AdditiveToricVarieties: A Macaulay2 package for working with additive complete toric varieties

TL;DR

This paper presents AdditiveToricVarieties, a Macaulay2 package that identifies additive group actions on complete toric varieties, providing new algorithms and classifications for smooth Fano toric varieties up to dimension 6.

Contribution

It introduces algorithms for detecting additive actions on toric varieties and classifies smooth Fano cases up to dimension 6, expanding computational tools in algebraic geometry.

Findings

Every smooth complete toric variety of Picard rank two is additive.

Algorithms successfully determine additive actions on various toric varieties.

Complete classification of smooth Fano toric varieties of dimension up to 6 with additive actions.

Abstract

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. More precisely, we implement algorithms, based on results by Arzhantsev, Dzhunusov and Romaskevich, to determine whether a complete toric variety admits an action of the commutative unipotent group and whether it is unique or not. We also observe that every smooth complete toric variety of Picard rank two is additive. We apply our methods to the class of smooth Fano toric varieties and notably determine all such varieties of dimension up to 6 admitting an additive action.