Filling the nest gaps
type: Item
instance of: scholarly article
title: Filling the nest gaps
zbMATH Open document ID: 0873.51002
published in: Finite Fields and their Applications
publication date: 1996-05-08T00:00:00Z
review text: One construction technique for constructing translation planes involves the replacement of \(\tau\)-nests in the geometry \(\text{PG}(3,q)\). One of the authors, \textit{G. L. Ebert} [Eur. J. Comb. 10, No. 4, 319-330 (1989; Zbl 0691.05009)], used this method to construct translation planes whose translation complement contains a direct product of two cyclic homology groups of order \(q+1\) with affine axis. Essentially, a \(\tau\)-nest is a set \({\mathcal N}\) of \(\tau\) reguli in a regular spread with the property that each line of \(\text{PG}(3,q)\) lies in exactly 0 or 2 of the reguli. The nest is replaceable if and only if a new spread is obtained by replacing the lines of \({\mathcal N}\) with lines from the opposite reguli, \((q+1)/2\) from each. The authors' idea is to replace a \(((q+1)/2+i)\)-nest for suitable values of \(i\). They discuss bounds on \(i\) and the collineation groups of the resulting planes. In particular, the planes have in the translation complement the direct product of two cyclic homology groups of order \((q+1)/2\) with affine axis. (By the nature of the construction the translation planes have dimension 2 over their kernel \(GF(q)\).) An interesting question, at least in the reviewer's opinion, is the following. \textit{N. L. Johnson} and \textit{R. Pomareda} [J. Geom. 56, No. 1-2, 59-86 (1996; Zbl 0855.51004)] have characterized the planes described in the first paragraph. Does a similar characterization exist for the translation planes of the last paragraph? It should also be noted that the concept of \(\tau\)-nest has a close connection with the concept of weighted circle introduced by \textit{F. A. Sherk} [J. Geom. 45, No. 1/2, 158-173 (1992; Zbl 0766.51018)]. The authors' construction appears to give a ``robust'' -- the authors' term -- class of nonisomorphic planes. However, they do not discuss the question of intersection with known classes of planes, except in certain small cases.
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 851861
zbMATH Keywords: nest
zbMATH Keywords: translation planes
author: R. D. Baker
author: Gary L. Ebert
reviewed by: Michael J. Kallaher
MaRDI profile type: Publication
full work available at URL: ffta.1996.0004
OpenAlex ID: W1969138479
DOI: 10.1006/FFTA.1996.0004
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description: scientific article; zbMATH DE number 851861