Vol. 36, No. 2, 1971

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On the hyperplane section through a rational point of an algebraic variety

Wei-Eihn Kuan

Vol. 36 (1971), No. 2, 393–405

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[X1,,Xn]. Let P be a rational normal point on V∕k. It is proved that (1) for a generic hyperplane Hu through P,(p,Hu) is a prime ideal and (p,Hu) is quasi-absolutely (absolutely irreducible) if p is quasi-absolutely (absolutely irreducible). (2) It is nol true in general that V Hu is normal at P; however, V Hu is normal at P if the local ring of V∕k at P is also Cohen-Macaulay (Theorem 8).

Mathematical Subject Classification 2000
Primary: 14A10
Received: 16 December 1969
Published: 1 February 1971
Wei-Eihn Kuan