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- W2352208974 abstract "The Kleiman-Mori cone plays important roles in the birational geometry. In this paper, we construct complete varieties whose Kleiman-Mori cones have interesting properties. First, we construct a simple and explicit example of complete non-projective singular varieties for which Kleiman's ampleness criterion does not hold. More precisely, we construct a complete non-projective toric variety X and a line bundle L on X such that L is positive on NE(X) n f0g. Next, we construct complete singular varieties X with NE(X) = N1(X) ' R k for any k. These explicit examples seem to be missing in the literature. The Kleiman-Mori cone plays important roles in the birational ge- ometry. In this paper, we construct complete varieties whose Kleiman- Mori cones have interesting properties. First, we construct a simple and explicit example of complete non-projective singular varieties for which Kleiman's ampleness criterion does not hold. More precisely, we construct a complete non-projective toric variety X and a line bundle L on X such that L is positive on NE(X) n f0g. Denition 1.1. Let V be a complete algebraic scheme dened over an algebraically closed eld k. We say that Kleiman's ampleness criterion holds for V if and only if the interior of the nef cone of V coincides with the ample cone of V. Note that Kleiman's original statements are very sharp. We recom- mend the readers to see (Kl, Chapter IV x2 Theorems 1, 2). Of course, our example is not quasi-divisorialin the sense of Kleiman (see (Kl, Chapter IV x2 Denition 4 and Theorem 2)). We do not repeat the denition of quasi-divisorial since we do not use it in this paper. Note that if X is projective or Q-factorial then X is quasi-divisorial in the sense of Kleiman. Next, we construct complete singular varieties X" @default.
- W2352208974 created "2016-06-24" @default.
- W2352208974 creator A5038762670 @default.
- W2352208974 date "2004-01-01" @default.
- W2352208974 modified "2023-09-27" @default.
- W2352208974 title "ON THE KLEIMAN-MORI CONE (PRIVATE NOTE)" @default.
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- W2352208974 hasPublicationYear "2004" @default.
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