The Stacks project

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

\[ X_ n \to \ldots \to X_1 \to X \]

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. $\square$

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