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).