Blowing up a rational point of a smooth surface

By [F1] the local ring $A={\mathcal{O}}_{S,p}$ is regular, and by [F2] it has dimension and embedding dimension two, so there are regular parameters $\bar{x},\bar{y}$; lifting them along $R\to R_{{\mathfrak{m}}_p}=A$ and clearing denominators gives $x,y\in R$ with these images, and the failure loci of the conditions below are closed subsets of the affine scheme $U$ not containing $p$, so $U$ may be shrunk while keeping $p$. Arrange that (i) $U$ is connected and $R$ is a domain: every local ring of the smooth surface $S$ is a domain by [F1] and [F2], so a connected affine open neighbourhood of $p$ has domain ring; (ii) $(x,y)$ generates the ideal of $p$ on $U$, which holds at $p$ because the images generate ${\mathfrak{m}}_p$ and the locus where the two coherent ideals differ is closed and avoids $p$; (iii) $(x,y)$ and $(y,x)$ are regular sequences in $R$, namely $x$ and $y$ are nonzerodivisors and each is a nonzerodivisor modulo the other: this holds at $p$ because $\bar{x},\bar{y}$ is a regular system of parameters in the Cohen-Macaulay ring $A$ by [F2], and each failure is the support of the kernel of multiplication on a coherent module, a closed subset avoiding $p$. Thus a sufficiently small affine neighbourhood and functions as in the statement exist.

Let $Z=V(xv-yu)\subseteq U{\times }_k{\mathbb{P}}_k^1$. By [F7] the two charts of the projective factor give $Z\cap \{u\mathrm{\ne }0\}=\mathrm{Spec}R[T]\mathrm{/}(xT-y)$ with $T=v\mathrm{/}u$, and $Z\cap \{v\mathrm{\ne }0\}=\mathrm{Spec}R[U_1]\mathrm{/}(y{U}_1-x)$ with $U_1=u\mathrm{/}v$; on the overlap both $T$ and $U_1$ are invertible and $T{U}_1=1$, so $Z$ is obtained by gluing these two affine charts along $R[T,T^{-1}]\mathrm{/}(xT-y)$. On the other hand, by [F4] the standard charts of ${\mathrm{Bl}}_{(x,y)}U$ are $\mathrm{Spec}R[(x,y)\mathrm{/}x]$ and $\mathrm{Spec}R[(x,y)\mathrm{/}y]$ with overlap $R[(x,y)\mathrm{/}x][x\mathrm{/}y]$. Since $(x,y)$ is a regular sequence in the domain $R$ by step 1.1, the homomorphism $R[T]\mathrm{/}(xT-y)\to R[(x,y)\mathrm{/}x]$, $T\mapsto y\mathrm{/}x$, is an isomorphism: it is surjective with kernel the $x$-power torsion by [F4], and a coefficient comparison in a relation $xg=(xT-y)h$, using that $y$ is a nonzerodivisor modulo $x$, shows $g\in (xT-y)$, so no nonzero torsion exists; symmetrically $R[U_1]\mathrm{/}(y{U}_1-x)\cong R[(x,y)\mathrm{/}y]$ via $U_1\mapsto x\mathrm{/}y$. These identifications carry $T\mapsto y\mathrm{/}x$ and $U_1\mapsto x\mathrm{/}y$, matching the ratio identifications of the blowup charts, so by [F6] they glue to an isomorphism $Z\to {\mathrm{Bl}}_{(x,y)}U$ over $U$, canonical because both sides are determined by the same chart data.

By [F5] the restriction of the blowup of $S$ at $p$ to the open $U$ is ${\mathrm{Bl}}_{(x,y)}U$, so step 2.1 identifies it with the incidence subscheme $V(xv-yu)$ and gives the two charts $\mathrm{Spec}R[T]\mathrm{/}(xT-y)=\mathrm{Spec}R[y\mathrm{/}x]$ and $\mathrm{Spec}R[U_1]\mathrm{/}(y{U}_1-x)=\mathrm{Spec}R[x\mathrm{/}y]$ with $T{U}_1=1$, the descriptions displayed in the statement. Base change to $\mathrm{Spec}A$ replaces $R$ by $A$: the formulas $\mathrm{Spec}A[T]\mathrm{/}(xT-y)$ and $\mathrm{Spec}A[U_1]\mathrm{/}(y{U}_1-x)$ are exactly the local charts of [F8], and over this affine base the structure-sheaf pushforward is $A$ with vanishing higher direct images by [F9].

Smoothness and the local structure of $E$. Since $S$ is smooth over $k$ and $p$ is $k$-rational, [F8] gives that ${\mathrm{Bl}}_{p}S$ is smooth over $k$ with exceptional curve $E\cong {\mathbb{P}}_k^1$ and ${\mathcal{O}}_E(E)=\mathcal{O}(-1)$, and that the local rings on $E$ have dimension one at its generic point and two at closed points. The two charts of step 3.1 cover ${\pi }^{-1}(U)$ and are open subschemes of ${\mathrm{Bl}}_{p}S$; smoothness is local on the source by [F1], so each chart is a smooth surface over $k$. The centre is a single point, so by [F11] the blowup is an isomorphism away from $p$, and the chart descriptions of steps 2.1 and 3.1 glue to the global blowup. In the model $S={\mathbb{A}}_k^2$, $p=0$ with the coordinate functions $x,y$, [F10] gives literally $\mathrm{Spec}k[x,T]$ and $\mathrm{Spec}k[y,U_1]$ inside ${\mathbb{A}}_k^2{\times }_k{\mathbb{P}}_k^1$.

Steps 1.1-4.1 prove the statement: a sufficiently small affine neighbourhood $U=\mathrm{Spec}R$ with regular parameters $x,y$ generating the ideal of $p$ exists, over $U$ the blowup is the incidence subscheme $V(xv-yu)\subseteq U{\times }_k{\mathbb{P}}_k^1$ with charts $\mathrm{Spec}R[T]\mathrm{/}(xT-y)=\mathrm{Spec}R[y\mathrm{/}x]$ and $\mathrm{Spec}R[U_1]\mathrm{/}(y{U}_1-x)=\mathrm{Spec}R[x\mathrm{/}y]$ glued by $T{U}_1=1$, the base change to $\mathrm{Spec}{\mathcal{O}}_{S,p}$ is obtained by replacing $R$ by ${\mathcal{O}}_{S,p}$, the charts are smooth surfaces over $k$, the local rings on $E$ have the asserted dimensions, and the plane model has the two affine-plane charts inside ${\mathbb{A}}_k^2{\times }_k{\mathbb{P}}_k^1$.