Lemma 54.15.4. Let $X$ be a Noetherian scheme. Let $Y_ i \subset X$, $i = 1, \ldots , n$ be an integral closed subschemes of dimension $1$ each 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 $Y'_ i \subset X_ n$ of $Y_ i$ in $X_ n$ are pairwise disjoint regular curves.

Proof. It follows from Lemma 54.15.2 that we may assume $Y_ i$ is a regular curve for $i = 1, \ldots , n$. For every $i \not= j$ and $p \in Y_ i \cap Y_ j$ we have the invariant $m_ p(Y_ i \cap Y_ j)$ (54.15.2.1). If the maximum of these numbers is $> 1$, then we can decrease it (Lemma 54.15.3) by blowing up in all the points $p$ where the maximum is attained. If the maximum is $1$ then we can separate the curves using the same lemma by blowing up in all these points $p$. $\square$

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