A proof of the linearity conjecture for k-blocking sets in PG(n,p3), p prime

In this paper, we show that a small minimal k-blocking set in PG(n,q3), q=ph, h⩾1, p prime, p⩾7, intersecting every (n−k)-space in points, is linear. As a corollary, this result shows that all small minimal k-blocking sets in PG(n,p3), p prime, p⩾7, are Fp-linear, proving the linearity conjecture (see Sziklai, 2008 [9]) in the case PG(n,p3), p prime, p⩾7.