Return to Question

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 $X$ gives 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 $3$.

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.

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.

Let $X$ be a smooth complex 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 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 $X$ gives 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 $3$.

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.

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 $X$ gives 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 $3$.

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.