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I have the reducible variety $X=\mathbb{V}(x_1x_2)\subset\mathbb{A}^2$, which is a pair of lines intersecting transversely, and I would like to compute the blow-up at the origin.

The Rees ring of the ideal $(x_1,x_2)\subset k[x_1,x_2]/(x_1x_2)$ is the graded ring $$R:=\frac{k[x_1,x_2]}{(x_1x_2)}\big[x_1t,\,x_2t\big],$$ where $\deg(t)=1$ and $\deg(x_1)=\deg(x_2)=0$. The blow-up of $X$ is $\mathrm{Proj}(R)$. On the affine patch $x_1t\neq 0$, the ring of functions on the blow-up is the degree $0$ piece of $R[1/(x_1t)]$, which is generated (as a $k$-algebra) by $x_1$ and $\frac{x_2}{x_1}$. The relation $x_1x_2=0$ can be written in terms these generators: $x_1x_2=x_1^2\frac{x_2}{x_1}$, so the affine patch $x_1t\neq 0$ is isomorphic to $$ \mathrm{Spec}\;\frac{k\left[x_1,\frac{x_2}{x_1}\right]}{(x_1^2\frac{x_2}{x_1})}=\mathrm{Spec}\;\frac{k[a,b]}{(a^2b)}, $$ where I have written $a=x_1$, $b=\frac{x_2}{x_1}$. This is a problem, because the blow-up of something reduced should still be reduced and this isn't. Where did I go wrong?

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Let $S=k[x,y]$ with $xy=0$, I prefer it this way. Then, $R=S[xt,yt]$ as you said. Now you localize in $xt$, this means $R_{xt}=S[xt,yt,(xt)^{-1}]$. Note that $y\cdot xt = 0$, so $y = y \cdot xt \cdot (xt)^{-1} = 0$ as well. Hence, $R_{xt} = k[x,xt,(xt)^{-1}]$. If you take the degree $0$ part of this ring with respect to the Rees grading, then you should get $(R_{xt})_0=k[x]$, which means that affine patch of the blow-up is just a line.

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