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 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.