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

## TL;DR

This paper extends the classical P_{12}-theorem, which classifies algebraic surfaces based on plurigenera, to surfaces over fields of positive characteristic, providing new insights into their classification.

## Contribution

It proves that the P_{12}-theorem holds in positive characteristic and describes the growth of plurigenera for certain elliptic surfaces, establishing sharpness of the results.

## Key findings

- P_{12} classification holds in characteristic p > 0
- Growth of plurigenera for elliptic surfaces characterized
- Results are sharp, with limit cases analyzed

## Abstract

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.