I've been thinking about weird rings recently, and I couldn't answer the following question to myself:
What are the sections of the inclusion $\mathbb{C}\rightarrow \mathbb{C}[[x,y]]^{alg}[\frac{1}{xy}]$ (the $alg$ in the superscript means that I only take those formal power series that are algebraic over $\mathbb{C}(x,y)$; though if you have an answer offhand for the ring of all power series, I suppose that would be interesting too)?
In other words, what are the possible values that $x$ and $y$ can take so that it gives us a well-defined section?
P.S. I put this under algebraic geometry because I'm given to believe that this has something to do with something called the etale stalk.