Gorenstein Algebras and Uniqueness of Additive Actions

We study induced additive actions on projective hypersurfaces, i.e. regular actions of the algebraic group ${\mathbb{G}}_a^m$ with an open orbit that can be extended to a regular action on the ambient projective space. We prove that if a projective hypersurface admits an induced additive action, then it is unique if and only if the hypersurface is non-degenerate. We also show that for any $n\ge 2$, there exists a non-degenerate hypersurface in ${\mathbb{P}}^n$ of each degree $d$ from $2$ to $n$.