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:
- Does anybody know a collection of similar interesting examples?
- What general techniques can be used to see properties of the given spectra?
