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Compactifying the relative Jacobian over families of reduced curves


Author: Eduardo Esteves
Journal: Trans. Amer. Math. Soc. 353 (2001), 3045-3095
MSC (2000): Primary 14H40, 14H60; Secondary 14D20
DOI: https://doi.org/10.1090/S0002-9947-01-02746-5
Published electronically: January 18, 2001
MathSciNet review: 1828599
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Abstract: We construct natural relative compactifications for the relative Jacobian over a family $X/S$ of reduced curves. In contrast with all the available compactifications so far, ours admit a Poincaré sheaf after an étale base change. Our method consists of studying the étale sheaf $F$ of simple, torsion-free, rank-1 sheaves on $X/S$, and showing that certain open subsheaves of $F$ have the completeness property. Strictly speaking, the functor $F$ is only representable by an algebraic space, but we show that $F$ is representable by a scheme after an étale base change. Finally, we use theta functions originating from vector bundles to compare our new compactifications with the available ones.


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  • 1. V. Alexeev, Compactified Jacobians, Available at http://xxx.lanl.gov/abs/alg-geom/9608012, August, 1996.
  • 2. A. Altman, A. Iarrobino and S. Kleiman, Irreducibility of the compactified Jacobian, Real and complex singularities, Oslo 1976 (Proc. Ninth Nordic Summer School), Sijthoff and Noordhoff, 1977, pp. 1-12.MR 58:16650
  • 3. A. Altman and S. Kleiman, Compactifying the Picard scheme, Adv. Math. 35 (1980), 50-112. MR 81f:14025a
  • 4. E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of algebraic curves, Vol. I, Grundlehren der mathematischen Wissenschaften, vol. 267, Springer-Verlag, 1985. MR 86h:14019
  • 5. M. Artin, Algebraization of formal moduli. I, Global Analysis, Papers in honor of K. Kodaira, University of Tokyo Press, Princeton University Press, 1969, pp. 21-71. MR 41:5369
  • 6. S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 21, Springer Verlag, 1990. MR 91i:14034
  • 7. L. Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc. 7 (1994), 589-660. MR 95d:14014
  • 8. C. D'Souza, Compactification of generalized Jacobians, Proc. Indian Acad. Sci. Sect. A Math. Sci. 88 (1979), 419-457. MR 81h:14004
  • 9. E. Esteves, Very ampleness for Theta on the compactified Jacobian, Math. Z. 226 (2) (1997), 181-191. MR 98k:14037
  • 10. E. Esteves, Separation properties of theta functions, Duke Math. J. 98 (3) (1999), 565-593. MR 2000e:14048
  • 11. E. Esteves, M. Gagné and S. Kleiman, Abel maps and presentation schemes, Available at http://xxx.lanl.gov/abs/math.AG/9911069, November, 1999. To appear in a special issue, dedicated to R. Hartshorne.
  • 12. G. Faltings, Stable $G$-bundles and projective connections, J. Algebraic Geom. 2 (1993), 507-568. MR 94i:14015
  • 13. D. Gieseker, Moduli of curves, Tata Inst. Fund. Res. Lecture Notes, Springer-Verlag, 1982.
  • 14. A. Grothendieck with J. Dieudonné, Éleménts de Géométrie Algébrique III-1, IV-4, Inst. Hautes Études Sci. Publ. Math. 11, 32 (1961, 1967). MR 36:177c; MR 39:220
  • 15. A. Grothendieck, Fondements de la géometrie algébrique, Semináire Bourbaki, exp. 232 (1961/62). MR 26:3566
  • 16. M. Homma, Personal communication.
  • 17. J. Igusa, Fibre systems of Jacobian varieties, Amer. J. Math. 78 (1956), 171-199. MR 18:935d
  • 18. L. Illusie, Conditions de finitude relatives, Lecture Notes in Mathematics, vol. 225, Springer-Verlag, 1971, pp. 222-273.
  • 19. M. Ishida, Compactifications of a family of generalized Jacobian varieties, Proc. Internat. Sympos. Algebraic Geometry, Kyoto, January, 1977 (M. Nagata, ed.), Kinokuniya, Tokyo, 1978, pp. 503-524. MR 81h:14025
  • 20. F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves I: Preliminaries on ``det'' and ``Div'', Math. Scand. 39 (1976), 19-55. MR 55:10465
  • 21. S. Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. Math. 101 (1975), 88-110. MR 51:510
  • 22. A. Mayer and D. Mumford, Further comments on boundary points, Unpublished lecture notes distributed at the Amer. Math. Soc. Summer Institute, Woods Hole, 1964.
  • 23. T. Oda and C.S. Seshadri, Compactifications of the generalized Jacobian variety, Trans. Amer. Math. Soc. 253 (1979), 1-90. MR 82e:14054
  • 24. R. Pandharipande, A compactification over $\overline{M}_{g}$ of the universal moduli space of slope-semi-stable vector bundles, J. Amer. Math. Soc. 9 (1996), 425-471. MR 96f:14014
  • 25. C.S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Astérisque 96 (1982). MR 85b:14023
  • 26. C.S. Seshadri, Vector bundles on curves, Contemp. Math. 153 (1993), 163-200. MR 95b:14008
  • 27. C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47-129. MR 96e:14012

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Additional Information

Eduardo Esteves
Affiliation: Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro RJ, Brazil
Email: esteves@impa.br

DOI: https://doi.org/10.1090/S0002-9947-01-02746-5
Received by editor(s): December 15, 1997
Received by editor(s) in revised form: May 2, 2000
Published electronically: January 18, 2001
Additional Notes: Research supported by an MIT Japan Program Starr fellowship, by PRONEX, Convênio 41/96/0883/00 and CNPq, Proc. 300004/95-8.
Article copyright: © Copyright 2001 American Mathematical Society

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