An explicit semi-factorial compactification of the Néron model
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.
scientific article; zbMATH DE number 6361364
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