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

TL;DR

This paper proves the linearity conjecture for small minimal k-blocking sets in projective spaces over finite fields of order p^3, establishing their Fp-linearity for primes p >= 7.

Contribution

It demonstrates that small minimal k-blocking sets in PG(n, p^3) are linear, confirming the linearity conjecture in this specific case.

Findings

Small minimal k-blocking sets are linear in PG(n, p^3).

All such sets are Fp-linear for p >= 7.

The linearity conjecture is proved for this case.

Abstract

In this paper, we show that a small minimal k-blocking set in PG(n, q3), 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 Fp-linear, proving the linearity conjecture (see [7]) in the case PG(n, p3), p prime, p >= 7.