Twisted derivations and distinct sets of lines that cover the same pairs of points
type: Item
instance of: scholarly article
title: Twisted derivations and distinct sets of lines that cover the same pairs of points
zbMATH Open document ID: 0839.51007
DOI: 10.1007/BF01263530
author: Klaus Metsch
published in: Geometriae Dedicata
publication date: 1996-06-19T00:00:00Z
review text: A well known construction of projective planes is the method of derivation. Given a projective plane \(\pi = (P,L)\) of order \(n = m^2\) and a set \(D \subseteq l \in L\) of \(m + 1\) collinear points such that for each pair \(p,q \in P \backslash l\) with \((p \vee q) \wedge l \in D\), there exists a Baer subplane of \(\pi\) containing \(p,q\) and all points of \(D\), then we replace the \(m^2 (m + 1)\) lines of \(L \backslash \{l\}\) intersecting \(l\) in a point belonging to \(D\) with the above mentioned Baer subplanes of \(\pi\). The resulting incidence structure is a projective plane on the same point set \(P\) which has all but \(m^2 (m + 1)\) lines in common with \(\pi\). Denote by \(L_1\) the set of the \(m^2 (m + 1)\) removed lines, and by \(L_2\) the set of the new lines. Then \(L_1\) and \(L_2\) cover the same pairs of points, i.e. two points \(p,q \in P\) are contained in an element of \(L_1\) if and only if they are contained in an element of \(L_2\). For \(0 \leq i \leq n + 1\) denote by \(v_i\) the number of points of \(\pi\) which are contained in exactly \(i\) elements of \(L_1\) (or \(L_2)\). These numbers are called the parameters of \(L_1\); in our case we have \(v_0 = n - m\), \(v_{m + 1} = n^2\), \(v_n = m + 1\) and \(v_i = 0\) for the remaining \(i\)'s. The author proves: Given two non-isomorphic projective planes \(\pi\), \(\pi'\) of order \(n = m^2\) on the same point set \(P\) such that \(\pi'\) is obtained from \(\pi\) by replacing \(m^2 (m + 1)\) lines of \(\pi\) such that the set of these lines has the parameters of a derivation, then this line replacement is actually a derivation. More generally, the author considers two non-isomorphic projective planes \(\pi\), \(\pi'\) on the same point set \(P\) and denotes by \(s\) the number of lines of \(\pi\) which are not lines of \(\pi'\). He assumes that the order \(n\) of \(\pi\) is a square \(n = m^2 > 81\). Under these assumptions, he proves the following theorem: \(s \geq m^2 (m + 1)\); …
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 51A20
Mathematics Subject Classification ID: 05D05
zbMATH DE Number: 806206
zbMATH Keywords: derivation
zbMATH Keywords: line replacements
MaRDI profile type: Publication
cites work: Blocking Sets in Finite Projective Planes
cites work: Q5545191
cites work: Ovoids and translation planes revisited
cites work: A Class of Non-Desarguesian Affine Planes
cites work: Nets with critical deficiency
description: scientific article; zbMATH DE number 806206