Suppose $X$ is a proper closed subset of $\mathbb{P}^n\_k$, $P$ a point not in $X$ and a hyperplane $H$. Denote $m\_{P,H}$ the supremum of intersections of straight lines from $P$ to points in $H$ with $X$.


Is $m\_{P,H}$ finite? What is the relation between $m\_{P,H}$ and $X$? 



Intersection of lines and a closed variety in projective space
If you know that your $X$ is an intersection of hypersurfaces of degree $d$, then $m\_P\leq d$ for all $P$ (note that your $H$ does not play any role). Indeed there exists a hypersurface $V$ of degree $d$ containing $X$ but not $P$; then for any line $L$ passing through $P$, $\#X\cap L\leq \#V\cap L\leq d$. In particular you can always take $d=\deg(X)$ (just consider cones).


