Equivariant Koszul Cohomology of Canonical Curves

Abstract

This paper investigates the representation-theoretic structure of the Koszul cohomology of a smooth projective variety $X$ over an algebraically closed field $k$, admitting an action of a finite group $G$ of order coprime to ${\rm char}(k)$. Properties of $G$-equivariant functors are employed to show that the associated Koszul complex is a complex of $kG$-modules, and to generalize known dimension formulas to identities between virtual representations. In the case of canonical curves, explicit formulas are obtained by combining the theory of equivariant Euler characteristics and equivariant Riemann-Roch theorems with that of generating functions for Schur functors.