  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.
