In this section we study what happens when we blow up a nonsingular point on a surface. We hesitate the formally define such a morphism as a quadratic transformation as on the one hand often other names are used and on the other hand the phrase â€śquadratic transformationâ€ť is sometimes used with a different meaning.

Lemma 54.3.1. Let $(A, \mathfrak m, \kappa )$ be a regular local ring of dimension $2$. Let $f : X \to S = \mathop{\mathrm{Spec}}(A)$ be the blowing up of $A$ in $\mathfrak m$ wotj exceptional divisor $E$. There is a closed immersion

$r : X \longrightarrow \mathbf{P}^1_ S$

over $S$ such that

1. $r|_ E : E \to \mathbf{P}^1_\kappa$ is an isomorphism,

2. $\mathcal{O}_ X(E) = \mathcal{O}_ X(-1) = r^*\mathcal{O}_{\mathbf{P}^1}(-1)$, and

3. $\mathcal{C}_{E/X} = (r|_ E)^*\mathcal{O}_{\mathbf{P}^1}(1)$ and $\mathcal{N}_{E/X} = (r|_ E)^*\mathcal{O}_{\mathbf{P}^1}(-1)$.

Proof. As $A$ is regular of dimension $2$ we can write $\mathfrak m = (x, y)$. Then $x$ and $y$ placed in degree $1$ generate the Rees algebra $\bigoplus _{n \geq 0} \mathfrak m^ n$ over $A$. Recall that $X = \text{Proj}(\bigoplus _{n \geq 0} \mathfrak m^ n)$, see Divisors, Lemma 31.32.2. Thus the surjection

$A[T_0, T_1] \longrightarrow \bigoplus \nolimits _{n \geq 0} \mathfrak m^ n, \quad T_0 \mapsto x,\ T_1 \mapsto y$

of graded $A$-algebras induces a closed immersion $r : X \to \mathbf{P}^1_ S = \text{Proj}(A[T_0, T_1])$ such that $\mathcal{O}_ X(1) = r^*\mathcal{O}_{\mathbf{P}^1_ S}(1)$, see Constructions, Lemma 27.11.5. This proves (2) because $\mathcal{O}_ X(E) = \mathcal{O}_ X(-1)$ by Divisors, Lemma 31.32.4.

To prove (1) note that

$\left(\bigoplus \nolimits _{n \geq 0} \mathfrak m^ n\right) \otimes _ A \kappa = \bigoplus \nolimits _{n \geq 0} \mathfrak m^ n/\mathfrak m^{n + 1} \cong \kappa [\overline{x}, \overline{y}]$

a polynomial algebra, see Algebra, Lemma 10.106.1. This proves that the fibre of $X \to S$ over $\mathop{\mathrm{Spec}}(\kappa )$ is equal to $\text{Proj}(\kappa [\overline{x}, \overline{y}]) = \mathbf{P}^1_\kappa$, see Constructions, Lemma 27.11.6. Recall that $E$ is the closed subscheme of $X$ defined by $\mathfrak m\mathcal{O}_ X$, i.e., $E = X_\kappa$. By our choice of the morphism $r$ we see that $r|_ E$ in fact produces the identification of $E = X_\kappa$ with the special fibre of $\mathbf{P}^1_ S \to S$.

Part (3) follows from (1) and (2) and Divisors, Lemma 31.14.2. $\square$

Lemma 54.3.2. Let $(A, \mathfrak m, \kappa )$ be a regular local ring of dimension $2$. Let $f : X \to S = \mathop{\mathrm{Spec}}(A)$ be the blowing up of $A$ in $\mathfrak m$. Then $X$ is an irreducible regular scheme.

Proof. Observe that $X$ is integral by Divisors, Lemma 31.32.9 and Algebra, Lemma 10.106.2. To see $X$ is regular it suffices to check that $\mathcal{O}_{X, x}$ is regular for closed points $x \in X$, see Properties, Lemma 28.9.2. Let $x \in X$ be a closed point. Since $f$ is proper $x$ maps to $\mathfrak m$, i.e., $x$ is a point of the exceptional divisor $E$. Then $E$ is an effective Cartier divisor and $E \cong \mathbf{P}^1_\kappa$. Thus if $g \in \mathfrak m_ x \subset \mathcal{O}_{X, x}$ is a local equation for $E$, then $\mathcal{O}_{X, x}/(g) \cong \mathcal{O}_{\mathbf{P}^1_\kappa , x}$. Since $\mathbf{P}^1_\kappa$ is covered by two affine opens which are the spectrum of a polynomial ring over $\kappa$, we see that $\mathcal{O}_{\mathbf{P}^1_\kappa , x}$ is regular by Algebra, Lemma 10.114.1. We conclude by Algebra, Lemma 10.106.7. $\square$

Lemma 54.3.3. Let $(A, \mathfrak m, \kappa )$ be a regular local ring of dimension $2$. Let $f : X \to S = \mathop{\mathrm{Spec}}(A)$ be the blowing up of $A$ in $\mathfrak m$. Then $\mathop{\mathrm{Pic}}\nolimits (X) = \mathbf{Z}$ generated by $\mathcal{O}_ X(E)$.

