Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the Kleiman-Mori cone

Osamu Fujino

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 3 June 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1117805146

Digital Object Identifier
doi:10.3792/pjaa.81.80

Mathematical Reviews number (MathSciNet)
MR2143547

Zentralblatt MATH identifier
1093.14025

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

Keywords
Ampleness projectivity toric geometry Kleiman-Mori cone

Citation

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. https://projecteuclid.org/euclid.pja/1117805146


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