Moduli of nodal curves on K3 surfaces

We consider modular properties of nodal curves on general K3 surfaces. Let Kp be the moduli space of primitively polarized K3 surfaces (S,L) of genus p⩾3 and Vp,m,δ→Kp be the universal Severi variety of δ-nodal irreducible curves in |mL| on (S,L)∈Kp. We find conditions on p,m,δ for the existence of an irreducible component V of Vp,m,δ on which the moduli map ψ:V→Mg (with g=m2(p−1)+1−δ) has generically maximal rank differential. Our results, which for any p leave only finitely many cases unsolved and are optimal for m⩾5 (except for very low values of p), are summarized in Theorem 1.1 in the introduction.