The generic Green-Lazarsfeld secant conjecture

Abstract

Generalizing the well-known Green Conjecture on syzygies of canonical curves, Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy if and only if L fails to be (p+1)-very ample. Via lattice theory for special K3 surfaces, Voisin's solution of the classical Green Conjecture and calculations on moduli stacks of pointed curves, we prove: (1) The Green-Lazarsfeld Secant Conjecture in various degree of generality, including its strongest possible form in the divisorial case in the universal Jacobian. (2) The Prym-Green Conjecture on the naturality of the resolution of a general Prym-canonical curve of odd genus.

24 pages. Final version, to appear in Inventiones Math

Cited by in corpus (9)

• Singular divisors and syzygies of polarized abelian threefolds
• The Prym-Green Conjecture for torsion bundles of high order
• Stable maps and singular curves on K3 surfaces
• Excess dimension for secant loci in symmetric products of curves
• Prym varieties and moduli of polarized Nikulin surfaces
• A note on Nikulin surfaces and their moduli spaces
• The Rank of Syzygies of Canonical Curves
• The resolution of paracanonical curves of odd genus
• Difference varieties and the Green-Lazarsfeld Secant Conjecture