Let $c_1, c_2$ be curves on a plane in $\mathbb P^3$ that intersect at a point $p$ with multiplicity $m \ge 1$ and $H$ be an ample divisor on $\mathbb P^3$. Let $\pi_1:X_1 \rightarrow \mathbb P^3 $ be the blow-up along $c_1$, $E_1$ be the exceptional divisor, $c'_2$ be the proper transform of $c_2$ and $l$ be the fiber over $p$.

Again let $\pi_2: X_2 \rightarrow X_1$ be the blow-up along $c'_2$, $E_2$ be the exceptional divisor and $l'$ be the proper transform of $l$.

Then the divisor $D:=n \pi^* H - E'_1 - E_2$ is not ample on $X_2$ for any number $n$, where $\pi = \pi_2 \circ \pi_1$ and $E'_1 = \pi_2^* E_1$. That's because the intersection number $D \cdot l' =1-m$ is not positive. Instead for any fixed positive integers $a, b$ with $a> m b$, the divisor $H':=n \pi^* H - a E'_1 - b E_2$ is ample on $X_2$ for sufficiently large $n$.

So I would like to ask if this is generally true. Let $X$ be a smooth projective variety and $c_1, c_2$ be distinct smooth irreducible subvarieties of $X$ that are codimension two. Let $H$ be an ample divisor on $X$. Let $\pi_1 : X_1 \rightarrow X$ be the blow-up along $c_1$, $E_1$ be the exceptional divisor and $c'_2$ be the proper transform of $c_2$. Let $\pi_2: X_2 \rightarrow X_1$ be the blow-up along $c'_2$ and $E_2$ be the exceptional divisor.

Is there some positive number $m$ such that, for any fixed positive integers $a, b$ with $a>m b$, is the divisor $H':=n \pi^* H - a E'_1 - b E_2$ ample on $X_2$ for sufficiently large $n$?