Projective Hypersurfaces of High Degree Admitting an Induced Additive Action

We study induced additive actions on projective hypersurfaces, i.e. effective 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. It is known that the degree of a hypersurface $X\subseteq {\mathbb{P}}^n$ admitting an induced additive action cannot be greater than $n$ and there is a unique such hypersurface of degree $n$. We give a complete classification of hypersurfaces $X\subseteq {\mathbb{P}}^n$ admitting an induced additive action of degrees from $n-1$ to $n-3$.