# $\text{Bl}_{\Delta}(X\times X)$ as the "Hilbert scheme" of ordered two points on $X$

$$\DeclareMathOperator\Bl{Bl}\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Sym{Sym}\newcommand\Sch{\mathit{Sch}}\newcommand\Sets{\mathit{Sets}}$$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 $$\Bl_{\Delta}(X\times X)$$?

• 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 $S\}$.
– abx
Dec 10, 2021 at 7:20
• By the way, these are called "nested Hilbert schemes", and there is an abundant literature about them.
– abx
Dec 10, 2021 at 7:30
• @abx Thanks! This helps a lot! Dec 10, 2021 at 8:02
• I have also heard the phrase “flag Hilbert schemes”, by analogy with flag varieties. Dec 10, 2021 at 14:15

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:
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.