By Jorge Vitório Pereira, Luc Pirio

This booklet takes an in-depth examine abelian kin of codimension one webs within the advanced analytic atmosphere. In its classical shape, internet geometry is composed within the research of webs as much as neighborhood diffeomorphisms. an important a part of the idea revolves round the inspiration of abelian relation, a specific type of sensible relation one of the first integrals of the foliations of an internet. major focuses of the publication contain what number abelian family can an internet hold and which webs are wearing the maximal attainable variety of abelian family members. The publication bargains whole proofs of either Chern’s sure and Trépreau’s algebraization theorem, together with the entire important must haves that transcend effortless advanced research or simple algebraic geometry. lots of the examples recognized brand new of non-algebraizable planar webs of maximal rank are mentioned intimately. A old account of the algebraization challenge for maximal rank webs of codimension one can also be presented.

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Example text

L notice that W L is tangent to a line ` at a point p if To determine the degree of W, L and only if one of the tangent spaces of W at p contains the line `. `/ PT P of `. W/ ; i D 0; : : : ; n 1: In order to deal with the case when L jW W W ! X; W/ where X Pn is an irreducible projective variety and W is an irreducible web6 of codimension 1 on X . X; W/ to PT Pn . X; W/ is a union of linear subspaces corresponding to the projectivizations of the conormal bundles in Pn of the leaves of W through x.

E. C/. The simplest example of an asymptotic web is certainly the asymptotic web of a smooth quadric Q in P3 . As is well-known, the latter is isomorphic to P1 P1 . The fibers of both natural projections to P1 are lines in P3 . Since any line contained in it is an asymptotic curve of Q, it is clear that its asymptotic web WQ is formed by these two families of lines. Asymptotic Webs: Alternative Approach When S P3 is a smooth projective surface the definition of the asymptotic web of S is amenable to a more intrinsic formulation.

3 Singular and Global Webs 25 variable of degree k. t // up to order n. x/g of gt is an irreducible plane curve of degree n. Moreover varying t 2 C one obtains a family of degree n curves which corresponds to a degree k curve €f on the space of degree n curves. The degree n curves through a generic point p 2 C2 determine a hyperplane H in the space of degree n curves. Because H intersects €f in k points, through a generic p 2 C2 pass k distinct curves of the family fGt g. Therefore this family of curves determines a k-web Wf on C2 .

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