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

top

Smith, Roy, and Varley, Robert. "Deformations of theta divisors and the rank 4 quadrics problem." Compositio Mathematica 76.3 (1990): 367-398. <http://eudml.org/doc/90062>.

@article{Smith1990,
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 = {http://eudml.org/doc/90062},
volume = {76},
year = {1990},
}

TY - JOUR
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 - http://eudml.org/doc/90062
ER -

References

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.