Duke Mathematical Journal

Sur les varietes abeliennes dont le diviseur theta est singulier en codimension 3

Olivier Debarre

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J., Volume 57, Number 1 (1988), 221-273.

First available in Project Euclid: 20 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14K10: Algebraic moduli, classification [See also 11G15]
Secondary: 14H40: Jacobians, Prym varieties [See also 32G20]


Debarre, Olivier. Sur les varietes abeliennes dont le diviseur theta est singulier en codimension $3$. Duke Math. J. 57 (1988), no. 1, 221--273. doi:10.1215/S0012-7094-88-05711-0. https://projecteuclid.org/euclid.dmj/1077306857

Export citation


  • [ACGH] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves, I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985.
  • [A-M] A. Andreotti and A. L. Mayer, On period relations for abelian integrals on algebraic curves, Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 189–238.
  • [Ba] W. Barth, Fortsetzung, meromorpher Funktionen in Tori und komplex-projektiven Räumen, Invent. Math. 5 (1968), 42–62.
  • [Be 1] A. Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149–196.
  • [Be 2] A. Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309–391.
  • [Be 3] A. Beauville, Sous-variétés spéciales des variétés de Prym, Compositio Math. 45 (1982), no. 3, 357–383.
  • [B-D] A. Beauville and O. Debarre, Une relation entre deux approches du problème de Schottky, Invent. Math. 86 (1986), no. 1, 195–207.
  • [D-M] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 75–109.
  • [Do] R. Donagi, The tetragonal construction, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 181–185.
  • [D-S] R. Donagi and R. C. Smith, The structure of the Prym map, Acta Math. 146 (1981), no. 1-2, 25–102.
  • [EGA] A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. (1961), no. 8, 222.
  • [F-L] W. Fulton and R. Lazarsfeld, On the connectedness of degeneracy loci and special divisors, Acta Math. 146 (1981), no. 3-4, 271–283.
  • [F-S 1] R. Friedman and R. Smith, Degenerations of Prym varieties and intersections of three quadrics, Invent. Math. 85 (1986), no. 3, 615–635.
  • [F-S 2] R. Friedman and R. Smith, The generic Torelli theorem for the Prym map, Invent. Math. 67 (1982), no. 3, 473–490.
  • [Gr] M. Green, Quadrics of rank four in the ideal of a canonical curve, Invent. Math. 75 (1984), no. 1, 85–104.
  • [Ha] J. Harris, Theta-characteristics on algebraic curves, Trans. Amer. Math. Soc. 271 (1982), no. 2, 611–638.
  • [Ig 1] J. I. Igusa, A desingularization problem in the theory of Siegel modular functions, Math. Ann. 168 (1967), 228–260.
  • [Ig 2] J. I. Igusa, Theta Functions, Grundlehren der Math. Wiss., vol. 194, Springer-Verlag, Berlin-New York, 1972.
  • [Ma] L. Masiewicki, Universal properties of Prym varieties with an application to algebraic curves of genus five, Trans. Amer. Math. Soc. 222 (1976), 221–240.
  • [Mu 1] D. Mumford, On the Kodaira Dimension of the Siegel Modular Variety, Algebraic geometry—open problems (Ravello, 1982), Springer Lecture Notes, vol. 997, Springer-Verlag, New York, 1983, pp. 348–375.
  • [Mu 2] D. Mumford, Prym Varieties I, Contributions to Analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325–350.
  • [Mu 3] D. Mumford, Theta characteristics of an algebraic curve, Ann. Sci. École Norm. Sup. (4) 4 (1971), 181–192.
  • [Mu 4] D. Mumford, Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29–100.
  • [Mu 5] D. Mumford, On the equations defining abelian varieties I, Invent. Math. 1 (1966), 287–354.
  • [Mu 6] D. Mumford, Abelian Varieties, Tata Studies in Math., vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, London, 1970.
  • [Mu 7] D. Mumford, An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24 (1972), 239–272.
  • [Na]1 Y. Namikawa, A new compactification of the Siegel space and degeneration of Abelian varieties. I, Math. Ann. 221 (1976), no. 2, 97–141.
  • [Na]2 Y. Namikawa, A new compactification of the Siegel space and degeneration of Abelian varieties. II, Math. Ann. 221 (1976), no. 3, 201–241.
  • [Re] S. Recillas, Jacobians of curves with $g\sp1\sb4$'s are the Prym's of trigonal curves, Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 1, 9–13.
  • [S-V] R. Smith and R. Varley, Components of the locus of singular theta divisors of genus $5$, Algebraic Geometry, Sitges (Barcelona) 1983, Lecture Notes in Math., vol. 1124, Springer-Verlag, Berlin-New York, 1985, pp. 338–416.
  • [Te] M. Teixidor i Bigas, For which Jacobi varieties is $\rm Sing\,\Theta$ reducible? J. Reine Angew. Math. 354 (1984), 141–149.
  • [Tj 1] A. N. Tjurin, Five lectures on three-dimensional varieties, Uspehi Mat. Nauk 27 (1972), no. 5, (167), 3–50.
  • [Tj 2] A. N. Tjurin, The intersection of quadrics, Uspehi Mat. Nauk 30 (1975), no. 6(186), 51–99.
  • [Tj 3] A. N. Tjurin, The geometry of the Poincaré theta divisor of a Prym variety, Math. USSR-Izv 9 (1975), 951–986.
  • [We 1] G. Welters, A theorem of Gieseker-Petri type for Prym varieties, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 671–683.
  • [We 2] G. Welters, The surface $C-C$ on Jacobi varieties and 2nd order theta functions, Acta Math. 157 (1986), no. 1-2, 1–22.
  • [We 3] G. Welters, Abel-Jacobi isogenies for certain types of Fano threefolds, Mathematical Centre Tracts, vol. 141, Mathematisch Centrum, Amsterdam, 1981.