Enriques' classification in characteristic $ p >0$ : the $P_{12}$-Theorem

The main goal of this paper is to show that Castelnuovo- Enriques' $P_{12}$-theorem also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ($char(k) = p > 0$). The $P_{12}$-theorem is a precise version of the rough classification of algebraic surfaces, in particular the conditions $P_{12} = 0$, $P_{12} = 1$,$P_{12} \geq 2$ are respectively equivalent to : Kodaira dimension $-\infty, 0 , \geq 1$. The result relies on a main theorem describing the growth of the plurigenera for properly-elliptic or properly quasi-elliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e. the families of surfaces which show that the results of the main theorem are sharp.