Stabilizer Posets and the Local Moduli of Fermat Curves

Abstract

The automorphism group of a Riemann surface naturally acts on the vector space of holomorphic quadratic differentials on it, which determines the canonical representation of the automorphism group. In light of the theory of linear quotient families developed by S. Takamura, the universal family over the moduli space around the Riemann surface is determined by this representation together with the automorphism group action on the Riemann surface — it is essential to determine the stabilizer poset of the former. Unfortunately this theory does not tell us how to determine the canonical representation and its stabilizer poset. The aim of this paper is to explicitly determine the canonical representations of Fermat curves of all genera together with the stabilizer poset for genus 3 case. We then describe, with a group-theoretic flavor, the local moduli of the Fermat curve of genus 3.