Classification of ovoids in \(PG(3,32)\)
type: Item
instance of: scholarly article
title: Classification of ovoids in \(PG(3,32)\)
zbMATH Open document ID: 0805.51004
DOI: 10.1007/BF01222671
author: Christine M. O'Keefe
author: Tim Penttila
author: Gordon Royle
published in: Journal of Geometry
publication date: 1994-09-27T00:00:00Z
review text: An ovoid in the projective space \(\text{PG}(3,q)\) is a set of \(a^ 2+1\) points, no three of which are collinear. For each prime power \(q\), an elliptic quadric provides an example which is the only possibility for odd \(q\). For even \(q\) the only other known examples is the Tits ovoid which exists only for nonsquare \(q>4\). Until recently a classification of ovoids in spaces of even order had been completed only for \(q=2\), 4 and 8. In general the classification of ovoids in \(\text{PG} (3,q)\) seems to depend on a classification of the ovals in \(\text{PG} (2,q)\), a very difficult problem in itself. However, both problems have been solved for \(q=16\), so the next case to consider is \(q=32\). Very recently the ovals in \(\text{PG} (2,32)\) were classified by \textit{T. Penttila} and \textit{G. F. Royle} [ibid., 151-158 (1994; see the paper below)], thus enabling the present authors to complete the ovoid classification problem for \(q=32\). An interesting mixture of theory and computer computation is required, with the end result being that the only ovoids of PG(3, 32) are the two well known examples. Hence by a famous result of P. Dembowski on inversive planes of even order, there are exactly two inverse planes of order 32.
reviewed by: Stanley E. Payne
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 621997
zbMATH Keywords: hyperoval
zbMATH Keywords: ovoid
zbMATH Keywords: ovals
zbMATH Keywords: inversive planes
MaRDI profile type: Publication
cites work: Inversive planes of even order
cites work: Q3847896
cites work: Q4937820
cites work: Q3694152
cites work: Ovoids of PG(3,16) are elliptic quadrics
cites work: Q3227205
cites work: Q3344217
cites work: Ovoids and Translation Ovals
full work available at URL: https://doi.org/10.1007/bf01222671
OpenAlex ID: W1981362197
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description: scientific article; zbMATH DE number 621997