A proof of the linearity conjecture for k-blocking sets in , p prime

In this paper, we show that a small minimal k-blocking set in PG ( n , q 3 ) , q = p h , h ⩾ 1 , p prime, p ⩾ 7 , intersecting every ( n − k ) -space in 1 ( mod 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 ⩾ 7 , are F p -linear, proving the linearity conjecture (see Sziklai, 2008 [9]) in the case PG ( n , p 3 ) , p prime, p ⩾ 7 .