An explicit semi-factorial compactification of the Néron model
type: Item
instance of: scholarly article
title: An explicit semi-factorial compactification of the Néron model
zbMATH Open document ID: 1303.14041
author: Jesse Leo Kass
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1507.08753
reviewed by: Noah Giansiracusa
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14K30
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 6361364
zbMATH Keywords: Jacobian
zbMATH Keywords: compactification
zbMATH Keywords: arithmetic surface
zbMATH Keywords: Néron model
MaRDI profile type: Publication
OpenAlex ID: W2022986324
arXiv ID: 1507.08753
cites work: Compactifying the Picard scheme
cites work: Compactified Jacobians, Abel maps and Theta divisors
cites work: Néron models and compactified Picard schemes over the moduli stack of stable curves
cites work: Compactified Jacobians of Néron type
cites work: The irreducibility of the space of curves of a given genus
cites work: Q3884235
cites work: Very ampleness for Theta on the compactified Jacobian
cites work: AUTODUALITY OF THE COMPACTIFIED JACOBIAN
cites work: On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field
cites work: Two ways to degenerate the Jacobian are the same
cites work: Desingularization of two-dimensional schemes
cites work: Fine compactified Jacobians
cites work: Q5628266
cites work: Compactifications of the Generalized Jacobian Variety
cites work: Semi-factorial models and Néron models
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: Q4312298
DOI: 10.1016/J.CRMA.2014.07.007
review text: The Jacobian \(J\) of a smooth curve \(C\) parameterizes degree zero line bundles on \(C\). Suppose that \(C\) degenerates to a possibly singular curve via a 1-parameter family. Algebraically, this is encoded by viewing \(C\) as the general fiber of a family of curves over a discrete valuation ring \(S\); the special fiber is the limit of the smooth curve \(C\), and the total space of this family is an arithmetic surface \(\widetilde{C}\). The Jacobian \(J\) extends to the special fiber via a construction called the Néron model, resulting in a scheme (also denoted \(J\)) over \(S\). A key issue, however, is that the Néron model \(J/S\) need not be proper, so one naturally studies its various compactifications.NEWLINENEWLINEOn the one hand, a recent result of \textit{C. Pépin} [Math. Ann. 355, No. 1, 147--185 (2013; Zbl 1263.14046)] is that the Néron model \(J/S\) admits a ``semi-factorial'' model, meaning a compactification of \(J\), flat and projective over \(S\), such that every line bundle on the general fiber extends to a line bundle on the total space. On the other hand, assuming the curve \(C\) and its limit are sufficiently nice, there is a compactification \(\overline{J}\)/S, called the Altman-D'Souza-Kleiman family of compactified Jacobians parameterizing degree zero, rank one, torsion-free sheaves on \(\widetilde{C}/S\) (the locus corresponding to line-bundles is precisely \(J/S\)). We therefore have two natural compactifications of the Jacobican. The purpose of the present short note is to prove that, when the Picard rank of the Jacobian of \(C\) is one (i.e., the theta divisor generates the torsion-free part of the Néron-Severi group of \(J\)), then the compactified Jacobian \(\overline{J}\) enjoys Pépin's desirable property of being a semi-factorial model. Projectivity was already known, and flatness is a short lemma, so the main work (though still only one page) is to show that line bundles extend.
description: scientific article; zbMATH DE number 6361364