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 ?**