Abstract

In this work, we present an algorithm running over SAGE, which allows users to deal with group actions on Riemann surfaces up to topological equivalence. Our algorithm allows us to study the equisymmetric stratification of the branch locus (Formula presented.) of the moduli space (Formula presented.) of compact Riemann surfaces of genus (Formula presented.) corresponding to group actions with orbit genus 0. That is, it works for actions on surfaces of any genus in the case the genus of the quotient surface is zero, except for obvious hardware constraints. Our approach is toward studying inclusions and intersections of (closed) strata of (Formula presented.) We apply our algorithm to describe part of the geometry of the branch locus (Formula presented.) in terms of equisymmetric stratification. We also use it to compute all group actions up to topological equivalence for genus 5–10, this completes the lists. Finally, we add an optimized version of an algorithm, which allows us to identify Jacobian varieties of CM-type. As a byproduct, we obtain a Jacobian variety of dimension 11 which is isogenous to (Formula presented.) where Ei and (Formula presented.) are elliptic curves with complex multiplication.