Koszul cohomology and applications to moduli

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.

26 pages

References in corpus (3)

• Birational aspects of the geometry of M_g
• Effective divisors on \bar{M}_g, curves on K3 surfaces and the Slope Conjecture
• The Kodaira dimension of the moduli space of Prym varieties

Cited by in corpus (13)

• The Kodaira dimensions of ${\overline{\mathcal{M}}}_{22}$ and ${\overline{\mathcal{M}}}_{23}$
• Degenerations of Grassmannians via lattice configurations
• A tropical proof of the Brill-Noether Theorem
• Computer aided Unirationality Proofs of Moduli Spaces
• The Strong Maximal Rank Conjecture and higher rank Brill--Noether theory
• Max Noether's Theorem for Integral Curves
• On the strong maximal rank conjecture in genus 22 and 23
• The maximal rank conjecture and rank two Brill-Noether theory
• Recent Developments in Brill-Noether Theory
• The non-abelian Brill-Noether divisor on ${\overline{\mathcal{M}}}_{13}$ and the Kodaira dimension of ${\overline{\mathcal{R}}}_{13}$
• Birational geometry of moduli of curves with an $S_3$-cover
• Geometric invariant theory of syzygies, with applications to moduli spaces
• On Green and Green-Lazarfeld conjectures for simple coverings of algebraic curves