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The Stacks project

Lemma 54.6.4. Let S, s_ i, S_ i be as in (54.6.0.1) and assume f : X \to S corresponds to g_ i : Y_ i \to S_ i under F. Then there exists a factorization

X = Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S

of f where Z_{j + 1} \to Z_ j is the blowing up of Z_ j at a closed point z_ j lying over \{ s_1, \ldots , s_ n\} if and only if for each i there exists a factorization

Y_ i = Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_ i

of g_ i where Z_{i, j + 1} \to Z_{i, j} is the blowing up of Z_{i, j} at a closed point z_{i, j} lying over s_ i.

Proof. Let's start with a sequence of blowups Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = S. The first morphism Z_1 \to S is given by blowing up one of the s_ i, say s_1. Applying F to Z_1 \to S we find a blowup Z_{1, 1} \to S_1 at s_1 is the blowing up at s_1 and otherwise Z_{i, 0} = S_ i for i > 1. In the next step, we either blow up one of the s_ i, i \geq 2 on Z_1 or we pick a closed point z_1 of the fibre of Z_1 \to S over s_1. In the first case it is clear what to do and in the second case we use that (Z_1)_{s_1} \cong (Z_{1, 1})_{s_1} (Lemma 54.6.3) to get a closed point z_{1, 1} \in Z_{1, 1} corresponding to z_1. Then we set Z_{1, 2} \to Z_{1, 1} equal to the blowing up in z_{1, 1}. Continuing in this manner we construct the factorizations of each g_ i.

Conversely, given sequences of blowups Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = S_ i we construct the sequence of blowing ups of S in exactly the same manner. \square


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