We give a classification of finite group actions on a $K3$ surface giving rise to $K3$ quotients, from the point of view of their fixed points. It is shown that except two cases, each such group gives rise to a unique type of fixed point set.

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 $K_{g-l^{\text{'}}-2,1}(C,K_C)=0,\forall l^{\text{'}}\le l$. We propose a new approach, which allows up to prove this result for generic curves $C$ of genus $g\left(C\right)$ and gonality $\text{gon(C)}$ in the range $$\frac{g\left(C\right)}{3}+1\le \text{gon(C)}\le \frac{g\left(C\right)}{2}+1.$$