Deformations of theta divisors and the rank 4 quadrics problem

Roy Smith; Robert Varley

Compositio Mathematica (1990)

  • Volume: 76, Issue: 3, page 367-398
  • ISSN: 0010-437X

How to cite


Smith, Roy, and Varley, Robert. "Deformations of theta divisors and the rank 4 quadrics problem." Compositio Mathematica 76.3 (1990): 367-398. <>.

author = {Smith, Roy, Varley, Robert},
journal = {Compositio Mathematica},
keywords = {theta divisor in the Jacobian of a non-hyperelliptic smooth curve; rank-4 double point; rank-4 quadrics conjecture; generic constructive Torelli theorem; infinitesimal deformation theory for the singularities of theta divisors},
language = {eng},
number = {3},
pages = {367-398},
publisher = {Kluwer Academic Publishers},
title = {Deformations of theta divisors and the rank 4 quadrics problem},
url = {},
volume = {76},
year = {1990},

AU - Smith, Roy
AU - Varley, Robert
TI - Deformations of theta divisors and the rank 4 quadrics problem
JO - Compositio Mathematica
PY - 1990
PB - Kluwer Academic Publishers
VL - 76
IS - 3
SP - 367
EP - 398
LA - eng
KW - theta divisor in the Jacobian of a non-hyperelliptic smooth curve; rank-4 double point; rank-4 quadrics conjecture; generic constructive Torelli theorem; infinitesimal deformation theory for the singularities of theta divisors
UR -
ER -


