Equigeneric and equisingular families of curves on surfaces

2014/10/15 by Thomas Dedieu, Dedieu, Thomas, Edoardo Sernesi +1 · 2 citations
Mathematics · #14B07 (secondary) #14H10 (primary) #14H20 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14B07 #msc:14H10 #msc:14H20

paper · pdf · doi:10.48550/arxiv.1410.4221

final version, to appear in Publicacions Matemàtiques

arxiv created 2015/07/30 · arxiv updated 2015/07/31

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.

Cited by

• IVHS of nodal plane curves
• Severi varieties on Enriques surfaces of base change type and on rational elliptic surfaces

Related

• Obstructions to deforming maps from curves to surfaces
• Modular properties of nodal curves on K3 surfaces
• Nodal curves with general moduli on K3 surfaces
• Knotting of algebraic curves in complex surfaces
• Deformation of singular curves on surfaces