Definition 89.4.1. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a closed point. By Decent Spaces, Lemma 68.14.6 we can represent $x$ by a closed immersion $i : \mathop{\mathrm{Spec}}(k) \to X$. The blowing up $X' \to X$ of $X$ at $x$ means the blowing up of $X$ in the closed subspace $Z = i(\mathop{\mathrm{Spec}}(k)) \subset X$.
89.4 Dominating by quadratic transformations
We define the blowup of a space at a point only if $X$ is decent.
In this generality the blowing up of $X$ at $x$ is not necessarily proper. However, if $X$ is locally Noetherian, then it follows from Divisors on Spaces, Lemma 71.17.11 that the blowing up is proper. Recall that a locally Noetherian algebraic space is Noetherian if and only if it is quasi-compact and quasi-separated. Moreover, for a locally Noetherian algebraic space, being quasi-separated is equivalent to being decent (Decent Spaces, Lemma 68.14.1).
Lemma 89.4.2. 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 $F$. Then there exists a factorization of $f$ where $Z_{j + 1} \to Z_ j$ is the 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 blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $u_ i$.
\[ Y = Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X \]
\[ Y_ i = Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = U_ i \]
Proof. A blowing up is a representable morphism. Hence in either case we inductively see that $Z_ j \to X$ or $Z_{i, j} \to U_ i$ is representable. Whence each $Z_ j$ or $Z_{i, j}$ is a decent algebraic space by Decent Spaces, Lemma 68.6.5. This shows that the assertions make sense (since blowing up is only defined for decent spaces). To prove the equivalence, let's start with a sequence of blowups $Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X$. The first morphism $Z_1 \to X$ is given by blowing up one of the $x_ i$, say $x_1$. Applying $F$ to $Z_1 \to X$ we find a blowup $Z_{1, 1} \to U_1$ at $u_1$ is the blowing up at $u_1$ and otherwise $Z_{i, 0} = U_ i$ for $i > 1$. In the next step, we either blow up one of the $x_ i$, $i \geq 2$ on $Z_1$ or we pick a closed point $z_1$ of the fibre of $Z_1 \to X$ over $x_1$. In the first case it is clear what to do and in the second case we use that $(Z_1)_{x_1} \cong (Z_{1, 1})_{u_1}$ (Lemma 89.3.3) 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 $Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = U_ i$ we construct the sequence of blowing ups of $X$ in exactly the same manner. $\square$
Lemma 89.4.3. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $T \subset |X|$ be a finite set of closed points $x$ such that (1) $X$ is regular at $x$ and (2) the local ring of $X$ at $x$ has dimension $2$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals such that $\mathcal{O}_ X/\mathcal{I}$ is supported on $T$. Then there exists a sequence where $X_{j + 1} \to X_ j$ is the blowing up of $X_ j$ at a closed point $x_ j$ lying above a point of $T$ such that $\mathcal{I}\mathcal{O}_{X_ n}$ is an invertible ideal sheaf.
\[ X_ m \to X_{m - 1} \to \ldots \to X_1 \to X_0 = X \]
Proof. Say $T = \{ x_1, \ldots , x_ r\} $. Pick elementary étale neighbourhoods $(U_ i, u_ i) \to (X, x_ i)$ as in Section 89.3. For each $i$ the restriction $\mathcal{I}_ i = \mathcal{I}|_{U_ i} \subset \mathcal{O}_{U_ i}$ is a quasi-coherent sheaf of ideals supported at $u_ i$. The local ring of $U_ i$ at $u_ i$ is regular and has dimension $2$. Thus we may apply Resolution of Surfaces, Lemma 54.4.1 to find a sequence
\[ X_{i, m_ i} \to X_{i, m_ i - 1} \to \ldots \to X_1 \to X_{i, 0} = U_ i \]
of blowing ups in closed points lying over $u_ i$ such that $\mathcal{I}_ i \mathcal{O}_{X_{i, m_ i}}$ is invertible. By Lemma 89.4.2 we find a sequence of blowing ups
\[ X_ m \to X_{m - 1} \to \ldots \to X_1 \to X_0 = X \]
as in the statement of the lemma whose base change to our $U_ i$ produces the given sequences. It follows that $\mathcal{I}\mathcal{O}_{X_ n}$ is an invertible ideal sheaf. Namely, we know this is true over $X \setminus \{ x_1, \ldots , x_ n\} $ and in an étale neighbourhood of the fibre of each $x_ i$ it is true by construction. $\square$
Lemma 89.4.4. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $T \subset |X|$ be a finite set of closed points $x$ such that (1) $X$ is regular at $x$ and (2) the local ring of $X$ at $x$ has dimension $2$. Let $f : Y \to X$ be a proper morphism of algebraic spaces which is an isomorphism over $U = X \setminus T$. Then there exists a sequence where $X_{i + 1} \to X_ i$ is the blowing up of $X_ i$ at a closed point $x_ i$ lying above a point of $T$ and a factorization $X_ n \to Y \to X$ of the composition.
\[ X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X \]
Proof. By More on Morphisms of Spaces, Lemma 76.39.5 there exists a $U$-admissible blowup $X' \to X$ which dominates $Y \to X$. Hence we may assume there exists an ideal sheaf $\mathcal{I} \subset \mathcal{O}_ X$ such that $\mathcal{O}_ X/\mathcal{I}$ is supported on $T$ and such that $Y$ is the blowing up of $X$ in $\mathcal{I}$. By Lemma 89.4.3 there exists a sequence
\[ X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X \]
where $X_{i + 1} \to X_ i$ is the blowing up of $X_ i$ at a closed point $x_ i$ lying above a point of $T$ such that $\mathcal{I}\mathcal{O}_{X_ n}$ is an invertible ideal sheaf. By the universal property of blowing up (Divisors on Spaces, Lemma 71.17.5) we find the desired factorization. $\square$
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)