Higher-order tangent lines to smooth hypersurfaces

We prove that for every smooth hypersurface $X\subset {\mathbb{P}}^n$ of degree $d\ge n\ge 5$ over an algebraically closed field of characteristic zero, the space of pointed lines $(\mathrm{\ell },p)$ having intersection multiplicity at least six with $X$ at $p$ has the expected dimension $2n-7$. The proof uses deformations of ruled surfaces preserving prescribed contact with $X$. As an application, we show that a smooth hypersurface of degree $d\ge 6$ in ${\mathbb{P}}^5$ contains at most $225d^3-1370d^2+1800d$ planes. This bound has optimal order of growth in $d$.