On Normality of Projective Hypersurfaces with an Additive Action
Ivan Arzhantsev, Ivan Beldiev, Yulia Zaitseva
TL;DR
This work analyzes projective hypersurfaces $X$ in $\mathbb{P}^n$ that admit induced additive actions by a vector group. It derives a practical normality criterion based on the top two homogeneous parts $f_d$ and $f_{d-1}$ of the defining equation, and links $X$ to its boundary $X_0$ via a decomposition $f=\sum_{k=1}^d z_0^{d-k} f_k$ with $f_d$ governing the boundary. The authors establish a universal boundary-cone construction showing that any base hypersurface $Z$ can arise as the cone over $X_0$, and develop a rich Young-diagram–driven framework for generating non-degenerate hypersurfaces through $H$-pairs. They further elucidate maximal-degree hypersurfaces, proving degree $d$ is at most $n$ with a unique such hypersurface, normal only when $n\le 2$, and provide explicit equations in these extreme cases. Overall, the paper ties additive actions to local algebras and combinatorial data, delivering new normal examples and a systematic method to study the geometry of projective hypersurfaces with additive symmetries.
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.