On the linearity of higher-dimensional blocking sets

A small minimal k-blocking set B in PG(n, q), q = pt, p prime, is a set of less than 3(qk + 1)/2 points in PG(n, q), such that every (n - k)-dimensional space contains at least one point of B and such that no proper subset of B satisfies this property. The linearity conjecture states that all small minimal k-blocking sets in PG(n, q) are linear over a subfield Fpe of Fq. Apart from a few cases, this conjecture is still open. In this paper, we show that to prove the linearity conjecture for k- blocking sets in PG(n, pt), with exponent e and pe \geq 7, it is sufficient to prove it for one value of n that is at least 2k. Furthermore, we show that the linearity of small minimal blocking sets in PG(2, q) implies the linearity of small minimal k-blocking sets in PG(n, pt), with exponent e, with pe \geq t/e + 11.