Flocks of infinite hyperbolic quadrics
A \textit{flock} of a hyperbolic quadric \(H\) in \(PG(3,K)\), where \(K\) is a field, is a set of mutually disjoint conics whose union covers \(H\). When \(K\) is finite, these flocks have been classified, but little has been done in the infinite case. Over a finite field it is known that a flock of a hyperbolic quadric corresponds to a translation plane with a spread in \(PG(3,q)\) which is the union of a set of reguli mutually intersecting in two lines. These are \textit{Bol} translation planes, i.e., translation planes that admit a left coordinatizing quasifield \(Q\) that satisfies the Bol axiom: \(a(b \cdot ac) = (a \cdot ba)c\) for all \(a, b, c \in Q\). Since all finite Bol translation planes are nearfield planes, the planes corresponding to flocks of hyperbolic quadrics are all nearfield planes. In this paper the author investigates the corresponding situation when \(K\) is infinite. He shows that the correspondence between flocks and translation planes with spreads in \(PG(3,K)\) composed of a set of reguli that share two components is still valid. As there are constructions of infinite Bol planes that are not nearfield planes with the requisite type of spread, there are flocks of hyperbolic quadrics which give rise to non-nearfield translation planes. In an attempt to find flock planes that are not Bol planes, the author completely determines the set of André planes which produce the type of translation plane corresponding to a flock of a hyperbolic quadric. These, however, all turn out to be Bol planes. But, there are translation planes due to \textit{R. Riesinger} [J. Geom 45, No. 1/2, 137-157 (1992; Zbl 0787.51005)] having the desired type of spread (and so, are flock planes) which are not Bol planes.
scientific article; zbMATH DE number 987621
scientific article