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Exceptional Webs

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An Invitation to Web Geometry

Part of the book series: IMPA Monographs ((IMPA,volume 2))

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Abstract

This last chapter is devoted to planar webs of maximal rank. More specifically, it surveys the current state of the art concerning exceptional planar webs.

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Notes

  1. 1.

    Euler’s dilogarithm is the function \(\mathbf{L}\text{i}_{2}(z) =\sum _{ n=0}^{\infty }z^{n}/n^{2}\). The series converges for | z |  < 1 and has analytic continuations along all paths contained in \(\mathbb{C}\setminus \{0, 1\}\).

  2. 2.

    Here only differential fields over \(\mathbb{C}\) will be considered. Of course, it is possible to deal with more general fields.

  3. 3.

    Recall the convention about germs used throughout. Here \((\mathbb{C}^{n}, 0)\) must be seen as a small open subset containing the origin.

  4. 4.

    Beware that algebraic here means that they are locally dual to plane curves. In the cases under scrutiny they are dual to products of lines.

  5. 5.

    These are the 5-webs mentioned in Example 2.2.4.

  6. 6.

    In this definition, ℜ m stands for the real part if n is odd and for the imaginary part otherwise; B k is the k-th Bernoulli number: B 0 = 1, \(B_{1} = -1/2\), \(B_{2} = 1/6\), etc.

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Authors and Affiliations

  1. Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

    Jorge Vitório Pereira

  2. Institut de Recherches Mathématiques de Rennes IRMAR, UMR 6625 du CNRS, Université Rennes 1, Rennes, France

    Luc Pirio

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  1. Jorge Vitório Pereira
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Pereira, J.V., Pirio, L. (2015). Exceptional Webs. In: An Invitation to Web Geometry. IMPA Monographs, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-14562-4_6

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