The Hyperplane Sections of Normal Varieties

A New Criterion for Affineness

“…Let be an irreducible curve on , then we may choose such that is irreducible [10] and ∩ contains more than two points [10]. Let 1 and 2 be two distinct points on ∩ , then there is a regular function on such that ( 1 ) ̸ = ( 2 ) since is an affine curve.…”

“…If has an irreducible complete curve , choose a hypersurface such that = ∩ is irreducible [10] and intersects at more than 2 distinct points. Let 1 , 2 ∈ ∩ , 1 ̸ = 2 .…”

“…Then, − is a finite set and a general hypersurface does not contain any point in − . Let be a hypersurface away from − such that intersects at more than two distinct points and = ∩ is irreducible [10]. Then, is an affine variety and ∩ = ∩ .…”

Constructive proof of Hilbert’s theorem on ascending chains

“…This was written from a simple classical point of view (we may also remark that in [3] the base field k had been extended by an indeterminate r to a base field kir), but this played no role in the lemma). From a constructivist point of view, we can say that the left-hand side can be constructed if Sn ' n ft X P"C,-…”

Prime Elements and Prime Sequences in Polynomial Rings

“…Armed with this elementary fact, one can produce an inductive proof of (2) by using the Bertini theorem trick-attributed by Swan to Murthy-contained in the proof of [Sn,1.5]. One should, however, use Seidenberg's Bertini theorem [Sg,Theorem 12] in place of [Sn,1.1]. (2) is recorded here for expository purposes only; the details of the proof, which shed no light on the more elementary proof of (1), are therefore omitted.…”