This paper studies modular properties of nodal curves on general \(K3\) surfaces.
Let \((S,L)\) be a smooth, primitively polarized \(K3\) surface of genus \(p\geq 2\), with \(L\) a globally generated, indivisible line bundle with \(L^2=2p-2\). Let \({\mathcal{K}}_p\) denote the moduli space (or stack) of smooth primitively polarized \(K3\) surfaces of genus \(p\) which is of dimension \(19\).
For \(m\geq 1\), the arithmetic genus \(p(m)\) of the curves in \(|mL|\) is given by \(p(m)=m^2(p-1)+1\). Let \(\delta\) be an integer such that \(0\leq \delta\leq p(m)\). Consider the quasi-projective scheme \({\mathcal{V}}_{p,m,\delta}\) called the \((m,\delta)\)-universal Severi variety. There is the projection \({\mathcal{V}}_{p,m,\delta} \to {\mathcal{K}}_p\) whose fiber over \((S,L)\) is the variety \(V_{m,\delta}(S)\) called the Severi variety of \(\delta\)-nodal irreducible curves in \(|mL|\). The variety \(V_{m,\delta}(S)\) is smooth of dimension \(g:=p(m)-\delta\) (the geometric genus of any curve in \({\mathcal{V}}_{m,\delta}\)). There is the moduli map \(\psi_{m,\delta}: {\mathcal{V}}_{m,\delta}\to M_g\) where \(M_g\) is moduli space of genus \(g\)-curves. The purpose of this paper is to 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 \(\psi: {\mathcal{V}}\to M_g\) (with \(g=p(m)-\delta)\) has generically maximal rank differential.
The results are summarized in the following theorem.
{Theorem}. (A) For the following values of \(p\geq 3, m\) and \(g\), there is an irreducible component \({\mathcal{V}}\) of \({\mathcal{V}}_{m,\delta}\) such that the moduli map \({\mathcal{V}}\to M_g\) is dominant:

\(\bullet\)

\(m=1\) and \(0\leq g\leq 7\);

\(\bullet\)

\(m=2\), \(p\geq g-1\) and \(0\leq g\leq 8\);

\(\bullet\)

\(m=3\), \(p\geq g-2\) and \(0\leq g\leq 9\);

\(\bullet\)

\(m=4\), \(p\geq g-3\) and \(0\leq g\leq 10\);

\(\bullet\)

\(m\geq 5\), \(g\geq g-4\) and \(0\leq g\leq 11\).

(B) For the following values of \(p, m\) and \(\delta\), there is an irreducible component \({\mathcal{V}}\) of \({\mathcal{V}}_{m,\delta}\) such that he moduli map \({\mathcal{V}}\to M_g\) is generically finite into its image:

\(\bullet\)

\(m=1\) and \(p\geq g\geq 15\);

\(\bullet\)

\(2\leq m\leq 4\), \(p\geq 15\) and \(g\geq 16\);

\(\bullet\)

\(m\geq 5\), \(p\geq 7\) and \(g\geq 11\).

To prove this, it suffices to exhibit some specific curves in the universal Severi variety such that a component of the fiber of the moduli map at that curve has the right dimension, i.e., \(\min\{0,22-2g\}\). To do this, consider partial compactifications \(\overline{\mathcal{K}}_p\) and \(\overline{\mathcal{V}}_{m,\delta}\) and prove the assertion for curves in the boundary.
The partial compactification \(\overline{\mathcal{K}}_p\) is obtained by adding to \({\mathcal{K}}_p\) a divisor parametrizing pairs \((S,T)\) where \(S\) is a reducible \(K3\) surface of genus \(p\) that can be realized in \({\mathbb{P}}^p\) as the union of two rational normal scrolls intersecting along an elliptic normal curve \(E\), and \(T\) is the zero scheme of a section of the first cotangent sheaf \(T_S^1\), consisting of \(16\) points on \(E\). These \(16\) points together with subtle deformation argument of nodal curves establishes the assertion for \(m=1\), and for \(m>1\), specialization argument of curves in the Severi variety to suitable unions of curves used for \(m=1\) plus other types of limit curves establishes the assertions.