Green's generic syzygy conjecture for curves of even genus lying on a $K\,3$ surface
Claire Voisin
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 ${\rm Cliff}\,C\gt l$ 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 ${\rm gon(C)}$ in the range $$\frac{g\,(C)}{3}+1\leq {\rm gon(C)}\leq\frac{g\,(C)}{2}+1.$$