What is the quotient space $\mathbb{C}[x,y,z]/(x^2+y^2)$ and more generally how do I determine such spaces (if possible include some reference). Such quotients appear a lot in algebraic geometry and I struggle to understand what these spaces are and "look like". Can you provide a couple of examples?
Note that I am a physicist lacking formal training in abstract algebra. I just want to understand these quotient spaces made out of rings modded out with some polynomials.
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Geometrically it is defined by equation $x^2 + y^2=0$ in three dimensional space $\mathbb{C}^3$. Notice $x^2+y^2 =(x+iy)(x-iy)$. So it is the Union of two (complex) plane given by equations $x+iy=0$ and $x-iy=0$
Algebraically, it is the same as $\mathbb{C} [x,y,z]$, besides that when you see $x^2+y^2$ you replace it by $0$
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I struggle to understand what these spaces are and "look like".
Take the union of two transversely intersecting planes in 3-dimensional real space, such as $x=0$ and $y=0$. This is visualizable. Now allow complex coordinates.
This is equivalent to the space in the question, up to a linear change of coordinates (though the linear changes of variable will have complex coefficients).
What you are considering is the function field determined by a variety. In your case, you are considering the variety determined by $x^2 + y^2$, so the function field is naturally $\mathbf{C}(x,y,z)$, where $x,y,z$ are transcendental subject to the relation $x^2 + y^2 = 0$. By looking for the key term "function field" either in common algebraic textbooks (Hartshorne is my reference) or online, you should be able to find a plethora of supplemental material. You should note that this has origins in complex analysis where the function field is just meromorphic functions. Using this fact, you have a natural generalization in scheme theory, namely considering sheaf of meromorphic functions which Hartshorne discusses in great detail in Chapter 2.