Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 ?

share|improve this question
    
Perhaps you are thinking of the pseudoeffective cone. –  Jason Starr May 9 at 13:20
    
Could you tell me the definition of a simplicial cone? Thanks –  Sándor Kovács May 10 at 17:46
    
A simplicial cone is a cone of dimension d with d rays. for some integer d. –  F. C. May 10 at 20:31
    
Thanks!!!!!!!!!! –  Sándor Kovács May 10 at 23:33
add comment

1 Answer 1

up vote 8 down vote accepted

The Kähler cone of a del Pezzo surface of degree 6 is not simplicial: see section 6 of these notes.

share|improve this answer
    
Indeed. Thanks for the example. –  F. C. May 9 at 16:53
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.