  The prolongation g^{(k)} of a linear Lie algebra g \subset gl(V) plays an
important role in the study of symmetries of G-structures. Cartan and
Kobayashi-Nagano have given a complete classification of irreducible linear Lie
algebras g \subset gl(V) with non-zero prolongations.
  If g is the Lie algebra aut(\hat{S}) of infinitesimal linear automorphisms of
a projective variety S \subset \BP V, its prolongation g^{(k)} is related to
the symmetries of cone structures, an important example of which is the variety
of minimal rational tangents in the study of uniruled projective manifolds.
From this perspective, understanding the prolongation aut(\hat{S})^{(k)} is
useful in questions related to the automorphism groups of uniruled projective
manifolds. Our main result is a complete classification of irreducible
non-degenerate nonsingular variety with non zero prolongations, which can be
viewed as a generalization of the result of Cartan and Kobayashi-Nagano. As an
application, we show that when $S$ is linearly normal and Sec(S) \neq P(V), the
blow-up of P(V) along S has the target rigidity property, i.e., any deformation
of a surjective morphism Y \to Bl_S(PV) comes from the automorphisms of
Bl_S(PV).

Algebraic Geometry