Blocking sets of certain line sets related to a hyperbolic quadric in PG(3, q)

Abstract

For a fixed hyperbolic quadric 𝓗 in PG(3, q), let 𝔼 (respectively 𝕋, 𝕊) denote the set of all lines of PG(3, q) which are external (respectively tangent, secant) with respect to 𝓗. We characterize the minimum size blocking sets of PG(3, q) with respect to each of the line sets 𝕊, 𝕋 ∪ 𝕊 and 𝔼 ∪ 𝕊.

Blocking sets of tangent and external lines to a hyperbolic quadric in PG(3,q)

“…When q > 2 is even, the minimum size (E ∪ S)-blocking sets were determined in [13, Theorem 1.3] using the properties of generalized quadrangles. For L ∈ {S, T ∪ S, E ∪ S}, the minimum size L-blocking sets are described in [12] for all q. We shall prove the following in this paper.…”

Blocking sets of tangent and external lines to an elliptic quadric in PG(3, q)

“…The following result was proved in [17,Lemma 2.4], also see [6,Proposition 3.1]. PROPOSITION 1.5 [6,17] Let x be a point of PG(2, q) and let L be the set of all lines of PG(2, q) not containing x. If A is an L-blocking set in PG(2, q), then |A| ≥ q and equality holds if and only if A = L\{x} for some line L through x.…”

“…The following result was proved in [17,Lemma 2.4], also see [6,Proposition 3.1]. PROPOSITION 1.5 [6,17] Let x be a point of PG(2, q) and let L be the set of all lines of PG(2, q) not containing x.…”

“…We note that the minimum size blocking sets in PG(3, q) of similar line sets with respect to a hyperbolic quadric have been characterized in the papers [5,6,9,10,[16][17][18].…”

Blocking sets of tangent lines to a hyperbolic quadric in PG(3, 3)

“…When q > 2 is even, the minimum size (E [ S)-blocking sets were determined in [12,Theorem 1.3] using the properties of generalized quadrangles. For L 2 {S, T [ S, E [ S}, the minimum size L-blocking sets are described in [11] for all q. When q is even, the minimum size (E [ T )-blocking sets are characterized in [10,Proposition 1.5].…”