The Stacks project

42.53 Blowing up at infinity

Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals. Denote $X' \to X$ the blowing up with center $Z$. Let $b : W \to \mathbf{P}^1_ X$ be the blowing up with center $\infty (Z)$. Denote $E \subset W$ the exceptional divisor. There is a commutative diagram

\[ \xymatrix{ X' \ar[r] \ar[d] & W \ar[d]^ b \\ X \ar[r]^\infty & \mathbf{P}^1_ X } \]

whose horizontal arrows are closed immersion (Divisors, Lemma 31.33.2). Denote $E \subset W$ the exceptional divisor and $W_\infty \subset W$ the inverse image of $(\mathbf{P}^1_ X)_\infty $. Then the following are true

  1. $b$ is an isomorphism over $\mathbf{A}^1_ X \cup \mathbf{P}^1_{X \setminus Z}$,

  2. $X'$ is an effective Cartier divisor on $W$,

  3. $X' \cap E$ is the exceptional divisor of $X' \to X$,

  4. $W_\infty = X' + E$ as effective Cartier divisors on $W$,

  5. $E = \underline{\text{Proj}}_ Z(\mathcal{C}_{Z/X, *}[S])$ where $S$ is a variable placed in degree $1$,

  6. $X' \cap E = \underline{\text{Proj}}_ Z(\mathcal{C}_{Z/X, *})$,

  7. $E \setminus X' = E \setminus (X' \cap E) = \underline{\mathop{\mathrm{Spec}}}_ Z(\mathcal{C}_{Z/X, *}) = C_ ZX$,

  8. there is a closed immersion $\mathbf{P}^1_ Z \to W$ whose composition with $b$ is the inclusion morphism $\mathbf{P}^1_ Z \to \mathbf{P}^1_ X$ and whose base change by $\infty $ is the composition $Z \to C_ ZX \to E \to W_\infty $ where the first arrow is the vertex of the cone.

We recall that $\mathcal{C}_{Z/X, *}$ is the conormal algebra of $Z$ in $X$, see Divisors, Definition 31.19.1 and that $C_ ZX$ is the normal cone of $Z$ in $X$, see Divisors, Definition 31.19.5.

We now give the proof of the numbered assertions above. We strongly urge the reader to work through some examples instead of reading the proofs.

Part (1) follows from the corresponding assertion of Divisors, Lemma 31.32.4. Observe that $E \subset W$ is an effective Cartier divisor by the same lemma.

Observe that $W_\infty $ is an effective Cartier divisor by Divisors, Lemma 31.32.11. Since $E \subset W_\infty $ we can write $W_\infty = D + E$ for some effective Cartier divisor $D$, see Divisors, Lemma 31.13.8. We will see below that $D = X'$ which will prove (2) and (4).

Since $X'$ is the strict transform of the closed immersion $\infty : X \to \mathbf{P}^1_ X$ (see above) it follows that the exceptional divisor of $X' \to X$ is equal to the intersection $X' \cap E$ (for example because both are cut out by the pullback of the ideal sheaf of $Z$ to $X'$). This proves (3).

The intersection of $\infty (Z)$ with $\mathbf{P}^1_ Z$ is the effective Cartier divisor $(\mathbf{P}^1_ Z)_\infty $ hence the strict transform of $\mathbf{P}^1_ Z$ by the blowing up $b$ maps isomorphically to $\mathbf{P}^1_ Z$ (see Divisors, Lemmas 31.33.2 and 31.32.7). This gives us the morphism $\mathbf{P}^1_ Z \to W$ mentioned in (8). It is a closed immersion as $b$ is separated, see Schemes, Lemma 26.21.11.

