Equivariant Koszul Cohomology of Canonical Curves

TL;DR

This paper explores the structure of Koszul cohomology for smooth projective varieties with group actions, deriving explicit formulas for canonical curves using equivariant representation theory and Riemann-Roch theorems.

Contribution

It introduces a representation-theoretic framework for equivariant Koszul cohomology and derives explicit formulas for canonical curves combining several advanced theories.

Findings

Derived dimension formulas as identities between virtual representations.

Established properties of G-equivariant functors in Koszul complexes.

Obtained explicit formulas for canonical curves using equivariant Riemann-Roch.

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.