Let $X$ a variety. Say $X=\operatorname{Spec} A$.

Consider two ideals of $A$, say $I$ and $J$, with equal radical ; and consider the blow-ups of X with centre $I$ and $J$, say $Y_I$ and $Y_J$. **How can I decide if the blow-up map $Y_I \to X$ factors through $Y_J$ ?**

For example, if $X = \operatorname{Spec}k[x,y]$ is the affine plane. Then we can consider the blow-ups with centre $I_1 = (x,y)$, $I_2 = (x,y^2)$, $I_3=(x^2,y^2)$ and $I_4=(x^2,xy,y^2)=I_1^2$. We have $I_4\subset I_3\subset I_2\subset I_1$. Let $Y_i$ denote the corresponding blow-ups.

It is well-known that $Y_1$ and $Y_4$ are isomorphic. However $Y_2$ and $Y_3$ are two other different varieties. Using the universal property of the blow-up we see that $Y_1\to X$ factors through $Y_3$ but not through $Y_2$.

**What kind of property should I look at to predict this kind of factorization, without computing the actual blow-up ?**