Hassett–Tschinkel correspondence: Modality and projective hypersurfaces

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