Suppose that $C\subset \mathbb{R}^3$ is an irreducible real-algebraic curve that meets a plane infinitely many times. Is it true that $C$ must be contained in this plane?
I'm working on a paper that's loosely related to algebraic geometry, but I am probably lacking even basic knowledge about algebraic curves.
This response is based on a comment from paul blart math cop. One definition of dimension is the maximal length $d$ of a chain $V_0\subsetneq...\subsetneq V_d=V$ of irreducible closed subsets. Since an algebraic curve has degree $1$ and points are closed, there are no irreducible algebraic varieties of intermediate size.
Assume the intersection of $C$ with the plane $P$ contains infinitely many points. Because every algebraic set may be written as the union of finitely many irreducible algebraic sets, $C\cap P$ must contain an irreducible algebraic variety with more than one point. But because $C$ had dimension $1$, this must be the whole curve, proving the claim.