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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

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