  Small codimensional embedded manifolds defined by equations of small degree
are Fano and covered by lines. They are complete intersections exactly when the
variety of lines through a general point is so and has the right codimension.
This allows us to prove the Hartshorne Conjecture for manifolds defined by
quadratic equations and to obtain the list of such Hartshorne manifolds. Using
the geometry of the variety of lines through a general point, we characterize
scrolls among dual defective manifolds. This leads to an optimal bound for the
dual defect, which improves results due to Ein. We discuss our conjecture that
every dual defective manifold with cyclic Picard group should also be secant
defective, of a very special type, namely a local quadratic entry locus
variety.
