Generalized Pellegrino caps

A cap in a projective or affine geometry is a set of points with the property that no line meets the set in more than two points. Barwick, et al. (Conics and caps. J. Geom., 100:15–28, 2011) provide a construction of caps in PG(4, q) by “lifting” arbitrary caps of PG(2, q2), such as conics. In this article, we extend this construction by considering when the union of two or more conics in AG(2, q2) can be lifted to a cap of AG(4, q) using a similar coordinate transformation. In particular, the authors investigate a family of caps of size 2(q2 + 1) in AG(4, q) for all prime powers q > 2, of which the celebrated Pellegrino 20-cap in AG(4, 3) is the smallest example.