On sets of type \((m,m+q)_2\) in \(\mathrm{PG}(3,q)\)
A set \(S\) of points of \(\mathrm{PG}(3,q)\) is a set of type \((m,m+q)_2\) if every plane of \(\mathrm{PG}(3,q)\) meets \(S\) in \(m\) or \(m+q\) points and there exists planes of both types. The notation \(H_m(q)\) refers to a set of type \((m,m+q)_2\) of size \(m(q+1)\). An \(H_m(q)\) is called \textit{irreducible} if \(m=1\) or if it does not contain an \(H_n(q)\) with \(1 \leq n \leq m-1\). Let \(\mathcal{P}\) be a partial spread of \(\mathrm{PG}(3,q)\) of size \(m\), then the set \(K\) of all points on the lines of \(\mathcal{P}\) is an \(H_m(q)\). The complement of \(K\), i.e. the set of holes with relation to \(\mathcal{P}\) is an \(H_{q^2+1-m}(q)\), not containing a line if and only if \(\mathcal{P}\) is a maximal partial spread. The main results of this paper are classification results of \(H_n(q)\)'s for small values of the parameters \(n\) and \(q\), and it turns out that the classification of maximal partial spreads for these values of \(q\) plays an important role. The main results concerning sets \(H_m(q)\) are summarized in the following theorem. Note that this result classifies \textit{previously known} structures, as is clearly indicated in the paper. Theorem. Up to projective equivalence, the only irreducible sets \(H_4(q)\) exist for \(q \in \{4,5,9\}\), and are {\parindent=0.7cm\begin{itemize}\item[--] two irreducible \(H_4(4)\), each of them the set of holes of one of the 13 maximal partial spreads of \(\mathrm{PG}(3,4)\) size \(13\); \item[--] the unique irreducible \(H_4(5)\), which is the set of holes of one of the maximal partial spreads of \(\mathrm{PG}(3,4)\) size \(22\); \item[--] the unique irreducible \(H_4(9)\) which is the set of points of a Baer subplane. \end{itemize}} Assume that \(q+1 \mid (m+q)(q^2+1)\), then the notation \(K_m\) refers to a set of type \((m,m+q)_2\) of size \(\frac{(m+q)(q^2+1)}{q+1}\). A set \(K_m\) is irreducible if \(m< q+1\) or it does not contain a set \(K_n\) with \(n <m\). The final section of the paper is devoted to some particular results on irreducible sets \(K_m\), \(m = \frac{q+3}{2}\).
scientific article; zbMATH DE number 6818637
scientific article