A characterisation of spreads ovally-derived from Desarguesian spreads
Let \({\mathcal S}\) be a spread of \(\Sigma= \text{PG} (2n-1,2)\). That is, \({\mathcal S}\) consists of \(2^ n+1\) pairwise skew \((n-1)\)-dimensional subspaces of \(\Sigma\). An \((n-1)\)-dimensional subspace \(\sigma\) of \(\Sigma\) is called a transversal to \({\mathcal S}\) if \(| \sigma\cap \gamma| \leq 1\) for all \(\gamma\in {\mathcal S}\). The two members of \({\mathcal S}\) disjoint from a transversal to \({\mathcal S}\) are called the carriers of the transversal. In a previous paper [Ann. Discrete Math. 52, 279-296 (1992; Zbl 0770.51003)] the authors showed that the translation plane \(\pi({\mathcal S})\) associated with \({\mathcal S}\) admits a translation oval if and only if \({\mathcal S}\) admits a transversal. In the paper under review the authors consider the possibility of transversal covers in the following sense. A transversal cover of \({\mathcal S}\), relative to a pair \(\{X, Y\}\) of distinct members of \({\mathcal S}\), is a collection of pairwise skew transversals to \({\mathcal S}\), all with carrier set \(\{X,Y\}\), that partition all the points of \(\Sigma\) not lying on \(X\) or \(Y\). Thus a transversal cover \({\mathcal O}\) of \({\mathcal S}\) is a replacement set for the partial spread \({\mathcal S}\setminus \{X,Y\}\), and the corresponding replaced spread \({\mathcal O}\cup \{x,y\}\) is said to be ovally derived from \({\mathcal S}\). The question of oval derivation was considered in a more general context by \textit{E. F. Assmus jun.} and \textit{J. D. Key} in J. Geom. 37, No. 1/2, 3-16 (1990; Zbl 0705.51011)]. The authors prove that if \({\mathcal S}'\) is ovally derived from a Desarguesian spread, then the associated translation plane \(\pi({\mathcal S}')\) is a generalized André plane. However, not all generalized André planes are ovally derived from Desarguesian planes. It is shown that a generalized André plane is ovally derivable from a Desarguesian plane if and only if it admits \(\lambda\) conics, where \(\lambda\)-conics are naturally generalizations of the translation ovals found by \textit{R. H. F. Denniston} in Ars. Comb. 7, 221-222 (1979; Zbl 0431.51009). The proof is based upon the classification of translation ovals in the Desarguesian plane given by \textit{S. E. Payne} [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 5, 1(1971), 328-331 (1972; Zbl 0238.50018)]. In the case of near field planes the question is more easily resolved. Namely, it is shown that a nearfield spread with kernel \(\text{GF}(2^ s)\) and dimension \(r\) is ovally derived from a Desarguesian spread if and only if all prime divisors of \(r\) divide \(s\) and \(2^ s-1\) (i.e, \((2^ s,r)\) is a strong Dickson pair). It is shown that a two-dimensional André plane of (even) order \(q^ 2\) is ovally derived from a Desarguesian plane if and only if \(q\) is a square.
scientific article; zbMATH DE number 567472