Blocking sets of external lines to a conic in \(\text{PG}(2,q)\), \(q\) odd
type: Item
instance of: scholarly article
title: Blocking sets of external lines to a conic in \(\text{PG}(2,q)\), \(q\) odd
zbMATH Open document ID: 1111.51007
author: Angela Aguglia
author: Gabor Korchmáros
published in: Combinatorica
publication date: 2007-01-08T00:00:00Z
review text: A point set \(B\subseteq \text{PG}(2,q)\) is a blocking set for a family \(\mathcal L\) of subsets of \( \text{PG}(2,q)\) if it meets any element of \({\mathcal L}\) in at least one point. Blocking sets for the set \(\mathcal F\) of all the lines of the plane have been widely investigated. An interesting problem is to determine minimal blocking sets for suitable subsets of \(\mathcal F\). In this paper, the authors investigate minimal blocking sets for the set of all the external lines to a conic \(\mathcal C\) in \( \text{PG}(2,q)\), \(q\) odd. They show that such blocking sets \(B\) always contain at least \(q-1\) points. For \(q\geq 9\) and \(q=3\), equality holds if only if \(B\) is linear, that is \(B\) consists of the \(q-1\) points of a chord \(r\) of \(\mathcal C\) not on \(\mathcal C\). For \(q=5\) or \(q=7\) they prove that, up to projectivities, there is just one more sporadic example of blocking set of size \(q-1\). The proof of these results is based on the investigation of the linear system of all polynomial vanishing at every internal point of \(\mathcal C\) and the study of the involutions of the group \( \text{PGL}(2,q)\).
reviewed by: Luca Giuzzi
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 5082371
zbMATH Keywords: conics
zbMATH Keywords: blocking sets
zbMATH Keywords: involutions of \( \text{PG}(2,q)\)
MaRDI profile type: Publication
full work available at URL: s00493-006-0016-z
OpenAlex ID: W1976459503
DOI: 10.1007/S00493-006-0016-Z
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description: scientific article; zbMATH DE number 5082371