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arXiv · 2310.07523

Bi-algebraic geometry of strata of differentials in genus zero

Frederik Benirschke
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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 Andr\'e-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. Additionally, we produce infinitely many new linear varieties in strata of genus zero, including infinitely many new examples of meromorphic Teichm\"uller curves.

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Frederik Benirschke. 2023-10-11. Bi-algebraic geometry of strata of differentials in genus zero. https://doi.org/10.1112/s0010437x2400770x

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