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On the Weights of Root Subgroups of Affine Toric Varieties

Immanuel van Santen

Abstract

Let $X$ be an affine toric variety and let $D(X)$ be the set of weights of all root subgroups. It is known that $D(X)$ together with its embedding into the character group determines $X$ as a toric variety. In this article we prove that $X$ is already determined by the abstract set $D(X)$ together with some additional combinatorial data.

On the Weights of Root Subgroups of Affine Toric Varieties

Abstract

Let be an affine toric variety and let be the set of weights of all root subgroups. It is known that together with its embedding into the character group determines as a toric variety. In this article we prove that is already determined by the abstract set together with some additional combinatorial data.
Paper Structure (13 sections, 20 theorems, 107 equations)

This paper contains 13 sections, 20 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Notation and setup concerning affine toric varieties
  3. General notions
  4. The Weyl group
  5. Decomposition of the affine toric variety
  6. Weights of root subgroups
  7. The subsets $H_{E, \rho, d}, H^{\pm}_{E, \rho}$ and $B_{E, \rho, D}$ of $S_{\rho}$
  8. General results on affine toric varieties
  9. Properties of the subsets HErd, H+-Er, BErD of S_r
  10. Combinatorics of $D(X)$ and the proof of the main theorem
  11. Reduction to the case when $\sigma^\vee$ is strongly convex
  12. The case when $\sigma^\vee$ is strongly convex
  13. The case when a bijection $D(X) \to D(X')$ extends to a group isomorphism $M \to M'$

Key Result

Lemma 3.1

Let $\rho$ be an extremal ray of $\sigma$. Then the group generated by $\sigma^\vee \cap \rho^\bot \cap M$ is equal to $\rho^\bot \cap M$.

Theorems & Definitions (49)

  • Remark 1.1
  • Lemma 3.1: see ReSa2023Maximal-commutativ
  • Lemma 3.2: see ReSa2023Maximal-commutativ
  • Lemma 3.3
  • proof : Proof of Lemma \ref{['lem.Finiteness_condition']}
  • Lemma 3.4
  • proof
  • Claim 1
  • Lemma 3.5
  • proof
  • ...and 39 more