Additive actions on toric projective hypersurfaces

TL;DR

This paper classifies all projective toric hypersurfaces that admit an additive group action with an open orbit, expanding understanding of symmetries in algebraic geometry.

Contribution

It provides a complete classification of toric hypersurfaces that admit additive actions, a specific type of symmetry in algebraic varieties.

Findings

Identifies all toric hypersurfaces with additive actions

Characterizes the structure of such hypersurfaces

Enhances understanding of group actions on algebraic varieties

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{G}_a$ be the additive group of $\mathbb{K}$. We say that an irreducible algebraic variety $X$ of dimension $n$ over the field $\mathbb{K}$ admits an additive action if there is a regular action of the group $\mathbb{G}_a^n = \mathbb{G}_a \times \ldots \times \mathbb{G}_a$ ($n$ times) on $X$ with an open orbit. In this paper we find all projective toric hypersurfaces admitting additive action.