Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field

The component through $p$ is open: regular local rings are domains, so distinct irreducible components of the Noetherian regular scheme cannot meet. Choose a domain affine neighborhood of $p$ in that component. Its dimension is two, and the maximal-ideal height theorem gives $\dim A=2$ for $A={\mathcal{O}}_{S,p}$. Choose regular parameters $x,y$. They form a regular sequence; $A\mathrm{/}(x)$ and $A\mathrm{/}(y)$ are one-dimensional regular local domains.

Flat localization of the base identifies the part over $\mathrm{Spec}A$ with the blowup of $(x,y)$. On the $x$-chart, put $C=A[T]\mathrm{/}(xT-y)$. If $xg=(xT-y)h$, reduction modulo $x$ gives $\bar{y}\bar{h}=0$ in the domain $(A\mathrm{/}(x))[T]$, so $h=x{h}_1$; cancellation of $x$ gives $g=(xT-y)h_1$. Hence $C$ has no $x$-power torsion, and the chart algebra theorem identifies $C=A[y\mathrm{/}x]$. It has $C_x=A_x$, exceptional ideal $xC$ and quotient $C\mathrm{/}xC=\kappa [T]$. The second chart is $A[U]\mathrm{/}(yU-x)$ by the same argument, with overlap $U=T^{-1}$. Localization of the base does not change local rings at points over $p$.

Outside $V(x)$ the local rings of $C$ are prime localizations of $A$, hence regular. A prime of $A[T]$ lying over a point of $V(x)$ in $C$ is either $Q=\mathfrak{m}A[T]$ or $Q=(\mathfrak{m},h)$, where $\bar{h}$ is a monic irreducible polynomial over $\kappa$. The ambient local ring $A[T{]}_Q$ is regular. Its maximal ideal is generated respectively by $x,y$ or by $x,y,h$. The prime chains $(0)⊊(x)⊊\mathfrak{m}A[T]$ and, in the second case, their extension by $Q$, together with the embedding-dimension bound, give dimensions two and three. These generators therefore form a cotangent basis. The class of $xT-y$ is $\bar{T}\bar{x}-\bar{y}$, which is nonzero because the coefficient of $\bar{y}$ is $-1$, even when $\bar{T}=0$. Quotienting by this parameter gives regular local rings of dimension one at the generic exceptional point and two at its closed points. The same proof works in the $y$-chart. These computations also show the local chart rings have dimension two, without asserting $\dim A_x=2$.

Away from $p$ the structural morphism is an isomorphism: on the complement of the exceptional ideal in each standard chart its denominator is inverted and the chart becomes the corresponding base principal open, compatibly with the ratio transitions. Thus all local rings of $S^{\mathrm{\prime }}$ are regular. Its charts over finite-type affine bases are finitely generated algebras, so $S^{\mathrm{\prime }}$ is finite type over $k$. Its irreducible components are disjoint and open, as for $S$. No component has generic point in $E$, since the local rings computed there have positive dimension, while a component's generic local ring has dimension zero. Every component consequently meets the unchanged open $S\setminus \{p\}$ and shares the function field of a two-dimensional component of $S$. By the affine-domain dimension formula every nonempty affine open in it has dimension two. This gives dimension two for the component itself: any finite strict chain of irreducible closed subsets remains strict after intersecting an affine open meeting its smallest member, since such an open contains every member's generic point. Therefore $S^{\mathrm{\prime }}$ is pure of dimension two.

The exceptional subscheme is canonically ${\mathrm{Proj}}_{\kappa }{\mathrm{gr}}_{\mathfrak{m}}A$. The multiplication map ${\mathrm{Sym}}_{\kappa }(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)\to {\mathrm{gr}}_{\mathfrak{m}}A$ is an isomorphism: choose any cotangent basis and apply the associated-graded theorem. Thus $E$ is canonically ${\mathbb{P}}_{\kappa }(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)$; the chosen basis $x,y$ identifies it with ${\mathbb{P}}_{\kappa }^1$. The center ideal is $\mathcal{O}(-E)\cong \mathcal{O}(1)$, so $E$ is effective Cartier and its normal line bundle is the restricted negative twist, ${\mathcal{O}}_{{\mathbb{P}}_{\kappa }^1}(-1)$. The projective-line coordinate identification depends on the chosen basis.

If $S$ is smooth over $k$ and $p$ is rational, then for every field extension $K\mathrm{/}k$, $S_K$ is smooth, regular and pure of dimension two, and $p_K$ is a rational closed point. Steps 1.1–4.2 apply over $K$. Flat base change identifies $(S^{\mathrm{\prime }}{)}_K$ with this point blowup, so it is regular for every $K$. The finite-type $k$-algebras of the charts are finitely presented, and they are flat over $k$ because vector spaces are free. Hence the geometric-regularity definition proves smoothness. In the affine-plane model the quotients eliminate $y$ or $x$, giving literal affine planes. For general $p$ only regularity is asserted; only $E$, not the whole blowup, carries the indicated residue-field structure.