On the Kleiman-Mori cone

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