A Néron model of the universal jacobian
Annales Henri Lebesgue, Volume 4 (2021), pp. 1727-1766.


KeywordsNéron models, jacobians, moduli of curves


Jacobians of degenerating families of curves are well-understood over 1-dimensional bases due to work of Néron and Raynaud; the fundamental tool is the Néron model and its description via the Picard functor. Over higher-dimensional bases Néron models typically do not exist, but in this paper we construct a universal base change ˜ g,n ¯ g,n after which a Néron model N g,n / ˜ g,n of the universal jacobian does exist. This yields a new partial compactification of the moduli space of curves, and of the universal jacobian over it. The map ˜ g,n ¯ g,n is separated and relatively representable. The Néron model N g,n / ˜ g,n is separated and has a group law extending that on the jacobian. We show that Caporaso’s balanced Picard stack acquires a torsor structure after pullback to a certain open substack of ˜ g,n .


[BH16] Biesel, Owen; Holmes, David Fine compactified moduli of enriched structures on stable curves (2016) (https://arxiv.org/abs/1607.08835v1)

[BLR90] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 21, Springer, 1990 | Article | Zbl 0705.14001

[Cap08] Caporaso, Lucia Néron models and compactified Picard schemes over the moduli stack of stable curves, Am. J. Math., Volume 130 (2008) no. 1, pp. 1-47 | Article | MR 2382140 | Zbl 1155.14023

[Chi15] Chiodo, Alessandro Néron models of Pic 0 via Pic 0 (2015) (http://arxiv.org/abs/1509.06483)

[DM69] Deligne, Pierre; Mumford, David The irreducibility of the space of curves of given genus, Publ. Math., Inst. Hautes Étud. Sci. (1969) no. 36, pp. 75-109 | Article | Numdam | MR 0262240 | Zbl 0181.48803

[Est01] Esteves, Eduardo Compactifying the relative Jacobian over families of reduced curves, Trans. Am. Math. Soc., Volume 353 (2001) no. 8, pp. 3045-3095 | Article | MR 1828599 | Zbl 0974.14009

[Ful93] Fulton, William Introduction to toric varieties. The 1989 William H. Roever lectures in geometry, Annals of Mathematics Studies, Princeton University Press, 1993 no. 131 | Zbl 0813.14039

[Hol17] Holmes, David Quasi-compactness of Néron models, and an application to torsion points, Manuscr. Math., Volume 153 (2017) no. 3-4, pp. 323-330 | Article | MR 3662048 | Zbl 1426.11061

[Hol19] Holmes, David Néron models of jacobians over base schemes of dimension greater than 1, J. Reine Angew. Math., Volume 747 (2019), pp. 109-145 | Article | MR 3905131 | Zbl 1423.14203

[Hol21] Holmes, David Extending the double ramification cycle by resolving the Abel–Jacobi map, J. Inst. Math. Jussieu, Volume 20 (2021) no. 1, pp. 331-359 | Article | MR 4205785 | Zbl 1462.14031

[Jon96] de Jong, Aise J. Smoothness, semi-stability and alterations, Publ. Math., Inst. Hautes Étud. Sci., Volume 83 (1996), pp. 51-93 | Numdam | MR 1423020 | Zbl 0916.14005

[Knu83] Knudsen, Finn F. The projectivity of the moduli space of stable curves. II. The stacks M g,n , Math. Scand., Volume 52 (1983) no. 2, pp. 161-199 | Article | MR 702953 | Zbl 0544.14020

[KP19] Kass, Jesse Leo; Pagani, Nicola The stability space of compactified universal Jacobians, Trans. Am. Math. Soc., Volume 372 (2019) no. 7, pp. 4851-4887 | Article | MR 4009442 | Zbl 1423.14187

[Liu02] Liu, Qing Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, 6, Oxford University Press, 2002 (translated from the French by Reinie Erné, Oxford Science Publications) | MR 1917232 | Zbl 0996.14005

[Mai98] Mainó, Laila Moduli space of enriched stable curves (1998) (Ph. D. Thesis) | MR 2697466

[Mel09] Melo, Margarida Compactified Picard stacks over the moduli space of curves with marked points (2009) (Ph. D. Thesis)

[Sta13] Stacks Project Stacks Project, 2013 (http://stacks.math.columbia.edu)