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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

1. Krull, W., Parameter spezializierung in Polynomringen, Arch. Math. 1 (1948), 56–64.Google Scholar
2. Krull, W., Parameter spezializierung in Polynomringen. II: Das Grund polynom, Arch. Math. 1 (1948), 129–137.Google Scholar
3. Lang, S., Introduction to algebraic geometry (Interscience, New York, 1964).Google Scholar
4. Nishimura, H. and Nakai, Y., On the existence of a curve connecting given points on an abstract variety, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 28 (1954), 267–270.Google Scholar
5. Seidenberg, A., The hyperplane section of normal varieties, Trans. Amer. Math. Soc. 69 (1950), 357–386.Google Scholar
6. Weil, A., Foundations of algebraic geometry, Amer. Math. Soc. Colloq. Publ., Vol. 29 (Amer. Math. Soc, Providence, R.I., 1962).Google Scholar