Consider the complex blow-up of the diagonal $\triangle\subset Sym^2(\mathbb{C}^n)$. It is known that the blown-up space is smooth when $n=1,2$. I was wondering if that is still smooth for $n\geq3$? Moreover, what if $\mathbb{C}^n$ is replaced by other complex variety? I'd be grateful if anyone could show me some hint or reference...
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7$\begingroup$ If true for $\Bbb{C}^n$, it will be true for any manifold -- this is a local question. $\endgroup$– abxJul 25, 2020 at 7:49
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$\begingroup$ Another way to see it: a quotient of a smooth space by a free action is still smooth. $\endgroup$– Dmitry VaintrobJul 25, 2020 at 20:00
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$\begingroup$ @DmitryVaintrob Do you mean blow up the product space first and then do the S_2-action? I don't see how that is free could you explain more? Thank you! $\endgroup$– Eric YuanJul 26, 2020 at 15:55
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1$\begingroup$ @EricYuan, good point! What I wrote above is incorrect (e.g. in the one-dimensional case the blow-up doesn't change the variety) $\endgroup$– Dmitry VaintrobJul 26, 2020 at 18:04
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Consider the quotient map $\pi:\mathrm{X}\times\mathrm{X}\rightarrow\mathrm{S}^2\mathrm{X}=(\mathrm{X}\times\mathrm{X})/\mathrm{S}_2$, and a point $p$ of $\mathrm{X}\times\mathrm{X}$. Then the tangent space of $\mathrm{S}^2\mathrm{X}$ at $\pi(p)$ equals the tangent space of $\mathrm{T}_p(\mathrm{X}\times\mathrm{X})/(\mathrm{S}_2)_p$ at the origin (see for example here), where $(\mathrm{S}_2)_p$ is the stabiliser of $p$. For $\mathrm{X}=\mathbf{A}^n$ you can also write down explicit equations for the blow up; another possibility is to prove that the blow up is isomorphic to the Hilbert scheme of two points $\mathrm{Hilb}^2\mathrm{X}$.
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$\begingroup$ @abx as the op is asking for a hint (not a complete explanation), I would say it does answer the question. But I guess I should have said that the answer to his question is yes. $\endgroup$– SamuelJul 26, 2020 at 10:01
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