Caps on Classical Varieties and their Projections

A 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.