Crossing the Transcendental Divide: From Translation Surfaces to Algebraic Curves

We study constructing an algebraic curve from a Riemann surface given via translation surface, which is collection of finitely many polygons in the plane with sides identified by translation. use theory discrete surfaces to give algorithm for approximating Jacobian variety whose polygon can be decomposed into squares. first implement case L shaped where already known. The also implemented any genus specific examples Jenkins–Strebel representatives, dense family that, until now, lived squarely on analytic side transcendental divide between and curves. Using theta functions, we numerical experiments resulting conjectures up 5.