Skip to main content
SpringerLink
Log in

The Sp 3 -grassmannian and duality for prime Fano threefolds of genus 9

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract.

By a result of Mukai, the non-abelian Brill-Noether locus X=M C (2, K : 3F) of type II, defined by a stable rank 2 vector bundle F of invariant 3 over a plane quartic curve C, is a prime Fano 3-fold X=X 16 of degree 16. The associate ruled surface S X=P(F) is uniquely defined by X, and we see that for the general X=X 16, S X is isomorphic to the Fano surface of conics on X. The argument uses the geometry of the Sp 3 -grassmannian and the double projection from a line on X 16.

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

Similar content being viewed by others

References

  1. Clemens, H., Griffiths, Ph.: The intermediate jacobian of the cubic threefold. Ann. Math. 95, 281–356 (1972)

    MATH  Google Scholar 

  2. Donagi, R.: On the geometry of grassmannians. Duke Math. J. 44(4), 795–837 (1977)

    MATH  Google Scholar 

  3. Gruson, L., Peskine, Ch.: Courbes de l'espace projectif: varietes de secantes. In: Enumerative geometry and classical algebraic geometry. Progress in Mathematics, Vol. 24, Birkhauser, Boston-Basel-Stuttgart, 1982, pp. 1–31

  4. Fulton, W., Harris, J.: Representation theory. Springer-Verlag, 1991

  5. Hartshorne, R.: Algebraic geometry. Springer-Verlag, 1977

  6. Iskovskikh, V.A.: Fano threefolds II. Math. USSR, Izv. 12, 469–506 (1978)

    Google Scholar 

  7. Iskovskikh, V.A.: Double projection from a line on Fano threefolds of the first kind. Math. USSR Sbornik 66(1), 265–284 (1990)

    MATH  Google Scholar 

  8. Iskovskikh, V.A., Prokhorov, Yu.G.: Algebraic geometry V: Fano varieties. Encycl. Math. Sci. 47, 1–245 (1999)

    MATH  Google Scholar 

  9. Iliev, A., Ranestad, K.: Geometry of the lagrangian grassmannian with applications to Brill-Nether loci. e-print math.AG/0209169

  10. Kimura, T.: The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces. Nagoya Math. J. 85, 1–80 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Kimura, T., Sato, M.: A classification of irreducible prehomogeneous vector spaces and their relative invariants. Nagoya Math. J. 65, 1–155 (1977)

    MATH  Google Scholar 

  12. Landsberg, J.M., Manivel, L.: The Projective Geometry of Freudenthal's Magic Square. J. Algebra 239, 477–512 (2001)

    Article  MathSciNet  Google Scholar 

  13. Landsberg, J.M., Manivel, L.: Representation theory and projective geometry. e-print math.AG/0203260

  14. Lange, H., Narasimhan, M.S.: Maximal Subbundles of rank two vector bundles on curves. Math. Ann. 266, 55–72 (1983)

    MathSciNet  MATH  Google Scholar 

  15. Mukai, S.: Curves, K3 Surfaces and Fano 3-folds of Genus ≤ 10. Algebraic geometry and commutative algebra in honor of M. Nagata, Kinokuniya, Tokyo, 1987, pp. 357–387

  16. Mukai, S.: Biregular classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Natl. Acad. Sci. USA 86, 3000–3002 (1989)

    MATH  Google Scholar 

  17. Mukai, S.: Non-Abelian Brill-Noether theory and Fano 3-folds. Sugaku Expositions 49, 1–24 (1997). e-print math.AG/9704015

    MATH  Google Scholar 

  18. Mukai, S.: Curves and Symmetric Spaces. Proc. Japan Acad. Ser. A 68, 7–10 (1992)

    MathSciNet  MATH  Google Scholar 

  19. Pragacz, P., Ratajski, J.: Pieri type formula for isotropic grassmannians: the operator approach. Manuscripta Math. 79(2), 127–151 (1993)

    MATH  Google Scholar 

  20. Seiler, W.K.: Deformations of ruled surfaces. J. reine angew. Math. 426, 203–219 (1992)

    MathSciNet  MATH  Google Scholar 

  21. Takeuchi, K.: Some birational maps of Fano 3-folds. Compositio Math. 71, 265–283 (1989)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., 8, 1113, Sofia, Bulgaria

    Atanas Iliev

Authors
  1. Atanas Iliev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Atanas Iliev.

Additional information

Partially supported by Grant MM-1106/2001 of the Bulgarian Foundation for Scientific Research

Rights and permissions

Reprints and permissions

About this article

Cite this article

Iliev, A. The Sp 3 -grassmannian and duality for prime Fano threefolds of genus 9. manuscripta math. 112, 29–53 (2003). https://doi.org/10.1007/s00229-003-0387-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-003-0387-z

Keywords

Search

Navigation