The Picard group of the compactified universal Jacobian
type: Item
instance of: scholarly article
title: The Picard group of the compactified universal Jacobian
zbMATH Open document ID: 1386.14107
author: Margarida Melo
author: Filippo Viviani
published in: Documenta Mathematica
publication date: 2014-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1007.4519
full work available at URL: 15.html
reviewed by: Jesse Leo Kass
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14L24
zbMATH DE Number: 6358800
zbMATH Keywords: Brauer group
zbMATH Keywords: Picard group
zbMATH Keywords: gm-gerbe
zbMATH Keywords: universal Jacobian stack and scheme
zbMATH Keywords: compactified universal Jacobian
MaRDI profile type: Publication
arXiv ID: 1007.4519
review text: This paper describes the group of line bundles, or the Picard group, of the moduli space of line bundles on smooth curves as well as some closely related moduli spaces. The Picard group had been studied before, notably in [\textit{J. Ebert} and \textit{O. Randal-Williams}, Doc. Math., J. DMV 17, 417--450 (2012; Zbl 1273.14058); \textit{C. Fontanari}, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 16, No. 1, 45--59 (2005; Zbl 1222.14055)], and [\textit{A. Kouvidakis}, J. Differ. Geom. 34, No. 3, 839--850 (1991; Zbl 0780.14004)], but this is the first paper to carefully treat stack-theoretic issues.NEWLINENEWLINERecall the moduli space of degree \(d\) line bundles on smooth curves of genus \(g\), denoted \(\mathcal{J}ac_{d,g}\) in the paper, exists as an algebraic stack. This stack has the property that every stabilizer group contains the multiplicative group \(\mathbb{G}_{m}\) (acting by scalar multiplication on line bundles), and a natural rigidification construction removes the \(\mathbb{G}_{m}\)s to produce a new stack \(\mathcal{J}_{d,g}\). The stabilizer groups of the new stack \(\mathcal{J}_{d,g}\) are finite groups, in fact \(\mathcal{J}_{d,g}\) is a Deligne-Mumford stack, but the stabilizers can be nontrivial, so \(\mathcal{J}_{d,g}\) is not an algebraic variety. The nontrivial stabilizer groups can, however, be removed to produce a third object, the coarse moduli space of \(\mathcal{J}_{d,g}\) which is a quasi-projective variety. In the literature, all of these spaces are often called the universal Jacobian, although in the present paper the term is reserved for \(\mathcal{J}ac_{d,g}\).NEWLINENEWLINEThe universal Jacobian is studied alongside certain moduli spaces parameterizing line bundles on (possible unstable) nodal curves that satisfy the balancedness condition (a numerical condition on the multidegree). Similar to the previous situation, one considers the moduli stack \(\overline{\mathcal{J}ac}_{d,g}\) of these obj…
description: scientific article; zbMATH DE number 6358800