COMPACTIFIED JACOBIANS AS MUMFORD MODELS

Karl Christ; Sam Payne; Tif Shen

doi:10.1090/tran/8875

COMPACTIFIED JACOBIANS AS MUMFORD MODELS

Karl Christ, Sam Payne, Tif Shen

Department of Mathematics

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

Abstract

We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g there is a unique compactified Jacobian encoding slope stability, and it is induced by the tropical break divisor decomposition.

Original language	English
Pages (from-to)	4605-4630
Number of pages	26
Journal	Transactions of the American Mathematical Society
Volume	376
Issue number	7
DOIs	https://doi.org/10.1090/tran/8875
State	Published - 1 Jul 2023

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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title = "COMPACTIFIED JACOBIANS AS MUMFORD MODELS",

abstract = "We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g there is a unique compactified Jacobian encoding slope stability, and it is induced by the tropical break divisor decomposition.",

author = "Karl Christ and Sam Payne and Tif Shen",

note = "Funding Information: Received by the editors May 16, 2022, and, in revised form, September 2, 2022. 2020 Mathematics Subject Classification. Primary 14H40, 14T20, 14G22 . The first author was partially supported by the Israel Science Foundation (grant No. 821/16). Funding Information: The second author was partially supported by NSF DMS-1702428, NSF DMS-2001502, and NSF DMS-2053261. Publisher Copyright: {\textcopyright} 2023 American Mathematical Society.",

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doi = "10.1090/tran/8875",

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AU - Payne, Sam

AU - Shen, Tif

N1 - Funding Information: Received by the editors May 16, 2022, and, in revised form, September 2, 2022. 2020 Mathematics Subject Classification. Primary 14H40, 14T20, 14G22 . The first author was partially supported by the Israel Science Foundation (grant No. 821/16). Funding Information: The second author was partially supported by NSF DMS-1702428, NSF DMS-2001502, and NSF DMS-2053261. Publisher Copyright: © 2023 American Mathematical Society.

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N2 - We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g there is a unique compactified Jacobian encoding slope stability, and it is induced by the tropical break divisor decomposition.

AB - We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g there is a unique compactified Jacobian encoding slope stability, and it is induced by the tropical break divisor decomposition.

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JO - Transactions of the American Mathematical Society

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COMPACTIFIED JACOBIANS AS MUMFORD MODELS

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