A classification of finite \(\{n-2,n-1\}\)-point-biregular spaces (Q1377816)

A linear space is an incidence structure \((P,{\mathcal L})\) such that any two distinct points \(p,q\in P\) are incident with exactly one line \(L\in{\mathcal L}\). A transversal of two lines \(L,K\in{\mathcal L}\) is a line which intersects both of these lines. The author considers finite linear spaces which have the property that the number \(\pi_x(L,K)\) of transversals through any point \(x\) of two disjoint lines \(L,K\) missing \(x\) belongs to given finite set \(H\) of nonnegative integers and does not depend on the lines \(L\) and \(K\). Such linear spaces are called \(H\)-point-regular. If \(H\) consists of a single integer, \textit{A. Beutelspacher} and \textit{A. Delandtsheer} [Eur. J. Comb. 2, 213-219 (1981; Zbl 0473.05014)], gave a complete classification of \(H\)-point-regular linear spaces. If \(\pi_x(L,K)\) ranges over two distinct integers, then the situation changes radically: \textit{A. Beutelspacher} and \textit{F. Nicholson} [Ric. Mat. 40, No. 2, 275-290 (1991; Zbl 0781.51004)], constructed many such spaces and a classification seems to be way out of reach. If, however, the number of transversals only depends on the lines \(L\) and \(K\) (and not on the point \(x\)), a classification has recently been given by the author [see Discrete Math. 129, No. 1-3, 3-18 (1994; Zbl 0809.51015)]. In this interesting and well written paper the author studies the case of proper \(H\)-point-biregular linear spaces, i.e. for every \(h\in H\) there is a pair of distinct, non-intersecting lines \(L\) and \(K\) and some point \(x\) not on \(L\) or \(K\), such that there exist exactly \(h\) transversals to \(L\) and \(K\) through \(x\). The author is able to give a complete list of proper\(\{n-2,n-1\}\)-point-biregular linear spaces: (1) Let \(P\) be a finite set with at least six elements and fix a point \(p\in P\). Let \(\{X,Y\}\) be a partition of \(P\setminus\{p\}\) with \(|X|\geq 2\) and \(|Y|\geq 3\). Setting \({\mathcal L}:=\{\{X\}\cup\{p\},\{Y\}\cup\{p\}\cup\setminus\{x,y\}\} x\in X, y\in Y\) one obtains a proper \(\{1,2\}\)-point-biregular space \((P,{\mathcal L})\). (2) Let \((P,{\mathcal L})\) be a finite projective plane, let \(L\in{\mathcal L}\), and take an additional element \(\infty\) not in \(P\). Setting \({\mathcal L}'={\mathcal L}\setminus\{L\}\), \(L_\infty=L\cup\{\infty\}\), and \({\mathcal L}_\infty=\{\{x,\infty\}\} x\in P\setminus L\), the pair \((P\cup\{\infty\}, {\mathcal L}'\cup\{L_\infty\}\cup{\mathcal L}_\infty)\) is a proper \(\{1,2\}\)-point-biregular space. (3) Let \((P,{\mathcal L})\) be a finite projective plane of order \(n\geq 3\). For some pair of distinct lines \(L\) and \(K\) set \(\{z\}=L\cap K\), \(S=(L\cup M)\setminus\{z\}\). Then \((P\setminus S, {\mathcal L}\setminus\{L,K\})\) is a proper \(\{n-2,n-1\}\)-point-biregular space. Throughout this self-contained paper, the arguments are elementary, but the proofs are far from being trivial.