Proof. Recall that $E = \mathbf{P}^1_\kappa$ has Picard group $\mathbf{Z}$ with generator $\mathcal{O}(1)$, see Divisors, Lemma 31.28.5. By Lemma 54.3.1 the invertible $\mathcal{O}_ X$-module $\mathcal{O}_ X(E)$ restricts to $\mathcal{O}(-1)$. Hence $\mathcal{O}_ X(E)$ generates an infinite cyclic group in $\mathop{\mathrm{Pic}}\nolimits (X)$. Since $A$ is regular it is a UFD, see More on Algebra, Lemma 15.121.2. Then the punctured spectrum $U = S \setminus \{ \mathfrak m\} = X \setminus E$ has trivial Picard group, see Divisors, Lemma 31.28.4. Hence for every invertible $\mathcal{O}_ X$-module $\mathcal{L}$ there is an isomorphism $s : \mathcal{O}_ U \to \mathcal{L}|_ U$. Then $s$ is a regular meromorphic section of $\mathcal{L}$ and we see that $\text{div}_\mathcal {L}(s) = nE$ for some $n \in \mathbf{Z}$ (Divisors, Definition 31.27.4). By Divisors, Lemma 31.27.6 (and the fact that $X$ is normal by Lemma 54.3.2) we conclude that $\mathcal{L} = \mathcal{O}_ X(nE)$. $\square$

Lemma 54.3.4. Let $(A, \mathfrak m, \kappa )$ be a regular local ring of dimension $2$. Let $f : X \to S = \mathop{\mathrm{Spec}}(A)$ be the blowing up of $A$ in $\mathfrak m$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

1. $H^ p(X, \mathcal{F}) = 0$ for $p \not\in \{ 0, 1\}$,

2. $H^1(X, \mathcal{O}_ X(n)) = 0$ for $n \geq -1$,

3. $H^1(X, \mathcal{F}) = 0$ if $\mathcal{F}$ or $\mathcal{F}(1)$ is globally generated,

4. $H^0(X, \mathcal{O}_ X(n)) = \mathfrak m^{\max (0, n)}$,

5. $\text{length}_ A H^1(X, \mathcal{O}_ X(n)) = -n(-n - 1)/2$ if $n < 0$.

Proof. If $\mathfrak m = (x, y)$, then $X$ is covered by the spectra of the affine blowup algebras $A[\frac{\mathfrak m}{x}]$ and $A[\frac{\mathfrak m}{y}]$ because $x$ and $y$ placed in degree $1$ generate the Rees algebra $\bigoplus \mathfrak m^ n$ over $A$. See Divisors, Lemma 31.32.2 and Constructions, Lemma 27.8.9. Since $X$ is separated by Constructions, Lemma 27.8.8 we see that cohomology of quasi-coherent sheaves vanishes in degrees $\geq 2$ by Cohomology of Schemes, Lemma 30.4.2.

Let $i : E \to X$ be the exceptional divisor, see Divisors, Definition 31.32.1. Recall that $\mathcal{O}_ X(-E) = \mathcal{O}_ X(1)$ is $f$-relatively ample, see Divisors, Lemma 31.32.4. Hence we know that $H^1(X, \mathcal{O}_ X(-nE)) = 0$ for some $n > 0$, see Cohomology of Schemes, Lemma 30.16.2. Consider the filtration

$\mathcal{O}_ X(-nE) \subset \mathcal{O}_ X(-(n - 1)E) \subset \ldots \subset \mathcal{O}_ X(-E) \subset \mathcal{O}_ X \subset \mathcal{O}_ X(E)$

The successive quotients are the sheaves

$\mathcal{O}_ X(-t E)/\mathcal{O}_ X(-(t + 1)E) = \mathcal{O}_ X(t)/\mathcal{I}(t) = i_*\mathcal{O}_ E(t)$

where $\mathcal{I} = \mathcal{O}_ X(-E)$ is the ideal sheaf of $E$. By Lemma 54.3.1 we have $E = \mathbf{P}^1_\kappa$ and $\mathcal{O}_ E(1)$ indeed corresponds to the usual Serre twist of the structure sheaf on $\mathbf{P}^1$. Hence the cohomology of $\mathcal{O}_ E(t)$ vanishes in degree $1$ for $t \geq -1$, see Cohomology of Schemes, Lemma 30.8.1. Since this is equal to $H^1(X, i_*\mathcal{O}_ E(t))$ (by Cohomology of Schemes, Lemma 30.2.4) we find that $H^1(X, \mathcal{O}_ X(-(t + 1)E)) \to H^1(X, \mathcal{O}_ X(-tE))$ is surjective for $t \geq -1$. Hence

$0 = H^1(X, \mathcal{O}_ X(-nE)) \longrightarrow H^1(X, \mathcal{O}_ X(-tE)) = H^1(X, \mathcal{O}_ X(t))$

is surjective for $t \geq -1$ which proves (2).

