Infinite nests of reguli (Q1388153)

Let \(K\) be a field. A nest of reguli is a set of reguli in PG\((3,K)\) such that any line of the union of the set is contained in exactly two reguli of the set. The nest is called replaceable if there exists a collection of mutually disjoint lines of PG\((3,K)\) which cover the same points as the lines contained in the union of the nest. If all these lines are contained in a spread of PG\((3,K)\) this replacement process yields a new spread. A field is called full if it admits exactly one quadratic extension. In the paper under review, the author assumes that \(K\) is a full field of characteristic not 2 and that the unique quadratic extension field \(L\) of \(K\) is also full. He shows that then there exist some replaceable nests in PG\((3,K)\) whose reguli are contained in the pappian spread associated with the plane over \(L\). Each of these nests is invariant under a certain automorphism group of the pappian spread and hence this group also acts on the translation plane associated with the spread obtained by the replacement process. The author also gives a characterization of these group replaceable spreads. Most of the results are infinite versions of known results in the finite case. Note that every finite field satisfies the assumptions made on \(K\).