The closure of double ramification loci via strata of exact differentials

TL;DR

This paper describes the geometric closure of double ramification loci within the moduli space of curves by linking rational functions, their exact differentials, and strata of meromorphic differentials, providing new insights into their structure and relation to Teichmüller dynamics.

Contribution

It introduces a novel geometric description of the closure of double ramification loci using exact differentials and recent boundary results, connecting rational functions with meromorphic differential strata.

Findings

Closure of double ramification loci characterized geometrically.

Double ramification loci realized as linear subvarieties of differential strata.

New connections established between rational functions, differentials, and Teichmüller dynamics.

Abstract

Double ramification loci, also known as strata of $0$-differentials, are algebraic subvarieties of the moduli space of smooth curves parametrizing Riemann surfaces such that there exists a rational function with prescribed ramification over $0$ and $\infty$. We describe the closure of double ramification loci inside the Deligne-Mumford compactification in geometric terms. To a rational function we associate its exact differential, which allows us to realize double ramification loci as linear subvarieties of strata of meromorphic differentials. We then obtain a geometric description of the closure using our recent results on the boundary of linear subvarieties. Our approach yields a new way of relating the geometry of loci of rational functions and Teichm\"uller dynamics. We also compare our results to a different approach using admissible covers.