Lines on projective varieties and applications

TL;DR

This paper reviews properties of the variety of lines on projective varieties, proves a key coincidence for quadratic varieties, and explores extensions of projective manifolds with applications to homogeneous manifolds.

Contribution

It provides a detailed proof linking the variety of lines to the second fundamental form for quadratic varieties and investigates manifold extensions using line geometry.

Findings

Variety of lines coincides with the base locus of the second fundamental form in quadratic cases

Extension techniques for embedded projective manifolds are developed

Applications to homogeneous manifolds are demonstrated

Abstract

The first part of this note contains a review of basic properties of the variety of lines contained in an embedded projective variety and passing through a general point. In particular we provide a detailed proof that for varieties defined by quadratic equations the base locus of the projective second fundamental form at a general point coincides, as a scheme, with the variety of lines. The second part concerns the problem of extending embedded projective manifolds, using the geometry of the variety of lines. Some applications to the case of homogeneous manifolds are included.