Deformation to the normal cone

Deformation to the normal cone appears in several places including Intersection theory and Verdier specialisation of construtible sheaves or D-modules. I'd like to understand what happens when we iterate the process: deform to the cone of $A_1 \subset X$ and then to the cone of $C_{A_1\cap A_2} A_2 \subset C_{A_1} X$. In particular is it commutative (can we exchange $A_1$ and $A_2$)?

Deformation to the normal cone: Let $A$ be a closed subscheme of $X$ with ideal $I \subset O_X$. One has the space of deformation to the normal cone $D_A X = Spec_X( R_I O_X)$ where $(R_I O_X)\subset O_X[t^{\pm 1}]$ is the Rees algebra of $I$ defined by $$R_I O_X = \bigoplus_{k\in \mathbb{Z}} I^{-k} t^k$$ with $I^{k} = O_X$ if $k \leq 0$. The projection $t: D_A X \to \mathbb{A}^1$ is flat. For $t \neq 0$ the fiber is $X$ while for $t =0$ the fiber is the normal cone $$C_A X = Spec \left( \bigoplus I^{k}/I^{k+1} \right)$$ hence the name. In a more geometric fashion, $D_A X$ is the complementary of $\mathbb{P}(C_AX) \subset \mathbb{P}(C_{A}X \oplus 1)$ inside the blow-up $B_{A\times 0} X\times \mathbb{A}^1$. The projection onto $X\times \mathbb{A}^1$ being given by the inclusion $O_X[t] \to R_I O_X$.

The construction is fonctorial. For $f:X\to Y$, $B\subset Y$ and $A = X\times_Y B$, we have $$f^* \left( \bigoplus I_B^{-k} t^k \right) \to \bigoplus I_A^{-k} t^k$$ inducing a morphism $D(f):D_A X \to X\times_Y D_B Y$ compatible with the projections onto $\mathbb{A}^1$. If $f$ is a closed embedding then so is $D(f)$.

Multiple deformations: Now consider $A_1,A_2 \subset X$ with ideals $I_1,I_2$.

We can form the multi-Rees algebra $$R_{(I_1,I_2)} O_X := \bigoplus_{k_1,k_2\in \mathbb{Z}} (I_1^{-k_1} \cap I_2^{-k_2}) t_1^{k_1} t_2^{k_2} \subset O_X[t_1^{\pm 1},t_2^{\pm 1}]$$ Its Spec aver $X$ is a space $D_{(A_1,A_2)} X$ together with a morphism $D_{(A_1,A_2)} X \to X\times \mathbb{A}^2$ given by the coordinates $(t_1,t_2)$.

We also have a closed immersion $D_{A_1 \cap A_2} A_2 \subset D_{A_1} X$ with ideal $$J_2 = \bigoplus_{k\in \mathbb{Z}} (I_2 \cap I_1^{-k}) t^{k}$$ So we can form the Rees algebra $R_{J_2} R_{I_1} O_X$ and its Spec over $D_{A_1} X$, we simply denote by $D_{A_2} D_{A_1} X$. It also has a canonical morphism $$D_{A_2} D_{A_1} X \to D_{A_1}X \times \mathbb{A}^1 \to X\times \mathbb{A}^1 \times \mathbb{A}^1.$$

Question 1: Is it the same thing to

1. Deform simultaneously using $D_{(A_1,A_2)} X$

2. Deform to the normal cone of $A_1$ in $X$ and then to the normal cone of $C_{A_1\cap A_2} A_2$ in $C_{A_1} X$

Question 2: Do we have a canonical isomorphism $$R_{(I_1,I_2)} O_X = R_{J_2} R_{I_1} O_X?$$

Does is it induce an isomorphism of $(X\times \mathbb{A}^2)$-schemes compatible with the $\mathbb{G}_m^2$-actions coming from the gradings?

Question 3: Do we have a canonical isomorphim $$D_{(A_1,A_2)} X|_{t_1 = 0} = D_{C_{A_1\cap A_2} A_2} C_{A_1} X$$ i.e. $$R_{(I_1,I_2)} O_X / (t_1) = R_{Gr_{I_1} I_2} Gr_{I_1} O_X$$

Question 4 What can we say if $A_1$ and $A_2$ are transverse subvarieties? What changes for the deformation spaces?

Note: I know that in this case the canonical morphism $C(i_2) : D_{A_1\cap A_2} A_2 \to A_2 \times_{X} D_{A_1} X$ is an isomorphism.

Question 5 Does any one know any good reference where basic functorial properties of Rees algebras are detailed?

• Have you worked out the example where $X$ is $\mathbb{C}^3$, with $A_1=\{ x=y=0 \}$ and $A_2 = \{ y=z=0 \}$? That's the standard example of two blowups not commuting. I've been meaning to do it for you, but it doesn't look like I'll get to it. – David E Speyer Mar 1 '11 at 0:45
• I don't know the universal property of the deformation space so I tried to deal with the Rees Algebras. From what I've worked out, it seems that the bigradings aren't the same. Perhaps I'm looking at the wrong projection $D_{A_2} D_{A_1} X \to X \times \mathbb{A}^2$. – AFK Mar 1 '11 at 1:32
• In my experience, from playing with the Rees algebras, no two degenerations like these ever nonobviously match. :( – Allen Knutson Mar 1 '11 at 5:33
• Added a few questions – AFK Mar 1 '11 at 18:01

Just like we have $$B_{\widetilde{I}_1} B_{I_2} X = B_{I_1I_2} X = B_{\widetilde{I}_2} B_{I_1} X$$ with $\widetilde{I}_j$ the total transform of the ideal $I_j \subset O_X$, we have $$D_{\widetilde{D}_1} D_{A_2} X = D_{(A_1,A_2)} X = D_{\widetilde{D}_2} D_{A_1} X$$ with $\widetilde{D}_j = A_i \times_X D_{A_j} X$ the total pullback of the normal cone deformation space to $A_i$.