Green's generic syzygy conjecture for curves of even genus lying on a \(K3\) surface (Q1871674)

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 \[ \frac{g(C)}{3}+1\leq \text{gon(C)}\leq\frac{g(C)}{2}+1. \]