Classifying additive smooth Fano toric varieties

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero. An irreducible algebraic variety $X$ over $\mathbb{K}$ of dimension $n$ is called additive if it admits a regular action of the additive group $(\mathbb{K}^n, +)$ with an open orbit, and uniquely additive if this action is unique up to isomorphism. Huang and the second author have previously determined all additive smooth Fano toric threefolds. Here we determine all additive and uniquely additive smooth Fano toric varieties of dimension up to $6$ by computational means, and give a detailed classification for dimension up to $4$. To this effect, we introduce the AdditiveToricVarieties package for Macaulay2, a software system for algebraic geometry and commutative algebra, with methods for working with additive group actions on complete toric varieties. Our work relies on results by Arzhantsev, Dzhunusov and Romaskevich, who relate the existence and uniqueness of such actions to conditions on the Demazure roots of the fans corresponding to the toric varieties. We also prove that every smooth complete toric variety of Picard rank two is additive.