Strict transform by normalization

Question

I am trying to understand the following definition in C. Sabbah's paper (Quelques remarques sur la géométrie des espaces conormaux), page 186, Numdam link.

Let $\phi\colon X\to \mathbb{C}^2$ be a map such that

1. X is irreducible
2. $\phi^{-1}(t)$ is of dimension $\dim X-2$ for $t\neq 0$.
3. $\phi$ factors through a closed imbedding $X\to M$ and a smooth morphism $M\to \mathbb{C}^2$.

Now, let $(C,0)\subset (\mathbb{C}^2,0)$ be a germ of irreducible curve, and let $p:\hat{C}\to C$ be a normalization. Sabbah used the following phrase:

"The strict transform of $X$ by $p$" (I denote it by $X_C$ here.)

My question is, what is the definition of this phrase? In the appendix, he was trying to show that, let $D_C$ be the fiber of $X_C$ over $0$, then $[D_C]$ is independent of the choice of $C$ in $H_*(|\phi^{-1}(0)|)$.

Answer 1

I think I got what it means by strict transform in the paper. In the Stacks Project 31.33, the strict transform is defined as follows

Let $f\colon X\to B$ be a morphism, let $p\colon \hat{B}\to B$ be the blowup of $B$ along a subscheme $Z$, then the strict transform $\hat{X}$ of $X$ by $p$ is just the blowup of $X$ along $f^{-1}(Z)$.

In this situation, this $X_C$ is just the strict transform of $\phi^{-1}(C)$ by $p$.