Lemma 54.6.5. 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 $F$. Assume every quasi-compact open of $S$ has finitely many irreducible components. Then there exists a factorization

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

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 $s_ i$.

## Comments (0)