Monomial algebras and Gan\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {G}_a^n$$\end{document}-equivariant e

Abstract

An induced additive action on a projective variety X⊆Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\subseteq \mathbb {P}^{n}$$\end{document} is a regular action of the group Gan\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {G}_a^n$$\end{document} on X with an open orbit that can be extended to a regular action on Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}^{n}$$\end{document}. 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.