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Threefolds of order one in the six-quadric

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Abstract

Consider the smooth quadric Q 6 in ℙ7. The middle homology group H 6(Q 6, ℤ) is isomorphic to ℤ ⊕ ℤ, with a basis given by two classes of linear subspaces. We classify all threefolds of bidegree (1, p) inside Q 6.

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References

  1. L. Borisov, S. Salamon, and J. Viaclovsky, “Orthogonal Complex Structures in Euclidean Spaces,” Preprint (Univ. Wisconsin and Politechn. Torino, 2008).

    Google Scholar 

  2. É. Cartan, The Theory of Spinors (Dover Publ., New York, 1981).

    MATH  Google Scholar 

  3. W. Fulton, Intersection Theory, 2nd ed. (Springer, Berlin, 1998), Ergebn. Math. Grenzgeb. 3. Flg. 2.

    MATH  Google Scholar 

  4. P. Griffiths and J. Harris, Principles of Algebraic Geometry (J. Wiley & Sons, New York, 1994).

    MATH  Google Scholar 

  5. M. Gross, “The Distribution of Bidegrees of Smooth Surfaces in Gr(1,P3),” Math. Ann. 292(1), 127–147 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Hartshorne, Algebraic Geometry (Springer, New York, 1977), Grad. Texts Math. 52.

    MATH  Google Scholar 

  7. Z. Ran, “Surfaces of Order 1 in Grassmannians,” J. Reine Angew. Math. 368, 119–126 (1986).

    MATH  MathSciNet  Google Scholar 

  8. S. Salamon and J. Viaclovsky, “Orthogonal Complex Structures on Domains in ℝ4,” Math. Ann. (in press); arXiv: 0704.3422.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Lev Borisov & Jeff Viaclovsky

Authors
  1. Lev Borisov
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  2. Jeff Viaclovsky
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Correspondence to Lev Borisov.

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Borisov, L., Viaclovsky, J. Threefolds of order one in the six-quadric. Proc. Steklov Inst. Math. 264, 18–29 (2009). https://doi.org/10.1134/S0081543809010039

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  • DOI: https://doi.org/10.1134/S0081543809010039

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