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
-
Deform simultaneously using $D_{(A_1,A_2)} X$
-
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?
Best Answer
Ok so the solution is to consider the product of ideals instead of their intersection.
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 $A_i$.
With this definition, we have nice functoriality properties w/r to maps and direct products.