Equations of linear subvarieties of strata of differentials

Abstract

We investigate the closure M of a linear subvariety M of a stratum of meromorphic differentials in the multi-scale compactification constructed in [BCG + 19]. Given the existence of a boundary point of M of a given combinatorial type, we deduce that certain periods of the differential are pairwise proportional on M , and deduce further explicit linear defining relations. These restrictions on linear defining equations of M allow us to rewrite them as explicit analytic equations in plumbing coordinates near the boundary, which turn out to be binomial. This in particular shows that locally near the boundary M is a toric variety, and allows us to prove existence of certain smoothings of boundary points and to construct a smooth compactification of the Hurwitz space of covers of P 1 . As applications of our techniques, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright [Wri15] to the case of real linear subvarieties of meromorphic strata.

The closure of double ramification loci via strata of exact differentials

“…By taking the closure of a double ramification locus or more generally of a Hurwitz space of covers of P 1 inside the moduli space of multi scale differentials one can produce smooth compactifications as we have seen in [BDG20]. It follows from our discussion in Section 5.1 that this yields birational models different from stacks of admissable covers and it seems worthwhile to investigate the relationship between the two.…”

“…Since exact differentials are described by the vanishing of all absolute periods, this requires an analysis of periods near the boundary of ΞM g,n (µ). The results obtained in [Ben20] and further refined in [BDG20] describe the boundary of subvarieties of strata of meromorphic differentials given by linear equations on periods in ΞM g,n (µ), and can thus be applied to the case of exact differentials. The idea of studying Hurwitz spaces via exact differentials has also appeared in [Sav17,Sec.…”

“…Furthermore, being an exact differential is equivalent to the vanishing of all absolute periods, a condition that is linear in period coordinates of the stratum. We can then describe the boundary of DR c g ( µ) in terms of twisted rational functions using the main result from [Ben20] and some further consequences from [BDG20]. A twisted rational function is a twistable rational function where we additionally mark all the critical points of the rational function, and such that the associated exact differential is a twisted differential in the sense of [BCGGM18].…”

The closure of double ramification loci via strata of exact differentials

“…Since exact differentials are described by the vanishing of all absolute periods, this requires an analysis of periods near the boundary of ΞM g,n (µ). The results obtained in [Ben20] and further refined in [BDG20] describe the boundary of subvarieties of strata of meromorphic differentials given by linear equations on periods in ΞM g,n (µ), and can thus be applied to the case of exact differentials.…”

“…Furthermore, being an exact differential is equivalent to the vanishing of all absolute periods, a condition that is linear in period coordinates of the stratum. We can then describe the boundary of DR c g ( µ) in terms of twisted rational functions using the main result from [Ben20] and some further consequences from [BDG20]. A twisted rational function is a twistable rational function where we additionally mark all the critical points of the rational function, and such that the associated exact differential is a twisted differential in the sense of [BCGGM18].…”