The equivariant Hilbert series of the canonical ring of Fermat curves

TL;DR

This paper analyzes the action of the automorphism group on the canonical ring of Fermat curves, explicitly determining the classes in the Grothendieck group and describing the equivariant Hilbert series as a rational function, with computational implementation.

Contribution

It explicitly computes the classes of sections in the Grothendieck group and describes the equivariant Hilbert series for Fermat curves under certain conditions, providing a computational tool.

Findings

Explicit classes in the Grothendieck group for Fermat curves.

Rational form of the equivariant Hilbert series.

Sage program for computing the Hilbert series.

Abstract

We consider a Fermat curve $F_n:x^n+y^n+z^n=1$ over an algebraically closed field $k$ of characteristic $p\geq0$ and study the action of the automorphism group $G=\left(\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\right)\rtimes S_3$ on the canonical ring $R=\bigoplus H^0(F_n,\Omega_{F_n}^{\otimes m})$ when $p>3$, $p\nmid n$ and $n-1$ is not a power of $p$. In particular, we explicitly determine the classes $[H^0(F_n,\Omega_{F_n}^{\otimes m})]$ in the Grothendieck group $K_0(G,k)$ of finitely generated $k[G]$-modules, describe the respective equivariant Hilbert series $H_{R,G}(t)$ as a rational function, and use our results to write a program in Sage that computes $H_{R,G}(t)$ for an arbitrary Fermat curve.