Equations of linear subvarieties of strata of differentials

Abstract

Strata of exact differentials are moduli spaces for differentials on Riemann surfaces with vanishing absolute periods. Our main result is that classes of closures of strata of exact differentials inside the moduli space of multi-scale differentials lie in the divisorial tautological ring. By relating exact differentials to rational functions we obtain a new proof that classes of Hurwitz spaces are tautological and a new method for computations.

The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials

“…Working with the normalization of the closure in some ambient compactification is usually unsuitable for intersection theory computations. Here, however, thanks to the work of Benirschke-Dozier-Grushevsky ( [BDG22]) and some minor upgrades we are able to work with this closure.…”

“…Rather we consider the normalization of the closure of a linear submanifold inside the moduli space of multi-scale differentials ΞM g,n (µ). We recall from [BDG22] the basic properties of such closures. The goal of this section is to make precise and to explain the following two slogans:…”

“…Decomposition of the logarithmic tangent bundle. We now define a Γ-adapted basis, combining [BDG22] and [CMZ22] with the goal of giving a decomposition of the logarithmic tangent bundle that is inherited by a linear submanifold, if the Γ-adapted basis is suitably chosen.…”

“…[i] j be a dual horizontal-crossing cycle, i.e. i is the top level (in the sense of [BDG22]) of this cycle, α…”

“…We summarize the relevant parts of [BDG22]. Equations of H are interpreted as homology classes and we say that a horizontal node is crossed by an equation, if the corresponding vanishing cycles has non-trivial intersection with the equation.…”