Moduli of polarized Enriques surfaces -- computational aspects

TL;DR

This paper explores the classification and arithmetic properties of moduli spaces of polarized Enriques surfaces, identifying 87 key groups and their relationships, and providing computational tools for their analysis.

Contribution

It determines the exact number of conjugacy classes of arithmetic groups associated with these moduli spaces and computes their Tits buildings using algorithms on quadratic forms.

Findings

Exactly 87 conjugacy classes of arithmetic groups identified

All moduli spaces are dominated by a degree 1240 polarized Enriques surface space

Computational methods for Tits building construction are developed

Abstract

Moduli spaces of (polarized) Enriques surfaces can be described as open subsets of modular varieties of orthogonal type. It was shown by Gritsenko and Hulek that there are, up to isomorphism, only finitely many different moduli spaces of polarized Enriques surfaces. Here we investigate the possible arithmetic groups and show that there are exactly $87$ such groups up to conjugacy. We also show that all moduli spaces are dominated by a moduli space of polarized Enriques surfaces of degree $1240$. Ciliberto, Dedieu, Galati, and Knutsen have also investigated moduli spaces of polarized Enriques surfaces in detail. We discuss how our enumeration relates to theirs. We further compute the Tits building of the groups in question. Our computation is based on groups and indefinite quadratic forms and the algorithms used are explained.