Homology groups of translation planes and flocks of quadratic cones. I: The structure (Q2471474)

A hyperbolic fibration of \(\text{PG}(3,q)\) is a collection of \(q-1\) hyperbolic quadrics and two lines in \(\text{PG}(3,q)\) that partition the points of \(\text{PG}(3,q)\). By selecting on each of the \(q-1\) quadrics one of the associated reguli one gets a spread of \(\text{PG}(3,q)\). A hyperbolic fibration is called regular if the two lines of the fibration are conjugate with respect to the polarity associated with any of the quadrics of the fibration. A regular hyperbolic fibration is said to have a constant half if one of the two lines in the fibration intersects each quadric in the fibration in the same pair of conjugate points when extended to \(\text{GF}(q^ 2)\). The author proves that a hyperbolic fibration is regular and has a constant half if and only if the translation planes associated with the fibration admit an affine homology group with a regulus in \(\text{PG}(3,q)\) as an orbit of components. By work of \textit{R. D. Baker, G. L. Ebert} and \textit{T. Penttila} [Des. Codes Cryptography 34, 295--305 (2005; Zbl 1079.51504)], regular hyperbolic fibrations with a constant half are known to be a equivalent to flocks of quadratic cones, and the author gives a new explanation for this correspondence. The author's results also cover the infinite case.