Equations of linear subvarieties of strata of differentials

We investigate the closure M of a linear subvariety M of a stratum of meromorphic differentials in the multiscale compactification constructed by Bainbridge, Chen, Gendron, Grushevsky and Möller. 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 P¹. As applications of our techniques, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright to the case of real linear subvarieties of meromorphic strata.