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$\DeclareMathOperator\Bl{Bl}\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Sym{Sym}\newcommand\Sch{\mathit{Sch}}\newcommand\Sets{\mathit{Sets}}$Let $X$ be a smooth variety. The Hilbert scheme of two points $X^{[2]}$ on $X$ can be obtained by blowup the diagonal of the symmetric product $\Sym^2X:=(X\times X)/\mathbb Z_2$: $$X^{[2]}\cong\Bl_{\Delta}(\Sym^2X)\to \Sym^2X.$$

By definition, this space represents the Hilbert functor \begin{gather*} \Hilb^2_X:\Sch^\text{op}\to \Sets \\ T\mapsto \{{\text{flat families of subschemes of $X$ of length 2 over $T$}}\}/{\sim}. \end{gather*} I'm interested in the space $\Bl_{\Delta}(X\times X)$, which is a double cover of $X^{[2]}$ branched along the exceptional divisor. In fact it is obtained as the fiber product square: $\require{AMScd}$ \begin{CD} \Bl_{\Delta}(X\times X) @>>> \Bl_{\Delta}(\Sym^2X)\\ @VVV @VVV\\ X\times X @>>> \Sym^2X. \end{CD}

I'd like to know if there is a modular interpretation of the double cover $\Bl_{\Delta}(X\times X)$. To me, it should be something like the "Hilbert scheme" of ordered two points: Its general point parameterizes two points with an order; When the two points collide and become a fat point, it forget the order. However, I don't know how to formulate the modularity more precisely.

So perhaps here is what I want to ask: Is there a moduli functor $\mathcal{M}:\Sch^\text{op}\to \Sets$ representable by the variety $\Bl_{\Delta}(X\times X)$?

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    $\begingroup$ The functor $\mathscr{M}$ is given by $\mathscr{M}(S)=\{$closed subschemes $Z_1\subset Z_2\subset S\times X$, with $Z_i$ finite flat of degree $i$ over $S\}$. $\endgroup$
    – abx
    Dec 10, 2021 at 7:20
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    $\begingroup$ By the way, these are called "nested Hilbert schemes", and there is an abundant literature about them. $\endgroup$
    – abx
    Dec 10, 2021 at 7:30
  • $\begingroup$ @abx Thanks! This helps a lot! $\endgroup$
    – AG learner
    Dec 10, 2021 at 8:02
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    $\begingroup$ I have also heard the phrase “flag Hilbert schemes”, by analogy with flag varieties. $\endgroup$ Dec 10, 2021 at 14:15

1 Answer 1

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Let $X$ be a smooth variety, and $X[n]$ the variety obtained from $X^n = X\times\dots \times X$ by blowing-up the diagonals in order of increasing dimension.

The variety $X[n]$ is a wonderful compactification (the added boundary divisor is simple normal crossing) of the configuration space of $n$ ordered points on $X$. It is known as the Fulton-MacPherson configuration space and it has been constructed in various ways here:

W. Fulton, R. MacPherson, A Compactification of Configuration Spaces, Annals of Mathematics, Second Series, Vol. 139, No. 1 (Jan., 1994), pp. 183-225.

In particular, in Theorem $4$ of the paper above you can find the precise definition of the moduli functor you are looking for.

When $n = 2$ the space $X[2]$ is just the blow-up of $X\times X$ along the diagonal. In particular, when $X = C$ is a curve then $C[2] = C\times C$ and $C[3]$ is $C\times C\times C$ blown-up along the small diagonal since the three bigger diagonals are divisors.

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