Images of conics under derivation
type: Item
instance of: scholarly article
title: Images of conics under derivation
zbMATH Open document ID: 0854.51008
DOI: 10.1016/0012-365X(94)00096-2
author: Christine M. O'Keefe
author: Arlene A. Pascasio
published in: Discrete Mathematics
publication date: 1996-01-01T00:00:00Z
review text: Let \(L_\infty\) be a line of the desarguesian projective plane \(PG(2,q^2)\) of order \(q^2\), \(q > 2\), so that \(AG(2,q^2) = PG(2,q^2) \setminus L_\infty\) is an affine plane of the same order. A derivation set \(\mathcal D\) of \(AG(2,q^2)\) is a set of \(q + 1\) points of \(L_\infty\) such that for any two affine points \(P\) and \(Q\) for which the ideal point of the line \(PQ\) is in \(\mathcal D\) there is a Baer subplane containing \(P\), \(Q\) and whose ideal points coincide with \(\mathcal D\). A new incidence structure \({\mathcal D} AG(2,q^2)\) is defined as follows: the points are just the points of \(AG(2,q^2)\); a line is either a line of \(AG(2,q^2)\) with ideal point not in \(\mathcal D\) or the points of a Baer subplane in \(AG(2,q^2)\) whose ideal points coincide with those of \(\mathcal D\). Incidence is the natural containment relation. Then \({\mathcal D}AG(2,q^2)\) is an affine plane of order \(q^2\) whose projective completion is the Hall plane \(\text{Hall} (q^2)\), of the same order. If \(q \geq 3\), and \(\mathcal C\) is a nondegenerate conic of \(AG(2,q^2)\), then \(\mathcal C\) is contained in a unique nondegenerate conic \(\overline {\mathcal C}\) in \(PG(2,q^2)\). Moreover, as a pointset, \(\mathcal C\) is called a derived conic in \(\text{Hall} (q^2)\). Many authors have considered the question as to when is this derived conic also an arc (no three points collinear) in \(\text{Hall} (q^2)\), and if so to determine how it extends to a complete arc. In general, given any conic \(\mathcal C\) in \(AG(2,q^2)\), the line \(L_\infty\) may be a secant, tangent or external line to \(\overline {\mathcal C}\). In the paper under review the cases considered are when \(L_\infty\) is secant to \(\overline {\mathcal C}\), \(q \geq 3\), and for \(q\) even, \(q > 2\), when \(L_\infty\) is tangent to \(\overline {\mathcal C}\). In the former case, the authors treat the case where the two points of \(\overline{\mathcal C}\) on \(L_\infty\) are in \(\…
reviewed by: Stanley E. Payne
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51A35
zbMATH DE Number: 896541
zbMATH Keywords: conics
zbMATH Keywords: derivation
zbMATH Keywords: ovals
zbMATH Keywords: Hall plane
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 896541