Let $X$ be a smooth projective variety such that $-K\_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that:


* $-K\_Y$ is big,
* there exists an effective $\mathbb{Q}$-divisor $D\subset Y$ such that $-(K\_Y+D)$ is ample.



Big divisors and small transformations
Let $f:X\dashrightarrow Y$ be a small map. Let $D$ be a big divisor on $D$. Then for $m >>0$ the map $\phi\_{mD}:Y\dashrightarrow Z$, induced by $mD$ is birational. Now, if $\Delta:=f^{\*}D$, then the map induced by $m\Delta$ is $\phi\_{mD}\circ f$, which is birational as well. Therefore $\Delta$ is big. We conclude that $D$ is big if and only if $\Delta$ is big. In particular $-K\_Y$ is big if and only if $-K\_X$ is big.


In you case, $-K\_X$ ample $\Rightarrow$ $-K\_X$ big $\Rightarrow$ $-K\_Y$ big.


By Corollary 2.2.7. in Lazarsfeld, *"Positivity in Algebraic Geometry I"*, a divisor $D$ is big if and only if there exists an ample divisor $A$ and a positive integer $m$ such that $mD = A+N$, where $N$ is effective. Therefore $-(-D+\frac{1}{m}N) = \frac{1}{m}A$. In you case $D = -K\_Y$ and you get that $-(K\_Y+\frac{1}{m}N) = \frac{1}{m}A$ is ample.


