The Stacks project

54.4 Dominating by quadratic transformations

Using the result above we can prove that blowups in points dominate any modification of a regular $2$ dimensional scheme.

Let $X$ be a scheme. Let $x \in X$ be a closed point. As usual, we view $i : x = \mathop{\mathrm{Spec}}(\kappa (x)) \to X$ as a closed subscheme. The blowing up $X' \to X$ of $X$ at $x$ is the blowing up of $X$ in the closed subscheme $x \subset X$. Observe that if $X$ is locally Noetherian, then $X' \to X$ is projective (in particular proper) by Divisors, Lemma 31.32.13.

Lemma 54.4.1. Let $X$ be a Noetherian scheme. Let $T \subset X$ be a finite set of closed points $x$ such that $\mathcal{O}_{X, x}$ is regular of dimension $2$ for $x \in T$. 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

\[ X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X \]

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.

Proof. Say $T = \{ x_1, \ldots , x_ r\} $. Set

\[ n_ i = \text{length}_{\mathcal{O}_{X, x_ i}}(\mathcal{O}_{X, x_ i}/I_ i) \]

This is finite as $\mathcal{O}_ X/\mathcal{I}$ is supported on $T$ and hence $\mathcal{O}_{X, x_ i}/I_ i$ has support equal to $\{ \mathfrak m_{x_ i}\} $ (see Algebra, Lemma 10.62.3). We are going to use induction on $\sum n_ i$. If $n_ i = 0$ for all $i$, then $\mathcal{I} = \mathcal{O}_ X$ and we are done.

Suppose $n_ i > 0$. Let $X' \to X$ be the blowing up of $X$ in $x_ i$ (see discussion above the lemma). Since $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}) \to X$ is flat we see that $X' \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i})$ is the blowup of the ring $\mathcal{O}_{X, x_ i}$ in the maximal ideal, see Divisors, Lemma 31.32.3. Hence the square in the commutative diagram

\[ \xymatrix{ \text{Proj}(\bigoplus \nolimits _{d \geq 0} \mathfrak m_{x_ i}^ d) \ar[r] \ar[d] & X' \ar[d] \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}) \ar[r] & X } \]

is cartesian. Let $E \subset X'$ and $E' \subset \text{Proj}(\bigoplus \nolimits _{d \geq 0} \mathfrak m_{x_ i}^ d)$ be the exceptional divisors. Let $d \geq 1$ be the integer found in Lemma 54.3.5 for the ideal $\mathcal{I}_ i \subset \mathcal{O}_{X, x_ i}$. Since the horizontal arrows in the diagram are flat, since $E' \to E$ is surjective, and since $E'$ is the pullback of $E$, we see that

\[ \mathcal{I}\mathcal{O}_{X'} \subset \mathcal{O}_{X'}(-dE) \]

(some details omitted). Set $\mathcal{I}' = \mathcal{I}\mathcal{O}_{X'}(dE) \subset \mathcal{O}_{X'}$. Then we see that $\mathcal{O}_{X'}/\mathcal{I}'$ is supported in finitely many closed points $T' \subset |X'|$ because this holds over $X \setminus \{ x_ i\} $ and for the pullback to $\text{Proj}(\bigoplus \nolimits _{d \geq 0} \mathfrak m_{x_ i}^ d)$. The final assertion of Lemma 54.3.5 tells us that the sum of the lengths of the stalks $\mathcal{O}_{X', x'}/\mathcal{I}'\mathcal{O}_{X', x'}$ for $x'$ lying over $x_ i$ is $< n_ i$. Hence the sum of the lengths has decreased.

By induction hypothesis, there exists a sequence

\[ X'_ n \to \ldots \to X'_1 \to X' \]

of blowups at closed points lying over $T'$ such that $\mathcal{I}'\mathcal{O}_{X'_ n}$ is invertible. Since $\mathcal{I}'\mathcal{O}_{X'}(-dE) = \mathcal{I}\mathcal{O}_{X'}$, we see that $\mathcal{I}\mathcal{O}_{X'_ n} = \mathcal{I}'\mathcal{O}_{X'_ n}(-d(f')^{-1}E)$ where $f' : X'_ n \to X'$ is the composition. Note that $(f')^{-1}E$ is an effective Cartier divisor by Divisors, Lemma 31.32.11. Thus we are done by Divisors, Lemma 31.13.7. $\square$

Lemma 54.4.2. Let $X$ be a Noetherian scheme. Let $T \subset X$ be a finite set of closed points $x$ such that $\mathcal{O}_{X, x}$ is a regular local ring of dimension $2$. Let $f : Y \to X$ be a proper morphism of schemes which is an isomorphism over $U = X \setminus T$. Then 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 $T$ and a factorization $X_ n \to Y \to X$ of the composition.

Proof. By More on Flatness, Lemma 38.31.4 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 54.4.1 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, Lemma 31.32.5) we find the desired factorization. $\square$

Lemma 54.4.3. Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is regular and has dimension $2$. Let $Y$ be a proper scheme over $S$. Given an $S$-rational map $f : U \to Y$ from $X$ to $Y$ there exists a sequence

\[ X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X \]

and an $S$-morphism $f_ n : X_ n \to Y$ such that $X_{i + 1} \to X_ i$ blowing up of $X_ i$ at a closed point not lying over $U$ and $f_ n$ and $f$ agree.

Proof. We may assume $U$ contains every point of codimension $1$, see Morphisms, Lemma 29.42.5. Hence the complement $T \subset X$ of $U$ is a finite set of closed points whose local rings are regular of dimension $2$. Applying Divisors, Lemma 31.36.2 we find a proper morphism $p : X' \to X$ which is an isomorphism over $U$ and a morphism $f' : X' \to Y$ agreeing with $f$ over $U$. Apply Lemma 54.4.2 to the morphism $p : X' \to X$. The composition $X_ n \to X' \to Y$ is the desired morphism. $\square$


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 0BFS. Beware of the difference between the letter 'O' and the digit '0'.