# Strict transform by normalization

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 $$H_*(|\phi^{-1}(0)|)$$.

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 $$p$$.