  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.
