Poincaré maps and Bol’s theorem

The usual Poincaré map is defined for a four-web admitting three independent Abelian relations, and gives a natural projective model of the web where the leaves are straight lines. This model thus proves that the web is associated with a quartic curve in the dual projective plane.

We shall present a generalization of this situation to the case of five-webs for which every extracted three-web carries an Abelian relation. This generalization takes into account not only the Abelian relations of the web, but also the spaces of Abelian relations of the extracted sub-webs. This way we prove the following theorem of G. Bol:

The main tool used in the separation of cases is a careful investigation of the configuration formed by the spaces of relations of the extracted sub-webs in the space of relations of the whole web. (PDF file)