Local algebras and unipotent group actions

B.Hassett and Yu.Tschinkel (Int. Math. Res. Notices 20 (1999), 1211-1230) introduced a remarkable correspondence between locally transitive actions of a commutative unipotent algebraic group G and finite-dimensional local algebras. We develop Hassett-Tschinkel's correspondence and calculate
modality of locally transitive G-actions on projective spaces, classify actions of modality one and characterize locally transitive G-actions on projective hypersurfaces of given degree. In particular, actions on projective quadrics are studied. The talk is based on a joint work with Elena Sharoyko.