Flocks of hyperbolic quadrics and translation planes admitting affine homologies (Q1118809)

\textit{J. A. Thas} [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 59(1975), 83-85 (1976; Zbl 0359.50023)] showed how to construct translation planes of order \(q^ 2\) having GF(q) in their kernel from flocks in PG(3,q). The work of \textit{H. Gevaert} and the author [Geom. Dedicata 27, No.3, 301-317 (1988; Zbl 0653.51012)] and \textit{H. Gevaert}, the author, and \textit{J. A. Thas} [Simon Stevin 62, 51-62 (1988; Zbl 0659.51007)] determines the planes obtained from flocks of quadratic cones while \textit{J. A. Thas} [Eur. J. Comb. 8, 441-452 (1987; Zbl 0646.51019)] shows that flocks of elliptic quadrics always give desarguesian planes. This article determines those translation planes arising from flocks of hyperbolic quadrics in PG(3,q). These planes, besides having order \(q^ 2\) and containing GF(q) in their kernel, admit an affine homology group H with axis 1 and coaxis m, say, such that some orbit (under H) of components together with 1 and m form a regulus in PG(3,q). The author also algebraically describes the spreads; by this description he explicitly gives the correspondence between the flocks and the planes.