On webs of maximum rank
type: Item
instance of: scholarly article
title: On webs of maximum rank
zbMATH Open document ID: 0677.53017
DOI: 10.1007/BF00184156
author: John B. Little
published in: Geometriae Dedicata
publication date: 1989-01-01T00:00:00Z
review text: Let W(d,n,r) be a d-web given on a differentiable manifold of dimension nr by d foliations of codimension r which are in general position. If the web W(d,n,r) is of maximum r-rank and \(d>r(n-1)+2\), then the web normals are (r-1)-dimensional generators of a rational normal scroll in the projectivized tangent space to the web domain. This theorem was formulated by \textit{S. S. Chern} and \textit{P. Griffiths} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 539-557 (1978; Zbl 0402.57001)] and proved by them in the case \(r=2\). In the paper under review the author, using the same method, gives the proof of this theorem in the cases \(r\geq 2\). This theorem allows to prove that webs of maximum rank, as in the theorem, are almost Grassmannizable. The latter webs were introduced by \textit{M. A. Akivis} in [Sov. Math., Dokl. 21, 707-709; translation from Dokl. Akad. Nauk SSSR 252, 267-270 (1980; Zbl 0479.53015)]. The last result (partially) gives an affirmative answer on a problem posed by the reviewer, whether every web W(d,n,r) of maximum r-rank is almost Grassmannizable. \textit{S. S. Chern} and \textit{P. Griffiths} [see Jahresber. Dtsch. Math.- Ver. 80, 13-110 (1978; Zbl 0386.14002)] also proved that so-called normal webs W(d,n,1) of maximum 1-rank are algebraizable, if \(n\geq 3\), \(d\geq 2n+1\), i.e. they are equivalent to the algebraic webs W(d,n,1) arising naturally from projective algebraic varieties. \textit{S. S. Chern} [see his paper ``Wilhelm Blaschke and web geometry'' in Wilhelm Blaschke Gesammelte Werke, Vol. 5 (1985; Zbl 0656.53003), pp. 21-24] wrote that ``the determination of all webs of maximum rank will remain a fundamental problem in web geometry and the non-algebraic ones, if there are any, will be most interesting''. It was known, that in the plane there exists at least one exceptional (non-algebraizable) web W(d,2,1) of maximum 1- rank - the famous Bol 5-web of rank 6 [see \textit{G. Bol}, Abh. Math. Semin. Univ. Hamb. 11, 387-393 (1936; …
Mathematics Subject Classification ID: 53A60
zbMATH DE Number: 4109521
zbMATH Keywords: web
zbMATH Keywords: normal scroll
zbMATH Keywords: webs of maximum rank
zbMATH Keywords: almost Grassmannizable
zbMATH Keywords: exceptional webs
reviewed by: Vladislav V. Goldberg
MaRDI profile type: Publication
cites work: Q4741937
cites work: Web geometry
cites work: Abel's theorem and webs
cites work: Q4187012
cites work: Q3938164
cites work: Variations on a theorem of Abel
cites work: Residues and zero-cycles on algebraic varieties
cites work: Q3919817
cites work: Rational equivalence of O-cycles on surfaces
cites work: RATIONAL EQUIVALENCE OF ZERO-CYCLES
full work available at URL: https://doi.org/10.1007/bf00184156
OpenAlex ID: W2039524541
description: scientific article; zbMATH DE number 4109521