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
$3\left({q}^{nk}+1\right)\u22152$ 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
$\left(nk\right)$spaces
or certain Baer cones. The latter ones are cones with vertex a
$t$space, where
$max\left\{1,n2k1\right\}\le t<nk1$, and with a
$2\left(\left(nk\right)t1\right)$dimensional
Baer subgeometry as a base. Bokler showed that nontrivial minimal blocking sets in
PG$\left(n,q\right)$ with respect
to
$k$spaces and
of size at most
$\left({q}^{nk+1}1\right)\u2215\left(q1\right)+$
$\sqrt{q}\left({q}^{nk}1\right)\u2215\left(q1\right)$
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 nontrivial minimal blocking sets of
PG$\left(n,q\right)$ with respect
to
$k$spaces and of
size less than
$3\left({q}^{nk}+1\right)\u22152$
are Baer cones.
