Publications at the Riemann Center

Complete moduli of cubic threefolds and their intermediate Jacobians

authored by
Sebastian Casalaina-Martin, Samuel Grushevsky, Klaus Hulek, Radu Laza
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 (Formula presented.) 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 (Formula presented.) to the second Voronoi toroidal compactification of (Formula presented.) — 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.

Organisation(s)
Institut für Algebraische Geometrie
Riemann Center for Geometry and Physics
External Organisation(s)
University of Colorado Boulder
Stony Brook University (SBU)
Type
Artikel
Journal
Proceedings of the London Mathematical Society
Volume
122
Pages
259-316
No. of pages
58
ISSN
0024-6115
Publication date
01.02.2021
Publication status
Veröffentlicht
Peer reviewed
Yes
ASJC Scopus Sachgebiete
Mathematik (insg.)
Electronic version(s)
https://arxiv.org/abs/1510.08891 (Access: Offen)
https://doi.org/10.1112/plms.12375 (Access: Offen)