Abstract
A curve, that is, a connected, reduced, projective scheme of dimension 1 over an algebraically closed field, admits two types of compactifications of its (generalized) Jacobian: the moduli schemes of P-quasistable torsion-free, rank-1 sheaves and Seshadri’s moduli schemes of S-equivalence classes of semistable torsion-free, rank-1 sheaves. Both are constructed with respect to a choice of polarization. The former are fine moduli spaces which were shown to be complete; here we show that they are actually projective. The latter are just coarse moduli spaces. Here we give a sufficient condition for when these two types of moduli spaces are equal.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Altman A., Kleiman S.: Compactifying the Jacobian. Bull. Amer. Math. Soc. 82, 947–949 (1976)
Altman A., Kleiman S.: Compactifying the Picard scheme. II. Amer. J. Math. 101, 10–41 (1979)
Altman A., Kleiman S.: Compactifying the Picard scheme. Adv. Math. 35, 50–112 (1980)
Altman, A., Iarrobino, A., Kleiman, S.: Irreducibility of the compactified Jacobian. In: Real and complex singularities, pp. 1–12. Oslo (1976), Proceedings of the 9th Nordic Summer School, Sijthoff and Noordhoff (1977)
Caporaso L.: A compactification of the universal Picard variety over the moduli space of stable curves. J. Amer. Math. Soc. 7, 589–660 (1994)
Caporaso L., Esteves E.: On Abel maps of stable curves. Michigan Math. J. 55, 575–607 (2007)
Caporaso L., Coelho J., Esteves E.: Abel maps of Gorenstein curves. Rend. Circ. Mat. Palermo 57(2), 33–59 (2008)
D’Souza, C.: Compactification of generalized Jacobians, Thesis. Tata Institute, Bombay (1973)
D’Souza C.: Compactification of generalized Jacobians. Proc. Indian Acad. Sci. Sect. A Math. Sci. 88, 419–457 (1979)
Esteves E.: Separation properties of theta functions. Duke Math. J. 98, 565–593 (1999)
Esteves E.: Compactifying the relative Jacobian over families of reduced curves. Trans. Amer. Math. Soc. 353, 3045–3095 (2001)
Igusa J.: Fiber systems of Jacobian varieties. Amer. J. Math. 78, 171–199 (1956)
Mayer, A., Mumford, D.: Further comments on boundary points. American Mathematical Society Summer Institute, Woods Hole, Massachusetts (1964)
Mumford D., Fogarty J.: Geometric invariant theory. Springer, Berlin Heidelberg (1982)
Oda T., Seshadri C.S.: Compactifications of the generalized Jacobian variety. Trans. Amer. Math. Soc. 253, 1–90 (1979)
Pandharipande R.: A compactification over \({\overline{\mathcal {M}_{g}}}\) of the universal moduli space of slope-semistable vector bundles. J. Amer. Math. Soc. 9, 425–471 (1996)
Seshadri, C.S.: Fibrés vectoriels sur les courbes algébriques. Astérisque vol. 96, Soc. Math. France, Montrouge (1982)
Simpson C.: Moduli of representations of the fundamental group of a smooth projective variety I. Inst. Hautes Études Sci. Publ. Math. 79, 47–129 (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Eduardo Esteves is Supported by CNPq, Processos 301117/04-7 and 470761/06-7, by CNPq/FAPERJ, Processo E-26/171.174/2003, and by the Institut Mittag–Leffler (Djursholm, Sweden).
Rights and permissions
About this article
Cite this article
Esteves, E. Compactified Jacobians of curves with spine decompositions. Geom Dedicata 139, 167–181 (2009). https://doi.org/10.1007/s10711-008-9322-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-008-9322-5