Lines on projective varieties and applications

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.

15 pages. One example removed; one remark and some references added; typos corrected

References in corpus (7)

• Varieties with quadratic entry locus, II
• Varieties with quadratic entry locus, I
• On projective varieties $n$-covered by curves of degree $\delta$
• Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras
• Manifolds covered by lines, defective manifolds and a restricted Hartshorne Conjecture
• Fubini-Griffiths-Harris rigidity of homogeneous varieties
• Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity

Cited by in corpus (2)

• On special quadratic birational transformations of a projective space into a hypersurface
• Quadro-quadric special birational transformations from projective spaces to smooth complete intersections