The Stacks project

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

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

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. $\square$