# Functorial properties of blow-up

Let $X, Y$ be projective algebraic surfaces with isolated singularities. Suppose they are diffeomorphic to each other. Denote by $\phi$ the diffeomorphism from $X$ to $Y$. Then does there exists a blow up $X', Y'$ of $X, Y$, respectively such that there exists a diffeomorphism $\phi':X' \to Y'$ which commutes with $\phi$ (via the natural maps from $X'$, $Y'$ to $X$, $Y$ respectively arising from blow up)?

What happens if $dim X=dim Y >2$? What happens if we assume that $X, Y$ lie as fibers over closed points of a family parametrized by a quasi-projective variety $B$ which is simply connected under the analytic topology (the underlying field is always $\mathbb{C}$)?

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what is a diffeomorphism of singular varieties? – YangMills Jan 16 '13 at 14:58
Perhaps you know this already, but the universal property of blow-ups should give you what you need in the case that $\phi$ is algebraic. See e.g. Hartshorne Corollary II.7.15. – Daniel Loughran Jan 16 '13 at 22:58
you don't say whether your surfaces are blown up at smooth or at singular points, nor whether the blowups are smooth. without these restrictions, your question can probably be answered as below for the smooth case. – roy smith Jan 18 '13 at 21:19
this seems to be a question of whether blowing up is a diffeomorphism invariant. I.e. we all think blowing up means replacing a point by the tangent vectors at that point, and these should be diffeomorphism invariant. if DIFFEOMORPHISM INDUCES A LINEAR MAP (oops), on the tangent space hence tangent cone at a point, it should induce a map on blowups. I would consult Whitney for a discussion of variops notions of tangent vectors, but I will guess yes. – roy smith Jan 19 '13 at 6:23
of course, the obvious definition of diffeomorphism near a singular point is a function that extends to a differentiable function nearby on some embedded copy, and has such an inverse. – roy smith Jan 19 '13 at 6:26

Let $f\colon X \to Y$ be a smooth map between smooth varieties. Suppose that $X_0\subset X$ and $Y_0\subset Y$ are smooth subvarieties, such that $X_0=f^{-1}(Y_0)$. If $f$ induces a fiber-wise injection from the normal bundle of $X_0$ to the normal bundle of $Y_0$, then $f$ induces a map from the blow-up of $X$ and $X_0$ to the blow-up of $Y$ at $Y_0$.
This does not apply to your example, where $X_0$ consists of isolated singularities. But I would think that in any situation where you can sensibly say that $f$ induces a monomorphism of normal bundles'', $f$ will induce a map of blow-ups.