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Varieties cut out by quadrics: Scheme-theoretic versus homogeneous generation of ideals

  • Lawrence Ein
  • David Eisenbud
  • Sheldon Katz
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1311)

Abstract

In this note we consider cases in which a curve in ℙr which is scheme theoretically the intersection of quadrics necessarily has homogeneous ideal generated by quadrics. The first case in which this does not happen is for a general elliptic octic in ℙ5; we give a proof of this using the surjectivity of the period map for K3 surfaces.

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References

  1. Bayer, D., and Stillman, M.: Macaulay, a computer algebra system. Available free from the authors for the Macintosh, VAX, Sun, and many other computers (1986).Google Scholar
  2. Beauville,A.: Introduction à l'application des périodes. In Géométrie des Surfaces K3: Modules et Périodes. Société Math. de France, Astérisque vol. 126 (1985).Google Scholar
  3. Castelnuovo, G.: Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica, Rend. Circ. Mat. Palermo 7 (1893) 89–110.CrossRefzbMATHGoogle Scholar
  4. Eisenbud, D., and Harris, J.: On Varieties of minimal degree (a centennial account). To appear in Proceedings of the Summer Institute on Algebraic Geometry, Bowdoin, 1985, ed. S. Bloch, Amer. Math. Soc., Providence R.I. (1987).Google Scholar
  5. Eisenbud, D., Koh, J., and Stillman, M.: Determinantal equations for curves of high degree. Preprint (1986). Am. J. Math., to appear.Google Scholar
  6. Fulton, W.: Intersection Theory. Springer-Verlag, New York, (1984).CrossRefzbMATHGoogle Scholar
  7. Green, M.: Koszul cohomology and the geometry of projective varieties. J. Diff. Geom. 19 (1984) 125–171.MathSciNetzbMATHGoogle Scholar
  8. Hartshorne, R.: Algebraic Geometry. Springer-Verlag, New York (1987).zbMATHGoogle Scholar
  9. Mattuck, A.: Symmetric products and Jacobians. Am. J. Math. 83 (1961) 189–206.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Mayer, A.: Families of K3 surfaces. Nagoya Math. J. 48 (1972) 1–17.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Mérindol, J.Y.: Propriétés élémentaires des surfaces K3. In Géométrie des Surfaces K3: Modules et Périodes. Société Math. de France, Astérisque vol. 126 (1985).Google Scholar
  12. Mori, S.: On the degree and genera of curves on smooth quartic surfaces in ℙ3. Nagoya Math. J. 96 (1984) 127–132.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Morrison, D.R.: On K3 surfaces with large Picard number. Invent.Math. 75 (1984) 105–121.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Peskine, C., and Szpiro, L.: Liaison des variétés algébriques, Inv. Math. 26 (1973) 271–302.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Saint-Donat, B.: Projective models of K3 surfaces, Am. J. Math. 96 (1974) 602–639.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Schreyer, F.-O.: Syzygies of canonical curves and special linear series. Math. Ann. 275 (1986) 105–137.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Lawrence Ein
  • David Eisenbud
  • Sheldon Katz

There are no affiliations available

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