Visualizing the "pinched plane" Exercise 12.5.I of Ravi Vakil's (free, online) algebraic geometry textbook _The Rising Sea_ studies the spectrum $X$ of the ring $A=k[x^3,x^2,xy,y]$, an example of a scheme that is regular in codimension 1 but is not normal. (These facts can be shown using the normalization map $\mathbb A^2\to X$ induced by the inclusion $A\to k[x,y]$.) Vakil calls this scheme the "pinched plane". But what is it about $X$ that makes it "pinched"? Is there a good way to visualize or understand the geometry of this scheme? (This blog post has names that didn't make the cut, like "crumpled" and "knotted".)

The mental image is that you consider the plane $\mathbb A^2$ as a napkin on a table, pinch that napkin with three fingers at its center and delicately lift your fingers one inch from the table while firmly holding the napkin.
The lifted napkin is then a representation of $X$, with the non normal point literally at your finger tips :-)
The normalization of $X$ is the map sending a point of the original napkin to the corresponding lifted point.
For the technicalities, see the answer here .