Suppose $n \ge 3$ and $X$ is a path-connected subset of $\Bbb{CP}^n$ under the manifold topology. If for every complex hyperplane $H \subset \Bbb{CP}^n$, $H \cap X$ is a degree $k$ algebraic hypersurface in $H$ (could be singular or reducible), is it true that $X$ itself is a degree $k$ algebraic hypersurface in $\Bbb{CP}^n$? This question arose when I was constructing an algebra-free definition of quardric surface (where $n=3$ and $k=2$ respectively), and I have no idea how to prove or disprove this.

**Appendix**: The $n=2$ case can be easily proved to be false as stated in the comment. However the reason might be that a "hypersurface" in $\Bbb{CP}^1$ is just a set of finite points, thus no enough rigidity occurs. I wonder whether things change in higher dimensions.

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fixednumber of points (possibly with multiplicity), say $k$. Is there a $X$ with this property that is not an algebraic curve of degree $k$? $\endgroup$2more comments