## Details

Original language | English |
---|---|

Pages (from-to) | 259-316 |

Number of pages | 58 |

Journal | Proceedings of the London Mathematical Society |

Volume | 122 |

Issue number | 2 |

Early online date | 17 Aug 2020 |

Publication status | Published - 1 Feb 2021 |

## 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.

## Keywords

- math.AG, 14J30, 14J10, 14K10, 14H40, 14K25, 14K25 (primary), 14H40, 14K10, 14J10, 14J30

## ASJC Scopus subject areas

## Cite this

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**Complete moduli of cubic threefolds and their intermediate Jacobians.**/ Casalaina-Martin, Sebastian; Grushevsky, Samuel; Hulek, Klaus et al.

In: Proceedings of the London Mathematical Society, Vol. 122, No. 2, 01.02.2021, p. 259-316.

Research output: Contribution to journal › Article › Research › peer review

*Proceedings of the London Mathematical Society*, vol. 122, no. 2, pp. 259-316. https://doi.org/10.1112/plms.12375

*Proceedings of the London Mathematical Society*,

*122*(2), 259-316. https://doi.org/10.1112/plms.12375

}

TY - JOUR

T1 - Complete moduli of cubic threefolds and their intermediate Jacobians

AU - Casalaina-Martin, Sebastian

AU - Grushevsky, Samuel

AU - Hulek, Klaus

AU - Laza, Radu

N1 - Funding Information: Research of the first author was supported in part by NSF grant DMS‐11‐01333 and a Simons Foundation Collaboration Grant for Mathematicians (317572). Research of the second author was supported in part by NSF grants DMS‐12‐01369 and DMS‐15‐01265, and a Simons Fellowship in mathematics. Research of the third author was supported in part by DFG grant Hu‐337/6‐2. Research of the fourth author was supported in part by NSF grants DMS‐12‐00875 and DMS‐12‐54812.

PY - 2021/2/1

Y1 - 2021/2/1

N2 - 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.

AB - 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.

KW - math.AG

KW - 14J30, 14J10, 14K10, 14H40, 14K25

KW - 14K25 (primary)

KW - 14H40

KW - 14K10

KW - 14J10

KW - 14J30

UR - http://www.scopus.com/inward/record.url?scp=85089452351&partnerID=8YFLogxK

U2 - 10.1112/plms.12375

DO - 10.1112/plms.12375

M3 - Article

VL - 122

SP - 259

EP - 316

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 2

ER -