The linearity Conjecture

The Linearity Conjecture states that in the projective plane PG(2,$q$), $q=p^h$, every minimal blocking set (w.r.t. lines) is a linear point set over some subfield of ${\mathbb{F}}_q$. This paper contains the proof of the Linearity Conjecture for $p^e\mathrm{\ne }3,4$, where $e$ is the exponent of the blocking set. The text contains 3 concatenated parts. In Part 0, we give a guide for reading Parts 1 and 2. Part 1 develops a great part of the method, and the case $p^e=2$ is solved. In Part 2 we prove the Linearity Conjecture for $p^e\ge 5$.