Blocking Structures of Hermitian Varieties

Abstract

Given a hermitian variety H(d,q2) and an integer k? (d?1)/2, a blocking set with respect to k-subspaces is a set of points of H(d,q2) that meets all k-subspaces of H(d,q2). If H(d,q2) is naturally embedded in PG(d,q2), then linear examples for such a blocking set are the ones that lie in a subspace of codimension k of PG(d,q2). Up to isomorphism there are k+1 non-isomorphic minimal linear blocking sets, and these have different cardinalities. In this paper it is shown for 1? k<? (d?1)/2? that all sufficiently small minimal blocking sets of H(d,q2) with respect to k-subspaces are linear. For 1? k<? d/2??3, it is even proved that the k+1 minimal linear blocking sets are smaller than all minimal non-linear ones.