## Publicacions Matemàtiques

### Equigeneric and equisingular families of curves on surfaces

#### 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 (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

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

Dates
Revised: 30 July 2015
First available in Project Euclid: 22 December 2016

https://projecteuclid.org/euclid.pm/1482375629

Digital Object Identifier
doi:10.5565/PUBLMAT_61117_07

Mathematical Reviews number (MathSciNet)
MR3590119

Zentralblatt MATH identifier
1374.14025

#### Citation

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. https://projecteuclid.org/euclid.pm/1482375629