Double cone with countably many cones instead of two

Question

Is there a notion of a singularity of a real "algebraic variety" which looks locally like a double cone but with a countable infinite number of cones which meet in the same point? A short example or reference would be highly appreciated.

I am self-studying algebraic geometry at the moment and the literature/videos do not treat this topic and I am lacking the necessary intuition.

Answer 1

Any set in $\Bbb R^n$ cut out by polynomial equations $f_1,\cdots,f_m$ has some strong finiteness properties: the number of irreducible components must be finite, as these correspond to minimal prime ideals of the ring $\Bbb R[x_1,\cdots,x_n]/\sqrt[\Bbb R]{(f_1,\cdots,f_m)}$, which is noetherian (being a quotient of the noetherian ring $\Bbb R[x_1,\cdots,x_n]$) and hence has finitely many minimal prime ideals. So infinitely many cones meeting at a point in $\Bbb R^n$ cannot be cut out by polynomial equations.

Answer 2

A semialgebraic set has finitely many connected components.

Assume you have $X$ semialgebraic with such a conic point, then $(X\cap B) \backslash\{P\}$ would be still semialgebraic, and have infinitely many components, contradiction.

Real algebraic geometry studies such sets.