ABSTRACT
In this text, we shall use the device that connects translation planes of order q2 with spreads in PG(3; K), for K a …eld isomorphic to GF (q), with ovoids of the hyperbolic quadric in PG(5; K), the ‘Klein Quadric.’Basically, the same theory is available for hyperbolic quadrics of standard type in PG(3; K), where K is a …eld admitting a quadratic extension, and spreads in PG(3; K). Let (x1; x2; x3; y1; y2; y3); for xi; yi 2 K;where i = 1; 2; 3, and K is a …eld admitting a quadratic extension. Let x1y3 x2y2 + x3y1 = 0 de…ne a non-degenerate hyperbolic quadric Q in a 6-dimensional vector space V6 over K (or in PG(5; K)). We note that (1; a; b; c; d;), where = ad cb, then is a point of Q; the other point (1-dimensional K–subspace) has the form (0; 0; 0; 0; 0; 1). Let x and y be 2-vectors over K. We consider the ‘Klein map’K:
K : (1; a; b; c; d;)! y = x a b c d
;
K : (0; 0; 0; 0; 0; 1)! x = 0:
Then the image of Q under the Klein map is a bijection to the set of all 2-dimensionalK–subspaces of a 4-dimensionalK-space with vectors (x; y). Indeed, two mutually disjoint 2-dimensional K–subspaces map back to points on the quadric that are not incident to a line of the quadric. When K is isomorphic to GF (q), an ‘ovoid’of the quadric is a set of q2 +1 points no two of Q are incident with a line of the quadric. An analogous de…nition is available in the in…nite case.
