The Varieties of Tangent Lines to Hypersurfaces in Projective Spaces

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.

Key Words:

• Fano scheme
• Hypersurface
• Tangent line

2010 Mathematics Subject Classification:

• Primary 14C05
• Secondary 14M15

ACKNOWLEDGMENT

Partially supported by Grant-in-Aid for Young Scientists (B) 20740014, Japan Society for the Promotion of Science.

Notes

Communicated by L. Ein.