Special secants to curves

Question

Let $X\subset\mathbb{P}^n$ be a smooth nondegenerate (i.e. not contained in any hyperplane) curve over $\mathbb{C}$. Is it possible that every collection of $n-3$ points on $X$ lies on a $n-1$-secant (i.e. a linear space spanned by $n-1$ points on $X$) that has dimension at most $n-3$? If yes, can $X$ even be chosen to be projectively normal?

For example, for $n=4$, is there a curve $X\subset\mathbb{P}^4$ such that every $p\in X$ lies on a trisecant line?

My hope and intuition is that this cannot happen but I don't have a proof.

Answer 1

I will give an example for $n = 4$. Let $$ S = \mathbb{P}_{\mathbb{P}^1}(\mathcal{O}(-1) \oplus \mathcal{O}(-2)) \subset \mathbb{P}^4 $$ be a cubic scroll and let $$ X \subset S $$ be a curve which is a 3-section of the morphism $S \to \mathbb{P}^1$. Then every point of $X$ lies on a trisecant line (the ruling of the scroll).