Lemma 51.3.1. 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 $\mathfrak m$. 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 30.32.2. 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 26.11.5. This proves (2) because $\mathcal{O}_ X(E) = \mathcal{O}_ X(-1)$ by Divisors, Lemma 30.32.4.

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 10.105.1. 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 26.11.6. 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 30.14.2. $\square$

Comment #4246 by Dario Weißmann on

One could mention in the statement that $E$ is the exeptional divisor of the blowup

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