Bi-graded Koszul modules, K3 carpets, and Green's conjecture

TL;DR

This paper extends Koszul module theory to bi-graded cases, proving a vanishing theorem that confirms the Canonical Ribbon Conjecture and Green's conjecture in various characteristics, advancing understanding of algebraic curves.

Contribution

It introduces a bi-graded extension of Koszul modules and proves a vanishing theorem that verifies key conjectures in algebraic geometry over different fields.

Findings

Confirmed the Canonical Ribbon Conjecture over fields of characteristic zero or at least the Clifford index.

Validated Eisenbud and Schreyer's conjecture on characteristics where Green's conjecture holds.

Extended results to positive characteristics, showing Green's conjecture for generic curves of each gonality.

Abstract

We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the Canonical Ribbon Conjecture of Bayer and Eisenbud holds over a field of characteristic zero or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics results due to Aprodu and Voisin asserting that Green's Conjecture holds for generic curves of each gonality.