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 (this is due to Arbarello and Cornalba, Zariski, and Harris), 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.
"Equigeneric and equisingular families of curves on surfaces." Publ. Mat. 61 (1) 175 - 212, 2017. https://doi.org/10.5565/PUBLMAT_61117_07