Jorge V. Pereira
Webs and abelian relations I
Web geometry studies the superposition of foliations. A key concept on the subject is the one of abelian relation. It consists of finite collections of first integrals for the foliations defining the web which satisfy a functional equation of a particularly simple form. The set of abelian relations of a given web forms a finite dimensional vector space. S.-S. Chern proved in his Phd thesis (generalizing previous results by G. Bol) that for smooth d-webs on n-dimensional manifolds the dimension of the space of abelian relations is bounded by the Castelnuovo number (which gives the bound for the genus of non-degenerate degree d curves on n-dimensional projective spaces).
Webs with space of abelian relations having the maximal possible dimension(i.e. webs of maximal rank), have a rich structure. In dimension three and higher, results by Bol, Chern-Griffiths, and Trépreau guarantee that they are all defined by curves of maximal genus through projective duality. The situation in dimension two is considerably richer, and a complete picture is not available yet. It is perhaps worth mentioning, that the first example of a web of maximal rank on the plane not associated to a projective curves has among its abelian relations, Abel's equation for the dilog.
In this series of lectures, I will review some of the theory of abelian relations of codimension one webs. Although the theory is local in nature, all the known relevant examples are global. Thus I will also discuss
global properties of webs on projective surfaces.
Lecture #1. Webs and abelian relations. Chern's bound. Algebraizable webs. Projective duality.