Geodesic
A Note on a Generic Hyperplane Section of an Algebraic Variety
Canadian journal of mathematics, Tome 22 (1970) no. 5, pp. 1047-1054

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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 u 0 + u 1 X 1 + ... + unXn = 0, where u 0, u 1, ..., un are indeterminates over k. It is well known that:(1) if V is normal over k, then V ∩ H is normal over k(u 0, ..., un ) (see [6]), and(2) if P is in the intersection V ∩ H, then P is absolutely simple on V ∩ H over k(u 0, ..., 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(u 0, ..., un ) (Theorem 3), and(2′) if P is in V ∩ H, then P is normal on V ∩ H over k(u 0, ..., un ) if and only if P is normal on V over k (Theorem 2). 
Kuan, Wei-Eihn. A Note on a Generic Hyperplane Section of an Algebraic Variety. Canadian journal of mathematics, Tome 22 (1970) no. 5, pp. 1047-1054. doi: 10.4153/CJM-1970-121-9
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