Regularization of an integral curve on an arbitrary Noetherian ambient scheme

Apply [F1] to $Y$: this gives a finite sequence of closed points $y_0\in Y_0=Y$, $y_1\in Y_1,\dots ,y_{n-1}\in Y_{n-1}$ with $Y_i={\mathrm{Bl}}_{y_{i-1}}{Y}_{i-1}$, each $y_{i-1}$ non-regular in $Y_{i-1}$, and $Y_n$ regular. Since $Y_{i-1}$ is integral of dimension one, each $y_{i-1}$ is a closed point of $Y_{i-1}$ (Integral schemes).

Define the ambient sequence by $X_0=X$ and, inductively, $X_i={\mathrm{Bl}}_{z_{i-1}}{X}_{i-1}$, where $z_{i-1}$ is the image of $y_{i-1}$ under the closed immersion $Y_{i-1}\hookrightarrow X_{i-1}$ whose construction is the induction claim proved below. Each $z_{i-1}$ is a closed point of $X_{i-1}$ by [F4], and each $X_i$ is Noetherian by [F3].

Claim: for every $i$ the strict transform of $Y$ in $X_i$ is canonically isomorphic over $Y$ to $Y_i$; in particular $Y_i$ embeds in $X_i$ as a closed subscheme, so the induction of step 2.1 is legitimate. For $i=0$ this is the identity $Y_0=Y$. Assume it for $i-1$. The blowup $X_i\to X_{i-1}$ is the blowup at the point $z_{i-1}\in Y_{i-1}\subseteq X_{i-1}$; by [F2] the strict transform of $Y_{i-1}$ in $X_i$ is the blowup of $Y_{i-1}$ in the inverse image ideal of $z_{i-1}$, which is the maximal ideal of the point $y_{i-1}$, and this blowup is ${\mathrm{Bl}}_{y_{i-1}}{Y}_{i-1}=Y_i$. Since strict transforms may be iterated (the strict transform of $Y$ in $X_i$ equals the strict transform of the strict transform $Y_{i-1}$ of $Y$ in $X_{i-1}$, by the saturation description of [F2]), the strict transform of $Y$ in $X_i$ is $Y_i$.

Taking $i=n$, the strict transform of $Y$ in $X_n$ is canonically $Y_n$, which is regular by step 1.1; hence the finite sequence of blowups in closed points constructed in step 2.1 has the required property, and by construction its centers are the images of the intrinsic centers, so the canonical identification of strict transforms with the intrinsic blowups holds at every stage. This proves both assertions.