GREEN’S GENERIC SYZYGY CONJECTURE FOR CURVES OF EVEN GENUS LYING ON A K3 SURFACE

CLAIRE VOISIN

Abstract. We consider the generic Green conjecture on syzygies of a canonical curve, and particularly the following reformulation thereof: For a smooth projective curve C of genus g in characteristic 0, the condition Cliff C > l is equivalent to the fact that K_{g−l′−2,1}(C, K_{C}) = 0, ∀l′ ≤ l. We propose a new approach, which allows up to prove this result for generic curves C of genus g(C) and gonality gon(C) in the range

\frac{g(C)}{3} + 1 ≤ gon(C) ≤ \frac{g(C)}{2} + 1.