Point blowups of regular surfaces stay regular, with rational exceptional curves at two-dimensional local rings

Take a Noetherian affine neighbourhood $U=\mathrm{Spec}R$ of $p$, with center ideal $\mathfrak{n}$, and put $A=R_{\mathfrak{n}}={\mathcal{O}}_{S,p}$. The blowup restricts to the blowup on $U$ (Blowups restrict to open subschemes of the base). Flat base change along $R\to A$ identifies its further pullback with ${\mathrm{Bl}}_{\mathfrak{m}}\mathrm{Spec}A$, where $\mathfrak{m}=\mathfrak{n}A$ (Flat base change for blowups, and failure without flatness). The local ring of any point above $p$ is unchanged by this localization, because every element of $R\setminus \mathfrak{n}$ is already a unit there. Thus regularity above $p$ can be checked over $A$. Its dimension is one or two: $p$ cannot be a zero-dimensional irreducible component of the pure two-dimensional regular scheme, and the dimension is at most two. The two dimensions are treated separately below.

Suppose $\dim A=2$ and choose regular parameters $x,y$ generating $\mathfrak{m}$. The standard charts are $A[\mathfrak{m}\mathrm{/}x]$ and $A[\mathfrak{m}\mathrm{/}y]$ (Affine blowup standard charts and overlaps). By the power-torsion presentation of Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, the first is the quotient of $B^{\mathrm{\prime }}=A[T]\mathrm{/}(xT-y)$ by its $x$-power torsion. This torsion is zero. Indeed $A\mathrm{/}(x)$ is a regular one-dimensional local domain and $y$ is nonzero modulo $x$ by [F1]. If $xg=(xT-y)h$ in $A[T]$, reduction modulo $x$ gives $\bar{y}\bar{h}=0$ in the domain $(A\mathrm{/}(x))[T]$, so $h=x{h}_1$; cancelling $x$ in the domain $A[T]$ gives $g=(xT-y)h_1$. Thus multiplication by $x$ is injective on $B^{\mathrm{\prime }}$, and $A[\mathfrak{m}\mathrm{/}x]=B^{\mathrm{\prime }}$. It is a domain because it embeds into $A[1\mathrm{/}x]$. The other chart is symmetrically $A[U]\mathrm{/}(yU-x)$.

In the case $\dim A=1$, the one-dimensional regular local ring $A$ is a discrete valuation ring [F3], so $\mathfrak{m}$ is principal generated by a uniformizer, the center ideal is invertible on a neighbourhood of $p$, and the blowup of an invertible ideal is an isomorphism: $\pi$ is an isomorphism (Blowing up an effective Cartier divisor does nothing). Then $E={\pi }^{-1}(p)$ is the closed point $p$ with its reduced structure $\mathrm{Spec}\kappa (p)$, and it is an effective Cartier divisor.

Consider a prime $Q$ of $B=A[T]\mathrm{/}(xT-y)$ whose contraction $\mathfrak{q}$ in $A$ is not $\mathfrak{m}$. If $x\mathrm{\notin }\mathfrak{q}$, the localized chart is $B{\otimes }_A{A}_{\mathfrak{q}}=A_{\mathfrak{q}}$ by solving $T=y\mathrm{/}x$. If $x\in \mathfrak{q}$, then $y\mathrm{\notin }\mathfrak{q}$; in the localized chart $y=xT$ is a unit, so $x$ is a unit and the chart is $A_{\mathfrak{q}}[1\mathrm{/}x]$. In either case $B_Q$ is a localization of the regular ring $A_{\mathfrak{q}}$ and is regular. This includes contraction $\mathfrak{q}=(0)$, where the local ring is a field.

For a prime $Q$ of $B$ over $\mathfrak{m}$, its preimage $P$ in $C=A[T]$ is either $\mathfrak{m}C$ or $(x,y,h(T))$, where $h$ lifts a monic irreducible polynomial of $\kappa [T]$. The regular local ring $C_P$ has dimension two in the first case and three in the second: the prime chains $(0)⊊(x)⊊\mathfrak{m}C$ and, in the second case, $\mathfrak{m}C⊊P$ give the lower bounds, while the displayed two or three maximal-ideal generators give the upper bounds on dimension. Consequently $x,y$, or $x,y,h$, form a cotangent basis by [F1]. The class of $xT-y$ in $P{C}_P\mathrm{/}(P{C}_P{)}^2$ is the residue of $T$ times the class of $x$ minus the class of $y$, and is nonzero because the coefficient of $y$ is $-1$. The parameter-quotient lemma therefore makes $B_Q=C_P\mathrm{/}(xT-y)$ regular. Its dimension is one for $P=\mathfrak{m}C$ (the generic point of the exceptional curve), and two for $P=(x,y,h)$.

Steps 3.1 and 3.2 prove regularity on the $x$-chart, and the symmetric proof proves it on the $y$-chart; step 1.1 transfers these local rings to all points above $p$. Away from $p$ the blowup is an isomorphism (The blowup is an isomorphism off the center), so $S^{\mathrm{\prime }}$ is regular everywhere. It is pure of dimension two as well. In the dimension-one case this follows from the identity in step 2.2. In the dimension-two case the local charts above $p$ are domains of dimension two: all their prime-local dimensions are at most two, and step 3.2 exhibits dimension-two closed local rings. The exceptional divisor is not an irreducible component, since its chart equation $x$ (or $y$) is a nonzerodivisor. Thus every component meeting the exceptional divisor is the strict transform of the corresponding component through $p$ and has dimension two; every other component is unchanged. The components of the original regular surface all have dimension two, so the same holds for $S^{\mathrm{\prime }}$. The dimension-one local ring at the generic point of $E$ does not contradict pure dimension two.

The exceptional curve is effective Cartier: the pulled-back center ideal ${\mathcal{I}}_p{\mathcal{O}}_{S^{\mathrm{\prime }}}$ is invertible with nonzerodivisor local generators, and $E=V({\mathcal{I}}_p{\mathcal{O}}_{S^{\mathrm{\prime }}})$ is the associated effective Cartier divisor (The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier). In the case $\dim A=2$, the normal cone theorem identifies $E$ with ${\mathrm{Proj}}_Z({\mathrm{gr}}_{{\mathcal{I}}_p}{\mathcal{O}}_S)$ over $Z=\{p\}$ (The exceptional divisor is the projectivized normal cone); on the one-point scheme $Z=\mathrm{Spec}\kappa (p)$ its graded pieces are ${\mathfrak{m}}^n\mathrm{/}{\mathfrak{m}}^{n+1}$, so $E\cong \mathrm{Proj}({\mathrm{gr}}_{\mathfrak{m}}A)$ and [F2] gives $E\cong \mathrm{Proj}\kappa (p)[X,Y]={\mathbb{P}}_{\kappa (p)}^1$ over $\kappa (p)$. In particular $E$ is a regular curve, so the blowup creates no new singular one-dimensional component.

Collecting: $S^{\mathrm{\prime }}$ is regular of pure dimension two and $E$ is effective Cartier by steps 4.1 and 5.1; in the two-dimensional case $E\cong {\mathbb{P}}_{\kappa (p)}^1$ by step 5.1; in the one-dimensional case $\pi$ is an isomorphism and $E=\mathrm{Spec}\kappa (p)$ by step 2.2; and $\pi$ restricts to an isomorphism off $E$ by step 4.1. Only regularity of $S$ and of the local charts was used, never smoothness over a field, so no smoothness claim is made.