The Green Conjecture for Exceptional Curves on a K3 Surface

We use the Brill–Noether theory to prove the Green conjecture for exceptional curves on K3 surfaces. Such curves count among the few ones having Clifford dimension ⩾3. We obtain our result by adopting an infinitesimal approach due to Pareschi, and using the degenerate version of the Hirschowitz–Ramanan–Voisin theorem obtained in the paper [4].