An upper bound for the X-ranks of points of P^n in positive characteristic

Let $Xsubset mathbb {P}^n$ be an integral and non-degenerate $m$-dimensional variety.For any $Pin mathbb {P}^n$ the $X$-rank $r_X(P)$ is the minimal cardinalityof $Ssubset X$ such that $Pin langle S angle$. Here we study thepairs $(X,P)$ such that $r_X(P) ge n+2-m$, i.e. $r_X(P)=n+2-m$. These pairs exist onlyin positive characteristic, with $X$ strange and $P$ a strange point of $X$.