ABSTRACT
Let be a translation plane with spread in PG(3; K) that produces a hyperbolic …bration with carriers x = 0; y = 0. We are assuming that K is a …eld that admits a quadratic extension K+. What we do in this chapter is show that the way that the reguli are represented as hyperbolic quadrics induces certain functions F and G on K+(+1) that will represent the partial ‡ock of a quadratic cone in PG(3; K). Recalling that K+(+1) corresponds to the set of determinants of the …eld of matrices representing K+, we see that there is an intrinsic problem of representation. In the …nite case, this never becomes an issue since K+(+1) is simply K isomorphic to GF (q). We begin with a general representation of the spread of as
x = 0; y = 0; y = x
u t
F (u; t) G(u; t)
;u; t 2 K,
for functions F and G on K K to K. Let
u;t = det
u t
F (u; t) G(u; t)
:
We have K+ as u t tf u+ gt
;u; t 2 K
.
