Polarities in planar spaces (Q1601366)

A linear space is a pair \((\mathcal S,\mathcal L)\), where \(\mathcal S\) is a set of points and \(\mathcal L\) is a set of subsets of \(\mathcal S\), called lines, such that any line contains at least two points and any two distinct points lie on exactly one line. A subspace of a linear space is a set of points containing the line through any two of its distinct points. A planar space is a triple \((\mathcal S,\mathcal L,\mathcal P)\), where \((\mathcal S,\mathcal L)\) is a linear space and \(\mathcal P\) is a set of proper subspaces of \((\mathcal S,\mathcal L)\), called planes, such that through every three non-collinear points there is exactly one plane and every plane contains at least three non-collinear points. A planar space is locally projective if for every pair of distinct planes their intersection is either a line or the empty set. Let \({\mathbb P}_3\) be a \(3\)-dimensional projective space and let \(\mathcal S\) be a subset of the point-set of \({\mathbb P}_3\). If no line of \({\mathbb P}_3\) meets \(\mathcal S\) in exactly one point, then \((\mathcal S,\mathcal L,\mathcal P)\), where \(\mathcal L=\{\ell \cap \mathcal S: \ell\) line of \({\mathbb P}_3 \}\) and \(\mathcal P=\{\pi \cap \mathcal S: \pi \) plane of \({\mathbb P}_3\}\) is a locally projective planar space which is said to be embedded in \({\mathbb P}_3.\) In the last thirty years many authors tried to work on the following problem: Is any locally projective planar space embedded in a \(3\)-dimensional projective space? The problem seems to be too difficult without further conditions on the planar space. All results on this problem have been obtained putting some additional arithmetical or geometrical conditions on the locally projective planar space. In the paper under review the author considers the previous problem assuming that the locally projective planar space is equipped with an involutorial permutation of the line-set. Let \((\mathcal S,\mathcal L,\mathcal P)\) be a locally projective planar space and suppose there exists an involutorial permutation \(f\) of the line-set satisfying the following conditions: (i) if \(L\) and \(M\) are coplanar lines, then \(f(L)\) and \(f(M)\) are coplanar lines; (ii) there are three non-coplanar lines through a common point whose images under \(f\) are three coplanar lines. The author proves that under these hypotheses \((\mathcal S,\mathcal L,\mathcal P)\) can be embedded in some \(3\)-dimensional projective space \({\mathbb P}_3\) and there exists a polarity \(\phi\) of \({\mathbb P}_3\) whose restriction to \((\mathcal S,\mathcal L,\mathcal P)\) gives \(f\). A second result obtained in the paper is that a locally projective planar space \((\mathcal S,\mathcal L,\mathcal P)\) is embedded in a \(3\)-dimensional projective space \({\mathbb P}_3\) also if \((\mathcal S,\mathcal L\,\mathcal P)\) is equipped with an involutorial permutation \(f\) of the line-set satisfying (i) and such that no line is fixed by \(f\). Finally, in the last section it is proved that, in the particular case when any two distinct planes intersect in a line, the two results of the paper follow also from two theorems of \textit{H. Havlicek} [ Discrete Math. 208/209, 319--324 (1999; Zbl 0943.51015)] and \textit{J. Kahn} [ Math. Z. 175 , 219--247 (1980; Zbl 0433.06013)], respectively.