Consider a smooth projective curve $C\subset\mathbb{P}^n$. Let $G(1,n)$ the Grassmannian of lines of $\mathbb{P}^n$. The variety $S_2(C)\subset G(1,n)$ parametrizing lines that are secant to $C$ (intersect $C$ in at least two points) has dimension two.

Let $S_3(C)\subset S_2(C)\subset G(1,n)$ be the variety parametrizing lines that intersect $C$ in al least three points. Is there a formula (perhaps depending on $n$, the genus and the degree of $C$) for the dimension of $S_3(C)$? Under which assumption do we have that $\dim(S_3(C)) = 0$?

Consider a smooth projective curve $C\subset\mathbb{P}^n$. Let $G(1,n)$ the Grassmannian of lines of $\mathbb{P}^n$. The variety $S_2(C)\subset G(1,n)$ parametrizing lines that are secant to $C$ (i.e., that intersect $C$ in at least two points) has dimension two.

Let $S_3(C)\subset S_2(C)\subset G(1,n)$ be the variety parametrizing lines that intersect $C$ in al least three points.

Question. Is there a formula (perhaps depending on $n$, the genus and the degree of $C$) for the dimension of $S_3(C)$? Under which assumption do we have that $\dim(S_3(C)) = 0$?