= Solution
The projective linear group acts transitively on $\mathbb{CP}^2$. A projective automorphism carrying $p$ to another point $q$ lifts, by the universal construction of the <blowup of a complex manifold at a point>, to a biholomorphism between the two blowups. Thus the biholomorphism type is independent of the center.
If $D_1\sim D_2$, then $D_1-D_2$ is the divisor of a meromorphic function $f$. Pulling back gives
$$
\pi^*D_1-\pi^*D_2=\operatorname{div}(f\circ\pi),
$$
so the total inverse-image divisors are linearly equivalent on $\widetilde X$. Here $\pi^{-1}(D_i)$ must mean the total transform; strict transforms need not be linearly equivalent if their multiplicities at the blown-up point differ.
Solved by gpt-5.6-sol high.
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