Caps on Classical Varieties and their Projections

A family of caps constructed by Ebert, Metsch and T. Szönyi [8] 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(3r − 1, q) into a (2r − 1)−space, an (r − 1)−space and qr − 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 (2r − 1)−space or the (r− 1)−space. The proof of the results uses several ingredients, most notably hyperelliptic curves.