Innovations in Incidence Geometry

Small point sets of $\mathrm{PG}(n,q)$ intersecting each $k$-space in $1$ modulo $\sqrt{q}$ points

Zsuzsa Weiner

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The main result of this paper is that point sets in PG(n,q), q=p2h, q81, p>2, of size less than 3(qnk+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 (nk)-spaces or certain Baer cones. The latter ones are cones with vertex a t-space, where max{1,n2k1}t<nk1, and with a 2((nk)t1)-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 (qnk+11)(q1)+ q(qnk1)(q1) 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 non-trivial minimal blocking sets of PG(n,q) with respect to k-spaces and of size less than 3(qnk+1)2 are Baer cones.

Article information

Innov. Incidence Geom., Volume 1, Number 1 (2005), 171-180.

Received: 10 December 2004
Accepted: 21 December 2004
First available in Project Euclid: 26 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx] 51E21: Blocking sets, ovals, k-arcs

blocking sets Baer subgeometries


Weiner, Zsuzsa. Small point sets of $\mathrm{PG}(n,q)$ intersecting each $k$-space in $1$ modulo $\sqrt{q}$ points. Innov. Incidence Geom. 1 (2005), no. 1, 171--180. doi:10.2140/iig.2005.1.171.

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