Let $\mathcal{F}$ be globally generated. This means there exists a short exact sequence

$0 \to \mathcal{G} \to \bigoplus \nolimits _{i \in I} \mathcal{O}_ X \to \mathcal{F} \to 0$

Note that $H^1(X, \bigoplus _{i \in I} \mathcal{O}_ X) = \bigoplus _{i \in I} H^1(X, \mathcal{O}_ X)$ by Cohomology, Lemma 20.19.1. By part (2) we have $H^1(X, \mathcal{O}_ X) = 0$. If $\mathcal{F}(1)$ is globally generated, then we can find a surjection $\bigoplus _{i \in I} \mathcal{O}_ X(-1) \to \mathcal{F}$ and argue in a similar fashion. In other words, part (3) follows from part (2).

For part (4) we note that for all $n$ large enough we have $\Gamma (X, \mathcal{O}_ X(n)) = \mathfrak m^ n$, see Cohomology of Schemes, Lemma 30.14.3. If $n \geq 0$, then we can use the short exact sequence

$0 \to \mathcal{O}_ X(n) \to \mathcal{O}_ X(n - 1) \to i_*\mathcal{O}_ E(n - 1) \to 0$

and the vanishing of $H^1$ for the sheaf on the left to get a commutative diagram

$\xymatrix{ 0 \ar[r] & \mathfrak m^{\max (0, n)} \ar[r] \ar[d] & \mathfrak m^{\max (0, n - 1)} \ar[r] \ar[d] & \mathfrak m^{\max (0, n)}/\mathfrak m^{\max (0, n - 1)} \ar[r] \ar[d] & 0\\ 0 \ar[r] & \Gamma (X, \mathcal{O}_ X(n)) \ar[r] & \Gamma (X, \mathcal{O}_ X(n - 1)) \ar[r] & \Gamma (E, \mathcal{O}_ E(n - 1)) \ar[r] & 0 }$

with exact rows. In fact, the rows are exact also for $n < 0$ because in this case the groups on the right are zero. In the proof of Lemma 54.3.1 we have seen that the right vertical arrow is an isomorphism (details omitted). Hence if the left vertical arrow is an isomorphism, so is the middle one. In this way we see that (4) holds by descending induction on $n$.

Finally, we prove (5) by descending induction on $n$ and the sequences

$0 \to \mathcal{O}_ X(n) \to \mathcal{O}_ X(n - 1) \to i_*\mathcal{O}_ E(n - 1) \to 0$

Namely, for $n \geq -1$ we already know $H^1(X, \mathcal{O}_ X(n)) = 0$. Since

$H^1(X, i_*\mathcal{O}_ E(-2)) = H^1(E, \mathcal{O}_ E(-2)) = H^1(\mathbf{P}^1_\kappa , \mathcal{O}(-2)) \cong \kappa$

by Cohomology of Schemes, Lemma 30.8.1 which has length $1$ as an $A$-module, we conclude from the long exact cohomology sequence that (5) holds for $n = -2$. And so on and so forth. $\square$

Lemma 54.3.5. Let $(A, \mathfrak m)$ be a regular local ring of dimension $2$. Let $f : X \to S = \mathop{\mathrm{Spec}}(A)$ be the blowing up of $A$ in $\mathfrak m$. Let $\mathfrak m^ n \subset I \subset \mathfrak m$ be an ideal. Let $d \geq 0$ be the largest integer such that

$I \mathcal{O}_ X \subset \mathcal{O}_ X(-dE)$

where $E$ is the exceptional divisor. Set $\mathcal{I}' = I\mathcal{O}_ X(dE) \subset \mathcal{O}_ X$. Then $d > 0$, the sheaf $\mathcal{O}_ X/\mathcal{I}'$ is supported in finitely many closed points $x_1, \ldots , x_ r$ of $X$, and

\begin{align*} \text{length}_ A(A/I) & > \text{length}_ A \Gamma (X, \mathcal{O}_ X/\mathcal{I}') \\ & \geq \sum \nolimits _{i = 1, \ldots , r} \text{length}_{\mathcal{O}_{X, x_ i}} (\mathcal{O}_{X, x_ i}/\mathcal{I}'_{x_ i}) \end{align*}

Proof. Since $I \subset \mathfrak m$ we see that every element of $I$ vanishes on $E$. Thus we see that $d \geq 1$. On the other hand, since $\mathfrak m^ n \subset I$ we see that $d \leq n$. Consider the short exact sequence

