The defect of a cubic threefold and applications to intermediate Jacobian fibrations

TL;DR

This paper investigates the defect of cubic threefolds with isolated singularities, linking it to topological, geometric, and Hodge-theoretic properties, and explores its implications for intermediate Jacobian fibrations.

Contribution

It computes the defect for singular cubic threefolds using topological data and relates it to geometric properties and intermediate Jacobian fibrations.

Findings

A cubic threefold is not $Q$-factorial iff it contains a plane or cubic scroll.

The defect can be expressed via topological and local invariants of singularities.

Connections established between defect, $Q$-factoriality, and intermediate Jacobian fibrations.

Abstract

The defect of a cubic threefold $X$ with isolated singularities is a global invariant that measures the failure of $\mathbb{Q}$-factoriality. We compute the defect for such cubics in terms of topological data about the curve of lines through a singular point. We express the mixed Hodge structure on the middle cohomology of $X$ in terms of both the defect and local invariants of the singularities. We then relate the defect to various geometric properties of $X$: in particular, we show that a cubic threefold is not $\mathbb{Q}$-factorial if and only if it contains either a plane or a cubic scroll. We relate the defect to existence of compactified intermediate Jacobian fibrations with irreducible fibers associated to a cubic fourfold.