Equations of linear subvarieties of strata of differentials

Benirschke, Frederik; Dozier, Benjamin; Grushevsky, Samuel

doi:10.2140/gt.2022.26.2773

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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 $\bar{M}$ 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 $\bar{M}$ 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.

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Keywords

Teichmüller dynamics, moduli of curves, flat surfaces

Mathematical Subject Classification

Primary: 37F34

Secondary: 14H15, 32G15

References

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

Volume 26, issue 6 (2022)