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Cubic fourfolds with symplectic automorphisms

Kenji Koike

Abstract

We determine projective equations of smooth complex cubic fourfolds with symplectic automorphisms by classifying 6-dimensional projective representations of Laza and Zheng's 34 groups. In particular, we determine the number of irreducible components for moduli spaces of cubic fourfolds with symplectic actions by these groups. We also discuss the fields of definition of cubic fourfolds in six maximal cases.

Cubic fourfolds with symplectic automorphisms

Abstract

We determine projective equations of smooth complex cubic fourfolds with symplectic automorphisms by classifying 6-dimensional projective representations of Laza and Zheng's 34 groups. In particular, we determine the number of irreducible components for moduli spaces of cubic fourfolds with symplectic actions by these groups. We also discuss the fields of definition of cubic fourfolds in six maximal cases.
Paper Structure (35 sections, 5 theorems, 249 equations, 2 tables)

This paper contains 35 sections, 5 theorems, 249 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Projective actions of finite groups and invariant hypersurfaces
  3. Schur covering groups
  4. Symplectic automorphisms of cubic 4-folds
  5. Invariants
  6. Moduli spaces
  7. $\mathrm{A}_6$-invariant cubic fourfolds
  8. Proof of Theorem 1.1
  9. Groups of small $r(G)$
  10. Groups of $r(G) \leq 14$
  11. The dihedral group $\mathrm{D}_8$
  12. Groups of $r(G) = 16$
  13. The dihedral groups $\mathrm{D}_{10}$ and $\mathrm{D}_{12}$
  14. The groups $\mathrm{A}_{3,3} = (\mathrm{S}_3 \times \mathrm{S}_3) \cap \mathrm{A}_6$
  15. The alternating group $\mathrm{A}_4$
  16. ...and 20 more sections

Key Result

Theorem 1.1

The moduli space of cubic fourfolds with $G \subset \mathrm{Aut}^s(X)$ has two irreducible components if $G$ is one of the following groups: For other cases, the moduli spaces are irreducible (see Table TBL-Schur in Appendix B ).

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 7 more