Lines on projective varieties and applications

The Hilbert scheme of lines passing through a general point of an embedded projective variety
has a natural embedding into the projectivezed tangent space at the point.
When it is not empty and inherits significant extrinsic properties of the original variety,
then interesting structure results appear by reconstructing the original variety from its variety of lines.

For example we shall prove that Hartshorne Conjecture on Complete Intersections holds for manifolds
defined by quadratic equations because its variety of lines through a general point is defined
by quadratic equations and it is a smooth complete intersection.

Other results and some open problems will be discussed by illustrating also in other settings
the above Principle.