Commutative algebraic monoid structures on affine spaces

TL;DR

This paper classifies commutative algebraic monoid structures on affine spaces up to dimension 3, constructs new series of such operations in higher dimensions, and explores their connection to additive actions on toric varieties.

Contribution

It provides a classification of commutative monoid structures on affine spaces up to dimension 3 and links these structures to additive actions on toric varieties.

Findings

Classification of monoid operations up to dimension 3

Construction of new monoid series in arbitrary dimensions

Connection established between monoids and additive actions on toric varieties

Abstract

We study commutative associative polynomial operations $\mathbb{A}^n\times\mathbb{A}^n\to\mathbb{A}^n$ with unit on the affine space $\mathbb{A}^n$ over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.