Topology of complex webs of codimension one and geometry of projective space curves
type: Item
instance of: scholarly article
title: Topology of complex webs of codimension one and geometry of projective space curves
zbMATH Open document ID: 0647.57018
DOI: 10.1016/0040-9383(87)90043-7
author: Isao Nakai
published in: Topology
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32L99
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
zbMATH DE Number: 4056589
zbMATH Keywords: foliations of codimension 1
zbMATH Keywords: webs of the dual projective space
zbMATH Keywords: b-web of a manifold
zbMATH Keywords: complex analytic webs of a complex manifold
zbMATH Keywords: algebraic curves
zbMATH Keywords: singular locus
zbMATH Keywords: rational and elliptic normal curves
zbMATH Keywords: singular plane curves
MaRDI profile type: Publication
full work available at URL: 0040-9383(87)90043-7
OpenAlex ID: W2018634808
description: scientific article; zbMATH DE number 4056589