Ampleness after repeated blow-ups

Question

Let $c_1, c_2$ be smooth 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 $a>m b$, the divisor $H':=n \pi^* H - a E'_1 - b E_2$ ample on $X_2$ for sufficiently large $n$?

Answer 1

In other words, you are asking whether $\{-aE'_1 - bE_2 \mid a > b > 0\}$ is contained in the relative ample cone for $X_2 \to X$. This is true. The reason is that the relative cone of effective curves is generated by the fiber $l_2$ of the second exceptional divisor and by the strict transform $l'_1$ of the fiber of the first exceptional divisor intersecting $c'_2$.

Clearly, $E_2\cdot l_2 = -1$ and $E'_1 \cdot l_2 = 0$ --- this gives condition $b > 0$ for relative ampleness. Similarly, $E'_1 \cdot l'_1 = -1$, and $E_2 \cdot l'_1 = 1$ (since smoothness of $c_2$ implies that the nontrivial intersections of $c_2$ with the fibers of $E_1$ are hyperplanes). This gives condition $a > b$. So, the relative ample cone is equal to the cone $\{a > b > 0\}$.