  B. Hassett and Yu. Tschinkel (1999) introduced a remarkable correspondence
between generically transitive actions of a commutative unipotent algebraic
group G and finite-dimensional local algebras. In this paper we develop
Hassett-Tschinkel correspondence and calculate modality of generically
transitive G-actions on projective spaces, classify actions of modality one,
and characterize generically transitive G-actions on projective hypersurfaces
of given degree. In particular, actions on degenerate projective quadrics are
studied.
