I have some problem understanding the Proj construction. I hope I can understand it better by the following example:
Let $I\subset R$ be an ideal, and consider ${\rm Proj}(\oplus I^k)$, which is the blow-up of ${\rm Spec} R$ at $V(I)$. Can we see the closed points of ${\rm Proj}(\oplus I^k)$? For example, let $R=\mathbb C[x,y]$ and $I=(x,y)$. Then for any maximal ideal $J\subset R$ which does not contain $I$, there is a corresponding maximal ideal $\oplus I^kJ$ which corresponds to a closed point away from the exceptional divisor. Then, what are the ideals corresponding to the closed points on the exceptional divisor? I see the exceptional divisor corresponds to $\oplus I^{k+1}$, so these ideals should contain it.
I think a problem to me is, I don’t know how to see the geometry side of an homogeneous ideal. For Spec, it is quite clear to me that the ideal $I \subset R$ can be interpreted as the zero locus $V(I)\subset {\rm Spec} R$, i.e. the common zeros of $f\in I$, and maximal ideals corresponds to minimal closed subschemes, which are just closed points. What is the corresponding version in the Proj construction?
Thanks in advance.