Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the Kleiman-Mori cone

Osamu Fujino

Full-text: Open access


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.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 5 (2005), 80-84.

First available in Project Euclid: 3 June 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 14E30: Minimal model program (Mori theory, extremal rays)

Ampleness projectivity toric geometry Kleiman-Mori cone


Fujino, Osamu. On the Kleiman-Mori cone. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 5, 80--84. doi:10.3792/pjaa.81.80.

Export citation


  • M. Eikelberg, The Picard group of a compact toric variety, Results Math. 22 (1992), nos. 1-2, 509–527.
  • A. Fujiki and S. Nakano, Supplement to “On the inverse of monoidal transformation”, Publ. Res. Inst. Math. Sci. 7 (1971/72), 637–644.
  • O. Fujino, On the Kleiman-Mori cone, preprint 2005, math.AG/0501055.
  • O. Fujino and H. Sato, Introduction to the toric Mori theory, Michigan Math. J. 52 (2004), no. 3, 649–665.
  • O. Fujino and H. Sato, An example of toric flops. available at my homepage. (http://www.
  • S. L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344.
  • J. Kollár, Rational curves on algebraic varieties, Springer, Berlin, 1996.
  • S. Nakano, On the inverse of monoidal transformation, Publ. Res. Inst. Math. Sci. 6 (1970/71), 483–502.
  • S. Payne, A smooth, complete threefold with no nontrivial nef line bundles, preprint 2005, math. AG/0501204.