Small point sets of PG(n,q) intersecting each k-space in 1 modulo points
Zsuzsa Weiner · Innovations in incidence geometry · 2005
The main result of this paper is that point sets in PG(n, q), q = p 2h , q ≥ 81, p > 2, of size less than 3(q n-k + 1)/2 and intersecting each k-space in 1 modulo √ q points (such point sets are always minimal blocking sets with respect to k-spaces) are either (n-k)-spaces or certain Baer cones.The latter ones are cones with vertex a t-space, where max{-1, n -2k -1} ≤ t < nk -1, and with a 2((nk)t -1)-dimensional Baer subgeometry as a base.Bokler showed that non-trivial minimal blocking sets in PG(n, q) with respect to k-spaces and of size at most (q n-k+1 -1)/(q -1)+ √ q(q n-k -1)/(q -1) are such Baer cones.The corollary of the main result is that we improve on Bokler's bound.The improvement depends on the divisors of h; for example, when q is a prime square, we get that the nontrivial minimal blocking sets of PG(n, q) with respect to k-spaces and of size less than 3(q n-k + 1)/2 are Baer cones.