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Mathematische Annalen

, Volume 342, Issue 2, pp 279–289 | Cite as

The reflexivity of a Segre product of projective varieties

  • Satoru Fukasawa
  • Hajime KajiEmail author
Article
  • 68 Downloads

Abstract

We study the reflexivity of a Segre product of a projective space \({\mathbb P^m}\) and a projective variety Y, and give a criterion for \({\mathbb P^m \times Y}\) to be reflexive in terms of m, the dimension of Y, the rank of the general Hessian of Y and the characteristic of the ground field. Our study is closely related to a question raised by Kleiman and Piene on the relationship between the conormal map and the Gauss map.

Mathematics Subject Classification (2000)

14N05 

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References

  1. 1.
    Fukasawa S. (2006). On Kleiman–Piene’s question for Gauss maps. Compos. Math. 142: 1305–1307 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Fukasawa S. and Kaji H. (2007). The separability of the Gauss map and the reflexivity for a projective surface. Math. Z. 256: 699–703 zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Fukasawa, S., Kaji, H.: Existence of a non-reflexive embedding with birational Gauss map for a projective variety. Math. Nachr. (to appear)Google Scholar
  4. 4.
    Hefez, A., Kleiman, S.: Notes on the duality of projective varieties, “Geometry Today”, Prog. Math. 60, pp. 143–183. Birkhäuser, Boston (1985)Google Scholar
  5. 5.
    Hefez A. and Thorup A. (1987). Reflexivity of Grassmannians and Segre varieties. Commun. Algebra 15: 1095–1108 zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kaji H. (1992). On the inseparable degrees of the Gauss map and the projection of the conormal variety to the dual of higher order for space curves. Math. Ann. 292: 529–532 zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kaji H. (2003). On the duals of Segre varieties. Geom. Dedicata 99: 221–229 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kleiman, S.: About the conormal scheme, Complete intersections (Acireale, 1983), Lecture Notes in Math., vol. 1092, pp. 161–197. Springer, Berlin (1984)Google Scholar
  9. 9.
    Kleiman, S., Piene, R.: On the inseparability of the Gauss map, In: Contemp. Math., vol 123, pp.~107–129. Amer. Math. Soc., Providence (1991)Google Scholar
  10. 10.
    Satake, I.: Linear Algebra, Pure and Applied Mathematics, vol. 29. Marcel Dekker, Inc., New York (1975)Google Scholar
  11. 11.
    Weyman J. and Zelevinsky A. (1994). Multiplicative properties of projectively dual varieties. Manuscripta Math. 82: 139–148 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics, School of Science and EngineeringWaseda UniversityShinjuku, TokyoJapan

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