Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Compactifications of the generalized Jacobian variety
HTML articles powered by AMS MathViewer

by Tadao Oda and C. S. Seshadri
Trans. Amer. Math. Soc. 253 (1979), 1-90
DOI: https://doi.org/10.1090/S0002-9947-1979-0536936-4

Abstract:

The generalized Jacobian variety of an algebraic curve with at most ordinary double points is an extension of an abelian variety by an algebraic torus. Using the geometric invariant theory, we systematically compactify it in finitely many different ways and describe their structure in terms of torus embeddings. Our compactifications include all known good ones.
References
    M. Artin, Lectures on deformations of singularities, Tata Inst. of Fundamental Research, No 54, Bombay, 1976.
  • Claude Berge, Graphs, North-Holland Mathematical Library, vol. 6, North-Holland Publishing Co., Amsterdam, 1985. Second revised edition of part 1 of the 1973 English version. MR 809587
  • P. R. Bryant, Graph theory applied to electrical networks, Graph Theory and Theoretical Physics, Academic Press, London, 1967, pp. 111–137. MR 0233627
  • P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240, DOI 10.1007/BF02684599
  • P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143–316 (French). MR 0337993
  • J. Dieudonné and A. Grothendieck, Éléments de géométrie algébrique, Publ. Math. Inst. Hautes Études Sci., Nos. 4, 8, 11, 17, 20, 24, 28, 32 (1960-1967).
  • T. J. Dickson, On Voronoi reduction of positive definite quadratic forms, J. Number Theory 4 (1972), 330–341. MR 319900, DOI 10.1016/0022-314X(72)90068-6
  • C. D’Souza, Compactification of generalized Jacobian, Thesis, TIFR and Bombay Univ. 1974; Astérisque (to appear).
  • D. Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2) 106 (1977), no. 1, 45–60. MR 466475, DOI 10.2307/1971157
  • —, Stable vector bundles on degenerating families of curves (to appear). FGA. A. Grothendieck, Fondements de la géométrie algébrique, Extraits du Séminaire Bourbaki 1957-1962, Paris, 1962.
  • W. J. Haboush, Reductive groups are geometrically reductive, Ann. of Math. (2) 102 (1975), no. 1, 67–83. MR 382294, DOI 10.2307/1970974
  • Frank Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London 1969. MR 0256911, DOI 10.21236/AD0705364
  • A. J. Hoffman and J. B. Kruskal, Integral boundary points of convex polyhedra, Linear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, pp. 223–246. MR 0085148
  • I. Heller and C. B. Tompkins, An extension of a theorem of Dantzig’s, Linear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, pp. 247–254. MR 0081871
  • Jun-ichi Igusa, Fibre systems of Jacobian varieties. II. Local monodromy groups of fibre systems, Amer. J. Math. 78 (1956), 745–760. MR 84849, DOI 10.2307/2372466
  • Stacy G. Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. (2) 101 (1975), 88–110. MR 364255, DOI 10.2307/1970987
  • Masaki Maruyama, Stable vector bundles on an algebraic surface, Nagoya Math. J. 58 (1975), 25–68. MR 396576, DOI 10.1017/S0027763000016688
  • Tadao Oda and Katsuya Miyake, Almost homogeneous algebraic varieties under algebraic torus action, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 373–381. MR 0379501
  • A. L. Mayer, Compactification of the variety of moduli of curves, Lectures 2 & 3, Seminar on degeneration of algebraic varieties (mimeographed notes), Inst. for Advanced Study, Princeton, N. J., 1969/70. D. Mumford, Further comments on boundary points (mimeographed notes), AMS Summer School at Woods Hole, 1964.
  • David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602, DOI 10.1007/978-3-662-00095-3
  • —, Abelian varieties, Oxford Univ. Press, Bombay, 1970.
  • David Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239–272. MR 352106
  • David Mumford, Curves and their Jacobians, University of Michigan Press, Ann Arbor, Mich., 1975. MR 0419430
  • G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518, DOI 10.1007/BFb0070318
  • D. Mumford and P. Newstead, Periods of a moduli space of bundles on curves, Amer. J. Math. 90 (1968), 1200–1208. MR 234958, DOI 10.2307/2373296
  • Iku Nakamura, On moduli of stable quasi abelian varieties, Nagoya Math. J. 58 (1975), 149–214. MR 393049, DOI 10.1017/S002776300001672X
  • Yukihiko Namikawa, A new compactification of the Siegel space and degeneration of Abelian varieties. I, Math. Ann. 221 (1976), no. 2, 97–141. MR 480537, DOI 10.1007/BF01433145
  • M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. MR 184252, DOI 10.2307/1970710
  • M. Raynaud, Spécialisation du foncteur de Picard, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 27–76 (French). MR 282993, DOI 10.1007/BF02684651
  • C. A. Rogers, Packing and covering, Cambridge Tracts in Mathematics and Mathematical Physics, No. 54, Cambridge University Press, New York, 1964. MR 0172183
  • C. S. Seshadri, Space of unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 85 (1967), 303–336. MR 233371, DOI 10.2307/1970444
  • —, Mumford’s conjecture for ${\text {GL(2)}}$ and applications, (Internat. Colloq. TIFR, Bombay, 1968), Algebraic Geometry, Oxford Univ. Press, London, 1969, pp. 347-371.
  • C. S. Seshadri, Theory of moduli, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 263–304. MR 0396565
  • G. Voronoi, Nouvelles applications des paramétres continues et théorie des formes quadratiques. I, II-1, II-2, J. Reine Angew. Math. 133 (1908), 97-178; 134 (1908), 198-287; 136 (1909), 67-181.
  • A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129–152. MR 369747, DOI 10.1007/BF01343949
  • A. Grothendieck, Séminaire de géométrie algébrique, Inst. Hautes Études Sci., 1960/61.
  • Masa-Nori Ishida, Compactifications of a family of generalized Jacobian varieties, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 503–524. MR 578869
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 14K30, 14D25
  • Retrieve articles in all journals with MSC: 14K30, 14D25
Bibliographic Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 253 (1979), 1-90
  • MSC: Primary 14K30; Secondary 14D25
  • DOI: https://doi.org/10.1090/S0002-9947-1979-0536936-4
  • MathSciNet review: 536936