Twisted hyperbolic flocks

Abstract

We give a generalization of the theory of flocks of hyperbolic quadrics in PG$(3,q)$ to what is called an $\alpha$-twisted hyperbolic flock in an arbitrary $3$-dimensional projective space over a field $K$. We obtain an equivalence between a set of translation planes with spreads in PG$(3,K)$ that admit affine homology groups such that the axis and coaxis and one orbit is a twisted regulus. Examples and generalizations are also given.