A Note on a Generic Hyperplane Section of an Algebraic Variety

Abstract

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).