On the Kleiman-Mori cone

Details On the Kleiman-Mori cone On the Kleiman-Mori cone

2005-01-05

arxiv.org/abs/math/0501055v1

Mathematics

Algebraic Geometry

8 pages

Scientific paper

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 $\bar {NE}(X)\setminus \{0\}$. Next, we construct complete singular varieties $X$ with $NE(X)=N_1(X)\simeq \mathbb R^k$ for any $k$. These explicit examples seem to be missing in the literature.

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