# Whether Hilbert schemes of 3 points on arbitrary smooth projective varieties are smooth

As is well known, the Hilbert scheme of two points on a given smooth projective variety X are blow up along diagonal of product of X and then quotient the Z2 action. It is smooth. My question is whether Hilbert schemes of 3 points on arbitrary smooth projective varieties are smooth. If so, why and how to describe the geometry of them?

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For surfaces, $\textrm{Hilb}^[n]$ is smooth for any $n$ (this is a classical result of Fogarty) –  Francesco Polizzi Sep 5 '12 at 11:48
A useful reference might be the article of Fantechi and Göttsche "The cohomology ring of the Hilbert scheme of 3 points on a smooth projective variety." J. Reine Angew. Math. 439 (1993), 147–158. –  ulrich Sep 5 '12 at 13:18

2. For a zero dimensional subscheme $\text{Spec} A$ of a smooth variety $X$, there is a natural functorial map from embedded deformations of $\text{Spec} A\subseteq X$ to abstract deformations of $\text{Spec} A$, and this map is smooth.
Fact 1 is easy. Fact 2 takes more work, but it follows from some elementary arguments about deformation theory for affine schemes. Combining these facts: $\text{Spec} A$ is a smooth point of $\text{Hilb}^3 X$ if and only if the miniversal abstract deformation ring of $\text{Spec} A$ is smooth if and only if, after any reembedding of $\text{Spec} A$ into $\mathbb A^2$ we have that $\text{Spec} A$ is a smooth point of $\text{Hilb}^3 \mathbb A^2$; the last statement is true by Fogarty.