The Stacks project

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. $\square$