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Cubic fourfolds containing highly singular hyperplane sections

Lisa Marquand, Sasha Viktorova

TL;DR

This work identifies five irreducible divisors in the moduli space $\\mathcal{C}$ of smooth cubic fourfolds, each determined by containing a hyperplane section with a highly singular $K$-type (including $E_6$, $D_6$, $D_5+A_1$, $D_4+A_2$, and $D_4+2A_1$) and a total Tjurina number of $6$. The authors establish the irreducibility (and codimension properties) of these loci, analyze the deformation theory of singular hyperplane sections, and show that these divisors are not Hassett divisors, using the Addington–Auel computational method to certify non-Hassett status via Frobenius eigenvalues. They also discuss a split in the $D_{D_4+2A_1}$ case according to the defect, and treat the $T_{333}$ locus as codimension $2$ outside Hassett divisors. The results yield explicit, computationally verifiable irrational cubic fourfolds outside Hassett loci and contribute to the broader understanding of the geometry of cubic fourfold moduli and their hyperplane-section singularities.

Abstract

We construct five irreducible divisors in the moduli space of complex cubic fourfolds parametrising smooth cubic fourfolds that contain highly singular hyperplane sections. We prove that each is not a Noether-Lefschetz (or Hassett) divisor, utilising the computational method developed by Addington-Auel.

Cubic fourfolds containing highly singular hyperplane sections

TL;DR

This work identifies five irreducible divisors in the moduli space of smooth cubic fourfolds, each determined by containing a hyperplane section with a highly singular -type (including , , , , and ) and a total Tjurina number of . The authors establish the irreducibility (and codimension properties) of these loci, analyze the deformation theory of singular hyperplane sections, and show that these divisors are not Hassett divisors, using the Addington–Auel computational method to certify non-Hassett status via Frobenius eigenvalues. They also discuss a split in the case according to the defect, and treat the locus as codimension outside Hassett divisors. The results yield explicit, computationally verifiable irrational cubic fourfolds outside Hassett loci and contribute to the broader understanding of the geometry of cubic fourfold moduli and their hyperplane-section singularities.

Abstract

We construct five irreducible divisors in the moduli space of complex cubic fourfolds parametrising smooth cubic fourfolds that contain highly singular hyperplane sections. We prove that each is not a Noether-Lefschetz (or Hassett) divisor, utilising the computational method developed by Addington-Auel.
Paper Structure (8 sections, 15 theorems, 52 equations, 2 tables)

This paper contains 8 sections, 15 theorems, 52 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of the divisors
  4. Deformation theory of Y
  5. Proof of Theorem \ref{['theorem: divisors']}
  6. Proof of main theorem
  7. The Addington-Auel method
  8. Proof of Theorem \ref{['theorem: main']}

Key Result

Theorem 1.1

The loci $D_{E_6}, D_{D_6}\subset \mathcal{C}$ are irreducible divisors in $\mathcal{C}.$ Further, both divisors are not Hassett divisors.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Corollary 2.6
  • ...and 17 more