Containment of two varieties with a lot of intersection

Question

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there a number $N$ (depending on numerical invariants of the embeddings of $X$ and $C$) so that if $X\cap C$ contains at least $N$ points, then $C\subset X$?

Answer 1

Assume that the variety $X$ and the curve $C$ have degree $a,b$ respectively. Take $N = ab$. Now, if you know that $X$ and $C$ intersect in $k$ points counted with multiplicity with $k >N$ then, by Bézout theorem you get $C\subseteq X$.

For instance, assume $X\subset\mathbb{P}^n$ is variety of degree three with two singular points of multiplicity two. Then the line $L$ joining the two points intersects $X$ at least in four points counted with multiplicity. Then, $L\subset X$.