Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta\_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair $(X,\Delta\_X)$ is klt. Let us take the image $\Delta\_Y=f(\Delta\_X)\subset Y$. Is the pair $(Y,\Delta\_Y)$ still klt or not?


I am actually interested in the following case. Let $f:\mathbb{P}^n\dashrightarrow\mathbb{P}^n$ be the standard Cremona, and let $p\_1,...,p\_{n+3}\in \mathbb{P}^n$ be general points. We take $p\_1,...,p\_{n+1}$ as the center of the Cremona. Let $X$ be the blow-up of $\mathbb{P}^n$ at $p\_1,...,p\_{n+3}$. Then $f$ induces a small birational map $\overline{f}:X\dashrightarrow X$. If $(X,\Delta)$ is a klt pair then is $(X,\overline{f}(\Delta))$ a klt pair as well ? 



Small birational maps and singularities of the pair
Here is a very simple example and you can generalize this to get higher dimensional and more complicated ones: 




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Let $X=\mathbb P^2$ and $\Delta=\alpha\cdot C$ where $C$ is a smooth proper (and hence irreducible) curve of degree $>1$ and $0<\alpha<1$ a rational number. Clearly $(X,\Delta)$ is klt. Now let $p\_1,p\_2\in X$ be such that the line connecting $p\_1$ and $p\_2$ is tangent to $C$. Consider a Cremona transformation where $p\_1$ and $p\_2$ are blown-up. 


The resulting $(Y,\Delta\_Y)$ is such that $Y\simeq \mathbb P^2$ and $\Delta\_Y=\alpha\cdot C\_Y$ where $C\_Y\subset \mathbb P^2$ is a curve with a singularity (in fact one with a non-ordinary singularity). This gives a counter-example right away if $\alpha>1/2$. Q.E.D.




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**Note**, that you can easily deal with an arbitrary $\alpha$: If the first pair you get is still klt because $\alpha$ was sufficiently small, then do the same thing again: choose a tangent line through the singular point, two points (not on $\Delta$) on that tangent and do a Cremona with those points. In each step the singularity gets worse and no matter what $\alpha$ is, in finitely many steps you get to a pair that is not klt. (In case you wonder, the goal is to get the multiplicity of the singular point, $m$, to satisfy that $\alpha > 1/m$).


