Full paper in PDF:

$%F. Santos, Optimal degree construction of real algebraic plane nodal curves with
prescribed
topology, I: the orientable case, Rev. Mat. Univ. Complut. Madrid 10 (1997), supplementary,
291–310.
%$

Optimal Degree Construction of Real
Algebraic Plane Nodal Curves with
Prescribed Topology, I: the Orientable Case

Francisco SANTOS

Dpto. de Matemáticas, Estadística y Computación

Universidad de Cantabria

E-39071 Santander — Spain

Universidad de Cantabria

E-39071 Santander — Spain

ABSTRACT

We study a constructive method to find an algebraic curve in the real projective plane with a (possibly singular) topological type given in advance. Our method works if the topological model to be realized has only double singularities and gives an algebraic curve of degree , where and are the numbers of double points and connected components of . This degree is optimal in the sense that for any choice of the numbers and there exist models which cannot be realized algebraically with lower degree. Moreover, we characterize precisely which models have this property. The construction is based on a preliminary topological manipulation of the topological model followed by some perturbation technique to obtain the polynomial which defines the algebraic curve. This paper considers only the case in which has an orientable neighborhood. The non-orientable case will appear in a separate paper.

1991 Mathematics Subject Classification: 14P25, 14Q05.