Let $X$ be a smooth projective variety over an algebraically closed field $k$. For any length $n$ ideal sheaf $P$ of $X$ (e.g $N$ different points $P=(P_1)(P_2)..(P_N)$), we can consider $Bl_{P}(X)$, the blow up of $X$ at $P$. Assume $Bl_{P}(X) \cong Bl_{Q}(X)$ for a length $n$ ideal sheaf $P$ and a length $m$ ideal sheaf $Q$ , what can we say about $P$ and $Q$ ? If neccessary, one can assume $P,Q$ are radical.

Note blow up at $I$ and $I^2$ are the same. So there is a trivial case i.e when there exists an automorphism $\phi:X \rightarrow X$ s.t $\phi^*(P^k)=Q^l$.

I am interested in some examples e.g projective spaces, abelian varieties, surfaces.