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Horospherical stacks and stacky coloured fans

Sean Monahan

TL;DR

The paper builds a comprehensive combinatorial framework for horospherical stacks by introducing stacky coloured fans as the central dictionary between stacks and coloured fans. It establishes precise correspondences between horospherical stacks and stacky coloured fans, and proves that morphisms of stacks correspond to maps of stacky coloured fans, including a robust isomorphism criterion. The authors introduce coloured fantastacks, a Cox-inspired construction, and the decolouration process to relate horospherical stacks to toroidal models, together with a complete description of good moduli spaces in combinatorial terms. Collectively, these results extend toric stack theory to a broader horospherical setting, enabling explicit constructions and computations of stacks, morphisms, and moduli spaces with practical implications for moduli problems and quotients in representation-theoretic contexts.

Abstract

We introduce a combinatorial theory of horospherical stacks which is motivated by the work of Geraschenko and Satriano on toric stacks. A horospherical stack corresponds to a combinatorial object called a stacky coloured fan. We give many concrete examples, including a class of easy-to-draw examples called coloured fantastacks. The main results in this paper are combinatorial descriptions of horospherical stacks, the morphisms between them, their decolourations, and their good moduli spaces.

Horospherical stacks and stacky coloured fans

TL;DR

The paper builds a comprehensive combinatorial framework for horospherical stacks by introducing stacky coloured fans as the central dictionary between stacks and coloured fans. It establishes precise correspondences between horospherical stacks and stacky coloured fans, and proves that morphisms of stacks correspond to maps of stacky coloured fans, including a robust isomorphism criterion. The authors introduce coloured fantastacks, a Cox-inspired construction, and the decolouration process to relate horospherical stacks to toroidal models, together with a complete description of good moduli spaces in combinatorial terms. Collectively, these results extend toric stack theory to a broader horospherical setting, enabling explicit constructions and computations of stacks, morphisms, and moduli spaces with practical implications for moduli problems and quotients in representation-theoretic contexts.

Abstract

We introduce a combinatorial theory of horospherical stacks which is motivated by the work of Geraschenko and Satriano on toric stacks. A horospherical stack corresponds to a combinatorial object called a stacky coloured fan. We give many concrete examples, including a class of easy-to-draw examples called coloured fantastacks. The main results in this paper are combinatorial descriptions of horospherical stacks, the morphisms between them, their decolourations, and their good moduli spaces.
Paper Structure (40 sections, 26 theorems, 30 equations)

This paper contains 40 sections, 26 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Notation for horospherical varieties
  3. Algebraic group notation
  4. Horospherical varieties
  5. Colours
  6. Coloured fans
  7. Horospherical morphisms and maps of coloured fans
  8. Expanding groups acting on horospherical varieties
  9. Horospherical stacks and stacky coloured fans
  10. Horospherical stacks
  11. Stacky coloured fans
  12. Stacky coloured fan $\to$ horospherical stack
  13. Horospherical stack $\to$ stacky coloured fan
  14. Examples
  15. Horospherical morphisms and maps of stacky coloured fans
  16. ...and 25 more sections

Key Result

Theorem 1

Let $\mathcal{X}_{\Sigma_i^c,\beta_i}$ be horospherical $G$-stacks for $i=1,2$, and suppose that $\pi:\mathcal{X}_{\Sigma_1^c,\beta_1}\to\mathcal{X}_{\Sigma_2^c,\beta_2}$ is a morphism of horospherical stacks. Then there exists a stacky coloured fan $(\Sigma_0^c,\beta_0)$ and a tower of stacky colou

Theorems & Definitions (107)

  • Theorem : \ref{['thm:horospherical morphisms']}
  • Theorem : \ref{['thm:horospherical isomorphisms']}
  • Proposition : \ref{['prop:non-toric condition']}
  • Theorem : \ref{['cor:good moduli space of a horospherical stack']}
  • definition 3.1: Horospherical stack
  • remark 3.2: Open $G$-orbit of a horospherical stack
  • remark 3.3: Horospherical varieties are horospherical stacks
  • remark 3.4: Toric stacks are horospherical stacks
  • remark 3.5: Alternative definition of horospherical stack
  • definition 3.6: Stacky coloured fan
  • ...and 97 more