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.
The authors are grateful to the NSF for partial support, and to the NSF and Brigham Young University for having supported the conference on Enumerative Geometry at Sundance, Utah, which provided a pleasant and congenial backdrop for work on this project.
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References
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).
Beauville,A.: Introduction à l'application des périodes. In Géométrie des Surfaces K3: Modules et Périodes. Société Math. de France, Astérisque vol. 126 (1985).
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.
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).
Eisenbud, D., Koh, J., and Stillman, M.: Determinantal equations for curves of high degree. Preprint (1986). Am. J. Math., to appear.
Fulton, W.: Intersection Theory. Springer-Verlag, New York, (1984).
Green, M.: Koszul cohomology and the geometry of projective varieties. J. Diff. Geom. 19 (1984) 125–171.
Hartshorne, R.: Algebraic Geometry. Springer-Verlag, New York (1987).
Mattuck, A.: Symmetric products and Jacobians. Am. J. Math. 83 (1961) 189–206.
Mayer, A.: Families of K3 surfaces. Nagoya Math. J. 48 (1972) 1–17.
Mérindol, J.Y.: Propriétés élémentaires des surfaces K3. In Géométrie des Surfaces K3: Modules et Périodes. Société Math. de France, Astérisque vol. 126 (1985).
Mori, S.: On the degree and genera of curves on smooth quartic surfaces in ℙ3. Nagoya Math. J. 96 (1984) 127–132.
Morrison, D.R.: On K3 surfaces with large Picard number. Invent.Math. 75 (1984) 105–121.
Peskine, C., and Szpiro, L.: Liaison des variétés algébriques, Inv. Math. 26 (1973) 271–302.
Saint-Donat, B.: Projective models of K3 surfaces, Am. J. Math. 96 (1974) 602–639.
Schreyer, F.-O.: Syzygies of canonical curves and special linear series. Math. Ann. 275 (1986) 105–137.
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Ein, L., Eisenbud, D., Katz, S. (1988). Varieties cut out by quadrics: Scheme-theoretic versus homogeneous generation of ideals. In: Algebraic Geometry Sundance 1986. Lecture Notes in Mathematics, vol 1311. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082908
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DOI: https://doi.org/10.1007/BFb0082908
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