Equigeneric and equisingular families of curves on surfaces

• Autores: Thomas Dedieu, E. Sernesi
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 61, Nº 1, 2017, págs. 175-212
• Idioma: inglés
• DOI: 10.5565/10.5565_PUBLMAT 61117 07
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• Resumen

  • 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 armatively 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.