Let $R=k[x\_1,..,x\_n]/I$ and let $X=Spec(R)$ be it's associated affine scheme. Suppose that $X$ has only one isolated singularity, say at the origin $\mathfrak{m}=\langle x\_1,...,x\_n\rangle$. Now, let $R\_{\mathfrak{m}}$ be the localization at $\mathfrak{m}$, which is a local ring with maximal ideal $\mathfrak{m}\_{\mathfrak{m}}$ that is not regular based on the assumption that $X$ is singular at $\mathfrak{m}$. Take $Bl\_{\mathfrak{m}\_{\mathfrak{m}}}(Spec(R\_{\mathfrak{m}}))=Proj(R\_{\mathfrak{m}}[\mathfrak{m}\_\mathfrak{m}t])$ to be the blowup of $Spec(R\_{\mathfrak{m}})$ at the point $\mathfrak{m}\_{\mathfrak{m}}$. 


My question is: If the local blowup $Bl\_{\mathfrak{m}\_{\mathfrak{m}}}(Spec(R\_{\mathfrak{m}\_\mathfrak{m}}))$ is nonsingular, does that imply that the blowup $Bl\_{\mathfrak{m}}(X)$ is also nonsingular? If not, is there an obvious counterexample? If so, can this be generalized to schemes with multiple isolated singularities or possibly to non-isolated singular loci?



Local blowup versus global blowup
Sure. This is true. 


Indeed, the formation of the blowup is easily seen to commute with localization (your Rees algebra $R\_{\mathfrak{m}}[\mathfrak{m}\_{\mathfrak m} t]$s formation certainly commutes with localization of $R$. This generalizes to any sort of situation you'd like.


In particular, if $\mathfrak{m}$ is the only singular point, then the singularities of the blowup $Bl\_{\mathfrak{m}}(X)$ certainly lie over $\mathfrak{m}$. Since all those points appear in the local blowup, the result you want holds. Indeed, the local blowup is obtained (say on charts) as a localization of the global blowup (and those charts are then reglued in the obvious way).


In terms of the greater generalization, if $Bl\_{\mathfrak{m\_m}}(R\_{\mathfrak{m}\_{\mathfrak{m}}})$ is regular, then the global blowup $f : Bl\_{\mathfrak{m}}(X) \to X$ is regular in a neighborhood of $f^{-1}(\mathfrak{m})$. I don't think you can say more than that though.


