On some one parameter families of genus 2 algebraic curves and half twists (Q878325)

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