# zbMATH — the first resource for mathematics

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.

##### MSC:
 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
Full Text: