Publicacions Matemàtiques

Equigeneric and equisingular families of curves on surfaces

T. Dedieu and E. Sernesi

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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.

Article information

Publ. Mat., Volume 61, Number 1 (2017), 175-212.

Received: 20 May 2015
Revised: 30 July 2015
First available in Project Euclid: 22 December 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H10: Families, moduli (algebraic)
Secondary: 14H20: Singularities, local rings [See also 13Hxx, 14B05] 14B07: Deformations of singularities [See also 14D15, 32S30]

Families of singular curves on algebraic surfaces equigeneric and equisingular deformations nodal curves


Dedieu, T.; Sernesi, E. Equigeneric and equisingular families of curves on surfaces. Publ. Mat. 61 (2017), no. 1, 175--212. doi:10.5565/PUBLMAT_61117_07.

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