2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128503We 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.18 pagesAlgebraic Geometry14H10 (14E30, 14H51, 14J10)text