I am attempting to understand the global Proj description of a blowup. The following example is giving me difficulty.
Start by taking $\mathbb{A}^2_{\mathbb{C}} = \text{Spec}(\mathbb{C}[x,y])$ and blow up the origin. We arrive at:
$$X = \text{Proj}(\mathbb{C}[x,y] \oplus (x,y) \oplus (x,y)^2 \oplus \dots) \cong \text{Proj}(\mathbb{C}[x,y][\tilde{x}, \tilde{y}]/(x\tilde{y} - y\tilde{x}))$$ where $\tilde{x}$ and $\tilde{y}$ are of grade 1.
Now in $X$ I want to blow up the ideal sheaf described by $\mathscr{I} = (x,\tilde{y})$. In the two standard affine patches of $X$, we have the ideal $(1)$ where $\tilde{y} \neq 0$ and the ideal $(x, \frac{\tilde{y}}{\tilde{x}})$ where $\tilde{x} \neq 0$. I can compute these locally and glue together, and in doing so I get $$\mathbb{C}[x,y][\tilde{x} : \tilde{y}][\tilde{\tilde{x}} : \tilde{\tilde{y}}]/(x\tilde{y} - y\tilde{x}, x\tilde{x}\tilde{\tilde{y}}-\tilde{y}\tilde{\tilde{x}})$$ where I am using the notation $[\tilde{x} : \tilde{y}][\tilde{\tilde{x}} : \tilde{\tilde{y}}]$ to describe $\mathbb{P}^1 \times \mathbb{P}^1$.
However as remarked above, I want to understand this globally using Proj instead of locally + patching. This is where my trouble is. According to the Proj construction of the blowup, I believe I am supposed to forget the grading on $\mathscr{O}_X$ (i.e. everything in $\mathscr{O}_X$ now has grade 0) and consider $$\text{Proj}(\mathscr{O}_X \oplus \mathscr{I} \oplus \mathscr{I}^2 \oplus \dots)$$ where recall that $\mathscr{I}$ is described by $(x, \tilde{y})$. So I consider the homomorphism $$\mathscr{O}_X[\tilde{\tilde{x}}, \tilde{\tilde{y}}] \rightarrow \mathscr{O}_X \oplus \mathscr{I} \oplus \mathscr{I}^2 \oplus \dots$$ defined by sending $\tilde{\tilde{x}} \rightarrow x$ and $\tilde{\tilde{y}} \rightarrow \tilde{y}$ (where we must remember both the images are in grade 1). The kernel of this map appears to be generated by the relation $x\tilde{\tilde{y}} - \tilde{y}\tilde{\tilde{x}}$. Immediately I am worried because this is not bihomogeneous, but because I forgot the grading on $\mathscr{O}_X$ (or at least, I think I'm supposed to), perhaps this is ok.
Looking at affine charts, I get relations like $\tilde{y} = x \cfrac{\tilde{\tilde{y}}}{\tilde{\tilde{x}}}$ which seem to be just ridiculous: $\tilde{y}$ is not a regular function, only $\frac{\tilde{y}}{\tilde{x}}$ is!
Can anyone help point out where I went wrong?