First let me give a precise formulation of the question; I'll give some background/motivation at the end.


If X is a projective variety which is Q-factorial (meaning X is normal, and some sufficiently high multiple of every Weil divisor is Cartier), then a *small Q-factorial modification* or *SQM* of X means a birational map φ: X --> Y (where Y is another Q-factorial projective variety) which is an isomorphism in codimension 1. (Examples: flips and flops.) The question is then this:



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> Is there an example of a Q-factorial projective variety X and φ: X --> Y an SQM of X such that the nef cone Nef(X) is rational polyhedral but Nef(Y) is not rational polyhedral? Or (highly unlikely I think) can one prove that this situation is impossible?
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And to push my luck:



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> Give sufficient conditions to ensure that in the above situation, Nef(X) rational polyhedral implies Nef(Y) rational polyhedral.
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Really I think only the first question has a hope of being answered positively, but it would be nice to know if I was wrong. 


**Background:** The question was prompted by a theorem of Hu and Keel ("Mori dream spaces and GIT", Michigan Math. J., 2000) which gives a characterisation of so-called *Mori dream spaces* --- certain varieties which behave very well with respect to the operations of the minimal model program. In particular, if X is a Mori dream space, then Nef(X) is rational polyhedral and so too is Nef(Y) for any SQM Y of X. It then seems natural to ask for an example where the first condition here holds but the second fails.


Hu--Keel's theorem gives one answer to my second question above, since Birkar--Cascini--Hacon--McKernan proved that any Fano variety (or more generally, any variety of Fano type) is a Mori dream space. But it would be great to know of any sufficient condition that applies outside the Fano domain. 


**Edit (July 23):** Balazs' answer settles the first question in the affirmative: flops can change the nef cone from finite to infinite. But it seems to remain open (and interesting to me) whether there are examples $X$ of this phenomenon with Kodaira dimension $kd(X)=-\infty$.



How much can small modifications change the nef cone? 
In the world of Calabi-Yau (as opposed to Fano) varieties, one does not expect miracles of Mori dream space type. A specific example which gives a positive answer to your first question is described in the paper <http://xxx.lanl.gov/abs/math/0102055> of Michael Fryers: a Calabi-Yau threefold (a degenerate quintic) which has some small resolutions having finite polyhedral nef cone, and some having an infinitely generated cone, which is locally rational polyhedral away from a single point on the boundary. The geometry, related to the Horrocks-Mumford bundle and abelian surfaces in projective four-space, is very beautiful.


