Partial linear complexes of PG\((3,q\)) (Q1849900)

A linear complex of PG\((3,q)\) is the set of singular lines of a symplectic polarity in PG\((3,q).\) It is also characterized as the unique set of \(q^3+q^2+q+1\) lines such that every plane contains exactly \(q+1\) concurrent lines and through every point there are exactly \(q+1\) (coplanar) lines. A partial linear complex of PG\((3,q)\) is a set \(B\) of lines of PG\((3,q)\) such that for every plane \(\pi\) the lines of \(B\) in \(\pi\) are concurrent and for every point \(P\) the lines of \(B\) through \(P\) are coplanar. Since every plane contains at most \(q+1\) lines of a partial linear complex, a partial linear complex has at most \(q^3+q^2+q+1\) lines. From a characterization theorem of \textit{M. J. de Resmini} [Ars Comb. 18, 99--102 (1984; Zbl 0552.51010)] it is known that a linear complex is the unique partial linear complex with this maximal number of lines. In the nice paper under review it is shown that the largest partial linear complex of PG\((3,q)\) that is not contained in a linear complex has \(q^3+3\) lines. An example of a partial linear complex with \(q^3+3\) lines is constructed in the following way: Consider a linear complex \(B\) and a line \(\ell\) not in \(B\). Then the lines of \(B\) in a plane \(\pi\) on \(\ell\) pass through a common point \(P\). If the plane \(\pi\) runs on the line \(\ell\), then the point \(P\) runs on a line \(h\) skew to \(\ell\). Hence the set \(S\) of the \((q+1)^2\) lines that meet \(\ell\) and \(h\) is a subset of \(B\). Let \(M\) be a subset of pairwise skew lines of \(S\). Remove from \(B\) the set \(S\setminus M\) and add the lines \(\ell\) and \(h\). This gives a partial linear complex with \(q^3+3\) lines that is not a subset of a linear complex. It is shown that this example is projectively unique for \(q\neq 3\). This is done first by translating the problem onto the Klein quadric via Plücker coordinates and then by proving seven lemmata.