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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The stability space of compactified universal Jacobians


Authors: Jesse Leo Kass and Nicola Pagani
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 14H10, 14H40, 14K10
DOI: https://doi.org/10.1090/tran/7724
Published electronically: June 21, 2019
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Abstract: In this paper we describe compactified universal Jacobians, i.e., compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank $ 1$ torsion-free sheaves on stable curves, using an approach due to Oda-Seshadri. We focus on the combinatorics of the stability conditions used to define compactified universal Jacobians. We explicitly describe an affine space, the stability space, with a decomposition into polytopes such that each polytope corresponds to a proper Deligne-Mumford stack that compactifies the moduli space of line bundles. We apply this description to describe the set of isomorphism classes of compactified universal Jacobians (answering a question of Melo) and to resolve the indeterminacy of the Abel-Jacobi sections (addressing a problem raised by Grushevsky-Zakharov).


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Additional Information

Jesse Leo Kass
Affiliation: Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, South Carolina 29208
Email: kassj@math.sc.edu

Nicola Pagani
Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, United Kingdom
Email: pagani@liv.ac.uk

DOI: https://doi.org/10.1090/tran/7724
Received by editor(s): May 17, 2018
Received by editor(s) in revised form: October 5, 2018, and October 12, 2018
Published electronically: June 21, 2019
Additional Notes: The first author was supported by a grant from the Simons Foundation (Award Number 429929) and by the National Security Agency under Grant Number H98230-15-1-0264. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein. This manuscript is submitted for publication with the understanding that the United States Government is authorized to reproduce and distribute reprints.
The second author was supported by the EPSRC First Grant EP/P004881/1 with the title “Wall-crossing on universal compactified Jacobians”.
Article copyright: © Copyright 2019 American Mathematical Society