A SAGE Package for n-Gonal Equisymmetric Stratification of

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 ${\mathcal{B}}_g$ of the moduli space ${\mathcal{M}}_g$ of compact Riemann surfaces of genus $g\ge 2,$ 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 ${\mathcal{B}}_g.$ We apply our algorithm to describe part of the geometry of the branch locus ${\mathcal{B}}_9,$ 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 $E_i^9\times E_{i\sqrt{3}}^2,$ where E_i and $E_{i\sqrt{3}}$ are elliptic curves with complex multiplication.

Keywords:

• group actions
• classification of actions
• equisymmetric stratification
• Riemann surfaces

1991 MATHEMATICS SUBJECT CLASSIFICATION:

• 14H15
• 30F10

Acknowledgments

We are deeply grateful to the anonymous referees for the time they took to carefully read our article and make several profound suggestions which we included and certainly improved the exposition. We also thank Camila Muñoz-Santander [29] for her help by performing computations with the aid of our algorithm which served as a way of testing it.

Additional information

Funding

This study is partially supported by ANID-FONDECYT grant 1180073.