linear section of codimension $k+1$ of a variety of dimension $k$

Question

Let $X$ be a variety of dimension $k$ and degree $d$. If $L$ is a linear subspace of codimension $k+1$ such that $|L\cap X|$ consists of a finite number of points, is there a way to find the maximum number of points for $|L\cap X|$ ? I was thinking the following: if $P \in L\cap X$, then I can find a linear subspace $L'$ of codimension $k$ containing $L$ and a tangent line at $P$, so if $L'\cap X$ consists of a finite number of points, then the intersection multiplicity at $P$ is at least $2$, hence $|L\cap X|\leq |L'\cap X|\leq d-1$. Of course, if I can construct a $L'$ containing more tangent lines, then I could get a bound lower than $d-1$. But my concern is that $L'$ could intersect $X$ in a curve. How can I rule out this case? Is there another approach to tackle this problem?

Answer 1

Let $X$ be a degree $d$ curve contained in a fixed plane $H=\mathbb{P}^2\subset\mathbb{P}^3$, take $L$ to be a generic line contained in $H$.