The generic Green-Lazarsfeld secant conjecture

TL;DR

This paper proves the Green-Lazarsfeld Secant Conjecture and the Prym-Green Conjecture, extending the understanding of syzygies and linear series on algebraic curves using advanced geometric and lattice-theoretic methods.

Contribution

It establishes the Secant Conjecture in various degrees and confirms the Prym-Green Conjecture for Prym-canonical curves, advancing the theory of syzygies in algebraic geometry.

Findings

Proved the Secant Conjecture in multiple cases including the divisorial case.

Confirmed the Prym-Green Conjecture for odd genus Prym-canonical curves.

Extended the understanding of syzygies and linear series on algebraic curves.

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.