Another look at Bruen chains
There is a well-known one-to-one correspondence [see \textit{J. André}, Math. Z. 60, 156-186 (1954; Zbl 0056.385)] between spreads of odd- dimensional projective spaces over finite fields and the class of finite translation planes. Therefore spreads have been studied as a way of constructing nondesarguesian planes. A standard method is to take a regular spread, find an ``interesting'' partial spread in it and (if possible) to change the lines of the partial spread with the lines of an other partial spread covering the same set of points. One example of an interesting replaceable partial spread is that of a regulus. \textit{R. H. Bruck} [Combin. Math. Appl., Proc. Conf. Univ. North Carolina 1967, 426- 514 (1969; Zbl 0206.234)] used this idea to define and begin the classification of the subregular spreads. Surprisingly, not many such spreads are known. To make more and larger replaceable partial spreads, \textit{A. Bruen} [Geom. Dedicata 7, 81-98 (1978; Zbl 0373.50006)] defined the notion of a chain of reguli in a regular spread. All the known chains in the projective 3-space over GF(q) were for \(5\leq q\leq 13\). The authors develop a method to combine chains in order to get a number of replaceable partial spreads for small values of q, and they produce a new chain for \(q=17\).
scientific article; zbMATH DE number 4042685
scientific article