Lines on algebraic varieties

Abstract

A variety $X$ is covered by lines if there exist a finite number of lines contained in $X$ passing through each general point. I prove two theorems. Theorem 1:Let $X^n\subset P^M$ be a variety covered by lines. Then there are at most $n!$ lines passing through a general point of $X$. Theorem 2:Let $X^n\subsetP^{n+1}$ be a hypersurface and let $x\in X$ be a general point. If the set of lines having contact to order $k$ with $X$ at $x$ is of dimension greater than expected, then the lines having contact to order $k$ are actually contained in $X$.