Moduli of polarized Enriques surfaces

TL;DR

This paper studies the structure and properties of moduli spaces of polarized Enriques surfaces, revealing their finite classification, Kodaira dimension behavior, and birational relations for certain polarizations.

Contribution

It provides a description of moduli spaces as orthogonal modular varieties and establishes finiteness, Kodaira dimension results, and birational equivalences for polarized Enriques surfaces.

Findings

Moduli spaces are open subsets of orthogonal modular varieties of dimension 10.

Finitely many isomorphism classes of these moduli spaces exist.

Kodaira dimension is negative for small degrees in many cases.

Abstract

In this paper we consider moduli spaces of polarized and numerically polarized Enriques surfaces. The moduli spaces of numerically polarized Enriques surfaces can be described as open subsets of orthogonal modular varieties of dimension 10. One of the consequences of our description is that there are only finitely many isomorphism classes of moduli spaces of polarized and numerically polarized Enriques surfaces. We use modular forms to prove for a number of small degrees that the Kodaira dimension of the moduli space of numerically polarized Enriques surfaces is negative. Finally we prove that there are infinitely many polarizatons for which the moduli space of numerically polarized Enriques surfaces is birational to the moduli space of unpolarized Enriques surfaces with a level 2 structure.