Moduli of nodal curves on K3 surfaces

We consider modular properties of nodal curves on general $K3$ surfaces. Let $\mathcal{K}_p$ be the moduli space of primitively polarized $K3$ surfaces $(S,L)$ of genus $p\geqslant 3$ and $\mathcal{V}_{p,m,\delta}\to \mathcal{K}_p$ be the universal Severi variety of $\delta$--nodal irreducible curves in $|mL|$ on $(S,L)\in \mathcal{K}_p$. We find conditions on $p, m,\delta$ for the existence of an irreducible component $\mathcal{V}$ of $\mathcal{V}_{p,m,\delta}$ on which the moduli map $\psi: \mathcal{V}\to \mathcal{M}_g$ (with $g= m^2 (p -1) + 1-\delta$) has generically maximal rank differential. Our results, which for any $p$ leave only finitely many cases unsolved and are optimal for $m\geqslant 5$ (except for very low values of $p$), are summarized in Theorem 1.1 in the introduction.