Enriques' characterization of Abelian surfaces in positive characteristic

TL;DR

This paper extends Enriques' characterization of Abelian surfaces to algebraically closed fields with characteristic p ≥ 7, showing specific conditions under which surfaces are birational to Abelian surfaces.

Contribution

It generalizes Enriques' classical result to positive characteristic p ≥ 7 and identifies the failure of this characterization for p ≤ 5.

Findings

Surfaces with h^1 = 2 and p_1 = p_2 = 1 are birational to Abelian surfaces in characteristic p ≥ 7.

The characterization does not hold for p ≤ 5, with a sharp alternative provided.

The result bridges classical complex geometry and positive characteristic algebraic geometry.

Abstract

Extending Enriques' characterization to algebraically closed fields of characteristic $p \geq 7$, we show that every smooth projective surface $X$ with $h^1(X, \mathcal{O}_X) = 2$ and $p_1(X) = p_2(X) = 1$ is birational to an Abelian surface. This characterization fails if $p \leq 5$, and we give a sharp alternative.