Arcs in cyclic affine planes
type: Item
instance of: scholarly article
title: Arcs in cyclic affine planes
zbMATH Open document ID: 1181.51008
author: Vincenzo Giordano
published in: Innovations in Incidence Geometry
publication date: 2009-06-23T00:00:00Z
review text: In a projective plane \(\Pi\) of order \(n,\) a \textit{\(k-\)arc} \(\mathcal K\) \((k>2)\) is a set of \(k\) points no three of which collinear. An arc not contained in a larger one is said to be complete. The \((n+1)-\)arcs are also called \textit{ovals}. An affine plane \(A\) of order \(n\) is \textit{cyclic} if it admits a cyclic group \(G\) fixing one point and acting regularly on the set of the remaining points. Any such plane can be constructed from an affine difference set of order \(n.\) Let \(\Pi\) be the projective closure of an affine cyclic plane \(A\) of order \(n\equiv 1 ~(mod.~ 4).\) Using the construction of \(A\) from an affine difference set, the author provides a new \(k-\)arc \(\mathcal K\) of \(\Pi\) with the following properties: (a) \(k=\frac{n+7}{2},\) (b) \(\mathcal K\) shares \( \frac{n+3}{2}\) points with a suitable oval of \(\Pi.\)
reviewed by: Guglielmo Lunardon
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 05B10
zbMATH DE Number: 5567863
zbMATH Keywords: arcs
zbMATH Keywords: ovals
zbMATH Keywords: cyclic planes
zbMATH Keywords: difference sets
MaRDI profile type: Publication
full work available at URL: iig.2008.6.203
OpenAlex ID: W2916861995
DOI: 10.2140/IIG.2008.6.203
description: scientific article; zbMATH DE number 5567863