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I am studying the secondary fan decomposition of the effective cone of a projective variety $X$. Let as assume that $X$ is a Mori Dream Space. As far as I understand passing from a cone of maximal dimension to another cone of maximal dimension corresponds to a small transformation of $X$ while passing from a cone to a cone of smaller dimension corresponds to a divisorial contraction.

It seems to me that the curves that are contracted when we meet a face are the curves having non positive intersection with any divisor in that face. I would like two know is this last fact is true.

For instance take X to be the blow-up of $\mathbb{P}^3$ in two points $p_1,p_2$. Let $H_1$ and $H_2$ be the strict transforms of planes passing through $p_1$ and $p_2$ respectively. What happens when we meet the two dimensional face spanned by $H_1$ and $H_2$?

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  • $\begingroup$ Proj of a line bundle in the interior of that face is the contraction of the line joining $p_1$ and $p_2$. Proj of a line bundle just on the other side of that face is the flop of that line. $\endgroup$
    – user5117
    Commented Nov 29, 2014 at 21:44
  • $\begingroup$ Then the picard number of the variety we get in the interior of the face is three. Is this correct? $\endgroup$
    – user61586
    Commented Nov 29, 2014 at 22:10
  • $\begingroup$ No, $X$ itself has Picard number 3; when you contract you get something of smaller Picard number. (This is true even though the contraction is small.) The reason is that any divisor on $X$ that has nonzero intersection number with the contracted curve must push down to a non $\mathbf Q$-Cartier divisor on the image. $\endgroup$
    – user5117
    Commented Nov 29, 2014 at 22:26

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