Additive actions on projective hypersurfaces with a finite number of orbits

TL;DR

This paper classifies projective hypersurfaces that admit a specific type of group action with finitely many orbits, expanding understanding of symmetries in algebraic geometry.

Contribution

It provides a complete classification of hypersurfaces with additive group actions that have finitely many orbits, a new result in the study of algebraic group actions.

Findings

Classified all hypersurfaces with additive actions and finite orbits

Identified conditions for the existence of such actions

Extended understanding of symmetries in projective hypersurfaces

Abstract

An induced additive action on a projective variety $X \subseteq \mathbb{P}^n$ is a regular action of the group $\mathbb{G}_a^m$ on $X$ with an open orbit, which can be extended to a regular action on the ambient projective space $\mathbb{P}^n$. In this work, we classify all projective hypersurfaces admitting an induced additive action with a finite number of orbits.