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The equation of the Kenyon-Smillie \((2, 3, 4)\)-Teichmüller curve. (English) Zbl 1404.32027

Summary: 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.

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces
14H10 Families, moduli of curves (algebraic)

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References:

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