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GIT stable cubic threefolds and certain fourfolds of $K3^{[2]}$-type

Lucas Li Bassi

TL;DR

The paper extends the Boissière–Camere–Sarti correspondence between smooth cubic threefold moduli and HK4-type manifolds with a non-symplectic order-3 automorphism to loci of singular cubics with isolated $A_i$-type singularities. It uses period maps, degeneracy lattices, and lattice-polarized moduli to obtain birational equivalences for $ix{ix{A1}}$, $ix{ix{A3}}$, and $ix{ix{A4}}$ to moduli spaces of $K3^{[2]}$-type manifolds with enlarged invariant lattices, all carrying a compatible non-symplectic automorphism of order three. The $A_2$ case, which presents additional challenges due to non-general degeneracy, motivates and develops the concept of Kähler cone sections of $K$-type, yielding a refined moduli framework and a birational description of $ix{A2}$ in terms of a specialized $( ho,j)$-polarized moduli space. Collectively, the results illuminate how nodal degenerations of cubic threefolds correspond to HK4-type moduli with larger invariant lattices, enriching the interplay between cubic hypersurfaces and irreducible holomorphic symplectic geometry.

Abstract

We study the behaviour on some nodal hyperplanes of the isomorphism, described in a paper of 2019 by Boissière, Camere and Sarti, between the moduli space of smooth cubic threefolds and the moduli space of hyperkähler fourfolds of $K3^{[2]}$-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square 6; along those hyperplanes the automorphism degenerates by jumping to another family. We generalize their result to singular nodal cubic threefolds having one singularity of type $A_i$ for $i=2, 3, 4$ providing birational maps between the loci of cubic threefolds where a generic element has an isolated singularity of the types $A_i$ and some moduli spaces of hyperkähler fourfolds of $K3^{[2]}$-type with non-symplectic automorphism of order three belonging to different families. In order to treat the $A_2$ case, we introduce the notion of Kähler cone sections of $K$-type generalizing the definition of $K$-general polarized hyperkähler manifolds.

GIT stable cubic threefolds and certain fourfolds of $K3^{[2]}$-type

TL;DR

The paper extends the Boissière–Camere–Sarti correspondence between smooth cubic threefold moduli and HK4-type manifolds with a non-symplectic order-3 automorphism to loci of singular cubics with isolated -type singularities. It uses period maps, degeneracy lattices, and lattice-polarized moduli to obtain birational equivalences for , , and to moduli spaces of -type manifolds with enlarged invariant lattices, all carrying a compatible non-symplectic automorphism of order three. The case, which presents additional challenges due to non-general degeneracy, motivates and develops the concept of Kähler cone sections of -type, yielding a refined moduli framework and a birational description of in terms of a specialized -polarized moduli space. Collectively, the results illuminate how nodal degenerations of cubic threefolds correspond to HK4-type moduli with larger invariant lattices, enriching the interplay between cubic hypersurfaces and irreducible holomorphic symplectic geometry.

Abstract

We study the behaviour on some nodal hyperplanes of the isomorphism, described in a paper of 2019 by Boissière, Camere and Sarti, between the moduli space of smooth cubic threefolds and the moduli space of hyperkähler fourfolds of -type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square 6; along those hyperplanes the automorphism degenerates by jumping to another family. We generalize their result to singular nodal cubic threefolds having one singularity of type for providing birational maps between the loci of cubic threefolds where a generic element has an isolated singularity of the types and some moduli spaces of hyperkähler fourfolds of -type with non-symplectic automorphism of order three belonging to different families. In order to treat the case, we introduce the notion of Kähler cone sections of -type generalizing the definition of -general polarized hyperkähler manifolds.
Paper Structure (17 sections, 30 theorems, 72 equations)

This paper contains 17 sections, 30 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Irreducible Holomorphic Symplectic Manifols
  3. Moduli Spaces and period maps
  4. Cubic threefolds
  5. Moduli space of cubic threefold as a ball quotient
  6. Motivating example
  7. Degeneracy lattices
  8. Singularity of type $A_1$
  9. Singularity of type $A_3$ and $A_4$
  10. Singularity of type $A_3$
  11. Singularity of type $A_4$
  12. Singularity of type $A_2$
  13. Kähler cone sections of $K$-type
  14. A moduli space for the singularity of type $A_2$
  15. Complex Reflection Groups and Springer Theory
  16. ...and 2 more sections

Key Result

Theorem 1.1

The $\Delta_3^{A_i}$ locus for $i=1, \dots, 4$ is birational to a $(10-i)$-dimensional moduli space of fourfolds of $K3^{[2]}$-type with Picard group of the generic member isometric to $R_i$ endowed with a non-symplectic automorphism of order three, having invariant lattice isometric to $T_i$. These

Theorems & Definitions (80)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Alternative definition
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6: Hodge theoretic Torelli Theorem Markman_surveyTorelli
  • Definition 2.7: amerikverbitsky2014, Mongardi_notes
  • Theorem 2.8: amerikverbitsky2014
  • Example 2.9: Numerical characterization in the $K3^{[2]}$-case
  • ...and 70 more