Small point sets of \(\text{PG}(n,p^{3h})\) intersecting each line in 1 mod \(p^{h}\) points
type: Item
instance of: scholarly article
title: Small point sets of \(\text{PG}(n,p^{3h})\) intersecting each line in 1 mod \(p^{h}\) points
zbMATH Open document ID: 1222.51006
author: Nóra V. Harrach
author: Klaus Metsch
author: Zsuzsa Weiner
author: Tamás Szőnyi
published in: Journal of Geometry
publication date: 2011-02-03T00:00:00Z
review text: An \((n - k)\)-blocking set \(B\) in \(PG(n, q)\) with \(0 < k < n\) is a set of points intersecting each \(k\)-subspace non-trivially. A theorem of Bose and Burton shows that the smallest such blocking sets must be \((n - k)\)-dimensional subspaces and these kinds of blocking sets are called trivial. A point \(p\) of \(B\) is essential to \(B\) if there is a subspace of dimension \(k\) through \(p\) that meets \(B\) in \(p\) alone. A blocking set \(B\) is called minimal if it is not trivial and each point of \(B\) is essential. Further, \(B\) is called small if its size satisfies \(|B| < \frac{3(q^{n - k} + 1)}{2}\). A point set \(S\) of \(PG(n, q^t)\) is said to be linear if there is a projective space \(PG(n', q^t)\) containing \(PG(n, q^t)\) such that \(S\) is the projection of the subgeometry \(PG(n', q) \subset PG(n', q^t)\) from a suitable vertex onto \(PG(n, q^t)\). The main result of this paper whose proof involves elaborate computations proves the following special case of the linearity conjecture: The point sets of \(PG(n, q)\) with \(q = p^{3h}\) and \(p \geq 7\) of size \(< \frac{3(q^{n - 1} + 1)}{2}\) intersecting each line \(1\pmod{\root{3}\of{q}}\) points are linear blocking sets.
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 94B25
zbMATH DE Number: 5844611
zbMATH Keywords: blocking sets
zbMATH Keywords: minimal blocking sets
zbMATH Keywords: projections
zbMATH Keywords: linear blocking sets
zbMATH Keywords: linearity conjecture
reviewed by: Sharad Sane
MaRDI profile type: Publication
full work available at URL: s00022-010-0051-1
OpenAlex ID: W2465847108
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DOI: 10.1007/S00022-010-0051-1
description: scientific article; zbMATH DE number 5844611