Advertisement

manuscripta mathematica

, Volume 117, Issue 4, pp 491–510 | Cite as

Cremona transformations and special double structures

  • Ivan PanEmail author
  • Francesco Russo
Article

Abstract

We consider Cremona Transformations on Open image in new window , whose base locus schemes are double Fossum-Ferrand structures supported on a smooth, irreducible positive dimensional subvariety. We show that if the codimension of the base locus is 2 or if its dimension is no greater than Open image in new window , then N=3 and such a transformation is a Cubo-Cubic Cremona Transformation not defined along a twisted cubic curve. We also prove that the same conclusion holds for such Cremona Transformations either assuming Hartshorne Conjecture on Complete Intersections or that they are defined by degree three homogeneous polynomials.

Keywords

Number Theory Algebraic Geometry Topological Group Complete Intersection Homogeneous Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abo, H., Decker, W., Sasakura, N.: An elliptic conic bundle in ℙ4 arising from a stable rank-3 vector bundle. Math. Z. 229, 725–741 (1998)Google Scholar
  2. 2.
    Alzati, A., Russo, F.: On the k-normality of projected algebraic varieties. Bull. Braz. Math. Soc. 33, 27–48 (2002)CrossRefGoogle Scholar
  3. 3.
    Alzati, A., Russo, F.: Some extremal contractions between smooth varieties arising from projective geometry. Proc. London Math. Soc. 89, 25–53 (2004)CrossRefGoogle Scholar
  4. 4.
    Badescu, L.: Projective Geometry and Formal Geometry. Monografie Matematyczne, vol. 65, Birkhäuser, 2004Google Scholar
  5. 5.
    Barth, W.: Transplanting cohomology classes in complex-projective space. Amer. J. Math. 92, 951–967 (1970)Google Scholar
  6. 6.
    Basili, B., Peskine, C.: Décomposition du fibré normal des surfaces lisses de ℙ4 et strucutures doubles sur les solides de ℙ5. Duke Math. J. 69, 87–95 (1993)CrossRefGoogle Scholar
  7. 7.
    Bertram, A., Ein, L., Lazarsfeld, R.: Vanishing theorems, a theorem of Severi and the equations defining projective varieties. Journ. A.M.S. 4, 587–602 (1991)Google Scholar
  8. 8.
    Crauder, B., Katz, S.: Cremona transfromations with smooth irreducible fundamental locus. Amer. J. Math. 111, 289–309 (1989)Google Scholar
  9. 9.
    Crauder, B., Katz, S.: Cremona transformations and Hartshorne’s conjecture. Amer. J. Math. 113, 289–309 (1991)Google Scholar
  10. 10.
    Corti, A.: Factoring birational maps of threefolds after Sarkisov. J. Algebraic Geom. 4, 223–254 (1995)Google Scholar
  11. 11.
    Ein, L., Shepherd-Barron, N.: Some special Cremona transformations. Amer. J. Math. 111, 783–800 (1989)Google Scholar
  12. 12.
    Faltings, G.: Algebraisation of some formal vector bundles. Annals of Math. 110, 501–514 (1979)Google Scholar
  13. 13.
    Faltings, G.: Ein Kriterium für vollständige Durchschnitte. Inv. Math. 62, 393–401 (1981)CrossRefGoogle Scholar
  14. 14.
    Ferrand, D.: Courbes Gauches et fibrés de Rang 2. C. R. Acad. Sci. Paris 281, 345–347 (1977)Google Scholar
  15. 15.
    Fossum, R.: Commutative Extensions by Canonical Modules are Gorenstein. Proc. A. M. S. 40, 395–400 (1973)Google Scholar
  16. 16.
    Fulton, W.: Intersection Theory. Erg. der Math. vol. 2, Springer-Verlag, 1984 and 1998Google Scholar
  17. 17.
    Fulton, W., Hansen, J.: A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings. Ann. of Math. 110, 159–166 (1979)Google Scholar
  18. 18.
    Hartshorne, R.: Ample subvarieties of algebraic varieties. L.N.M. 156, Springer-Verlag, 1970Google Scholar
