Remarks on families of singular curves with hyperelliptic normalizations ✩

We give restrictions on the existence of families of curves on smooth projective surfaces S of nonnegative Kodaira dimension all having constant geometric genus pg ⩾ 2 and hyperelliptic normalizations. In particular, we prove a Reider-like result that relies on deformation theory and bending-and-breaking of rational curves in Sym2(S). We also give examples of families of such curves.