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A Note on a Generic Hyperplane Section of an Algebraic Variety

Published online by Cambridge University Press:  20 November 2018

Wei-Eihn Kuan
Wei-Eihn Kuan*
Affiliation:
Michigan State University, East Lansing, Michigan
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Let V be an irreducible algebraic variety of dimension > 1 defined over a field k in an affine n-space over k, and let H be the generic hyperplane defined by u0 + u1X1 + … + unXn = 0, where u0, u1, …, un are indeterminates over k. It is well known that:

(1) if V is normal over k, then V ∩ H is normal over k(u0, …, un) (see [6]), and

(2) if P is in the intersection V ∩ H, then P is absolutely simple on V ∩ H over k(u0, …, un) if and only if P is absolutely simple on V over k (see [2; 5]).

In this paper we prove:

(1′) if V is factorial over k, then V ∩ H is also factorial over k(u0, …, un) (Theorem 3), and

(2′) if P is in V ∩ H, then P is normal on V ∩ H over k(u0, …, un) if and only if P is normal on V over k (Theorem 2).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 5 , 01 October 1970 , pp. 1047 - 1054
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Krull, W., Parameterspezialisierung in Polynomringen, Arch. Math. 1 (1948), 56–64.Google Scholar
2. Lang, S., Introduction to algebraic geometry (Interscience, New York, 1964).Google Scholar
3. Mumford, D., Introduction to modern algebraic geometry (preprint, Harvard University).Google Scholar
4. Nagata, M., Local rings (Interscience, New York, 1962).Google Scholar
5. Nakai, Y., Note on the intersection of an algebraic variety with the generic hyperplane, Mem. Coll. Sci. Univ. Kyoto, Ser. A Math. 26 (1951), 185–187.Google Scholar
6. Seidenberg, A., The hyperplane section of normal varieties, Trans. Amer. Math. Soc. 50 (1941), 357–386.Google Scholar
7. Zariski, O., The theorem of Bertini on the variable singular points of a linear system of varieties, Trans. Amer. Math. Soc. 56 (1944), 130–140.Google Scholar
8. Zariski, O., The concept of a simple point of an abstract algebraic variety, Trans. Amer. Math. Soc. 08 (1947), 1–52.Google Scholar
9. Zariski, O. and Samuel, P., Commutative algebra, Vol. I, The University series in Higher Mathematics (Van Nostrand, Princeton, NJ.—Toronto—London—New York, 1958).Google Scholar
10. Zariski, O. and Samuel, P., Commutative algebra, Vol. II, The University series in Higher Mathematics (Van Nostrand, Princeton, N.J.—Toronto—London—New York, 1960).Google Scholar