On the hyperplane section through a rational point of an algebraic variety
Wei-Eihn Kuan · Pacific Journal of Mathematics · 1971
Let V/k be an irreducible affine algebraic variety of dimension ^ 3 defined over an infinite field k with p as its prime ideal in k[X u -, X n ].Let P be a rational normal point on V/k.It is proved that (1) for a generic hyperplane H u through P, (p, Hu) is a prime ideal and (p, H u ) is quasi-absolutely (absolutely irreducible) if p is quasi-absolutely (absolutely irreducible).(2) It is not true in general that V Π H u is normal at P; however, V Π H u is normal at P if the local ring of V/k at P is also Cohen-Macaulay (Theorem 8).