Lemma 54.15.5. Let $X$ be a regular scheme of dimension $2$. Let $Z \subset X$ be a proper closed subscheme. There exists a sequence
of blowing ups in closed points such that the inverse image $Z_ n$ of $Z$ in $X_ n$ is an effective Cartier divisor.
Proof. Let $D \subset Z$ be the largest effective Cartier divisor contained in $Z$. Then $\mathcal{I}_ Z \subset \mathcal{I}_ D$ and the quotient is supported in closed points by Divisors, Lemma 31.15.8. Thus we can write $\mathcal{I}_ Z = \mathcal{I}_{Z'} \mathcal{I}_ D$ where $Z' \subset X$ is a closed subscheme which set theoretically consists of finitely many closed points. Applying Lemma 54.4.1 we find a sequence of blowups as in the statement of our lemma such that $\mathcal{I}_{Z'}\mathcal{O}_{X_ n}$ is invertible. This proves the lemma. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)