In this question https://mathoverflow.net/questions/48798/non-finitely-generated-subalgebra-of-a-finitely-generated-algebra I can find an example of a non finitely generated $\mathbb{C}$-algebra, namely $$B=\mathbb{C}[xy,xy^2,xy^3,\cdots]\subset\mathbb{C}[x,y].$$ My problem is that I failed to visualise it geometrically.
What is $X=\rm{Spec}\,\mathbb{C}[xy,xy^2,xy^3,\cdots]$?
I have tried to use the third answer to the question linked to interpret $X$ as a gluing scheme. Anyway, I am still not sure about what it should be, since it seems to me that I am identifying the whole plane $\mathbb{A}^2_{x,y}$ with the point $\star=\rm{Spec}\,\mathbb{C}$ and hence getting a point as well. Anyway, I am not able to say explicitely what the $\rm{Spec}$ of $B$ is, but it seems to me that I can find at least all the closed points lying on the coordinate-axes plus all the infinitesimal variations of the $y$-axis. It doesn't look like a point, so I am a bit confused.
Any help/suggestion will be much appreciated.
Thank you very much!