In this paper, we show that a small minimal $k$-blocking set in $\PG(n,q^3)$, $q=p^h$, $h\geq 1$, $p$ prime, $p\geq 7$, intersecting every $(n-k)$-space in $1\pmod{q}$ points, is linear. As a corollary, this result shows that all small minimal $k$-blocking sets in $\PG(n,p^3)$, $p$ prime, $p\geq 7$, are $\mathbb{F}_p$-linear, proving the linearity conjecture (see \cite{sziklai}) in the case $\PG(n,p^3)$, $p$ prime, $p\geq 7$.
A proof of the linearity conjecture for k-blocking sets in PG(n, p(3)), p prime
LAVRAUW, MICHEL;
2011
Abstract
In this paper, we show that a small minimal $k$-blocking set in $\PG(n,q^3)$, $q=p^h$, $h\geq 1$, $p$ prime, $p\geq 7$, intersecting every $(n-k)$-space in $1\pmod{q}$ points, is linear. As a corollary, this result shows that all small minimal $k$-blocking sets in $\PG(n,p^3)$, $p$ prime, $p\geq 7$, are $\mathbb{F}_p$-linear, proving the linearity conjecture (see \cite{sziklai}) in the case $\PG(n,p^3)$, $p$ prime, $p\geq 7$.
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