By Jorge Vitório Pereira, Luc Pirio

This booklet takes an in-depth examine abelian family members of codimension one webs within the complicated analytic environment. In its classical shape, internet geometry is composed within the examine of webs as much as neighborhood diffeomorphisms. an important a part of the speculation revolves round the proposal of abelian relation, a specific form of practical relation one of the first integrals of the foliations of an internet. major focuses of the booklet contain what percentage abelian kinfolk can an online hold and which webs are sporting the maximal attainable variety of abelian kinfolk. The ebook deals whole proofs of either Chern’s sure and Trépreau’s algebraization theorem, together with all of the helpful necessities that transcend straight forward complicated research or easy algebraic geometry. lots of the examples recognized modern of non-algebraizable planar webs of maximal rank are mentioned intimately. A ancient account of the algebraization challenge for maximal rank webs of codimension one can be presented.

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**Example text**

The abuse of terminology is justified by the fact that all these germs are germifications of the very same global singular 5-web (a concept to be introduced in Sect. 3) defined on P2 . Fig. C2 ; 0/ which would not be equivalent to any of the previous examples is doomed to failure. Indeed Bol proved the following. 6. If W is a smooth hexagonal k-web, k 3, then W is equivalent to the superposition of k pencils of lines or k D 5 and W is equivalent to B5 . A proof will not be presented here. For a recent exposition, with a fairly detailed sketch of proof, see [121].

C2 ; 0/ ! C; 0/. 0/ ¤ 0. 3 implies that W has rank equal to 6 if and only if vu axx ax2 byy by2 ! ax2 by2 / ! D0 where vu D by @x ax @y . tanh x tanh y/. But there are still other functions.

These three quadrics are the quadrics of Darboux of S at p. For more details see [81, pp. 141–144]. `i /p / 2 PTp S for i D 1; 2; 3. This is the Darboux 3-web of S . 1/j. For a generic point p in a generic surface S there are exactly five hyperplanes which intersect S along a curve which has a tacnode8 singularity at p. S; L/, where L is a line-bundle on S . When S is a germ of surface, any linebundle on it is trivial hence a linear system is nothing else than the projectivization of a finite dimensional vector space of germs of functions.