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
- Mathematics(all)
- General Mathematics
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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
}
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 -