Finite projective geometries that are incidence structures of caps

Abstract

A t-cap in a geometry is a set of t points no three of which are collinear. A quadric in a projective geometry PG( n, q) is a set of points x which satisfy x T Q x = 0, Q being an n + 1 by n + 1 matrix over GF( q). We show that in any Desarguesian projective geometry PG(2 n, q), q odd, there exists a family of ( q + 1)-caps which obey the axioms for the lines in a projective geometry, and thus PG(2 n, q) can be regarded as an incidence structure of caps. To arrive at this result, a special family of quadrics is considered, the intersections of whose members provide the caps in question.