Lemma 89.5.3. Let X, x_ i, U_ i \to X, u_ i be as in (89.3.0.1) and assume f : Y \to X corresponds to g_ i : Y_ i \to U_ i under F. Assume X satisfies the equivalent conditions of Morphisms of Spaces, Lemma 67.49.1. Then there exists a factorization
Y = Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X
of f where Z_{j + 1} \to Z_ j is the normalized blowing up of Z_ j at a closed point z_ j lying over \{ x_1, \ldots , x_ 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} = U_ i
of g_ i where Z_{i, j + 1} \to Z_{i, j} is the normalized blowing up of Z_{i, j} at a closed point z_{i, j} lying over u_ i.
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