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

- X is irreducible
- $\phi^{-1}(t)$ is of dimension $\dim X-2$ for $t\neq 0$.
- $\phi$ 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 $H_*(|\phi^{-1}(0)|)$.