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Mathematics > Algebraic Geometry

arXiv:1807.11364 (math)
[Submitted on 30 Jul 2018 (v1), last revised 16 Mar 2022 (this version, v5)]

Title:The logarithmic Picard group and its tropicalization

Authors:Samouil Molcho, Jonathan Wise
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Abstract:We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne-Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.
Comments: 8 figures, 86 pages; this version corrects a mistake in the proof of Theorem 4.4.1; comments welcome!
Subjects: Algebraic Geometry (math.AG)
MSC classes: 14A21, 14H10, 14C20, 14C22, 14D20, 14D23, 14H40, 14K30, 14T10, 14T90
Cite as: arXiv:1807.11364 [math.AG]
  (or arXiv:1807.11364v5 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1807.11364

Submission history

From: Jonathan Wise [view email]
[v1] Mon, 30 Jul 2018 14:15:22 UTC (72 KB)
[v2] Thu, 23 Aug 2018 17:00:50 UTC (76 KB)
[v3] Thu, 22 Jul 2021 17:18:51 UTC (89 KB)
[v4] Wed, 28 Jul 2021 23:33:49 UTC (89 KB)
[v5] Wed, 16 Mar 2022 20:53:51 UTC (96 KB)
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