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We describe a period map for those simply connected Enriques surfaces incharacteristic 2 whose canonical double cover is K3. The moduli stack for thesesurfaces has a Deligne-Mumford quotient that is an open substack of a $\mathbbP^1$-bundle over the period space. We also give some general results relatinglocal and global moduli for algebraic varieties and describe the difference intheir dimensions in terms of the failure of the automorphism group scheme to bereduced.

Moduli and periods of simply connected Enriques surfaces