The equation of the Kenyon-Smillie (2,3,4)-Teichmüller curve
Matteo Costantini - Institut für Mathematik, Goethe-Universität Frankfurt/Main, Robert-Mayer-Str. 6–8, 60325 Frankfurt, Germany (email) Abstract: 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.
Received: February 2016; Revised: August 2016; Available Online: December 2016. |