Lines highly tangent to a hypersurface

We study spaces of lines that meet a smooth hypersurface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper P Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {P}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to high order. As an application, we give a polynomial upper bound on the number of planes contained in a smooth degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> hypersurface in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper P Superscript 5"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">P</mml:mi> </mml:mrow> <mml:mn>5</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {P}^5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and provide a proof of a result of Landsberg without using moving frames.