Topology of complex webs of codimension one and geometry of projective space curves
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 {\S} 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 \textit{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 {\S} 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.
complex analytic webs of a complex manifold
scientific article; zbMATH DE number 4056589