Decompose a big divisor as nef big divisor and effective divisor Let $W_n$ be a set of a log pair having the following property:
For any $(X, D) \in W_n$
(1)$X$ has dimensional $n$ with tirvial canonical divisor (i.e.$K_X = 0$). Moreover, $X$ is a $\mathbb{Q}$-factorial variety with canonical singularities. (one can also assume $(X, \frac{1}{2} D)$ is klt).
(2) $D$ is an integral Weil divisor which is big.
Is it true that there exists a universal $\delta > 0$ (i.e. only depends on $n$), such that we can decompose $D$ in $\mathbb{Q}$-linearly equivalent as follows:
$$D \sim_\mathbb{Q} A + E,$$ where $A$ is a nef and big divisor with coefficients bigger than $ \delta$, and $E$ is an effective divisor?
Any suggestions related to the topic are welcome!!
 A: Let's see.. Over C, if X is Q-factorial so mD is cartier for some m. 
mD is also big cartier, so mD ~ A+E, ample + effective. 
Then m'mD ~ m'A+m'E ~Q D. 
Does that work?
edit: nvmind, was thinking of the wrong definition of Q-equivalence :P been awake too long, maybe you can use Fujita Approximation to say something related, possibly combined with some result on boundedness of volumes of your set of (X, D)
A: I think the answer is (almost, i.e. up to birational modification) yes but non-trivial see Theorem 1.3 of arXiv:1208.4150 "ACC for log canonical thresholds" by Christopher Hacon, James McKernan, Chenyang Xu.
If you assume that $D$ is integral and $(X,D/2)$ is klt, then since $K_X+D/2=D/2$ is big, the above theorem says that there exists a uniform positive integer (depending only on the dimension of $X$) such that $|m(K_X+D/2)|$ defines a birational map $\phi :X\to \mathbb P ^N=|m(K_X+D/2)|=|(m/2)D|$. If $\phi$ is a morphism (which can be arranged by replacing $X$ by a higher model), then $(m/2)D=\phi ^* \mathcal O _{\mathbb P ^N}(1)+E$ where $E$ is an effective divisor and of course $\phi ^* \mathcal O _{\mathbb P ^N}(1)$ is nef and big.
I am not sure (and skeptical about) how to do this without replacing $X$ by a higher model. 
