Nodal curves on surfaces of general type

We study the family of irreducible curves with $\delta$ nodes belonging to a free linear system $|C|$ with smooth general member on a surface $S$ such that $|K_S|$ is ample. Under the assumption that $C$ is numerically equivalent to $pK_S$, $p\ge 3$ rational, we give an upper bound on $\delta$ ensuring that the family is smooth of codimension $\delta$ in $|C|$ (plus another slightly different result of the same type). We give examples showing that the bound is sharp. Some related results concerning linear normality of nodal curves are proved.