Specialization of a generic hyperplane section through a rational point of an algebraic variety

Abstract

Let V/k be an irreducible algebraic variety of dimension ≥3, defined over a field k of characteristic O, passing through the origin (O) of the ambient affine n-space. Let p be the prime ideal of V in the polynomial ring k[X1, ..., Xn]. If V/k is k-normal at (O), then for almost all hyperplanes Ha: a1X1+...+anXn=0 with a1, ..., an e k the ideal (p, a1X1+...+anXn) is a prime ideal. If the generic hyperplane section of V through (O) is normal over the ground field of the generic section of V, then the section V ∩ Ha is k-normal at (O) for almost all Ha.