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?



Blowing-up an ideal generated by squares
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 $X\_{Y}$.


