Projective Spectrum of $K[X,Y]$ Let's assume that $K$ is algebraically closed.
I'm having some difficulties figuring out what $\text{proj}\;K[X,Y]$ is, where $K[X,Y]$ is interpreted as a graded ring.
Any hints? So far I have only figured out that $(X,Y)$ obviously cannot be in $\text{proj}\;K[X,Y]$ since it contains all the elements of $K[X,Y]$ without summands coming from the ground field. Also, $(aX + Y)$ is prime and thus should be in the projective spectrum if I am not mistaken. What about the ideals generated by higher powers of $X$ and $Y$?
However, I fail to see how the fact that $K$ is algebraically closed comes to play.
 A: The only homogeneous prime ideals generated by one element are of the form $(bX-aY)$ for $(a,b) \neq (0,0)$ (these are maximal) and $(0)$ (which is not maximal). All other principal ideals will not be prime because $K$ is algebraically closed, so you will be able to factor the polynomial. For instance, with $K = \mathbb{C}$, you have $(X^2+Y^2) = (X+iY)(X-iY)$.
All other prime ideals, that is to say those that are not generated by at most one element, will contain the irrelevant ideal $(X,Y)$.
Hence $\mathrm{Proj}(K[X,Y])$ is what you'd expect - it contains a point $(a:b) = (ka:kb)$ (for $0 \neq k \in K$) for each maximal ideal $(bX-aY) = (kaX-kaY)$, and so the closed points form a projective line over $K$, and then you also have the ideal $(0)$ which is the generic point of the whole projective line. It's a very similar answer to the case of $\mathrm{Spec}(K[X])$, where you have a point for each $(X-a)$ and the generic point $(0)$. The difference is that with Proj, we also get a point at infinity, and so we get the projective line over $K$ instead of the affine line over $K$. 
A: Another way to help understand the Proj construction better is to understand the homogenization / dehomogenization process that relates to the standard cover by affine opens.  For $\text{Proj } K[x,y] \cong \mathbb{P}^1$, there are two open sets $U_0, U_1 \cong \mathbb{A}^1$.  It is worthwhile to understand which homogeneous prime ideals in $\text{Proj } K[x,y]$ correspond to prime ideals in $U_0$ and which correspond to prime ideals in $U_1$.
This is along the lines of P.A.I.'s last paragraph, with the extra information that there are natural morphisms $\phi_0, \phi_1 : \text{Spec } K[t] \rightarrow \text{Proj } K[x,y]$ that map isomorphically onto $U_0, U_1$.
