On non-hyperelliptic flocks
type: Item
instance of: scholarly article
title: On non-hyperelliptic flocks
zbMATH Open document ID: 0827.51008
author: Guglielmo Lunardon
author: Laura Bader
published in: European Journal of Combinatorics
publication date: 1995-12-11T00:00:00Z
review text: Let \(K\) be the quadratic cone of \(PG (3,q)\) with vertex \(v = (0,0,0,1)\) and equation \(X_0 X_1 = X_2^2\). A flock \(F\) of \(K\) is a partition of \(K \backslash \{v\}\) into \(q\) disjoint irreducible conics. Associated to each flock \(F\) is a translation plane \(T(F)\) of order \(q^2\) and dimension at most two over its kernel. When \(T(F)\) is a semifield (resp. likeable) plane, \(F\) is called a semifield (resp. likeable) flock. For \(q\) even, the semifield and likeable flocks have been classified, so the authors restrict their attention to the case that \(q = p^r\), where \(p\) is an odd prime. If \(f\) and \(g\) are two functions of \(GF (q)\) into itself such that \(f(0) = g(0) = 0\), for each \(t \in GF (q)\) let \(\pi_t\) be the plane with equation \(tX_0 - f(t)X_1 + g(t)X_2 + X_3 = 0\). The conics \(C_t = K \cap \pi_t\) are irreducible and \(F(f,g) = \{C_t |t \in GF (q)\}\) is a flock of \(K\) iff \((g(t) - g(s))^2 + 4(t - s) (f(t) - f(s))\) is a non-square in \(GF (q)\) for all distinct \(t\) and \(s\). Conversely, for each flock \(F\) of \(K\) there are two functions \(f\) and \(g\) of \(GF (q)\) into itself such that \(F = F(f,g)\). If for a given flock \(F = F(f,g)\) there exists a polynomial \(h(t) \in GF (q) [t]\) such that \(g^2(t) + 4tf(t) = nh^2(t)\) \((n\) a non-square) holds as a polynomial identity, then \(F\) is called a non-hyperelliptic flock. The main result of this paper is: Theorem: (a) If \(F\) is a non-hyperelliptic semifield flock, then \(F\) is isomorphic to a linear flock, a Kantor flock constructed via a Knuth semified, or a Ganley flock. (b) If \(F\) is a nonhyperelliptic likeable flock, then \(F\) is isomorphic to a Fisher-Thas-Walker flock or a Kantor likeable flock. [While it is not typical to bring up such matters in a review such as this, I will make an exception to correct the unfortunate type preceding the proof of Theorem 1(b). ``We are not able to \(\ldots\)'' should read ``We are now able to \(\ldots\)''].
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 683264
zbMATH Keywords: flock
zbMATH Keywords: semifield flock
zbMATH Keywords: likeable flock
reviewed by: William E. Cherowitzo
MaRDI profile type: Publication
full work available at URL: eujc.1994.1045
OpenAlex ID: W2010078321
DOI: 10.1006/EUJC.1994.1045
description: scientific article; zbMATH DE number 683264