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.