Syzygies of curves on K3 surfaces

Green's conjecture predicts that the shape of the Betti tables of canonical curves are completely determined by the Clifford indices. We present a proof of Green's conjecture for any smooth curve on an arbitrary K3 surface. This result has a particular interest, due to Green's hyperplane section theorem. Our result implies that the shapes of Betti tables of projective K3 surfaces are determined by the Clifford indices of corresponding hyperplane sections. (This is joint work with G. Farkas)