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Recently I came upon this thread:

isolated non-normal surface singularity

There QiL'8 gave the example of the complex plane with two points identified. It is the spectrum of

$$\{f\in \mathbb{C}[x,y]~|~f(0,0) = f(0,1)\}$$

Similar to this construction (in dimension $1$) there is the nodal cubic, given by the spectrum of

$$A = \{f\in \mathbb{C}[x]~|~f(0) = f(1)\}$$

I showed that $A=\mathbb{C}[x(x-1),x(x-1)^2]$ and that this is isomorphic to the 'usual' representation $\mathbb{C}[x,y]/(y^2-x^2(x+1))$. Especially we see it is indeed a variety. Another example I know of is the cuspidal cubic $\{f\in\mathbb{C}[x]~|~f'(0)=0\}$. But I'm sure there are other beautiful examples out there.

The geometric meaning of the constructions is clear, but since the proofs involved some not very interesting calculations I am now looking for a general means to handle examples like this. For instance, I want to see directly that these algebras are finitely generated over $\mathbb{C}$ and I'd like to have tools to 'read off' generators. So here are my questions:

  1. Does anybody know a collection of similar interesting examples?
  2. What general techniques can be used to see properties of the given spectra?
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1 Answer 1

up vote 2 down vote accepted
+100

All these examples are pushouts of schemes along closed immersions, mentioned in the paper by Karl Schwede on Gluing schemes. As far as I know, it is not easy to verify when a fiber product of finite type algebras is again finite type (there are only some results of that type in Ferrands paper), let alone to read off generators.

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