Remarks on families of singular curves with hyperelliptic normalizations

TL;DR

This paper investigates the existence and properties of families of curves with hyperelliptic normalizations on smooth projective surfaces, providing restrictions, examples, and a Reider-like result using deformation theory.

Contribution

It introduces new restrictions on such families, offers a Reider-like theorem without vector bundles, and provides explicit examples of these curve families.

Findings

Restrictions on families of hyperelliptic-normalized curves

A Reider-like theorem proved without vector bundles

Examples illustrating the existence of such curve families

Abstract

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.