Higher-order tangent lines to smooth hypersurfaces

We prove that for every smooth hypersurface $X\subset\mathbb{P}^n$ of degree $d\geq n \geq 5$ over an algebraically closed field of characteristic zero, the space of pointed lines $(\ell,p)$ having intersection multiplicity at least six with $X$ at $p$ has the expected dimension $2n-7$. The proof us