Cubic fourfolds containing highly singular hyperplane sections

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.

💡 Research Summary

The paper investigates smooth complex cubic fourfolds X ⊂ ℙ⁵ that contain a hyperplane section Y = X ∩ H with singularities more severe than those occurring generically. For a general cubic fourfold the total Tjurina number τ(Y) of any hyperplane section satisfies τ(Y) ≤ 5; the authors focus on the next possible case τ(Y) = 6, which by a dimension count should define divisors in the 20‑dimensional moduli space C of smooth cubic fourfolds.

Using the classification of isolated hypersurface singularities on cubic threefolds, they identify all possible combinations of ADE‑type singularities with total Tjurina number 6 that contain at least one singularity of corank ≥ 1. The list consists of the single singularities E₆ and D₆, and the mixed types D₅ + A₁, D₄ + A₂, and D₄ + 2A₁. In the mixed case D₄ + 2A₁ the defect σ(Y) (the difference b₂ − b₄) can be either 0 or 1, leading to two distinct families. They also treat the unimodal corank‑3 singularity T₃₃₃ (τ = 8), which gives a codimension‑2 locus.

For each singularity type K they write down an explicit normal form f_K of the cubic threefold Y (Proposition 2.8). Adding a general linear term x₅·q(x₀,…,x₅) yields a cubic fourfold equation F_K = x₅·q + f_K. The set W_K of such cubic forms is a locally closed subset of the open set U ⊂ ℙ(H⁰(ℙ⁵,𝒪(3))) parametrising smooth cubics. The image D_K of the closure of W_K under the GIT quotient C = U//SL(6) is the candidate divisor.

The authors prove that a smooth cubic fourfold containing a K‑hyperplane section admits only finitely many such sections (Proposition 3.3) and compute dim W_K = 21 for all K, while the SL(6)‑orbit has dimension 35. Hence each D_K is an irreducible divisor (Theorem 3.2); the T₃₃₃‑locus D_T333 has codimension 2.

To show that these divisors are not Hassett (Noether–Lefschetz) divisors, they employ the computational method of Addington–Auel. This technique translates the existence of a special algebraic surface in X into a lattice‑theoretic condition on the primitive Hodge structure H⁴(X,ℤ)_prim. Using the code from AKPW24/25, they verify that for each K (except the component D₁_D₄+2A₁, which coincides with the known Hassett divisor C₈) the required lattice does not occur. Consequently D_E6, D_D6, D_D5+A₁, D_D4+A₂, D₀_D₄+2A₁, and the codimension‑2 locus D_T333 are new non‑Hassett divisors.

The paper also discusses implications for the compactified intermediate Jacobian fibration π : J_X → (ℙ⁵)⁎ constructed by Laza–Saccà–Voisin. For a very general cubic fourfold (i.e., one not lying in C₈, C₁₂, or D_T333) the fibration yields a smooth OG10‑type hyperkähler manifold whose Lagrangian fibers are compactified Prym varieties of reduced, irreducible planar curves. Corollary 1.4 states precisely that any X∈C \ (C₈∪C₁₂∪D_T333) is “very good” in the sense of