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

Abstract

The main result of this paper is that point sets in PG$\left(n,q\right)$, $q=p^{2h}$, $q\ge 81$, $p>2$, of size less than $3\left(q^{n-k}+1\right)∕2$ and intersecting each $k$-space in $1$ modulo $\sqrt{q}$ points (such point sets are always minimal blocking sets with respect to $k$-spaces) are either $\left(n-k\right)$-spaces or certain Baer cones. The latter ones are cones with vertex a $t$-space, where $max\left\{-1,n-2k-1\right\}\le t<n-k-1$, and with a $2\left(\left(n-k\right)-t-1\right)$-dimensional Baer subgeometry as a base. Bokler showed that non-trivial minimal blocking sets in PG$\left(n,q\right)$ with respect to $k$-spaces and of size at most $\left(q^{n-k+1}-1\right)∕\left(q-1\right)+$ $\sqrt{q}\left(q^{n-k}-1\right)∕\left(q-1\right)$ 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$\left(n,q\right)$ with respect to $k$-spaces and of size less than $3\left(q^{n-k}+1\right)∕2$ are Baer cones.