Abstract
In the present paper we consider the following question: does there exist a Néron model for families of Jacobians of curves with sections? By applying a construction of the author of universal compactified Jacobians over the moduli stack of reduced curves with markings and a result by J. Kass, we give a positive answer to the question holding for curves with planar singularities.
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References
Altman, A., Kleiman, S.: Compactifying the Picard scheme. Adv. Math. 35, 50–112 (1980)
Artin, M.: Algebraization of formal moduli: I. Global analysis, papers in honor of K. Kodaira, pp. 21–71. Princeton University Press, Princeton (1969)
Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergeb. Math. Grenzgeb. (3), vol. 21. Springer, Berlin (1990)
Busonero, S.: Compactified Picard schemes and Abel maps for singular curves, PhD thesis, Sapienza Università di Roma (2008)
Busonero, S., Melo, M., Stoppino, L.: On the complexity group of stable curves. Adv. Geom. 11(2), 241–272 (2011)
Caporaso, L.: Néron models and compactified Picard schemes over the Moduli stack of stable curves. Am. J. Math. 130(1), 1–47 (2008)
Caporaso, L.: Compactified Jacobians of Néron type. Rend. Lincei. Mat. Appl. 21, 1–16 (2010)
Chiodo, A.: Néron models of \(Pic^0\) via \(Pic^0\). arXiv:1509.06483 (Preprint)
Esteves, E.: Compactifying the relative Jacobian over families of reduced curves. Trans. Am. Math. Soc. 353, 3045–3095 (2001)
Esteves, E., Pacini, M.: Semistable modifications of families of curves and compactified Jacobians. Ark. Mat. 54(1), 55–83 (2016)
Hall, J.: Moduli of singular curves. arXiv:1011.6007 (Preprint)
Holmes, D.: A Néron model of the universal jacobian. arXiv:1412.2243 (Preprint)
Kass, J.L.: Two ways to degenerate the Jacobian are the same. Algebra Number Theory 7(2), 379–404 (2013)
Melo, M.: Compactified Picard stacks over \(\overline{\cal M}_g\). Math. Z. 263(4), 939–957 (2009)
Melo, M.: Compactified Picard stacks over the moduli stack of stable curves with marked points. Adv. Math. 226, 727–763 (2011)
Melo, M.: Compactifications of the universal Jacobian over curves with marked points. arXiv:1509.06177 (Preprint)
Melo, M., Rapagnetta, A., Viviani, F.: Fine compactified Jacobians of reduced curves. Trans. Math. arXiv:1406.2299 (To appear, Preprint)
Melo, M., Viviani, F.: Fine compactified Jacobians. Math. Nach. 285(8–9), 997–1031 (2012)
Néron, A.: Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Inst. Hautes Études Sci. Publ. Math. 21, 5–128 (1964)
Oda, T., Seshadri, C.S.: Compactifications of the generalized Jacobian variety. Trans. Am. Math. Soc. 253, 1–90 (1079)
Raynaud, M.: Spécialisation du foncteur de Picard. Inst. Hautes Études Sci. Publ. Math. 38, 27–76 (1970)
Acknowledgments
The author wishes to thank Alessandro Chiodo, David Holmes and specially Jesse Kass for reading and commenting on a preliminary version of the paper. The author also thanks the referee for his/her careful reading of the paper and for the suggestions he/she presented. This work was funded by a Rita Levi Montalcini Grant, funded by the Italian government through MIUR. The author is a member of CMUC (Center for Mathematics of the University of Coimbra)–UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.
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Melo, M. Universal Néron models for Jacobians of curves with marked points. Boll Unione Mat Ital 10, 321–334 (2017). https://doi.org/10.1007/s40574-016-0103-z
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DOI: https://doi.org/10.1007/s40574-016-0103-z