  1. [A] M. Artin, Lectures on deformations of singularities, Tata Institute of Fundamental Research, Bombay, 1976. Zbl0395.14003
  2. [A-H] E. Arbarello and J. Harris, Canonical curves and quadrics of rank 4, Comp. Math.43 (1981), 145-179. Zbl0494.14011MR622446
  3. [A-M] A. Andreotti and A. Mayer, On period relations for abelian integrals and algebraic curves, Ann. Scuola Norm. Sup. Pisa21 (1967), 189-238. Zbl0222.14024MR220740
  4. [B-V] F. Bardelli and L. Verdi, On osculating cones and the Riemann-Kempf singularity theorem for hyperelliptic curves, trigonal curves, and smooth plane quintics, Comp. Math.65 (1988), 177-199. Zbl0662.14015MR932643
  5. [C] C.H. Clemens, The local geometry of the Abel-Jacobi mapping, in Algebraic Geometry, Bowdoin1985, Proceedings of Symposia in Pure Mathematics, A.M.S., vol. 46-part 2. Zbl0646.14020
  6. [C-E] H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, 1956. Zbl0075.24305MR77480
  7. [Go] R. Godement, Topologie Algébrique et Théorie des Faisceaux, Hermann, Paris, 1958. Zbl0080.16201MR102797
  8. [Gr1] A. Grothendieck, Geometrie formelle et geometrie algebrique, Seminaire Bourbaki, Exp. 182, Mai 1959. Zbl0229.14005
  9. [Gr2] A. Grothendieck, Techniques de construction et theoremes d'existence en geometrie algebrique, IV: Les schemas de Hilbert, Seminaire Bourbaki, 1960/1961, no. 221. Zbl0236.14003
  10. [Gr3] A. Grothendieck, Local Cohomology, Lecture Notes in Math.41, Springer-Verlag, New York, 1967. Zbl0185.49202MR224620
  11. [Green] M. Green, Quadrics of rank four in the ideal of the canonical curve, Inv. Math.75 (1984), 84-104. Zbl0542.14018
  12. [H] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977. Zbl0367.14001MR463157
  13. [K1] G. Kempf, On the geometry of a theorem of Riemann, Ann. Math.98 (1973), 178-185. Zbl0275.14023MR349687
  14. [K2] G. Kempf, Schubert methods with an application to algebraic curves, Stichting Mathematisch Centrum, Amsterdam, 1971. Zbl0223.14018
  15. [K3] G. Kempf, Abelian Integrals, Monografias del Instituto de Matemáticas13, Universidad Nacional Autónoma de México, 1983. Zbl0541.14023MR743421
  16. [K4] G. Kempf, Deformations of symmetric products, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Annals of Math. Studies97, Princeton Univ. Press, 1981, 319-341. Zbl0465.14013MR624823
  17. [K5] G. Kempf, The equations defining a curve of genus 4, Proc. A.M.S.97, no. 2, (1986), 219-225. Zbl0595.14021MR835869
  18. [Ka] S. Katz, Infinitesimal deformations of manifolds with submanifolds, unpublished type-script, 4 pp. 
  19. [L-S] S. Lichtenbaum and M. Schlessinger, The cotangent complex of a morphism, Trans. A.M.S.128 (1967), 41-70. Zbl0156.27201MR209339
  20. [M1] D. Mumford, Geometric Invariant Theory, Springer-Verlag, New York, 1965. Zbl0147.39304MR214602
  21. [M2] D. Mumford, Abelian Varieties, Oxford Univ. Press, 1970. Zbl0223.14022MR282985
  22. [M3] D. Mumford, Prym Varieties I, in Contributions to Analysis, Academic Press, New York, 1974, 325-350. Zbl0299.14018MR379510
  23. [M4] D. Mumford, Curves and their Jacobians, Univ. of Michigan Press, Ann Arbor, 1975. Zbl0316.14010MR419430
  24. [Mat] T. Matsusaka, On a characterization of a Jacobian variety, Mem. Coll. Sci. Kyoto, Series A, 32 (1959), 1-19. Zbl0094.34103MR108497
  25. [Mayer] A. Mayer, unpublished manuscript on constructive approaches to the Schottky and Torelli problems, circa 1965, 42 pages. 
  26. [O] F. Oort, Finite group schemes, local moduli for abelian varieties, and lifting problems, Algebraic Geometry, Oslo1970, Wolters-Noordhoff, Groningen, 1972, 223-254. Zbl0239.14018
  27. [O-S] F. Oort and J. Steenbrink, The local Torelli problem for algebraic curves, Journées de Géométrie Algébrique d'Angers1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, 157-204. Zbl0444.14007MR605341
  28. [R] D.S. Rim, Formal deformation theory, Exp. VI, in Groupes de Monodromie en Géométrie Algébrique, Lecture Notes in Math.288, Springer-Verlag, New York, 1972, 32-132. Zbl0246.14001
  29. [S1] M. Schlessinger, Infinitesimal deformation of singularities, Harvard Thesis, 1964. 
  30. [S2] M. Schlessinger, Functors of Artin rings, Trans. A.M.S. 130 (1968), 208-222. Zbl0167.49503MR217093
  31. [S3] M. Schlessinger, Rigidity of quotient singularities, Invent. Math.14 (1971), 17-26. Zbl0232.14005MR292830
  32. [S-D] B. Saint-Donat, On Petri's analysis of the linear system of quadrics through a canonical curve, Math. Ann.206 (1973), 157-175. Zbl0315.14010MR337983
  33. [S-V1] R. Smith and R. Varley, Tangent cones to discriminant loci for families of hypersurfaces, Trans. A.M.S. 307 (1988), 647-674. Zbl0674.14026MR940221
  34. [S-V2] R. Smith and R. Varley, Deformations of singular points on theta divisors, Theta Functions— Bowdoin1987, Proceedings Symp. Pure Math. vol. 49, Part I, A.M.S., 1989, 571-579. Zbl0702.14001
  35. [V] R. Varley, unpublished summary of results on deformations of theta divisors, 1982, 3 pp. 
  36. [W] G. Welters, Polarized abelian varieties and the heat equations, Comp. Math.49 (1983), 173-194. Zbl0576.14042MR704390

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