Regulus-free spreads of PG\((3,q)\)
A spread in \(PG(3,q)\) is a set of mutually disjoint lines which cover the space. This amounts exactly to what we call a spread on a 4-dimensional vector space over \(F= GF(q)\): A set of 2-dimensional subspaces which are disjoint except for the zero vector which cover the vector space. Then there is a translation plane whose lines are the translates of the components of this vector space spread. In a similar way a regulus corresponds to a derivable set defined on the vector space. An affine plane (of order \(q^2)\) which contains a derivable set can be derived to form a new plane. A translation plane is non-derivable if it contains no derivable sets -- i.e. the spread in \(PG(3,q)\) contains no reguli. There are many known examples of regulus-free spreads (non-derivable planes), particularly if \(q\) is a prime power but not a prime. The authors show that such non-derivable planes exist if \(q\) is an odd prime \(p\) such that \(p\equiv 1\) mod 3. The planes in question are non-Desarguesian flag transitive planes. A flag transitive translation plane is transitive on the points of \(l_\infty\).
scientific article; zbMATH DE number 994655
scientific article