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

Fabián Levicán, Pedro Montero

TL;DR

The paper addresses the problem of classifying additive actions on complete toric varieties and provides a Macaulay2 package, AdditiveToricVarieties, to compute Demazure-root data and normalized additive actions. It implements algorithms based on results of Arzhantsev, Dzhunusov, and Romaskevich, encoding the equivalence between the existence of an additive action on $X_\\Sigma$ and the existence of a complete collection of Demazure roots $\ rak{R}$, with efficient querying via a preprocessed database for smooth Fano toric varieties up to dimension $6$. A key theoretical outcome is that every smooth complete toric variety with Picard number $ ho(X)=2$ is additive, with unique additivity occurring only for $X\\cong \mathbb{P}^1 \times \mathbb{P}^1$, and the paper applies these ideas to classify smooth Fano toric varieties of dimension up to $6$, including new results for dimensions greater than $3$. The package supplies practical tools and a dataset that aligns with GRDB and Øbro classifications, enabling rapid determination of additivity and, where relevant, the complete Demazure-root data for toric varieties and their projective-bundle realizations, via functions like $demazureRoots$, $additiveActions$, $listAdditiveSmoothFanoToricVarieties$, and $listUniquelyAdditiveSmoothFanoToricVarieties$.

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.

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

TL;DR

The paper addresses the problem of classifying additive actions on complete toric varieties and provides a Macaulay2 package, AdditiveToricVarieties, to compute Demazure-root data and normalized additive actions. It implements algorithms based on results of Arzhantsev, Dzhunusov, and Romaskevich, encoding the equivalence between the existence of an additive action on and the existence of a complete collection of Demazure roots , with efficient querying via a preprocessed database for smooth Fano toric varieties up to dimension . A key theoretical outcome is that every smooth complete toric variety with Picard number is additive, with unique additivity occurring only for , and the paper applies these ideas to classify smooth Fano toric varieties of dimension up to , including new results for dimensions greater than . The package supplies practical tools and a dataset that aligns with GRDB and Øbro classifications, enabling rapid determination of additivity and, where relevant, the complete Demazure-root data for toric varieties and their projective-bundle realizations, via functions like , , , and .

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.

Paper Structure

This paper contains 21 sections, 27 theorems, 21 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complete toric varieties
  4. Projective toric varieties
  5. Smooth Fano toric varieties
  6. Betti numbers
  7. Toric varieties with Picard number equal to two
  8. Projective bundles over Hirzebruch surfaces
  9. Package description
  10. demazureRoots
  11. additiveActions
  12. listAdditiveSmoothFanoToricVarieties
  13. listUniquelyAdditiveSmoothFanoToricVarieties
  14. Applications to smooth Fano toric varieties
  15. Surfaces
  16. ...and 6 more sections

Key Result

Proposition 2.2

Let $\rho \in \Sigma(1)$. Define the set Then, $\mathfrak{R}_{\rho}'$ is bounded. In particular, the set $\mathfrak{R}$ of Demazure roots of $\Sigma$ is finite.

Figures (3)

  • Figure 1: The polytope corresponding to $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(1,1))$.
  • Figure 2: The fan of the Hirzebruch surface $\Sigma_d$.
  • Figure 3: The fan of $\Sigma(d;a,b)$.

Theorems & Definitions (59)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 2.7
  • Proposition 2.8
  • Theorem 2.9
  • ...and 49 more