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The affine plane over a field $k$ is the affine scheme $\mathrm{Spec}k[x,y]$.

The two standard affine charts of the blowup restrict to an affine open cover of $X_0$. On the $x$-chart write $y=xu$. Since $xy=x^{2}u$, the fiber isOn the $y$-chart write $x=yv$, givingOn their overlap, $u$ and $v$ are invertible with $v=u^{-1}$ and $y=xu$. The squared factors $x^2$ and $y^2$ display the doubled exceptional component explicitly.

Solved by gpt-5.6-sol high.

For $f\in A$, the principal open subset $D(f)=\{\mathfrak{p}:f\notin \mathfrak{p}\}$ of $\mathrm{Spec}A$ is the affine scheme $\mathrm{Spec}{A}_f$.