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Let $X$ be a smooth complex projective algebraic variety and let $D$ be a $\mathbb{Q}$-Cartier pseudo-effective divisor on $X$. Lets say that $D$ is birationally nef if there exists a birational rational map $\pi \colon X \dashrightarrow Y$, with $Y$ a possibly singular projective algebraic variety such that $\pi_*(D)$ is a nef divisor.

$1$. The minimal model program conjectures that if $D=K_X+\Delta$ for a klt pair $(X,\Delta)$ then pseudo-effective implies birationally nef.

$2$. In dimension two it seems that pseudo-effective implies birationally nef as well by Zariski decomposition and Artin contractibility criterion.

$3$. I expect that a pseudo-effective divisor $D$ whose diminished base locus ${\rm Bs}_{-}(D)$ is dense on $X$ may give an example of a pseudo-effective divisor which is not birationally nef. Since it looks like for any map $\pi \colon X \dashrightarrow Y$ the diminished base locus ${\rm Bs}_{-}(\pi_{*}(D))$ will be non-trivial.

I am only aware of one example holding the condition on $3$, and in such case $D$ pushes-forward to an ample divisor on $\mathbb{P}^3$, so it does not give a counter-example.

Is there a natural family of examples of pseudo-effective divisors which are not birationally nef?

Remark: If what I am defining as birationally nef is already defined in the literature, please feel free to edit the question.

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    $\begingroup$ What do you mean by $\pi_*$? $\endgroup$
    – Chen Jiang
    Oct 28, 2017 at 23:40
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    $\begingroup$ divisorial push-forward $\endgroup$
    – Joaquín Moraga
    Oct 28, 2017 at 23:48
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    $\begingroup$ @ChenJiang: if $U$ is the locus where $\pi$ is defined, then $X\setminus U$ has codimension $\geq 2$. Thus, we get an isomorphism $\operatorname{Cl}(X) \stackrel\sim\to \operatorname{Cl}(U)$. Define $\pi_*$ as the pushforward along $U \to Y$, in the sense of codimension $1$ cycles (see e.g. Fulton's Intersection Theory, $\S1.4$). Since $X$ is smooth, you don't need to worry about Weil vs. Cartier divisors, but it seems that $\pi_* D$ is a priori only a $\mathbb Q$-Weil divisor on $Y$. $\endgroup$
    – R. van Dobben de Bruyn
    Oct 29, 2017 at 5:19
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    $\begingroup$ Are you sure about condition $3$? If your example is what I suspect, you can just push the divisor forward via the blow-down to $\mathbb P^3$ and the pushforward class is $\mathcal O(1)$. On the other hand, there are probably similar examples on CY3's for which $B_{-}(D)$ is not closed and there are no contractions possible. $\endgroup$
    – user47305
    Oct 30, 2017 at 14:59
  • $\begingroup$ @Mark Yes, you are right, that example does not give a counter-example, I will clarify this in the question itself. May you elaborate a little more about those similar examples with CY3's? $\endgroup$
    – Joaquín Moraga
    Oct 30, 2017 at 21:21

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