  R. Hartshorne conjectured and F. Zak proved that any n-dimensional smooth
non-degenerate complex algebraic variety X in a m-dimensional projective space
P satisfies Sec(X)=P if m<3n/2+2. In this article, I deal with the limiting
case of this theorem, namely the Severi varieties, defined by the conditions
m=3n/2+2 and Sec(X) different from P. I want to give a different proof of a
theorem of F. Zak classifying all Severi varieties: I will prove that any
Severi variety is homogeneous and then deduce their classification and the
following geometric property : the derivatives of the equation of Sec(X), which
is a cubic hypersurface, determine a birational morphism of P.

Algebraic Geometry