  We give restrictions on the existence of families of curves on smooth
projective surfaces $S$ of nonnegative Kodaira dimension all having constant
geometric genus $g \geq 2$ and hyperelliptic normalizations. In particular, we
prove a Reider-like result whose proof is ``vector bundle-free'' and relies on
deformation theory and bending-and-breaking of rational curves in $\Sym^2(S)$.
We also give examples of families of such curves.
