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Specialization of a generic hyperplane section through a rational point of an algebraic variety

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Summary

Let V/k be an irreducible algebraic variety of dimension ≥3, defined over a field k of characteristic O, passing through the origin (O) of the ambient affine n-space. Let p be the prime ideal of V in the polynomial ring k[X1, ..., Xn]. If V/k is k-normal at (O), then for almost all hyperplanes Ha: a1X1+...+anXn=0 with a1, ..., an ε k the ideal (p, a1X1+...+anXn) is a prime ideal. If the generic hyperplane section of V through (O) is normal over the ground field of the generic section of V, then the section V ∩ Ha is k-normal at (O) for almost all Ha.

References

  1. [1]

    I. S. Cohen -A. Seidenberg,Prime ideals and integral dependence, Bull. Amer. Math. Soc.,52 (1946), pp. 252–261.

  2. [2]

    W. E. Kuan,On the hyperplane section through a rational point of an algebraic variety, Pacific Jour. of Math.,36, no. 2 (1971), pp. 393–405.

  3. [3]

    M. Nagata,Local Rings, Interscience, New York, 1962.

  4. [4]

    E. Noether,Eliminationstheorie und allgemeines Idealtheorie, Math. Ann.,90 (1923), pp. 229–261.

  5. [5]

    J. Ohm,Algebraic cone with linear vertex, Ann. di Mat. pura e appl., IV,49 (1962), pp. 357–376.

  6. [6]

    A. Seidenberg,The hyperplane section of normal varieties, Trans. Amer. Math. Soc.,69 (1960), pp. 357–386.

  7. [7]

    R. J. Walker,Algebraic Curves, Dover publications, Inc., New York, 1962.

  8. [8]

    A. H. Wallace,Tangency and duality over arbitrary fields, Proc. London Math. Soc.,6 (1956), pp. 321–342.

  9. [9]

    O. Zariski,The theorem of Bertini on the variable singular points of a linear system of varieties, Trans. Amer. Math. Soc.,56 (1944), pp. 130–140.

  10. [10]

    O. Zariski,The concept of a simple point of an abstract algebraic variety, Trans. Amer. Math. Soc.,62 (1947), pp. 1–52.

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Entrata in Redazione il 1 ottobre 1971.

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Kuan, W. Specialization of a generic hyperplane section through a rational point of an algebraic variety. Annali di Matematica 94, 75–82 (1972). https://doi.org/10.1007/BF02413603

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Keywords

  • Prime Ideal
  • Rational Point
  • Algebraic Variety
  • Polynomial Ring
  • Section Versus