Monomial algebras and $\mathbb{G}_a^n$-equivariant embeddings into toric varieties

TL;DR

This paper explores additive group actions on projective toric varieties, especially surfaces, linking them to monomial algebras and providing classifications in low dimensions.

Contribution

It establishes that additive actions correspond to monomial algebras with variable-spanned subspaces on linearly normal toric varieties.

Findings

Additive actions on toric varieties relate to monomial algebras.

On toric surfaces, additive actions are classified in low-dimensional projective spaces.

Abstract

An induced additive action on a projective variety $X\subseteq\mathbb{P}^n$ is a regular action of the group $\mathbb{G}_a^n$ on $X$ with an open orbit that can be extended to a regular action on $\mathbb{P}^n$. Such actions are known to correspond to pairs $(A, U)$, where $A$ is a local algebra and $U$ is a generating subspace lying in the maximal ideal. This paper studies additive actions on projective toric varieties, with a particular focus on toric surfaces. We prove that for any linearly normal toric variety equipped with a torus-normalized additive action, the associated pair consists of a monomial algebra and a subspace spanned by variables. Also we describe pairs that correspond to additive actions on toric surfaces in low-dimensional projective spaces.