# Degree 1 elements in a graded ring from a blow-up perspective

This may be an elementary question but I hope this question will benefit others as much as myself.

Let $k^4 = Spec \; k[x_1, x_2, x_3, x_4]$.

Writing $Bl_{\mathcal{I}}(k^4)$ as $R = Proj(\oplus_{i\geq 0} I^i)$ where $\mathcal{I}$ is defined by $(x_1-x_2, x_3-x_4)$, what are the degree $1$ elements in $R$?

Are they just $(x_1 - x_2)k[x_1, x_2, x_3, x_4] \oplus (x_3 - x_4)k[x_1, x_2, x_3, x_4]$?



$\mathbf{Added}:$ By the way, if anyone is interested, I am blowing up a certain diagonal subset in affine $4$-space.

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Yes. The degree $n$ component of the blow-up algebra is just a copy of $I^n$. –  YBL Jun 10 '12 at 12:35