Homology groups of translation planes and flocks of quadratic cones, I. The structure

Abstract

The set of translation planes with spreads in $PG(3,q)$ admitting cyclic affine homology groups of order $q+1$ is shown to be equivalent to the set of flocks of quadratic cones in $PG(3,q)$. The analysis is general and considers analogous homology groups in $PG(3,K)$, for $K$ an arbitrary field and corresponding partial flocks of quadratic cones in $PG(3,K)$.