Cubic fourfolds containing highly singular hyperplane sections

Technical Summary: Cubic Fourfolds Containing Highly Singular Hyperplane Sections

Problem Statement
The paper addresses the classification of divisors within the moduli space $\mathcal{C}$ of smooth complex cubic fourfolds. A central theme in the study of $\mathcal{C}$ is the distinction between "special" cubic fourfolds (those containing a surface non-homologous to a complete intersection, parametrized by Hassett divisors) and "non-special" ones. While Hassett divisors are well-understood and linked to rationality constructions, there is a scarcity of known explicit equations for non-Hassett divisors. The authors aim to construct new irreducible divisors in $\mathcal{C}$ that parametrize smooth cubic fourfolds containing hyperplane sections with specific, highly singular configurations (specifically, singularities with total Tjurina number 6) and prove that these divisors are not Hassett divisors. This contributes to the broader goal of identifying explicit examples of irrational cubic fourfolds, as guaranteed by the theory of Hodge atoms for non-special fourfolds.

Methodology
The authors employ a combination of algebraic geometry, singularity theory, and computational number theory:

1. Geometric Construction and Deformation Theory:

   • The authors define loci $D_K \subset \mathcal{C}$ parametrizing cubic fourfolds containing a hyperplane section $Y$ with a specific singularity type $K$ (where $K$ represents combinations of ADE singularities or the $T_{333}$ singularity).
   • They utilize the classification of singularities of cubic threefolds (hyperplane sections) to identify combinations with total Tjurina number $\tau(Y) = 6$.
   • To prove these loci are divisors (codimension 1), they analyze the deformation theory of the singular hyperplane section $Y$ inside the ambient fourfold $X$. Specifically, they examine the global-to-local map $\phi: H^0(Y, \mathcal{O}(1)) \to \bigoplus_{p \in \text{Sing}(Y)} T^1_{Y_p}$.
   • Using explicit examples constructed via the computer algebra system MAGMA, they demonstrate that for general members of these loci, the map $\phi$ is injective. This injectivity implies that a general cubic fourfold in these loci contains only finitely many such hyperplane sections, confirming that the loci are indeed divisors (or codimension 2 for the $T_{333}$ case).

2. Computational Verification of Non-Speciality (Addington–Auel Method):

   • To prove that the constructed divisors are not Hassett divisors (i.e., the fourfolds are not special), the authors apply the computational method developed by Addington and Auel.
   • This method relies on the work of van Luijk for K3 surfaces. It involves selecting a specific cubic fourfold $X$ defined over $\mathbb{Q}$ within the divisor, reducing it modulo a prime (specifically $p=2$), and counting points over finite fields $\mathbb{F}_{2^m}$.
   • Using the Lefschetz trace formula, they compute the characteristic polynomial $\chi(t)$ of the Frobenius action on the primitive étale cohomology $H^4_{\text{ét, prim}}$.
   • According to Corollary 4.3, if $\chi(t)$ is irreducible over $\mathbb{Q}$ and not all its coefficients are integers, the cubic fourfold is non-special. The authors implement this using the CubicLib.m library to verify this condition for explicit representatives of each divisor.

Key Contributions and Results

• Construction of New Divisors: The paper constructs five irreducible divisors in $\mathcal{C}$:
  • $D_{E6}$ and $D_{D6}$: Parametrizing fourfolds with hyperplane sections having $E_6$ and $D_6$ singularities, respectively.
  • $D_{D5+A1}$ and $D_{D4+A2}$: Parametrizing fourfolds with hyperplane sections having $D_5+A_1$ and $D_4+A_2$ singularities.
  • $D_{D4+2A1}$: This locus splits into two irreducible components, $D^0_{D4+2A1}$ and $D^1_{D4+2A1}$, distinguished by the defect $\sigma(Y)$ of the singular hyperplane section.
• Codimension Results: The locus $D_{T333}$ (parametrizing fourfolds with a $T_{333}$ singularity, a unimodular singularity of corank 3 and Tjurina number 8) is proven to be an irreducible subvariety of codimension 2 in $\mathcal{C}$.
• Non-Hassett Classification:
  • The authors prove that $D_{E6}, D_{D6}, D_{D5+A1}, D_{D4+A2}$, and $D^0_{D4+2A1}$ are not Hassett divisors.
  • They show that $D^1_{D4+2A1}$ coincides with the known Hassett divisor $C_8$ (parametrizing fourfolds containing a plane).
  • The codimension 2 locus $D_{T333}$ is shown not to be contained in any Hassett divisor.
• Explicit Equations: The paper provides explicit polynomial equations for representatives of each divisor, which can be used to generate examples of irrational cubic fourfolds.

Significance
The paper claims significance primarily in expanding the known landscape of non-Hassett divisors in the moduli space of cubic fourfolds. Prior to this work, only a few non-Hassett divisors were explicitly known (e.g., those constructed by Ranestad–Voisin, Addington–Auel, and Costa–Harvey–Kedlaya). By identifying five new irreducible divisors and proving their non-speciality via computational methods, the authors provide new explicit sources of irrational cubic fourfolds.

Furthermore, the results have implications for the study of compactified intermediate Jacobian fibrations. The authors note that recent work by Druel and Mauri [DM26] extends the construction of hyperkähler compactifications to "very good" cubic fourfolds—those not containing a plane, a cubic scroll, or a hyperplane section with a corank 3 singularity. The authors' results imply that any cubic fourfold in $\mathcal{C} \setminus (C_8 \cup C_{12} \cup D_{T333})$ is "very good," thereby guaranteeing the existence of a smooth projective hyperkähler manifold of OG10 type with a Lagrangian fibration for these fourfolds. The paper does not speculate on future applications beyond these immediate geometric consequences.