Lemma 54.15.2. Let $X$ be a Noetherian scheme. Let $Y \subset X$ be an integral closed subscheme of dimension $1$ satisfying the equivalent conditions of Lemma 54.15.1. Then there exists a finite sequence

$X_ n \to X_{n - 1} \to \ldots \to X_1 \to X$

of blowups in closed points such that the strict transform of $Y$ in $X_ n$ is a regular curve.

Proof. Let $Y_ n \to Y_{n - 1} \to \ldots \to Y_1 \to Y$ be the sequence of blowups given to us by Lemma 54.15.1. Let $X_ n \to X_{n - 1} \to \ldots \to X_1 \to X$ be the corresponding sequence of blowups of $X$. This works because the strict transform is the blowup by Divisors, Lemma 31.33.2. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).