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I am working on a familly of toric varieties which seem to have the following property:

  • the closure of the Kähler cone is a simplicial cone (and even a smooth cone with respect to the natural lattice).

This has the interesting aspect that it provides a natural basis of the $H^2$ cohomology group.

I was wondering whether this is or not a trivial property.

Is the closed Kähler cone of any smooth toric variety a simplicial cone ?

I would guess that the answer is no, but I do not know a counter-example.

It is known that this cone is polyhedral for all toric varieties. But maybe one can find examples where it has more generators ? Do you know one ?

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  • $\begingroup$ Perhaps you are thinking of the pseudoeffective cone. $\endgroup$ – Jason Starr May 9 '14 at 13:20
  • $\begingroup$ Could you tell me the definition of a simplicial cone? Thanks $\endgroup$ – Sándor Kovács May 10 '14 at 17:46
  • $\begingroup$ A simplicial cone is a cone of dimension d with d rays. for some integer d. $\endgroup$ – F. C. May 10 '14 at 20:31
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The Kähler cone of a del Pezzo surface of degree 6 is not simplicial: see section 6 of these notes.

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