On Normality of Projective Hypersurfaces with an Additive Action

Abstract

We study projective hypersurfaces $X$ admitting an induced additive action, i.e., an effective action ${\mathbb{G}}_a^m\times X\to X$ of the vector group ${\mathbb{G}}_a^m$ with an open orbit that can be extended to an action on the ambient projective space. A criterion for normality of such a hypersurface $X$ is given. Also, we prove that for any projective hypersurface $Z$ there exists a hypersurface $X$ with an induced additive action such that the complement to the open ${\mathbb{G}}_a^m$-orbit in $X$ is a projective cone over $Z$. We introduce a construction that produces non-degenerate hypersurfaces with induced additive action from Young diagrams and study the properties of the hypersurfaces obtained in this way.