Lines in affine toric varieties

We prove that up to automorphisms of the target the affine line A1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{A}^1}$$\end{document} admits a unique embedding into the regular part of an affine simplicial toric variety of dimension at least 4 which is smooth in codimension 2. This is an analog of the well-known result on the existence of a linearization of any polynomial embedding A1↪An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{A}^1}\hookrightarrow{\mathbb{A}^n}$$\end{document} for n ≥ 1.