Rank one sheaves over quaternion algebras on Enriques surfaces

TL;DR

This paper investigates quaternion algebra structures on Enriques surfaces and explores the moduli space of rank one torsion-free modules over these algebras, revealing a double cover relationship with a Lagrangian subvariety.

Contribution

It establishes the existence of nontrivial quaternion algebras on Enriques surfaces and describes the structure of the associated moduli space of modules.

Findings

Existence of a nontrivial quaternion algebra on Enriques surfaces

The moduli scheme of rank one A-modules is an étale double cover

The moduli scheme relates to a Lagrangian subscheme on the K3 cover

Abstract

Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra A on X. Then we study the moduli scheme of torsion free A-modules of rank one. Finally we prove that this moduli scheme is an \'{e}tale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.