Euler-symmetric projective toric varieties and additive actions

TL;DR

This paper proves that projective toric varieties with additive group actions are exactly the Euler-symmetric varieties, extending the classification and understanding of their geometric properties.

Contribution

It establishes the equivalence between additive actions and Euler-symmetric structure specifically for projective toric varieties, complementing previous classifications.

Findings

Additive actions characterize Euler-symmetric projective toric varieties.

Any projective toric variety with an additive action is Euler-symmetric.

Discussion of properties of Euler-symmetric projective toric varieties.

Abstract

Let $\mathbb{G}_a$ be the additive group of the field of complex numbers $\mathbb{C}$. We say that an irreducible algebraic variety $X$ of dimension $n$ 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 2017 Baohua Fu and Jun-Muk Hwang introduced a class of Euler-symmetric varieties. They gave a classification of Euler-symmetric varieties and proved that any Euler-symmetric variety admits an additive action. In this paper we show that in the case of projective toric varieites the converse is also true. More precisely, a projective toric variety admitting an additive action is an Euler-symmetric variety with respect to any linearly normal embedding into a projective space. Also we discuss some properties of Euler-symmetric projective toric varieties.