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Web geometry is devoted to the study of families of foliations which are in general position. We restrict ourselves to the local situation, in the neighborhood of the origin in C2 , with d ≥ 1 complex analytic foliations of curves in general position. We are interested in the geometry of such configurations, that is, properties of planar d-webs which are invariant with respect to analytic local isomorphisms of C2 . The initiators of the subject are W. Blaschke, G. Thomsen and G. Bol in the 1930’s (cf. [B-B], [B] and for instance [H1]). . | Annals of Mathematics On planar web geometry through abelian relations and connections By Alain H enaut Annals of Mathematics 159 2004 425 445 On planar web geometry through abelian relations and connections By Alain Hénaut 1. Introduction Web geometry is devoted to the study of families of foliations which are in general position. We restrict ourselves to the local situation in the neighborhood of the origin in C2 with d 1 complex analytic foliations of curves in general position. We are interested in the geometry of such configurations that is properties of planar d-webs which are invariant with respect to analytic local isomorphisms of C2. The initiators of the subject are W. Blaschke G. Thomsen and G. Bol in the 1930 s cf. B-B B and for instance H1 . Methods used here extend some works by S. S. Chern and P. A. Griffiths cf. for instance G1 G2 C C-G which bring a resurgence of interest in web geometry closely related to basic results due to N. Abel S. Lie H. Poincare and G. Darboux. For recent results and applications of web geometry in various domains refer to I. Nakai s introduction all papers and references contained in W . Let O C x y be the ring of convergent power series in two variables. A germ of a nonsingular d-web W d in C2 0 is defined by a family of leaves which are germs of level sets Fi x y const. where Fị G O can be chosen to satisfy Fi 0 0 such that dFi 0 dFj 0 0 for 1 i j d from the assumption of general position. From the local inverse theorem the study of possible configurations for the different W d is interesting only for d 3. The classification of such W d is a widely open problem and the search for invariants of planar webs W d motivates the present work. Let F x y p a0 x y .pd a1 x y .pd-1 - ad x y be an element of O p without multiple factor not necessarily irreducible and such that a0 0. We denote by R 1 2 a0. A the p-resultant of F where A G O is its p-discriminant. In a neighborhood of x0 y0 G C2 such that R x0 y0 0 the Cauchy theorem
