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On the Hyperplane Sections Through two Given Points 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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1. Let k be an infinite field and let V/k be an irreducible variety of dimension ≧ 2 in a projective n-space Pn over k. Let P and Q be two k-rational points on V In this paper, we describe ideal-theoretically the generic hyperplane section of V through P and Q (Theorem 1) and prove that the section is almost always an absolutely irreducible variety over k1/pe if V/k is absolutely irreducible (Theorem 3). As an application (Theorem 4), we give a new simple proof of an important special case of the existence of a curve connecting two rational points of an absolutely irreducible variety [4], namely any two k-rational points on V/k can be connected by an irreducible curve.

I wish to thank Professor A. Seidenberg for his continued advice and encouragement on my thesis research.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 128 - 133
Copyright
Copyright © Canadian Mathematical Society 1970

References

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