Let $k$ be a field. I want to show that these two schemes are isomorphic:
the closed subscheme of $Spec(k[x,y]) \times_k Proj(k[u, v])$ given by $xv = yu$.
and $\underline{Proj}_{X}(O_X \oplus I \oplus I^2 \oplus ...)$ , where $X = \mathbb{A}^2_k, I = (x, y) \subseteq k[x,y]$ = $Proj(k[x, y] [It])$, where $k[x, y] [It]$ is the Rees algebra.
I tried covering them with affine open subsets, and found that the first scheme is covered by $D(u) = Spec(k[x, y, w ] / (xw - y))$ (setting $w = v/u$) and $D(v) = Spec(k[x,y, t] /( x - yt))$ (setting $t= u/v)$
When I try the same thing with the Proj, I get $D(xt) = Spec(k[x, y][It]_{xt : 0})$ and $D(yt) = Spec(k[x, y][It]_{yt : 0})$.
I can't get them to match. The problem is that the Rees algebra is a subring of the polynomial ring, and not a quotient.
I then tried taking the inclusion $k[x, y][It]$ into $k[x, y, t]$, and seeing if the induced map $U \subseteq Proj(k[x_0, y_0, t]) \rightarrow Proj(k[x, y][It])$ is useful (the $x_0$ and $y_0$ are to indicate that they have degree 0). But $Proj(k[x_0, y_0, t])$ is just $Spec(k[x, y])$ so it turned out not to be.