentials in terms of pointed stable $k$-differentials, for all $k$. The upshot is a global $k$-residue condition that can also be reformulated in terms of admissible covers of stable curves. Moreover, we study properties of $k$-differentials regarding their deformations, residues, and flat geometric structure.