The closure of double ramification loci via strata of exact differentials

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 ∞. 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 Teichmüller dynamics. We also compare our results to a different approach using admissible covers.