Plane curve diagrams and geometrical applications

We look at plane curve diagrams (f,alpha), which are given by a plane curve multigerm alpha : (R, S) -> R(2) and a function on it f : (R, S) -> R. We obtain a classification of all such diagrams, where alpha has e-codimension <= 2 and f has finite order. Then we define an equivalence between plane curves which we call Ah(alpha)-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 R(3).

FAPESP Fundaeao de Amparo a Pesquisa do Estado de Sao Paulo[01/00652-0]

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

CAPES Coordenacao de Aperfeioamento de Pessoal de Ensino Superior[BEX 3647056]

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Comision Interministerial de Ciencia y Tecnologia

Comision Interministerial de Ciencia y Tecnologia[MTM2006-06027]