Skip to main content
SpringerLink
Log in

On the geometry of quadrics and their degenerations

  • Published:
Commentarii Mathematici Helvetici

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Bibliography

  1. Abeasis, S.,On a remarkable class of subvarieties of a symmetric variety, to appear in Advances in Mathematics.

  2. Alguneid, A. R.,Complete quadric primals in four dimensional space. Proc. Math. Phys. Soc. Egypt4 (1952) no. 4, pp. 93–104 (1953).

    MathSciNet  MATH  Google Scholar 

  3. Ash, A., Mumford, D., Rapoport, W., andTai.Smooth Compactifications of Locally Symmetric Varieties. Math. Sci. Press (Boston Mass.) 1975.

    Google Scholar 

  4. Atiyah, M. F.,Convexity and commuting hamiltonians. Bull. Lon. Math. Socl.14 (1982) 1–15.

    MathSciNet  MATH  Google Scholar 

  5. Beilinson, A., Bernstein, J., andDeligne, P.,Faisceaux Pervers. InAnalyse et Topologie sur les Espaces Singuliers. Astérique # 100, Soc. Math. de France (1983).

  6. Bialynicki-Birula, A.,Some properties of the decomposition of algebraic varieties determined by actions of a torus. Bull. Acad. Polon. Sci.24 (1976) no. 9, pp. 667–674.

    MathSciNet  MATH  Google Scholar 

  7. Bialynicki-Birula, A.,Some theorems on actions of algebraic groups. Ann. of Math.98 (1973), 480–497.

    Article  MathSciNet  Google Scholar 

  8. Borel, A.,Cohomologie des Espaces Localement Compacts, d’après J. Leray, 1951. Springer Lecture Notes in Mathematics #2, Springer Verlag, N.Y. (1964).

    Google Scholar 

  9. Borel, A.,The Homology Theory of Fibre Bundles, 1954. Springer Lecture Notes in Mathematics #36, Springer Verlag, N.Y. (1967).

    Google Scholar 

  10. Borho, W. andMacPherson, R.,Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes. C.R. Acad. Sci. Paris. t.292 (1981) 707–710.

    MathSciNet  MATH  Google Scholar 

  11. Chasles, M.,Détermination du nombre des sections coniques qui doivent toucher cinq courbes données d’ordre quelconque, ou satisfaire à diverses autres conditions. C.R. Acad. Sci. Paris, Séance du 1 Fev. 1864. t58 (1864) 222–226.

    Google Scholar 

  12. Danilov, V. I.,The geometry of toric varieties. Russ. Math. Surveys33 (1978) 97–154.

    Article  MathSciNet  MATH  Google Scholar 

  13. DeConcini, C. andProcesi, C.,Complete symmetric varieties. InInvariant Theory, Springer Lecture Notes in Mathematics #996. Springer Verlag, N. Y. (1983) pp. 1–44.

    Chapter  Google Scholar 

  14. DeConcini, C. andProcesi, C.,Complete symmetric varieties II, Advanced Studies in Pure Mathematics VI, Academic Press, N.Y., 1985.

    Google Scholar 

  15. DeConcini, C. andProcesi, C.,Cohomology of compactifications of algebraic groups. Duke Math. Journal,53 (1986) 585–594.

    Article  MathSciNet  Google Scholar 

  16. DeConcini, C. andSpringer, A.,Betti numbers of complete symmetric varieties. inGeometry Today, Birkhäuser-Boston (1985).

    Google Scholar 

  17. Drechsler, K. andIhle, W.,Complete Quadrics. Math. Nachr.122 (1985) 175–185.

    MathSciNet  MATH  Google Scholar 

  18. Drechsler, K. andSterz, U., Die p-1-s-Kegelschnitte auf der Grundlage des mittelbaren Superoskulierens, Beiträge zur Algebra und Geometrie14 (1983), 87–91. (See also the bibliography in this paper).

    MathSciNet  MATH  Google Scholar 

  19. Flensted-Jensen, M.,Spherical functions on a real semisimple Lie group. A method of reduction to the complex case. J. of Funct. Anal.30, 106–146 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  20. Fulton, W., Kleiman, S., andMacPherson, R.,About the enumeration of contacts. InAlgebraic Geometry: Open Problems (C. Ciliberto, ed.) Springer Lecture Notes in Mathematics #997 (1983) 156–196.

