Gorenstein local algebras and additive actions on projective hypersurfaces

Let us say that an additive action on an algebraic variety $X$ is an effective regular action with an open orbit on $X$ of a commutative unipotent linear algebraic group. In 1999, Hassett and Tschinkel established a correspondence between artinian local commutative associative algebras with unit and additive actions on projective spaces. It turns out that this approach may be applied to the study of additive actions on other projective varieties. In this talk we discuss additive actions on projective hypersurfaces. The case of non-degenerate hypersurfaces corresponds to Gorenstein local algebras. We plan to present several results on existence and uniqueness of additive actions. The talk is based on a joint work with Julia Zaitseva.