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Moduli of polarized Enriques surfaces -- computational aspects

Mathieu Dutour Sikirić, Klaus Hulek

TL;DR

The possible arithmetic groups and show that there are exactly 87 such groups up to conjugacy are investigated and it is shown that all moduli spaces are dominated by a moduli space of polarised Enriques surfaces of degree 1240.

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.

Moduli of polarized Enriques surfaces -- computational aspects

TL;DR

The possible arithmetic groups and show that there are exactly 87 such groups up to conjugacy are investigated and it is shown that all moduli spaces are dominated by a moduli space of polarised Enriques surfaces of degree 1240.

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 such groups up to conjugacy. We also show that all moduli spaces are dominated by a moduli space of polarized Enriques surfaces of degree . 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.
Paper Structure (19 sections, 13 theorems, 59 equations, 1 figure, 6 tables)

This paper contains 19 sections, 13 theorems, 59 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Construction of the moduli spaces
  3. The computations
  4. Some basic facts and roots
  5. Enumerating polarizations of small degree
  6. Enumerating moduli spaces
  7. Degree of the polarization and number of moduli spaces
  8. Connection with the $\phi$-invariant
  9. The Tits building
  10. Classical cases
  11. The degree $2$ case
  12. Enriques realizations
  13. Reye congruences
  14. Algorithms for working with indefinite forms
  15. Integral group algorithms
  16. ...and 4 more sections

Key Result

Theorem 3.1

The minimal norm of integer vectors with trivial stabilizer in $U + E_8(-1)$ is $1240$ and in this degree there is a unique such vector.

Figures (1)

  • Figure 1: Coset graphs of the first $8$ coset graphs from Table \ref{['table_subgroups1']}. The red, blue, black and green dots correspond to the orbit arising from the orbit of $L_1$, $P_1$, $L_2$ and $P_2$.

Theorems & Definitions (16)

  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Lemma 3.6
  • Definition 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 4.4
  • ...and 6 more