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 $L\cap X\leq L'\cap X\leq d1$. Of course, if I can construct a $L'$ containing more tangent lines, then I could get a bound lower than $d1$. 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?

$\begingroup$ You should at least add the hypothesis that the span of $X$ is not a linear space of codimension $k1$. Otherwise your bound is wrong. $\endgroup$– Jason StarrAug 27 '14 at 14:38
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 $H$.