  21. Hiller, H.,Geometry of Coxeter Groups, Pitman Press, Boston 1982.

    MATH  Google Scholar 

  22. Jozefiak, T.,Ideals generated by minors of a symmetric matrix. Comm. Math. Helv.53 (1978), 595–607.

    MathSciNet  MATH  Google Scholar 

  23. Kempf, G., Knudsen, F., Saint-Donat, B. andMumford, D.,Toroidal Embeddings I. Springer Lecture Notes in Mathematics #339, Springer Verlag N.Y. (1972).

    Google Scholar 

  24. Kleiman, S.,Chasles’ enumerative theory of conics: a historical introduction. InStudies in Algebraic Geometry. Math. Assn. Amer. Studies in Mathematics #20 (A. Seidenberg, ed.) (1980) 117–138.

  25. Kleiman, S.,Problem 15: rigorous foundations of Schubert’s enumerative calculus. Proc. Symp. Pure Math.28 (1976) Amer. Math. Soc., Providence R.I., 1976.

    Google Scholar 

  26. Procesi, C., to appear.

  27. Rossmann, W.,The structure of semisimple symmetric spaces. Can. J. of Math.31, (1979), 157–180.

    MathSciNet  MATH  Google Scholar 

  28. Schubert, H.,Kalkül der Abzählenden Geometrie, Leipzig (1879), reprinted by Springer Verlag, Berlin (1979).

  29. Semple, J. G.,On complete quadrics (I), J. Lon. Math. Soc.23 (1948), 258–267.

    MathSciNet  Google Scholar 

  30. Severi, F.,Sui fondamenti della geometria numerativa e sulla teoria delle caratteristiche. Atti del R. Ist. Veneto,75 (1916) pp. 1122–1162.

    Google Scholar 

  31. Severi, F.,I fondamenti della geometria numerativa, Ann. di Mat. (4),19 (1940), 151–242.

    MathSciNet  Google Scholar 

  32. Steinberg, R.,A geometric approach to the representations of the full linear group over a Galois field. Trans. Amer. Math. Soc.71 (1951) 274–282.

    Article  MathSciNet  MATH  Google Scholar 

  33. Strickland, E.,Schubert type cells for complete quadrics, to appear.

  34. Study, E.,Über die Geometrie der Kegelschnitte, insbesondere deren Charakteristiken Problem. Math. Ann.26 (1886) 51–58.

    Google Scholar 

  35. Tyrrell, J. A.,Complete quadrics and collineations in S n . Mathematika3 (1956) 69–79.

    Article  MathSciNet  MATH  Google Scholar 

  36. Vainsencher, I.,Schubert calculus for complete quadrics. InEnumerative Geometry and Calssical Algebraic Geometry (P. LeBarz and Y. Hervier, ed.).Progress in Mathematics vol 24, Birkhauser Boston (1982), pp. 199–236.

    Google Scholar 

  37. Van der Waerden, B. L.,Topologische abzählende Geometrie. Math. Ann.102 (1930) 337–362.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dip. di Matematico, Univ. di Roma II, 00173, Roma, Italy

    C. DeConcini

  2. Northeastern University, 02115, Boston, Mass., USA

    M. Goresky

  3. 2-246 M.I.T., 02139, Cambridge, MA

    R. MacPherson

  4. Istituto Matematico “G. Castelnuovo”, Città Universitaria, I-00185, Roma

    C. Procesi

Authors
  1. C. DeConcini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Goresky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. MacPherson
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Procesi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

members of G.N.S.A.G.A. of C.N.R.

Partially supported by the National Science Foundation grant # DMS820-1680 and DMS 8601161.

Partially supported by the National Science Foundation grant # DMS850-2442.

Rights and permissions

Reprints and permissions

About this article

Cite this article

DeConcini, C., Goresky, M., MacPherson, R. et al. On the geometry of quadrics and their degenerations. Commentarii Mathematici Helvetici 63, 337–413 (1988). https://doi.org/10.1007/BF02566772

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02566772

Keywords

Search

Navigation