Conics and caps

In this article, we begin with arcs in PG(2, qn) and show that they correspond to caps in PG(2n, q) via the André/Bruck-Bose representation of PG(2, qn) in PG(2n, q). In particular, we show that a conic of PG(2, qn) that meets ℓ∞ in x points corresponds to a (qn + 1 - x)-cap in PG(2n, q). If x = 0, this cap is the intersection of n quadrics. If x = 1 or 2, this cap is contained in the intersection of n quadrics and we discuss ways of extending these caps. We also investigate the structure of the n quadrics. © 2011 Springer Basel AG.