The varieties of tangent lines to hypersurfaces in projective spaces

TL;DR

This paper studies the geometric structure of tangent lines to hypersurfaces in projective spaces, showing that for a general hypersurface, the set of point-line pairs with fixed intersection multiplicity forms a smooth variety.

Contribution

It proves that the set of pairs of points and lines with fixed intersection multiplicity on a hypersurface is a smooth variety for a general hypersurface.

Findings

The set of pairs forms a smooth variety.

The result holds for a general hypersurface.

Provides a geometric description of tangent lines.

Abstract

For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs forms a smooth variety for a general hypersurface.