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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".)

user134824
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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 .

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Georges Elencwajg
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