## Innovations in Incidence Geometry

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

Zsuzsa Weiner

#### Abstract

The main result of this paper is that point sets in PG$(n,q)$, $q=p2h$, $q≥81$, $p>2$, of size less than $3(qn−k+1)∕2$ 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, and with a $2((n−k)−t−1)$-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 $(qn−k+1−1)∕(q−1)+$ $q(qn−k−1)∕(q−1)$ 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 non-trivial minimal blocking sets of PG$(n,q)$ with respect to $k$-spaces and of size less than $3(qn−k+1)∕2$ are Baer cones.

#### Article information

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

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

https://projecteuclid.org/euclid.iig/1551206819

Digital Object Identifier
doi:10.2140/iig.2005.1.171

Mathematical Reviews number (MathSciNet)
MR2213957

Zentralblatt MATH identifier
1113.51003

Keywords
blocking sets Baer subgeometries

#### Citation

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. https://projecteuclid.org/euclid.iig/1551206819

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