The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve
Matteo Costantini André Kappes

We compute the algebraic equation of the universal family over the Kenyon-Smillie (2, 3, 4)-Teichmüller curve, and we prove that the equation is correct in two different ways. Firstly, we prove it in a constructive way via linear conditions imposed by three special points of the Teichmüller curve. Secondly, we verify that the equation is correct by computing its associated Picard-Fuchs equation. We also notice that each point of the Teichmüller curve has a hyperflex and we see that the torsion map is a central projection from this point.

keywords: Teichmüller curve Veech group is triangle group universal family of curves Picard-Fuchs equation

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