Moduli and periods of simply connected Enriques surfaces

TL;DR

This paper studies the moduli and period maps of certain simply connected Enriques surfaces in characteristic 2, revealing their structure via a $P^1$-bundle over the period space and analyzing local-global moduli relations.

Contribution

It introduces a period map for these Enriques surfaces, describes their moduli stack as an open substack of a $P^1$-bundle, and explores the relationship between local and global moduli dimensions.

Findings

The moduli stack has a Deligne-Mumford quotient as an open substack.

The period map relates Enriques surfaces to a period space via a $P^1$-bundle.

Differences in local and global moduli dimensions are linked to automorphism group scheme properties.

Abstract

We describe a period map for those simply connected Enriques surfaces in characteristic 2 whose canonical double cover is K3. The moduli stack for these surfaces has a Deligne-Mumford quotient that is an open substack of a $\mathbb P^1$-bundle over the period space. We also give some general results relating local and global moduli for algebraic varieties and describe the difference in their dimensions in terms of the failure of the automorphism group scheme to be reduced.