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The Quarterly Journal of Mathematics Advance Access published online on December 4, 2007
The Quarterly Journal of Mathematics, doi:10.1093/qmath/ham039
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© The author 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

PLANE CURVE DIAGRAMS AND GEOMETRICAL APPLICATIONS

F. S. Dias{dagger}

Universidade de São Paulo — ICMC, Av. do Trabalhador São Carlense, Centro, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil

J. Nuño-Ballesteros {ddagger}

Departament de Geometria i Topologia, Universitat de València, Campus de Burjassot, 46100 Burjassot, Spain

{dagger} Corresponding author. E-mail: scalco{at}icmc.usp.br

Received 28 November 2006; revised 11 June 2007
  Abstract

We look at plane curve diagrams (f,{alpha}), which are given by a plane curve multigerm {alpha} : (R, S) -> R2 and a function on it f:(R, S) -> R. We obtain a classification of all such diagrams, where {alpha} has Ae-codimension ≤ 2 and f has finite order. Then we define an equivalence between plane curves which we call Ah-equivalence and which is determined by the class of the diagram (h{alpha}, {alpha}). Here, h{alpha} denotes the height function of {alpha} with respect to its normal vector. This is an equivalence which not only takes into account the topology of the singularity of {alpha}, but also its flat geometry. Finally, we apply our results in order to obtain a classification of all the plane projections of a generic space curve {gamma} embedded in R3

{ddagger} E-mail: juan.nuno{at}uv.es


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