Equigeneric and equisingular families of curves on surfaces

Thomas Dedieu (1) , Edoardo Sernesi (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 ROMA TRE - Università degli Studi Roma Tre = Roma Tre University

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.

Domaines: Géométrie algébrique [math.AG]

Thomas Dedieu, Edoardo Sernesi. Equigeneric and equisingular families of curves on surfaces. Publicacions Matemàtiques, 2017, 61, pp.175-212. ⟨hal-01979020⟩