Caps on Classical Varieties and their Projections

AbstractA family of caps constructed by G. L. Ebert, K. Metsch and T. Szönyi results from projecting a Veronesian or a Grassmannian 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(3 r− 1, q) into a (2 r− 1)-space, an (r− 1)-space andqr− 1 cyclic caps, each of size (q2r− 1)/(q− 1). We also decide when one of our caps can be extended by a point from the (2 r− 1)-space or the (r− 1)-space. The proof of the results uses several ingredients, most notably hyperelliptic curves