Moduli of nodal curves on K3 surfaces

We consider modular properties of nodal curves on general K3 surfaces. Let K_p be the moduli space of primitively polarizedK3 surfaces (S,L) of genus p and V_{p,m,δ} → K_p 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 optimalfor m (except for very low values of p), are summarized in Theorem 1.1 in the introduction.