A geometric relationship between equivalent spreads
It is well known that a translation plane can be defined by a spread of a projective space \(P = PG(d, q)\) and vice versa. An \(n\)-spread of \(P = PG(d, q)\) is a set of \(n\)-dimensional subspaces partitioning the point set of \(P\). It may occur that an \(n\)-spread of \(PG(d, q)\) and an \(m\)-spread of \(PG(t, q)\) with \(n \neq m \) define the same translation plane. Two such spreads are called equivalent. In the article under review Mellinger provides a geometric algorithm to construct equivalent spreads. More precisely, if \(S_{m-1}\) is an \((m-1)\)-spread of \(PG(2m-1, q)\) defining a translation plane \(\pi\) of dimension \(n \neq m\) over its kernel, then the algorithm of Mellinger yields an \((n-1)\)-spread \(S_{n-1}\) of \(PG(2n-1, q)\) defining the same translation plane \(\pi\).
scientific article; zbMATH DE number 1975927