Equations of linear subvarieties of strata of differentials

Benirschke, Frederik¹ ; Dozier, Benjamin¹ ; Grushevsky, Samuel¹

¹ Department of Mathematics, Stony Brook University, Stony Brook, NY, United States

Geometry & topology, Tome 26 (2022) no. 6, pp. 2773-2830.

Résumé

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 $M$, 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 ${\mathbb{P}}^1$. 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.

Détail

Keywords: Teichmüller dynamics, moduli of curves, flat surfaces

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