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

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.