Let $K$ be an algebraically close field, and let $X\subset \mathbb{P}^2(k)$ be an irreducible curve of degree $d$. We know that a general line in $\mathbb{P}^2$ intersects $X$ in $d$ distinct points. Actually, if $X$ is not strange, we also know something stronger:
For any $P \in \mathbb{P}^2$, the general line in the $\mathbb{P}^1$ of lines contaning $P$ intersects $X$ in $d$ distinct points.
I was wondering if something analogous for curves in higher dimensional spaces holds. More precisely, I asked myself the following question:
Consider a curve (say,irreducible, smooth) $X\subset \mathbb{P}^n$ of degree $d>n$. What is the largest number $r$ such that for any set of points $\{P_1,\cdots,P_r\}$ in $\mathbb{P}^n$ with the property that a general hyperplane in the family of all hyperplanes containing the points $P_1,\cdots,P_r$ intersects $X$ in $d$ distinct points?
My (naive) guess is $r=n-1$, but I'm not sure this is correct.
$r=1$ in all cases. If $r=2$ (similar arguments for $r>2$) then take two points on some tangent line to the curve. Then any hyperplane containing these will also contain the tangent line and thus tangential to the curve, giving smaller number of points of intersection.