I'm trying to understand the following statement which I read somewhere without proof. Let $C$ be a smooth algebraic curve in $\mathbb{CP}^n$. Define $X_k$ to be the subvariety of $\mathbb{CP}^n$ consisting of points that lie on lines that intersect $C$ at least $k$ times. Then $X_3$ is at most $2$ dimensional.

Here's how I've been trying to understand this result. It is easy to see that $X_2$ is at most $3$-dimensional. Indeed, let $Y \subset C \times C \times \mathbb{CP}^n$ be the closure of the set of points $(v,v',w)$, where $v \neq v'$ and $w$ lies on the line through $v$ and $v'$. Then $Y$ is clearly at most $3$ dimensional and $X_2$ is the projection of $Y$ onto its third factor.

We have $X_3 \subset X_2$, and the desired statement would follow if we could prove that $X_3 \neq X_2$. However, this need not hold -- for instance, $C$ could be a high degree curve in some $\mathbb{CP}^2$ in $\mathbb{CP}^n$ (in that case, of course, the desired result is trivial!). How do I complete this proof?