Koszul cohomology and applications to moduli

TL;DR

This paper reviews recent advances in the study of syzygies of algebraic curves, proving key conjectures and applying Koszul cohomology to understand the geometry of moduli spaces, while proposing new conjectures in the field.

Contribution

It provides a complete proof of Green's Conjecture for hexagonal curves and introduces several new conjectures on syzygies and moduli space geometry.

Findings

Proved Green's Conjecture for arbitrary hexagonal curves.

Applied Koszul cycles to moduli space birational geometry.

Proposed new conjectures on syzygies and Prym-Green conjecture.

Abstract

We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of new results, including a complete solution to Green's Conjecture for arbitrary hexagonal curves. Finally, we propose several new conjectures on syzygies, including a Prym-Green conjecture for l-roots of trivial bundles as well as a strong Maximal Rank Conjecture for generic curves. To appear in the Proceedings of Clay Mathematical Institute.