The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BI5. Beware of the difference between the letter 'O' and the digit '0'.