  19. 19.
    Hartshorne, R.: Varieties of small codimension in projective space. Bull. A. M. S. 80, 1017–1032 (1974)Google Scholar
  20. 20.
    Hartshorne, R.: Algebraic Geometry. G.T.M. 52, Springer-Verlag, 1977Google Scholar
  21. 21.
    Hudson, H. P.: Cremona Transformations: in Plane and Space. Cambridge University Press, 1927Google Scholar
  22. 22.
    Hulek, K., Katz, S., Schreyer, F.O.: Cremona transformations and syzygies. Math. Z. 209, 419–443 (1992)Google Scholar
  23. 23.
    Hulek, K., Van de Ven, A.: The Horrocks-Mumford bundle and the Ferrand construction. Man. Mathematica 50, 313–335 (1985)CrossRefGoogle Scholar
  24. 24.
    Ionescu, P.: Embedded projective varieties of small invariants. Springer L.N.M. 1056, 142–186 (1984)Google Scholar
  25. 25.
    Ionescu, P.: Embedded projective varieties of small invariants, III. Springer L.N.M. 1417, 138–154 (1990)Google Scholar
  26. 26.
    Ionescu, P., Russo, F.: Varieties with quadratic entry locus, II. Preprint 2005Google Scholar
  27. 27.
    Kollár, J., Smith, K. E., Corti, A.: Rational and Nearly Rational Varieties. Cambridge Studies in Adv. Math., vol. 92, Cambridge: Cambridge University Press, 2004Google Scholar
  28. 28.
    Lascu, A. T., Mumford, D., Scott, D. B.: The self-intersection formula and the “formule-clef”. Math. Proc. Camb. Phil. Soc. 78, 117–123 (1975)Google Scholar
  29. 29.
    Mella, M., Russo, F.: Special Cremona transformation whose base locus has dimension at most three. Preprint 2005Google Scholar
  30. 30.
    Pan, I.: Une remarque sur la génération du groupe de Cremona. Bull. Braz. Math. Soc, 30, 95–98 (1999)Google Scholar
  31. 31.
    Peskine, C., Szpiro, L.: Liaison des variétés algébriques I. Inv. Math. 26, 271–302 (1974)CrossRefGoogle Scholar
  32. 32.
    Ranestad, K.: A geometric construction of elliptic conic bundles in ℙ4. Math. Z. 231, 771–781 (1999)Google Scholar
  33. 33.
    Roggero, M., Valabrega, P.: Speciality Lemma, Rank 2 Bundles and Gherardelli type Theorems for surfaces in ℙ4. Comp. Math. 139, 101–111 (2003)CrossRefGoogle Scholar
  34. 34.
    Schneider, M.: Vector bundles and low-codimensional submanifolds of projective space: a problem list. In: Topics in Algebra, Part 2, Commutative Rings and Algebraic Groups, Banach Center Publ. 26, PWN-Polish Scientific Publisher, Varsovie, 1990, pp. 209–222Google Scholar
  35. 35.
    Semple, J. G., Roth, L.: Introduction to Algebraic Geometry. Oxford University Press, 1949 and 1986Google Scholar
  36. 36.
    Semple, J. G., Tyrrell, J. A.: Specialization of Cremona transformations. Mathematika 15, 171–177 (1968)Google Scholar
  37. 37.
    Semple, J. G., Tyrrell, J. A. : The Cremona transformation of S6 by quadrics through a normal elliptic septimic scroll 1R7. Mathematika 16, 88–97 (1969)Google Scholar
  38. 38.
    Semple, J. G., Tyrrell, J. A.: The T2,4 of S6 defined by a rational surface 3F8. Proc. Lond. Math. Soc. 20, 205–221 (1970)Google Scholar
  39. 39.
    Severi, F.: Sulle intersezioni delle varietá algebriche e sopra i loro caratteri e singolaritá proiettive. Mem. Accad. Sci. Torino 52, 61–118 (1902)Google Scholar
  40. 40.
    Zak, F. L.: Tangents and secants of algebraic varieties. Translations of Mathematical Monografhs, vol. 127, Amer. Math. Soc. 1993Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidade Federal do Rio Grande do SulPorto AlegreBrasil
  2. 2.Departamento de MatemáticaUniversidade Federal de PernambucoRecifeBrasil

Personalised recommendations