$0 \to I\mathcal{O}_ X \to \mathcal{O}_ X \to \mathcal{O}_ X/I\mathcal{O}_ X \to 0$

Since $I\mathcal{O}_ X$ is globally generated, we see that $H^1(X, I\mathcal{O}_ X) = 0$ by Lemma 54.3.4. Hence we obtain a surjection $A/I \to \Gamma (X, \mathcal{O}_ X/I\mathcal{O}_ X)$. Consider the short exact sequence

$0 \to \mathcal{O}_ X(-dE)/I\mathcal{O}_ X \to \mathcal{O}_ X/I\mathcal{O}_ X \to \mathcal{O}_ X/\mathcal{O}_ X(-dE) \to 0$

By Divisors, Lemma 31.15.8 we see that $\mathcal{O}_ X(-dE)/I\mathcal{O}_ X$ is supported in finitely many closed points of $X$. In particular, this coherent sheaf has vanishing higher cohomology groups (detail omitted). Thus in the following diagram

$\xymatrix{ & & A/I \ar[d] \\ 0 \ar[r] & \Gamma (X, \mathcal{O}_ X(-dE)/I\mathcal{O}_ X) \ar[r] & \Gamma (X, \mathcal{O}_ X/I\mathcal{O}_ X) \ar[r] & \Gamma (X, \mathcal{O}_ X/\mathcal{O}_ X(-dE)) \ar[r] & 0 }$

the bottom row is exact and the vertical arrow surjective. We have

$\text{length}_ A \Gamma (X, \mathcal{O}_ X(-dE)/I\mathcal{O}_ X) < \text{length}_ A(A/I)$

since $\Gamma (X, \mathcal{O}_ X/\mathcal{O}_ X(-dE))$ is nonzero. Namely, the image of $1 \in \Gamma (X, \mathcal{O}_ X)$ is nonzero as $d > 0$.

To finish the proof we translate the results above into the statements of the lemma. Since $\mathcal{O}_ X(dE)$ is invertible we have

$\mathcal{O}_ X/\mathcal{I}' = \mathcal{O}_ X(-dE)/I\mathcal{O}_ X \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(dE).$

Thus $\mathcal{O}_ X/\mathcal{I}'$ and $\mathcal{O}_ X(-dE)/I\mathcal{O}_ X$ are supported in the same set of finitely many closed points, say $x_1, \ldots , x_ r \in E \subset X$. Moreover we obtain

$\Gamma (X, \mathcal{O}_ X(-dE)/I\mathcal{O}_ X) = \bigoplus \mathcal{O}_ X(-dE)_{x_ i}/I\mathcal{O}_{X, x_ i} \cong \bigoplus \mathcal{O}_{X, x_ i}/\mathcal{I}'_{x_ i} = \Gamma (X, \mathcal{O}_ X/\mathcal{I}')$

because an invertible module over a local ring is trivial. Thus we obtain the strict inequality. We also get the second because

$\text{length}_ A(\mathcal{O}_{X, x_ i}/\mathcal{I}'_{x_ i}) \geq \text{length}_{\mathcal{O}_{X, x_ i}}(\mathcal{O}_{X, x_ i}/\mathcal{I}'_{x_ i})$

as is immediate from the definition of length. $\square$

Lemma 54.3.6. Let $(A, \mathfrak m, \kappa )$ be a regular local ring of dimension $2$. Let $f : X \to S = \mathop{\mathrm{Spec}}(A)$ be the blowing up of $A$ in $\mathfrak m$. Then $\Omega _{X/S} = i_*\Omega _{E/\kappa }$, where $i : E \to X$ is the immersion of the exceptional divisor.

Proof. Writing $\mathbf{P}^1 = \mathbf{P}^1_ S$, let $r : X \to \mathbf{P}^1$ be as in Lemma 54.3.1. Then we have an exact sequence

$\mathcal{C}_{X/\mathbf{P}^1} \to r^*\Omega _{\mathbf{P}^1/S} \to \Omega _{X/S} \to 0$

see Morphisms, Lemma 29.32.15. Since $\Omega _{\mathbf{P}^1/S}|_ E = \Omega _{E/\kappa }$ by Morphisms, Lemma 29.32.10 it suffices to see that the first arrow defines a surjection onto the kernel of the canonical map $r^*\Omega _{\mathbf{P}^1/S} \to i_*\Omega _{E/\kappa }$. This we can do locally. With notation as in the proof of Lemma 54.3.1 on an affine open of $X$ the morphism $f$ corresponds to the ring map

$A \to A[t]/(xt - y)$

where $x, y \in \mathfrak m$ are generators. Thus $\text{d}(xt - y) = x\text{d}t$ and $y\text{d}t = t \cdot x \text{d}t$ which proves what we want. $\square$

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