Let $X$ be a smooth projective variety over $\mathbb C$ and $A\to X$ an ample line bundle.

Is there an integer $k_0$ such that for all line bundle $L\to X$, the tensor product $A^{\otimes k}\otimes L$ is ample for every $k\ge k_0$?

Of course the answer is yes if we let $k_0$ depend on $L$, but the point here is that $k_0$ must be an universal constant depending only on $X$ and $A$.

Thanks in advance.

EDIT: In fact the answer is NO in general: take for example $X=\mathbb P^n$ and $A=\mathcal O(1)$. Then for each $k$ there exists a line bundle $L$ on $\mathbb P^n$, namely $L=\mathcal O(-k-1)$ such that $A^{\otimes k}\otimes L$ is not ample. So,
**is there any reasonable condition on $X$ and $A$, or on the family in which we let the bundles $L$ vary, such that the statement holds true?**

My question was motivated by the following on positivity:

Let $X$ be a compact complex manifold endowed with a positive (in the sense that it carries a smooth hermitian metric whose Chern curvature tensor is positive definite) line bundle $A\to X$. For every pair of points $p,q\in X$ consider the blow-up $\sigma\colon\widetilde X\to X$ at this two point and let $E_p$, $E_q$ be the two exceptional divisors (if $p=q$ then consider $2E_p$). Is there a uniform $k_0>0$ (independent of $p$ and $q$) such that $$ \sigma^* A^{\otimes k}\otimes\mathcal O_{\widetilde X}(-E_p-E_q) $$ is positive whenever $k\ge k_0$?

**EDIT bis.** Here is a beginning of a possible proof. Le us take the case $p=q$ and suppose the contrary. Then, there is a sequence of points $(p_k)\subset X$ such that
$$
\sigma^* A^{\otimes k}\otimes\mathcal O_{\widetilde X}(-2E_{p_k})
$$
is not positive definite. Since $X$ is compact, after extracting a subsequence we may suppose that $p_k\to p\in X$. Let $k_0$ be a integer such that
$$
\sigma^* A^{\otimes k}\otimes\mathcal O_{\widetilde X}(-2E_{p})
$$
is positive definite for all $k>k_0$. This integer certainly exists, since $\mathcal O_{E_p}(-E_p)$ is positive.

If we are able to construct for all points $q$ nearby $p$ a smooth hermitian metric such that $$ \sigma^* A^{\otimes k}\otimes\mathcal O_{\widetilde X}(-2E_{q}) $$ is positive definite whenever $k\ge k_0$ we then get a contradiction and we are done.

For the moment I am not still able to do this last part. This should be achieved by a somehow clever use of a partition of unity on $\widetilde X$.