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?