Crossing the Transcendental divide: from Schottky groups to algebraic curves

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. We construct a family of non-hyperelliptic surfaces of genus at least 3 where we know the Riemann surface as well as properties of the canonical embedding, including a nontrivial symmetry group and a real structure with the maximal number of connected components (an M-curve). I will also share some numerical approximations where we try to detect the underlying algebraic curve through sampling. This is based on joint work with Ángel David Ríos Ortiz.