The Picard Rank of an Enriques Surface

TL;DR

This paper provides a concise proof that the Picard rank of an Enriques surface equals its second Betti number, using crystalline methods and the Tate-conjecture.

Contribution

It introduces a novel proof technique combining crystalline methods and the Tate-conjecture for Enriques surfaces.

Findings

Picard rank equals second Betti number for Enriques surfaces

Crystalline methods effectively used in algebraic geometry proofs

Tate-conjecture application confirms Picard rank result

Abstract

In this note, we use crystalline methods and the Tate-conjecture to give a short proof that the Picard rank of an Enriques surface is equal to its second Betti number.