The equivariant Hilbert series of the canonical ring of Fermat curves

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

Abstract

We consider a Fermat curve Fn:xn+yn+zn=0 over an algebraically closed field K of characteristic p≥0 and study the action of the automorphism group G=Z/nZ×Z/nZ⋊S3 on the canonical ring R=⨁H0(Fn,ΩFn⊗m) when p>3, p∤n and n−1 is not a power of p. In particular, we explicitly determine the classes [H0(Fn,ΩFn⊗m)] in the Grothendieck group K0(G,K) of finitely generated K[G]-modules, describe the respective equivariant Hilbert series HR,G(t) as a rational function, and use our results to write a program in Sage that computes HR,G(t) for an arbitrary Fermat curve.