Tangent sets in finite spaces
type: Item
instance of: scholarly article
title: Tangent sets in finite spaces
zbMATH Open document ID: 0863.51009
DOI: 10.1016/0012-365X(94)00374-R
author: Yanyan Li
published in: Discrete Mathematics
publication date: 1996-10-07T00:00:00Z
review text: Let \(K\) be a point set in \(PG (n,q)\). A set \(\varepsilon\) is called a tangent set of \(K\) if \(\varepsilon\) is not contained in a hyperplane and the line joining any two points of \(\varepsilon\) is a tangent of \(K\). Let us denote the cardinality of \(K\) by \(k\) and the cardinality of \(\varepsilon\) by \(e\). Some properties of such sets are investigated in the paper. The most important results are the following: (I) Let \(K\) be a nonsingular quadric in \(PG (n,p^r)\), \(p\) an odd prime. (a) If \(p\) does not divide \(n\), then either \(e= n+1\) or \(K\) has not tangent set; (b) If \(p\) divides \(n\), then either \(e= n+2\) or \(K\) has not tangent set. (II) Let \(K\) be a Hermitian surface in \(PG (3,q^2)\), \(q\) odd. (a) There exists a maximal \(\varepsilon\) with \(e= 2q +2\), and the points of \(\varepsilon\) lie on two skew lines; (b) If there exists \(\varepsilon\) such that any four of its points form a tetrahedron \(T\) compatible with \(K\), then \(e \leq 10\) (a rather long definition of such compatibility can be found in the text). (III) In \(P(2,q)\), if the number of lines joining two points of \(K\) is at least \(q+1\), then \(e< (q^2 + q+1)/2\). (IV) In \(P(2, q)\) let \(B (\varepsilon)\) be a blocking set of the lines meeting \(\varepsilon\) in at least two points and let \(L\) be the set of the lines joining at least two points of \(B (\varepsilon)\). If \(e\geq (q^2 + q+1)/2\), then the cardinality of \(L\) is greater than \(q\). (V) Let \(K\) be a set of points in \(PG(n,q)\), \(q>2\), and let \(\varepsilon\) be a tangent set of \(K\). If \(e> \theta(n)/2\), then \(K\) is a hyperplane. Some examples are given.
reviewed by: Krzysztof Witczyński
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 932530
zbMATH Keywords: finite geometry
zbMATH Keywords: tangent set
MaRDI profile type: Publication
cites work: Q3982810
cites work: An observation on certain point-line configurations in classical planes
cites work: Q3851217
cites work: Q3694152
cites work: Q4003762
cites work: Q3982819
cites work: Automorphisms of unitary block designs
cites work: Q3272923
full work available at URL: 0012-365x(94)00374-r
OpenAlex ID: W2073167422
description: scientific article; zbMATH DE number 932530