Affine Algebraic Varieties
نویسنده
چکیده
In this paper, we give new criteria for affineness of a variety defined over C. Our main result is that an irreducible algebraic variety Y (may be singular) of dimension d (d ≥ 1) defined over C is an affine variety if and only if Y contains no complete curves, Hi(Y,OY ) = 0 for all i > 0 and the boundary X − Y is support of a big divisor, where X is a projective variety containing Y . We construct three examples to show that a variety is not affine if it only satisfies two conditions among these three conditions. We also give examples to demonstrate the difference between the behavior of the boundary divisor D and the affineness of Y . If Y is an affine variety, then the ring Γ(Y,OY ) is noetherian. However, to prove that Y is an affine variety, we do not start from this ring. We explain why we do not need to check the noetherian property of the ring Γ(Y,OY ) directly but use the techniques of sheaf and cohomology. 2000 Mathematics Subject Classification: 14J10, 14J30, 32E10.
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