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2017 Equigeneric and equisingular families of curves on surfaces
T. Dedieu, E. Sernesi
Publ. Mat. 61(1): 175-212 (2017). DOI: 10.5565/PUBLMAT_61117_07

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.

Citation

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T. Dedieu. E. Sernesi. "Equigeneric and equisingular families of curves on surfaces." Publ. Mat. 61 (1) 175 - 212, 2017. https://doi.org/10.5565/PUBLMAT_61117_07

Information

Received: 20 May 2015; Revised: 30 July 2015; Published: 2017
First available in Project Euclid: 22 December 2016

zbMATH: 1374.14025
MathSciNet: MR3590119
Digital Object Identifier: 10.5565/PUBLMAT_61117_07

Subjects:
Primary: 14H10
Secondary: 14B07 , 14H20

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

Rights: Copyright © 2017 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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