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Topology of complex webs of codimension one and geometry of projective space curves. (English) Zbl 0647.57018
A b-web of a manifold M of codimension 1 is a configuration \({\mathcal W}\) of b foliations \({\mathcal F}_ 1,...,{\mathcal F}_ b\) of M of codimension 1. In § 1, we prove that the topological and analytic classifications are the same for complex analytic webs of a complex manifold M under the condition \(b\geq \dim M+1\) and a certain generic condition. This is a complex analytic version of J. P. Dufour’s theorem for \(C^{\infty}\)-webs [C. R. Acad. Sci., Paris, Sér. I 293, 509-512 (1981; Zbl 0486.58005) and Topology 22, 449-474 (1983; Zbl 0521.58012)]. In § 2, we apply this theorem to the d-webs \({\mathcal W}_ C\) of the dual projective space \({\mathbb{P}}_ n^{\vee}\) of codimension 1 generated by the dual hyperplanes \(x^{\vee}\in {\mathbb{P}}_ n^{\vee}\) of \(x\in C\) for algebraic curves \(C\subset {\mathbb{P}}_ n\) of degree d, and prove that the imbeddings \(C\subset {\mathbb{P}}_ n\) are determined by the topological structures of \({\mathcal W}_ C\) up to projective transformations if \(d\geq n+2\). The singular locus \(\Sigma\) (\({\mathcal W}_ C)\) of \({\mathcal W}_ C\) is closely related to the projective geometry of C and the dual variety and curve of C. In the final two sections, we investigate the structure of \({\mathcal W}_ C\) for the exceptional cases that \(C\subset {\mathbb{P}}_ n\) is of degree n, \(n+1\), e.g. rational and elliptic normal curves, and singular plane curves.

57R30 Foliations in differential topology; geometric theory
14B05 Singularities in algebraic geometry
32L99 Holomorphic fiber spaces
14H99 Curves in algebraic geometry
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
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