Ruled sets of plane-type \((m,h)_2\) in \(\mathrm{PG}(3,q)\) with \(m\leq q\)
type: Item
instance of: scholarly article
title: Ruled sets of plane-type \((m,h)_2\) in \(\mathrm{PG}(3,q)\) with \(m\leq q\)
zbMATH Open document ID: 1382.51006
author: V. Napolitano
published in: Journal of Geometry
publication date: 2017-12-14T00:00:00Z
review text: The author generalizes the result that a set \(K\) of points in \(\mathrm{PG}(d,q)\) intersected by any hyperplane in 1 or \(h\) points is either a line or \(d=3\) and \(K\) is an ovoid in two ways. The first is as follows. Let \(0<m<h\) and \(s \geq m+1\) be three integers. Let \(K\) be a set of \(k\) points of \(\mathrm{PG}(3,q)\) of plane-type \((m,h)_2\). If there exists a line \(\ell\) intersecting \(K\) in \(s\) points then either: (1) \(K\) is ruled, \(k=m(q+1)\), \(h-m=q\) and either \(K\) is a line or \(m \geq 2\) and \(K = \ell \cup K'\) is a set of points of \(\mathrm{PG}(3,q)\) of plane-type \((m-1,h-1)_2\) and of size \((m-1)(q+1)\) or (2) \(K\) is an ovoid. The second is as follows. Let \(K\) be a set of points of \(\mathrm{PG}(3,q)\) of plane type \((m,h)_2\) with \(m \geq 1\). If there exists a line intersecting \(K\) in at least \(m+1\) points then \(K\) contains at most \(m\) lines, and \(K\) contains exactly \(m\) lines if and only if \(K\) is the set of points on \(m\) pairwise skew lines.
reviewed by: Steven T. Dougherty
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 6818624
zbMATH Keywords: projective space
zbMATH Keywords: intersection numbers
zbMATH Keywords: ovoid
MaRDI profile type: Publication
full work available at URL: s00022-017-0388-9
OpenAlex ID: W2783721175
cites work: Q5492614
cites work: Sets of type \((3,h)\) in \(\mathrm{PG}(3,q)\)
cites work: On sets of type \((m,h)_2\) in \(\mathrm{PG}(3,q)\) with \(m \leq q\)
cites work: A combinatorial problem
DOI: 10.1007/S00022-017-0388-9
description: scientific article; zbMATH DE number 6818624