The Stacks project

Lemma 54.15.6. Let $X$ be a regular scheme of dimension $2$. Let $Z \subset X$ be a proper closed subscheme such that every irreducible component $Y \subset Z$ of dimension $1$ satisfies the equivalent conditions of Lemma 54.15.1. Then there exists a sequence

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

of blowups in closed points such that the inverse image $Z_ n$ of $Z$ in $X_ n$ is an effective Cartier divisor supported on a strict normal crossings divisor.

Proof. Let $X' \to X$ be a blowup in a closed point $p$. Then the inverse image $Z' \subset X'$ of $Z$ is supported on the strict transform of $Z$ and the exceptional divisor. The exceptional divisor is a regular curve (Lemma 54.3.1) and the strict transform $Y'$ of each irreducible component $Y$ is either equal to $Y$ or the blowup of $Y$ at $p$. Thus in this process we do not produce additional singular components of dimension $1$. Thus it follows from Lemmas 54.15.5 and 54.15.4 that we may assume $Z$ is an effective Cartier divisor and that all irreducible components $Y$ of $Z$ are regular. (Of course we cannot assume the irreducible components are pairwise disjoint because in each blowup of a point of $Z$ we add a new irreducible component to $Z$, namely the exceptional divisor.)

Assume $Z$ is an effective Cartier divisor whose irreducible components $Y_ i$ are regular. For every $i \not= j$ and $p \in Y_ i \cap Y_ j$ we have the invariant $m_ p(Y_ i \cap Y_ j)$ ( 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 (note that the “new” invariants $m_{q_ i}(Y'_ i \cap E)$ are always $1$). If the maximum is $1$ then, if $p \in Y_1 \cap \ldots \cap Y_ r$ for some $r > 2$ and not any of the others (for example), then after blowing up $p$ we see that $Y'_1, \ldots , Y'_ r$ do not meet in points above $p$ and $m_{q_ i}(Y'_ i, E) = 1$ where $Y'_ i \cap E = \{ q_ i\} $. Thus continuing to blowup points where more than $3$ of the components of $Z$ meet, we reach the situation where for every closed point $p \in X$ there is either (a) no curves $Y_ i$ passing through $p$, (b) exactly one curve $Y_ i$ passing through $p$ and $\mathcal{O}_{Y_ i, p}$ is regular, or (c) exactly two curves $Y_ i$, $Y_ j$ passing through $p$, the local rings $\mathcal{O}_{Y_ i, p}$, $\mathcal{O}_{Y_ j, p}$ are regular and $m_ p(Y_ i \cap Y_ j) = 1$. This means that $\sum Y_ i$ is a strict normal crossings divisor on the regular surface $X$, see Étale Morphisms, Lemma 41.21.2. $\square$

Comments (2)

Comment #5078 by comment_bot on

In the statement it would be useful to mention that is supported on a strict normal crossings divisor.

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