Work

The Defect of a Cubic Threefold and Applications to Intermediate Jacobian Fibrations

Year: 2025

Type: article

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.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnaf244

Authors: Lisa Marquand, Sasha Viktorova

Institutions: Courant Institute of Mathematical Sciences, New York University, KU Leuven

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Vlaamse regering

Award: Classification, symmetries and singularities at the frontiers of algebra, analysis and geometry.