# Understanding $Bl_{\mathcal{I}}(k^4)/S_2$ where $\mathcal{I}$ is defined by $(x_1-x_2,x_3-x_4)$

Let $k_4=Spec(k[x_1, x_2, x_3,x_4])$ and $\mathcal{I}$ is the ideal sheaf defined by $(x_1-x_2,x_3-x_4)$. Then

$$Bl_{\mathcal{I}}(k^4) = Proj (\oplus_{i\geq 0} I^i t^i)$$ where $I=(x_1-x_2,x_3-x_4)$.

Then considering $Bl_{\mathcal{I}}(k^4)/S_2$ where $S_2$ is the symmetric group on $2$ letters, how do we write or see $$Bl_{\mathcal{I}}(k^4)/S_2$$ as $Proj$? The $S_2$ action on $Bl_{\mathcal{I}}(k^4)$ is by interchanging $(x_1,x_3)\in k^2$ with $(x_2, x_4)\in k^2$?



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