Title: Billiards and Ramified Covers of Curves
Thomas Schmidt, Oregon State University

Abstract: Given a Euclidean polygon, one can glue together copies (made by reflections through its various sides) of the polygon, to construct a surface. Straight-line paths with reflection off of the sides of the polygon can then be studied as geodesics on the constructed flat surface. Moreover, the complex structure of the plane induces a complex structure on the surface; dz induces an abelian differential.

Conversely, integration of an abelian differential ω on an algebraic curve X induces a flat structure. Since SL(2, R) acts on the plane, there is a natural action of this group on the space of abelian diffentials on algebraic curves of fixed genus. The orbit of any fixed pair (X, ω) projects to the Riemann moduli space of algebraic curves. On rare occasions, this projection is itself an algebraic curve.

We report on joint work with Pascal Hubert on the projections into moduli space of the orbit of pairs (X, ω) when ω is the pull-back of an abelian differential by way of a ramified covering.

PDF of slides