Cubic Threefold Jacobians: A Special Subvariety of \(\mathcal{A}_5\) — E8 Intelligence Research

FINDING: The intermediate Jacobian of a smooth cubic threefold is a 5-dimensional principally polarized abelian variety (ppav) whose moduli locus is characterized by a singular odd 2-torsion point on its theta divisor; this locus is a special subvariety of the moduli space \( \mathcal{A}_5 \) of ppavs of dimension 5, and its closure in the Satake compactification relates to degenerations with nodes. MATH: - Cubic threefold \( X \subset \mathbb{P}^4 \), smooth, has intermediate Jacobian \( J(X) \) of dimension 5, with principal polarization \( \Theta \). - The period map \( \mathcal{M}_{\text{cubic}} \to \mathcal{A}_5 \) sends \( X \) to \( (J(X), \Theta) \). - The image locus is defined by the condition: there exists a point \( \eta \in J(X)[2] \) (2-torsion) such that \( \Theta \) has a singularity at \( \eta \) of odd multiplicity (i.e., \( \text{mult}_\eta \Theta \) is odd). - The monodromy group for the family of cubic threefolds is the Weyl group \( W(E_6) \), of order 51 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com