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Equations of linear subvarieties of strata of differentials

### Frederik Benirschke, Benjamin Dozier and Samuel Grushevsky

Geometry & Topology 26 (2022) 2773–2830
DOI: 10.2140/gt.2022.26.2773
##### Abstract

We investigate the closure $\overline{M}$ 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 $M\phantom{\rule{-0.17em}{0ex}}$, 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 $\overline{M}$ 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 to the case of real linear subvarieties of meromorphic strata.

##### Keywords
Teichmüller dynamics, moduli of curves, flat surfaces
##### Mathematical Subject Classification
Primary: 37F34
Secondary: 14H15, 32G15
##### Publication
Received: 14 December 2020
Accepted: 8 May 2021
Published: 13 December 2022
Proposed: Benson Farb
Seconded: Paul Seidel, David M Fisher
##### Authors
 Frederik Benirschke Department of Mathematics Stony Brook University Stony Brook, NY United States Benjamin Dozier Department of Mathematics Stony Brook University Stony Brook, NY United States Samuel Grushevsky Department of Mathematics Stony Brook University Stony Brook, NY United States