Let $X$ be a noetherian scheme and let $Y$ be a closed subscheme of $X$. 
What relation is there between $\mathrm{Bl} \_ {Y}(X)$ and $\mathrm{Bl} \_{ Y \_{\mathrm{red}}}(X)$ ?


Thanks.



Blow-up along a subscheme and along its associated reduced closed subscheme
There is no map from one blow up to the other, and definitely not an isomorphism. Please see my comments to J.C. Ottems answer. 


*However*, if you replace radical by integral closure, then everything is fine. 


Here's what I mean, if $I$ is an ideal and $J$ is its integral closure, then you always have an everywhere defined map


$$Bl\_J X \to Bl\_I X.$$


This need not be an isomorphism, indeed the integral closure of $(x^2, y^2)$ is $(x^2, xy, y^2)$. The blow up of the latter ideal is the normalization of the blow up of the former. 


The other way you can get a map is if $J = \sqrt I$, and *also* if we can write $I = J \cdot \mathfrak{a}$ for some other ideal $\mathfrak{a}$. Then the blow up of $I$ is always the blow up of $\mathfrak{a}$ pulled back to $Bl\_J X$.


In general, you should expect no relation between the blow up of two ideals with the same radical unless there is some integral closure relation between them and/or one ideal is the product of the other (and something else).


