The Stacks project

Lemma 54.6.4. Let $S, s_ i, S_ i$ be as in ( 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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BFY. Beware of the difference between the letter 'O' and the digit '0'.