Green’s generic syzygy conjecture for curves of even genus lying on a \(K3\) surface. (English) Zbl 1080.14525

Voisin, Claire

J. Eur. Math. Soc. (JEMS) 4, No. 4, 363-404 (2002).

Summary: 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 \(\text{Cliff} C>l\) is equivalent to the fact that \(K_{g-l'-2,1}(C,K_C)=0, \forall l'\leq l\).
We propose a new approach, which allows up to prove this result for generic curves \(C\) of genus \(g(C)\) and gonality \(\text{gon(C)}\) in the range \[ \frac{g(C)}{3}+1\leq \text{gon(C)}\leq\frac{g(C)}{2}+1. \]

MSC:

14H51	Special divisors on curves (gonality, Brill-Noether theory)
14J28	\(K3\) surfaces and Enriques surfaces