By Jorge Vitório Pereira, Luc Pirio

This booklet takes an in-depth examine abelian kinfolk of codimension one webs within the advanced 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 idea revolves round the notion of abelian relation, a specific form of practical relation one of the first integrals of the foliations of an online. major focuses of the ebook comprise what number abelian relatives can an online hold and which webs are sporting the maximal attainable variety of abelian kinfolk. The e-book bargains entire proofs of either Chern’s sure and Trépreau’s algebraization theorem, together with all of the useful necessities that transcend straight forward advanced research or easy algebraic geometry. lots of the examples identified modern of non-algebraizable planar webs of maximal rank are mentioned intimately. A historic account of the algebraization challenge for maximal rank webs of codimension one is additionally presented.

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**Sample text**

X; W/ D 0 for i D 0; : : : ; q 1. X; W/ in k points. X / k. With these definitions at hand we have the following Biduality Theorem. 8. X; W/ as the superposition of irreducible pairs and applies D to each factor. 4 Webs Attached to Projective Surfaces One particularly rich source of examples of webs on surfaces is the classical projective differential geometry widely practiced until the late thirties of the twentieth century. The simplest example is perhaps that of the asymptotic webs on surfaces in P3 which will now be described.

2. 6. 4), the variety I is irreducible and in particular connected. Moreover the natural projection from I to U is surjective and has finite fibers. Therefore I has dimension n D dim U . p1 ; : : : ; pn ; H / 2 J ” p1 ; : : : ; pn are contained in a Pn 2 : Since C is non-degenerated, one can choose n distinct points on it which span a Pn 1 . Thus J is a proper subset of I . The irreducibility of I implies dim J < dim I D n. Therefore the image of the projection to U is a proper subset, with complement parametrizing hyperplanes intersecting C with the requisite property.

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 . x/dx seen as a degree k polynomial in the variable t with coefficients in CŒx; y; dx; dy.