Caps on Classical Varieties and Their Projections

A family of caps constructed by Ebert, Metsch and T. Szonyi [8] results from projecting a Veronesian or a Grasmannian to a suitable lower-dimensional space. We improve on this construction by projecting to a space of much smaller dimension. More precisely we partition PG(3r \Gamma 1; q) into a (2r \Gamma 1)\Gammaspace, an (r \Gamma 1)\Gammaspace and q r \Gamma 1 cyclic caps, each of size (q 2r \Gamma 1)(q \Gamma 1): We also decide when one of 1 our caps can be extended by a point from the (2r \Gamma 1)\Gammaspace or the (r \Gamma 1)\Gammaspace. The proof of the results uses several ingredients, most notably hyperelliptic curves. 1 Introduction Let PG(N; q) be the projective space of dimension N over the finite field GF (q). A k--cap K in PG(N; q) is a set of k points, no three of which are collinear [13], and a k--cap is complete if it is maximal with respect to inclusion. The maximum value of k for which there exists a k--cap in PG(N; q) is denoted by m 2 (N; q). The number m..