I have been somewhat shaky as to how exactly we glue $\mathbb{P}_k^n$ out of affine schemes. The explanation of this always seems somewhat guided by knowing what we want...
Take for example $\mathbb{P}_k^1= \operatorname{Proj} k[u,v]$, then we use the standard procedure of gluing (via a unique gluing function) together the affine patches $D^{+}(u)$ and $D^+(v)$ along $D^+(vu)$. I don't have a problem with this, however...
Is it possible for there to exist some other set of affines that glue together to form $\operatorname{Proj} k[u,v]$?