Projective bundles
type: Item
instance of: scholarly article
title: Projective bundles
zbMATH Open document ID: 0803.51009
author: V. Pereyra
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 1994-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/224131
review text: It is well-known that there exist collections of \(q^ 2+q+1\) conics of \(\text{PG} (2,q)\) which mutually intersect in a unique point and thus form another projective plane of order \(q\) on the same point set; the authors call such a collection of conics a projective bundle. Motivated by the question, whether one may partition the \(q^ 5 - q^ 2\) conics of \(\text{PG} (2,q)\) into \(q^ 2(q-1)\) projective bundles, they construct collections of \(q^ 2(q - 1)/2\) disjoint projective bundles for any odd prime power \(q\); for even \(q\), they can only obtain \(q-1\) disjoint bundles. In the cases \(q=3\) or 4, better results (but no complete partitioning) are obtained. The authors also give the following interesting result: If \(D\) is a cyclic difference set for \(\text{PG} (2,q)\) and if \(r\) is relatively prime to \(q^ 2 + q + 1\), then \(Dr^{- 1}\) is a curve of degree \(r\).
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 617007
zbMATH Keywords: conics
zbMATH Keywords: projective plane
zbMATH Keywords: cyclic difference set
reviewed by: Dieter Jungnickel
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 617007