Commentarii Mathematici Helvetici


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Volume 82, Issue 2, 2007, pp. 413–449
DOI: 10.4171/CMH/97

Published online: 2007-06-30

On some one parameter families of genus 2 algebraic curves and half twists

Robert Silhol[1]

(1) Université de Montpellier II, France

In this paper we show that for certain families of surfaces in genus 2 there is an action of PSL2(ℤ) that can be expressed very naturally both in terms of Fenchel–Nielsen coordinates for the surfaces and in terms of equations of the associated algebraic curves. We also show that one of these families coincides with the SL2(ℝ) orbit of the translation surface tiled by three squares and that the above PSL2(ℤ) action is exactly induced by the natural action of SL2(ℤ) on this orbit.

Keywords: Hyperbolic surfaces, algebraic curves, translation surfaces, Teichmüller spaces, uniformization

Silhol Robert: On some one parameter families of genus 2 algebraic curves and half twists. Comment. Math. Helv. 82 (2007), 413-449. doi: 10.4171/CMH/97