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arXiv · 1410.4221

Equigeneric and equisingular families of curves on surfaces

Thomas Dedieu·Edoardo Sernesi
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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.

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Thomas Dedieu, Edoardo Sernesi. 2015-07-30. Equigeneric and equisingular families of curves on surfaces. https://arxiv.org/abs/1410.4221

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