On the vanishing of higher syzygies of curves

Abstract

The present paper is related to a conjecture made by Green and Lazarsfeld concerning 1-linear syzygies of curves embedded by complete linear systems of sufficiently large degrees. Given a smooth, irreducible, complex, projective curve $X$, we prove that the least integer $q$ for which the property $(M_q)$ fails for a line bundle $L$ on $X$ does not depend on $L$ as soon as its degree becomes sufficiently large. Consequently, this number is an invariant of the curve, and the statement of Green-Lazarsfeld's conjecture is equivalent to saying that this invariant equals the gonality of the curve. We verify the conjecture for plane curves, curves lying on Hirzebruch surfaces, and for generic curves having the genus sufficiently large compared to the gonality. We conclude the paper by proving that Green's canonical conjecture holds for curves lying on Hirzebruch surfaces.

to appear in Math. Z

Cited by in corpus (9)

• Green's Conjecture for curves on arbitrary K3 surfaces
• Green-Lazarsfeld's Conjecture for Generic Curves of Large Gonality
• Koszul cohomology and applications to moduli
• Lazarsfeld-Mukai bundles and applications
• Non-vanishing for Koszul cohomology of curves
• Lazarsfeld-Mukai bundles and applications. II
• Green's conjecture for binary curves
• Green's Conjecture for curves on rational surfaces with an anticanonical pencil
• The Rank of Syzygies of Canonical Curves