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Bi-algebraic geometry of strata of differentials in genus zero

Part of: Deformations of analytic structures Riemann surfaces Curves

Published online by Cambridge University Press:  30 June 2025

Frederik Benirschke
Frederik Benirschke*
Affiliation:
Mathematics Department, University of Chicago, Eckhart Hall 327, 5734 S University Ave, Chicago, IL 60637, USA benirschke@uchicago.edu
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Abstract

We study algebraic subvarieties of strata of differentials in genus zero satisfying algebraic relations among periods. The main results are Ax–Schanuel and André–Oort-type theorems in genus zero. As a consequence, one obtains several equivalent characterizations of bi-algebraic varieties. It follows that bi-algebraic varieties in genus zero are foliated by affine-linear varieties. Furthermore, bi-algebraic varieties with constant residues in strata with only simple poles are affine-linear. In addition, we produce infinitely many new linear varieties in strata of genus zero, including infinitely many new examples of meromorphic Teichmüller curves.

Keywords

moduli space of curvesstrata of differentialsbi-algebraic geometry

MSC classification

Primary: 14H10: Families, moduli (algebraic) 32G15: Moduli of Riemann surfaces, Teichmüller theory 30F30: Differentials on Riemann surfaces

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 3 , March 2025 , pp. 503 - 535
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence.

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