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

Abstract

In this paper we show that for certain families of surfaces in genus 2 there is an action of ${\mathrm{PSL}}_2(Z)$ 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 ${\mathrm{SL}}_2(R)$ orbit of the translation surface tiled by three squares and that the above ${\mathrm{PSL}}_2(Z)$ action is exactly induced by the natural action of ${\mathrm{SL}}_2(R)$ on this orbit.