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.