Mathematische Annalen

, Volume 292, Issue 1, pp 127–147

The distribution of bidegrees of smooth surfaces in Gr(1, P3)

  • Mark Gross


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of algebraic curves. Berlin Heidelberg New York: Springer 1985Google Scholar
  2. 2.
    Arrondo, E.: On congruences of lines in the projective space. Ph.D. Thesis, Universidad Complutense de Madrid, 1990Google Scholar
  3. 3.
    Arrondo, E., Sols, I.: Classification of smooth congruences of low degree. J. Reine Angew. Math.393, 199–219 (1989)Google Scholar
  4. 4.
    Arrondo, E., Peskine, C., Sols, I.: Characterization of the spinor bundles by the vanishing of intermediate cohomology. (preprint)Google Scholar
  5. 5.
    Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. (Ergeb. Math. Grenzbeg., 3. Folge, vol. 4) Berlin Heidelberg New York: Springer 1984Google Scholar
  6. 6.
    Bogomolov, F.: Holomorphic tensors and vector bundles on projective varieties. Math. USSR, Izv.13, 499–555 (1979)Google Scholar
  7. 7.
    Cossec, F., Dolgachev, I., Verra, A.: Unpublished manuscriptGoogle Scholar
  8. 8.
    Dolgachev, I., Reider, I.: On rank 2 vector bundles withc 12=10 andc 2=3 on Enriques surfaces. (Preprint)Google Scholar
  9. 9.
    Eisenbud, D., Harris, J.: Curves in projective space. Montréal: Les Presses de l'Université de Montréal 1982Google Scholar
  10. 10.
    Fano, G.: Nuove richerce sulle congrueze di Rette del 3° ordine prive di linea singolare. Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat.51, 1–80 (1902)Google Scholar
  11. 11.
    Fulton, W.: Intersection theory. (Ergeb. Math. Grenzgeb., 3. Folge, vol. 2) Berlin Heidelberg New York: Springer 1984Google Scholar
  12. 12.
    Goldstein, N.: Scroll surfaces in Gr(1,P 3). In: Collono, A. et al. (eds.) Conference on algebraic varieties of small dimension. Turin, 1985. (Rend. Semin. Mat., Torino, Fasc. Spec., pp. 69–75) Torino: Editrice Univ. Levrotto & Bella 1986Google Scholar
  13. 13.
    Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978Google Scholar
  14. 14.
    Gross, M.: Surfaces in the four-dimensional Grassmannian. Ph.D. Thesis, U.C. Berkeley, 1990Google Scholar
  15. 15.
    Gross, M.:Surfaces of bidegree (3,n) in Gr(1,P 3). (to appear 1990)Google Scholar
  16. 16.
    Hartshorne, R.: Algebraic geometry. Berlin Heidelberg New York: Springer 1977Google Scholar
  17. 17.
    Hartshorne, R., Rees, F., Thomas, E.: Nonsmoothing of algebraic cycles on Grassmann varieties. Bull. Am. Math. Soc.80 (no. 5), 847–851 (1974)Google Scholar
  18. 18.
    Hernández, R., Sols, I.: Line congruences of low degree. In: Aroca, J.-M. et al., (eds.) Géométrie algébrique et applications. II. Singularités et géométrie complexe, pp. 141–154. Paris: 1987 HermannGoogle Scholar
  19. 19.
    Jessop, C. M.: A treatise on the line complex. Cambridge: Cambridge University Press 1903; reprinted Chelsea, Publishing Company 1969Google Scholar
  20. 20.
    Kleiman, S.: Geometry on Grassmannians and applications to splitting bundles and smoothing cycles. Publ. Math., Inst. Hautes Étud. Sci.36, 281–298 (1969)Google Scholar
  21. 21.
    Miyaoka, Y.: The Chern classes and Kodaira dimension of a minimal variety. In: Oda, T. (ed.) Algebraic geometry. Sendai, 1985. (Adv. Stud. Pure Math., vol. 10, pp. 449–476 Tokyo: Kinokuniya 1987Google Scholar
  22. 22.
    Mumford, D.: Geometric invariant theory. Berlin Heidelberg New York: Springer 1965; 2nd ed. 1982Google Scholar
  23. 23.
    Pollack, A.: Codimension 2 subvarieties of Grassmannians. Ph.D. Thesis, Berkeley, 1978Google Scholar
  24. 24.
    Ran, Z.: Surfaces of order 1 in Grassmannians. J. Reine Angew. Math.368, 119–126 (1986)Google Scholar
  25. 25.
    Reid, M.: Bogomolov's Theoremc 12≦4c 2. In: Nagata, M. (ed.) Intl. Symp. on algebraic geometry, pp. 623–642. Kyoto. 1977. Tokyo: Kinokinuya 1978Google Scholar
  26. 26.
    Roth, L.: Some properties of line congruences. Proc. Camb. Philos. Soc.26, 190–200 (1931)Google Scholar
  27. 27.
    Verra, A.: Smooth surfaces of degree 9 inG(1, 3). Manuscr. Math.63, 417–435 (1988)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Mark Gross
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations