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

Abstract

Let Q + (3, q) be a hyperbolic quadric in PG(3, q) and T be the set of all lines of PG(3, q) which are tangent to Q + (3, q). If k is the minimum size of a T-blocking set in PG(3, q), then we prove that q 2 + 1  k  q 2 + q. When q = 3, we show that: (i) there is no T-blocking set of size 10, and (ii) there are exactly two T-blocking sets of size 11 up to isomorphism. By means of the computer algebra systems GAP [13] and Sage [9], we find that there exist no T-blocking sets of size q 2 + 1 for each odd prime power q  13.

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

“…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].…”

On minimum size blocking sets of the outer tangents to a hyperbolic quadric in PG(3,q)

“…If k is the minimum size of a T -blocking set in PG(3, q), then q 2 + 1 ≤ k ≤ q 2 + q by [10,Lemmas 2.1,2.2]. If q is even, then the T -blocking sets of size q 2 + 1 are precisely the ovoids of the generalized quadrangle W (q) associated with Q + (3, q).…”

“…In the q odd case, not much is known for the minimum size T -blocking sets. In PG (3,3), by [10,Theorem 1.1], there is no T -blocking set of size 10 and there are exactly two T -blocking sets of size 11 up to isomorphism. By means of the computer algebra systems GAP [18] and Sage [13], it was proved that there exist no T -blocking sets of size q 2 + 1 for each odd prime power q ≤ 13, see [10,Theorem 1.2].…”

“…The number of nonisomorphic T -blocking sets of size q 2 + q − 1 that arise in this fashion is one if q = 3, four if q = 5 and six if q ∈ {7, 11}. Besides these seventeen T -blocking sets, one additional T -blocking set is known whose size is strictly smaller than q 2 + q, namely a particular T -blocking set of size 11 in PG (3,3), see [10,Section 3]. Note that T -blocking sets of size q 2 + q always exist, namely tangent planes minus their tangency points.…”

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

“…As l ranges over all four secant lines of π disjoint from π ∩ B, the point a will range over all four points of B ∩ H. As none of the four tangent planes π a , a ∈ B ∩ H, contains points of B \ H, we thus have: For every point x of P G(3, 3) \ H, the conic C x in x ζ is an ovoid of H. The map x → C x from P G(3, 3) \ H to the set of ovoids of H is a bijection (see e.g. [7]). Any two distinct ovoids of H intersect in at most two points.…”