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I recently heard about the following problem:

Let $X$ be a projective variety with klt singularities and such that $-K_X$ is big. Is $X$ a Mori Dream Space ?

Now, $-K_X$ big if and only if $-K_X -\epsilon D = -(K_X+\epsilon D)$ is ample for some effictive divisor $D$ and some positive rational number $\epsilon >0$. By BCHM this should imply that $X$ is a Mori Dream Space. Therefore I am certainly missing something here. Is the issue about the singularities of the pair $(X,\epsilon D)$ ?

On the other hand if $X$ is a smooth rational surface $-K_X$ big implies that $X$ is a Mori Dream Space. Why the issue does not arise for surfaces ?

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up vote 4 down vote accepted

Yes, the issue concerns singularities of the pair $(X,\epsilon D)$.

Assume that $-K_X$ is big then $-K_X -\epsilon D = -(K_X+\epsilon D)$ is ample for some effictive divisor $D$ and some positive rational number $\epsilon >0$. If $(X,\epsilon D)$ is a klt pair (that is $X$ is a log Fano variety) then, by BCHM $X$ is a Mori Dream Space.

If $-K_X$ is also nef (that is $X$ is weak Fano) then one can choose $\epsilon$ arbitrairly small and $(X,\epsilon D)$ is klt. Then $X$ is log Fano and therefore a Mori Dream space.

So far we have

$X$ weak Fano $\Rightarrow$ $X$ log Fano $\Rightarrow$ $X$ Mori Dream Space.

Now, if $-K_X$ is big but not nef, one may find $\epsilon$ big enough to have $-(K_X+\epsilon D)$ ample but at the same time $\epsilon$ is not small enough in order to ensure that $(X,\epsilon D)$ is klt. This is true even for surfaces, as Artie wrote you may take a look to Section 3 of Therefore

$X$ log Fano $\Rightarrow$ $-K_X$ big. However, $-K_X$ big does not imply $X$ log Fano.

The fact is that in order to prove that $X$ is a Mori Dream Space one does not need to pass first from the fact that $X$ is log Fano. Indeed if $X$ is a Mori Dream Space then it is not necessarily log Fano. For instance the surfaces in Section 3 of are Mori Dream Spaces but not log Fano. Indeed, in, they prove that a rational surface $X$ such that $-K_X$ is big is a Mori Dream Space by showing that $Cox(X)$ is finitely generated.

In general, it is not true that $-K_X$ big implies that $X$ is a Mori Dream Space. If $X$ is a normal projective variety and $D$ is a $\mathbb{Q}$-divisor, we say that $(X, D)$ is a weak log Fano pair if $−(K_X + D)$ is nef and big. In particular, $-K_X-D = B$ is big and $-K_X = B+D$ is big as well being the sum of a big divisor and an effective divisor. Example $5.6$ of describes a smooth $3$-fold with $-K_X$ big, but $X$ is not a MDS. Namely $X$ is the blow-up of the vertex of the cone over $S$, where $S$ is $\mathbb{P}^2$ blown-up at $9$ general points.

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Your argument doesn't work, as shown by the examples, due to Chenyang Xu, in Section 3 of this paper by Testa, Várilly-Alvarado, and Velasco. (To see why the conclusion is still true for surfaces, read the paper!)

The issue as you say is with the singularieties of the pair: there is such a $D$ and such an $\epsilon$, but $(X,\epsilon D)$ need not be klt (or even lc), which you need in order to apply BCHM. Of course if $D$ is any fixed effective divisor, then $(X,\epsilon D)$ is klt for sufficiently small $\epsilon$, but if $-K_X$ isn't nef, then $-K_X-\epsilon D$ won't be ample for very small $\epsilon$.

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