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?


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.

  • $\begingroup$ This is sort of what I was thinking, but I wasn't totally convinced by my train of thought. To clarify, when we say that the blowup commutes with localization, we mean something like $Bl_{\mathfrak{m}_{\mathfrak{m}}}(R_{\mathfrak{m}})=Bl_{\mathfrak{m}}(X)_{f^{-1}(\mathfrak{m})}$ which is the localization of the blowup at the exceptional fiber? Or is it the localization at the fiber over $\mathfrak{m}$ in the strict transform? $\endgroup$
    – DavidWayne
    Jul 11 '13 at 21:50
  • $\begingroup$ Probably the best way to do it is as follows. Form the Cartesian product: $$Bl_m(X) \times_X \text{Spec} R_{\mathfrak m}.$$ It's not exactly localization in the usual sense. $\endgroup$ Jul 11 '13 at 22:11
  • $\begingroup$ I agree that this description of the localization makes the most sense geometrically, but I was having a hard time with the algebra in my argument. After reading more, I used what you said about localization commuting with the Rees algebra formation. Let $S=R-\mathfrak{m}$. Then $S$ is included into $R[\mathfrak{m}t]$ as $St^0$, and $R_{\mathfrak{m}}[\mathfrak{m}_{\mathfrak{m}}t] =R_S[\mathfrak{m}_St]=R[\mathfrak{m}t]_{St^0}$. Using this, it is more obvious to me exactly how blowups commute with localization, and I have an idea of how the details of the original question should work out. $\endgroup$
    – DavidWayne
    Jul 12 '13 at 3:59
  • $\begingroup$ Great! I guess the other way to do this is to deal with some sort of multicative system sheaf, form that product, and then take sheafy Spec. $\endgroup$ Jul 12 '13 at 6:43

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