On Normality of Projective Hypersurfaces with an Additive Action

TL;DR

This paper investigates projective hypersurfaces with additive group actions, providing criteria for their normality, constructing examples from Young diagrams, and exploring their geometric properties.

Contribution

It introduces a normality criterion for such hypersurfaces, constructs examples from Young diagrams, and relates hypersurfaces with additive actions to arbitrary projective hypersurfaces.

Findings

Normality criterion for hypersurfaces with additive actions

Existence of hypersurfaces with additive actions related to any projective hypersurface

Construction method using Young diagrams for non-degenerate hypersurfaces

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.