Equigeneric and equisingular families of curves on surfaces

TL;DR

This paper studies the deformation of integral curves on algebraic surfaces into nodal curves while maintaining genus, providing positive results for certain surfaces and examples where it fails.

Contribution

It establishes conditions under which integral curves can be deformed into nodal curves on various algebraic surfaces, expanding understanding of curve families on these surfaces.

Findings

Positive deformation results for Del Pezzo and Hirzebruch surfaces

Partial results for K3 surfaces

Counterexamples where deformation is not possible

Abstract

We investigate the following question: let $C$ be an integral curve contained in a smooth complex algebraic surface $X$; is it possible to deform $C$ in $X$ into a nodal curve while preserving its geometric genus? We affirmatively answer it in most cases when $X$ is a Del Pezzo or Hirzebruch surface, and in some cases when $X$ is a $K3$ surface. Partial results are given for all surfaces with numerically trivial canonical class. We also give various examples for which the answer is negative.