Monomial flocks of monomial cones in even characteristic
type: Item
instance of: scholarly article
title: Monomial flocks of monomial cones in even characteristic
zbMATH Open document ID: 0930.51006
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2000-02-06T00:00:00Z
review text: This article generalizes the concept of a flock of a quadratic cone and of an \(\alpha\)-flock of a translation oval cone to more general types of cones. A flock of a quadratic cone of \(PG(3,q)\) is a set of \(q\) planes not passing through the vertex which do not intersect each other at a point of this quadratic cone. One of the most interesting features of flocks is that they are related to a large variety of other structures. They give rise to certain generalized quadrangles of order \((q^2,q)\) and can be used to construct translation planes. In \textit{W. Cherowitzo, T. Penttila, I. Pinneri} and \textit{G. Royle} [Geom. Dedicata 60, No. 1, 17-37 (1996; Zbl 0855.51008)], a new connection was found. Flocks of quadratic cones give rise to a set of \(q+1\) ovals in a projective plane \(PG(2,q)\), \(q\) even; called a herd of ovals. This link, found earlier by Payne, led to the discovery of the Payne ovals and to the discovery of the Subiaco ovals. So, flocks of quadratic cones give rise to ovals, but not every oval is related to a flock of a quadratic cone. This led Cherowitzo in: [\textit{W. Cherowitzo}, Geom. Dedicata 72, No. 3, 221-246 (1998)] to define flocks of cones \(\Sigma_\alpha\) with base a translation oval. And this, with the goal of linking them to ovals and with the goal of finding new ovals. This approach was successful; these flocks of the quadratic cones \(\Sigma_\alpha\), also called \(\alpha\)-flocks, are indeed linked to an oval, and this approach made it possible to prove the existence of the infinite class of Cherowitzo ovals. But still there remain ovals which are not linked to such \(\alpha\)-flocks. So, with the goal of finding even more links with ovals, an extension of the definition of \(\alpha\)-flocks is presented in this article. Now, flocks of monomial cones are studied. A monomial cone in \(PG(3,q)\), \(q=2^h\), is a set of points \(\Sigma_\beta=\{(x,y,z,w)||y^\beta=xz^{\beta-1} \}\), where \((\beta,q-1)=(\beta-1,q-1)=1\). The auth…
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1266811
zbMATH Keywords: flocks
zbMATH Keywords: cones
zbMATH Keywords: ovals
author: William E. Cherowitzo
reviewed by: Leo Storme
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1266811