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On the Kleiman-Mori cone. (English) Zbl 1093.14025

Two examples of non-projective varieties with interesting properties are given here.
In the first, \(X\) is a complete non-projective singular toric variety for which Kleiman’s ampleness criterion fails. More precisely, there is a line bundle \(L\) on \(X\) which is positive on \(\overline{\text{NE}}(X)\setminus\{0\}\). Such examples are quite easy to find: the author has found many, and directs the reader to his web page for more. He also supplies an ampleness criterion that holds for every complete toric variety with arbitrary, not just \({\mathbb Q}\)-factorial, singularities: if \(L\) is a line bundle on a complete toric variety and \(L\cdot C>0\) for every torus-invariant integral curve \(C\), then \(L\) is ample.
The second example is of a complete singular toric variety \(X\) for which \(\text{NE}(X)=N_1(X)\cong{\mathbb R}^k\), for some (arbitrary) \(k\): thus, in particular, a line bundle is nef on \(X\) if and only if it is numerically equivalent to zero.

MSC:

14E30 Minimal model program (Mori theory, extremal rays)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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References:

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[9] S. Payne, A smooth, complete threefold with no nontrivial nef line bundles, preprint 2005, math. AG/0501204. · Zbl 1141.14313 · doi:10.3792/pjaa.81.174
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