  By an additive structure on a hypersurface S in projective space we mean an
effective action of commutative unipotent group on projective space which
leaves S invariant and acts on S with an open orbit. It is known that these
structures correspond to pairs (R,H) of local finite-dimensional algebra R and
a hyperplane H in the maximal ideal of R. We show when a projective
hypersurface of degree 3 has an additive structure and when structure is
unique.
