Computational Aspects of Equivariant Hilbert Series of Canonical Rings for Algebraic Curves

Charalambous, Hara and Karagiannis, Kostas and Karanikolopoulos, Sotiris and Kontogeorgis, Aristides

Abstract

We study computational aspects of the problem of decomposing finite group actions on graded modules arising in arithmetic geometry, in the context of ordinary representation theory. We provide an algorithm to compute the equivariant Hilbert series of automorphisms acting on canonical rings of projective curves, using the formulas of Chevalley and Weil. Further, we apply our results on Fermat curves, determine explicitly the respective equivariant Hilbert series and extend the computation to the short exact sequence that arises from Petri's Theorem. Finally, we implement the above computations in Sage.