Open Access
May 2005 On the Kleiman-Mori cone
Osamu Fujino
Proc. Japan Acad. Ser. A Math. Sci. 81(5): 80-84 (May 2005). DOI: 10.3792/pjaa.81.80


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.


Download Citation

Osamu Fujino. "On the Kleiman-Mori cone." Proc. Japan Acad. Ser. A Math. Sci. 81 (5) 80 - 84, May 2005.


Published: May 2005
First available in Project Euclid: 3 June 2005

zbMATH: 1093.14025
MathSciNet: MR2143547
Digital Object Identifier: 10.3792/pjaa.81.80

Primary: 14E30 , 14M25

Keywords: Ampleness , Kleiman-Mori cone , projectivity , toric geometry

Rights: Copyright © 2005 The Japan Academy

Vol.81 • No. 5 • May 2005
Back to Top