Normal spreads (Q1301604)

A \((t - 1)\)-spread in \(\Sigma = PG(n - 1, q)\) is a family \({\mathcal S}\) of mutually disjoint subspaces of dimension \(t - 1\) which partition the points of \(\Sigma\). A necessary and sufficient condition for the existence of a spread is \(t |n\), and so we assume \(n = rt\) for some integer \(r\) from now on. There is a natural \(2 - (q^{rt}, q^t, 1)\) translation design with parallelism associated with each \((t - 1)\)-spread, which is a translation plane when \(r = 2\). If \(r > 2\), a \((t - 1)\)-spread \({\mathcal S}\) is called normal (or geometric) if \({\mathcal S}\) induces a spread in any subspace generated by two of its elements. The associated design \(A({\mathcal S})\) is isomorphic to \(AG(r,q^t)\) if and only if \({\mathcal S}\) is normal. Normal spreads have many applications in finite geometry from the construction of packings to the classification of certain kinds of flag-transitive affine planes. R. C. Bose pointed out that taking the elements of a normal \((t - 1)\)-spread \({\mathcal S}\) as ``points'', the subspaces of dimension \(2t - 1\) joining two elements of \({\mathcal S}\) as ``lines'', and the natural inclusion as ``incidence'' yields an isomorphic copy of \(PG(r - 1, q^t)\). In the paper under review the author represents normal spreads by using Grassmannians. Let \(\Sigma = PG(rt - 1, q)\), and let \({\mathcal G} \subseteq \overline \Sigma = PG ( {{rt}\choose{t}} - 1, q)\) be the Grassmannian of \((t - 1)\)-subspaces of \(\Sigma\). Let \(g\) be the associated Grassmannian map. If \({\mathcal S}\) is a normal \((t - 1)\)-spread of \(\Sigma\), it is shown here that there exists an \((r^t - 1)\)-subspace \(\Delta\) of \(\overline \Sigma\) such that \(g ({\mathcal S}) = \Delta \cap {\mathcal G}\) is an algebraic variety \({\mathcal V}_{r,t}\) which is a \((q^{t(r - 1)} + q^{t(r - 2)} + \ldots + q^t + 1)\)-cap of \(\Delta \cong PG(r^t - 1, q)\). Moreover, the collineation group of \(\Delta\) stabilizing \({\mathcal V}_{r,t}\) acts 2-transitively on its points. As a corollary it is shown that \({\mathcal V}_{r,2}\) is the union of 3-dimensional elliptic quadrics, and for \(q\) even \({\mathcal V}_{r,3}\) is the union of ovoids of the hyperbolic quadric \(Q^+ (7,q)\). Now restrict to the case \(r = 3\) and \(t = 2\), so that the Bose model for \(PG(2,q^2)\) is obtained from a normal 1-spread \({\mathcal S}\) of \(PG(5,q)\) as previously described. Let \({\mathcal T}\) be the lines of \({\mathcal S}\) corresponding to some Bose subplane of \(PG(2, q^2)\). Then it is shown that there is a 5-dimensional subspace \(S\) of \(\Delta \cong PG(8,q)\) such that \(g({\mathcal T}) = S \cap {\mathcal V}_{3,2}\) is a Veronesean surface of \(S\). From this it follows that \({\mathcal V}_{3,2}\) is partitioned into \(q^2 - q + 1\) Veronesean surfaces. Similarly, it is shown that a Hermitian curve of \(PG(2,q^2)\) is represented on the variety \({\mathcal V}_{3,2}\) by the intersection of \({\mathcal V}_{3,2}\) with a certain hyperplane of \(\Delta\). If \(q \equiv 0,2 \pmod 3\), this hyperplane section of \({\mathcal V}_{3,2}\) is a ``unitary'' ovoid of \(Q^+ (7,q)\) [see \textit{W. M. Kantor}, Can. J. Math. 34, 1195-1207 (1982; Zbl 0493.51006) for an alternate description]. The final ingredient of this paper has to do with blocking sets. Let \(r = 3\) and consider the Bose model for \(\Pi = PG(2, q^t)\) obtained from a normal \((t - 1)\)-spread \({\mathcal S}\) of \(\Sigma = PG (3t - 1, q)\). If \(L\) is a \(t\)-subspace of \(\Sigma\) not contained in one of the \((2t - 1)\)-subspaces joining two elements of \({\mathcal S}\), it is shown that the lines of \({\mathcal S}\) meeting \(L\) correspond to a nontrivial blocking set of \(\Pi\). Such blocking sets are called linear. Many known blocking sets, including those of Rédei-type, turn out to be linear. A class of non-Rédei minimal blocking sets, which are also linear, has recently been constructed in [\textit{P. Polito} and \textit{O. Polverino}, Combinatorica 18, No. 1, 133-137 (1998; Zbl 0910.05017)]. This idea is then extended to construct a blocking set of \(PG(3,q^2)\) (i.e., meets every line but contains no hyperplane) of the smallest possible size.