Suppose that $\mathop{\mathrm{Spec}}(A) \subset X$ is an affine open and that $Z \cap \mathop{\mathrm{Spec}}(A)$ corresponds to the finitely generated ideal $I \subset A$. An affine neighbourhood of $\infty (Z \cap \mathop{\mathrm{Spec}}(A))$ is the affine space over $A$ with coordinate $s = T_0/T_1$. Denote $J = (I, s) \subset A[s]$ the ideal generated by $I$ and $s$. Let $B = A[s] \oplus J \oplus J^2 \oplus \ldots $ be the Rees algebra of $(A[s], J)$. Observe that

\[ J^ n = I^ n \oplus sI^{n - 1} \oplus s^2I^{n - 2} \ldots \oplus s^ nA \oplus s^{n + 1}A \oplus \ldots \]

as an $A$-submodule of $A[s]$ for all $n \geq 0$. Consider the open subscheme

\[ \text{Proj}(B) = \text{Proj}(A[s] \oplus J \oplus J^2 \oplus \ldots ) \subset W \]

Finally, denote $S$ the element $s \in J$ viewed as a degree $1$ element of $B$.

Since formation of $\text{Proj}$ commutes with base change (Constructions, Lemma 27.11.6) we see that

\[ E = \text{Proj}(B \otimes _{A[s]} A/I) = \text{Proj}((A/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots )[S]) \]

The verification that $B \otimes _{A[s]} A/I = \bigoplus J^ n/J^{n + 1}$ is as given follows immediately from our description of the powers $J^ n$ above. This proves (5) because the conormal algebra of $Z \cap \mathop{\mathrm{Spec}}(A)$ in $\mathop{\mathrm{Spec}}(A)$ corresponds to the graded $A$-algebra $A/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots $ by Divisors, Lemma 31.19.2.

Recall that $\text{Proj}(B)$ is covered by the affine opens $D_+(S)$ and $D_+(f^{(1)})$ for $f \in I$ which are the spectra of affine blowup algebras $A[s][\frac{J}{s}]$ and $A[s][\frac{J}{f}]$, see Divisors, Lemma 31.32.2 and Algebra, Definition 10.70.1. We will describe each of these affine opens and this will finish the proof.

The open $D_+(S)$, i.e., the spectrum of $A[s][\frac{J}{s}]$. It follows from the description of the powers of $J$ above that

\[ A[s][\textstyle {\frac{J}{s}}] = \sum s^{-n}I^ n[s] \subset A[s, s^{-1}] \]

The element $s$ is a nonzerodivisor in this ring, defines the exceptional divisor $E$ as well as $W_\infty $. Hence $D \cap D_+(S) = \emptyset $. Finally, the quotient of $A[s][\frac{J}{s}]$ by $s$ is the conormal algebra

\[ A/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots \]

This proves (7).

The open $D_+(f^{(1)})$, i.e., the spectrum of $A[s][\frac{J}{f}]$. It follows from the description of the powers of $J$ above that

\[ A[s][\textstyle {\frac{J}{f}}] = A[\textstyle {\frac{I}{f}}][\textstyle {\frac{s}{f}}] \]

where $\frac{s}{f}$ is a variable. The element $f$ is a nonzerodivisor in this ring whose zero scheme defines the exceptional divisor $E$. Since $s$ defines $W_\infty $ and $s = f \cdot \frac{s}{f}$ we conclude that $\frac{s}{f}$ defines the divisor $D$ constructed above. Then we see that

\[ D \cap D_+(f^{(1)}) = \mathop{\mathrm{Spec}}(A[\textstyle {\frac{I}{f}}]) \]

which is the corresponding open of the blowup $X'$ over $\mathop{\mathrm{Spec}}(A)$. Namely, the surjective graded $A[s]$-algebra map $B \to A \oplus I \oplus I^2 \oplus \ldots $ to the Rees algebra of $(A, I)$ corresponds to the closed immersion $X' \to W$ over $\mathop{\mathrm{Spec}}(A[s])$. This proves $D = X'$ as desired.

Let us prove (6). Observe that the zero scheme of $\frac{s}{f}$ in the previous paragraph is the restriction of the zero scheme of $S$ on the affine open $D_+(f^{(1)})$. Hence we see that $S = 0$ defines $X' \cap E$ on $E$. Thus (6) follows from (5).

Finally, we have to prove the last part of (8). This is clear because the map $\mathbf{P}^1_ Z \to W$ is affine locally given by the surjection

\[ B \to B \otimes _{A[s]} A/I = (A/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots )[S] \to A/I[S] \]

and the identification $\text{Proj}(A/I[S]) = \mathop{\mathrm{Spec}}(A/I)$. Some details omitted.

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