Complete moduli of cubic threefolds and their intermediate Jacobians. Issue 2 (17th August 2020)

• Record Type: Journal Article
• Authors: Casalaina‐Martin, Sebastian; Grushevsky, Samuel; Hulek, Klaus; Laza, Radu
• Is Part Of: Proceedings of the London Mathematical Society. Volume 122:Issue 2(2021)
• Page Start: 259
• Page End: 316
• Publication Date: 2020-08-17
• Subjects: 14H40 -- 14J10 -- 14J30 -- 14K10 -- 14K25 (primary); Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12375
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract: The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the moduli space of principally polarized abelian fivefolds. A better 'wonderful' compactification M ∼ of the space of cubic threefolds was constructed by the first and fourth authors — it has a modular interpretation, and divisorial normal crossing boundary. We prove that the intermediate Jacobian map extends to a morphism from M ∼ to the second Voronoi toroidal compactification of A 5 — the first and fourth author previously showed that it extends to the Satake compactification. Since the second Voronoi compactification has a modular interpretation, our extended intermediate Jacobian map encodes all of the geometric information about the degenerations of intermediate Jacobians, and allows for the study of the geometry of cubic threefolds via degeneration techniques. As one application, we give a complete classification of all degenerations of intermediate Jacobians of cubic threefolds of torus rank 1 and 2.