I am reading the paper Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink due to G. Guibert, F. Loeser, and M. Merle. The main tool, like a lot of papers in singularity theory, is the log-resolutions. Let $k$ be a field of characteristic zero, let $X$ be a smooth $k$-variety of dimension $d$ and $Z$ a closed subscheme of codimension $\geq 1$. Let $h: (Y,E) \longrightarrow (X,Z)$ be a log-resolution of the pair $(X,Z)$. We let $(E_i)_{i \in A}$ be irreducible components of $h^{-1}(Z) = E = \sum_{i\in A}N_i(Z)E_i$ ($N_i(Z)$: multiplicities) the exceptional divisor. For any subset $I$ of $A$, we define $$E_I = \cap_{i\in I}E_i, \ E_I^{\circ} = E_I \setminus \cup_{i \notin I}E_i.$$ For every $i \in A$, let $\nu_{E_i}$ be the normal bundle of $E_i$ in $Y$ (which makes sense because $E_i \longrightarrow Y$ is a regular imbedding) and $U_{E_i}$ the complement of the zero section of $\nu_{E_i}$. Let $\nu_{E_I}$ be the fiber product of the restrictions of $\nu_{E_i}$ to $E_I$. Let $U_I$ be the fiber product of the restrictions of $U_{E_i}$ to $E^{\circ}_I$.
Now suppose that $Z$ comes from the special fiber of a morphism $f:X \longrightarrow \mathbf{A}_k^1$ (in this case we write $N_i(f)$ instead of $N_i(Z)$). The authors claims that as long as there is some $i\in I$ with $N_i(f) > 0$, then $f \circ h$ induces a morphism $$\bigotimes_{i\in I} (\nu_{E_i})_{E_I}^{\otimes N_i(f)} \longrightarrow \mathbf{A}_k^1.$$ Compose the one above with $$\nu_{E_I} \longrightarrow \bigotimes_{i\in I} (\nu_{E_i})_{E_I}^{\otimes N_i(f)}, (y_i) \longmapsto (\otimes y_i^{\otimes N_i(f)})$$ and then restrict to $U_I$ gives a morphism $f_I: U_I \longrightarrow \mathbf{A}_k^1$.
My problem begins from here, as they claim that this $f_I$ can be obtained by deforming to normal cone of $E_I$ in $Y$. Consider the affine space $\mathbf{A}_k^I = \mathrm{Spec}(k[u_i]_{i\in I})$ and the subsheaf (of algebras) $$\mathcal{A}_I = \sum_{\mathbf{n} \in \mathbf{N}^{\left|I \right|}} \mathcal{O}_{Y \times \mathbf{A}_k^I}\left(-n_i(E_i \times_k \mathbf{A}_k^I) \right)\prod_{i\in I}u_i^{-n_i}$$ of $\mathcal{O}_{Y \times \mathbf{A}_k^I}[u_i^{-1}]_{i \in I}$. We denote by $CY_I$ the spectrum of $\mathcal{A}_I$. The natural inclusion $\mathcal{O}_{Y \times \mathbf{A}_k^I}$ in $\mathcal{A}_I$ induces a morphism $$\pi_I: CY_I \longrightarrow Y \times \mathbf{A}_k^I$$ and if we project onto the second factor, we get $p_I: CY_I \longrightarrow \mathbf{A}_k^I$.
My questions are:
- How $CY_I$ is related to the normal cone of $E_I$ in $Y$? I have a quite poor intuition for deforming to normal cone, so I do not see how to relate this one with the formal definition, especially when the expression of $\mathcal{A}_I$ is kind of weird to me, I do not know where it comes from.
- How to see that $\nu_{E_I}$ can be identified with $p_I^{-1}(0)$?
- How the image of $f \circ h$ by the inclusion $\mathcal{O}_{Y \times \mathbf{A}_k^I}$ in $\mathcal{A}_I$ can be divided by $\prod_{i \in I}u_i^{N_i(f)}$, which again, leads to a quotient $\widetilde{f}_I$ in $\mathcal{A}_I$ and the restriction of this quotient to $p^{-1}(0)$ coincides with $f_I$ defined before?
This is a bit lengthy question, but these confuse me a lot. Everything is ok if I wave my hands by writing coordinates like in differential geometry, but I really want an explanation in terms of rigirous algebraic geometry.
