Nests and nest accessories

Source abstract

The theory of finite translation planes is intimately tied to the theory of spreads of PG(3,q), where a spread is a partition of PG(3,q) into $q^2+1$ pairwise disjoint lines. A traditional method of generating spreads begins with the regular spread and identifies sets of lines therein that can be replaced with other lines to create a new spread, such as reguli, Bruen chains, and nests. This work provides computational algorithms to enumerate all possible replaceable sets consisting of reguli, chains and nests in the regular spreads of PG(3,q) for $q \le 11$. Additional results include a refinement to Baker and Ebert's (q-1)-nests which expands that infinite family, and counterexamples to the long-standing conjecture that a t-nest of reguli in PG(3,q) must be replaceable if $t \le q$.