# Anti-canonical divisor of a Fano variety

Let $X$ be a normal projective Fano variety, that is the anti-canonical divisor $-K_X$ is ample.

For any $m>0$ let us consider the complete linear system $|-mK_X|$ and the map $$f_{|-mK_X|}:X\dashrightarrow X_m\subseteq\mathbb{P}(|-mK_X|) = \mathbb{P}^{N_m}$$ where $X_m$ is the closure of $f_{|-mK_X|}(X)$ in $\mathbb{P}^{N_m}$.

Do there exist estimates for the smallest rational number $m>0$ such that $f_{|-mK_X|}:X\dashrightarrow X_m$ is birational ?

Of course I should mention that the existence of such effective birationality for $\epsilon$-lc Fano is proved by Birkar last year, but his method does not give a way to estimate the number $m$.
By the way, if you are interested in Fano varieties with klt singularities which is K-semistable (e.g. admitting Kahler--Einstein metrics), then in this case we have effective estimation of $m$ by my recent work, see proof of Theorem 1.5 of arXiv:1705.02740.