Lines highly tangent to a hypersurface

TL;DR

This paper investigates the properties of lines tangent to a hypersurface in projective space, providing bounds on certain geometric configurations and offering new proofs of existing results.

Contribution

It introduces bounds on the number of planes in hypersurfaces and offers a novel proof of Landsberg's result without moving frames.

Findings

Polynomial upper bound on the number of planes in hypersurfaces in P^5

Characterization of lines tangent to hypersurfaces at high order

Alternative proof of Landsberg's theorem

Abstract

We study spaces of lines that meet a smooth hypersurface X in P^n to high order. As an application, we give a polynomial upper bound on the number of planes contained in a smooth degree d hypersurface in P^5 and provide a proof of a result of Landsberg without using moving frames.