Blowing-up an ideal generated by squares

Question

Let $f_1,\dots,f_r$ be regular functions on a smooth projective variety $X$, and consider the ideals $I = (f_1^2,\dots,f_r^2)$ and $J = (f_1,\dots,f_r)$. Let $Y = Z(I)$ and $W = Z(J)$ be the subschemes of $X$ defined respectively by $I$ and $J$. Assume that $W$ is smooth.

Now, let $X_Y = Bl_YX$ and $X_W = Bl_WX$ be the blow-ups of $X$ respectively along $Y$ and $W$.

What is the relation between $X_Y$ and $X_W$? For instance, does there exist a morphism from one to the other?

Answer 1

If $\alpha \in \mathbb{N}^{r}$ satisfies $|\alpha| = r+1$ then $\exists i, \alpha_i \geq 2$ so that $f^{\alpha} \in f_i^2 J^{r-1}$.

Thus $J^{r+1} = I J^{r-1}$. In particular $I$ becomes invertible on $X_W$, so that $X_W \rightarrow X$ uniquely factors through $X_{Y}$.