On the Largest Caps Contained in the Klein Quadric of PG(5, q), q Odd

This article studies the largest caps, of cardinality q3+q2+q+1, contained in the Klein quadric of PG(5, q), q odd. Presently, there are three examples of such caps known. They all are the intersection of the Klein quadric with a suitably chosen singular quadric with its vertex a line L and base a non-singular three-dimensional elliptic quadric. In this paper, we show that a (q3+q2+q+1)-cap contained in the Klein quadric of PG(5, q), q odd, q>3138, always is the intersection of the Klein quadric with another quadric, thus showing that such caps are a V43 variety of dimension 3 and of degree 4. This result also implies that a (q3+q2+q+1)-cap contained in the Klein quadric of PG(5, q), q odd, q>3138, defines a quadratic line complex of PG(3, q).