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Polymatroids and moduli of points in flags

Patricio Gallardo, Javier González-Anaya, José Luis González

Abstract

We introduce and study different compactifications of the moduli space of $n$ distinct weighted labeled points in a flag of affine spaces. We construct these spaces via the weighted and generalized Fulton-MacPherson compactifications of Routis and Kim-Sato. For certain weights, our compactifications are toric and isomorphic to the polypermutohedral and polystellahedral varieties, which arise in the work of Crowley-Huh-Larson-Simpson-Wang and Eur-Larson on polymatroids, a generalization of matroids. Moreover, we show that these toric compactifications have a fibration structure, with fibers isomorphic to the Losev-Manin space, and are related to each other via a geometric quotient.

Polymatroids and moduli of points in flags

Abstract

We introduce and study different compactifications of the moduli space of distinct weighted labeled points in a flag of affine spaces. We construct these spaces via the weighted and generalized Fulton-MacPherson compactifications of Routis and Kim-Sato. For certain weights, our compactifications are toric and isomorphic to the polypermutohedral and polystellahedral varieties, which arise in the work of Crowley-Huh-Larson-Simpson-Wang and Eur-Larson on polymatroids, a generalization of matroids. Moreover, we show that these toric compactifications have a fibration structure, with fibers isomorphic to the Losev-Manin space, and are related to each other via a geometric quotient.

Paper Structure

This paper contains 18 sections, 34 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Polymatroids, polypermutohedra and polystellahedra
  5. Wonderful compactifications
  6. Fulton-MacPherson compactifications and related constructions
  7. Moduli of points in a flag and polypermutohedral varieties
  8. Weighted Fulton-MacPherson compactification of points in a flag
  9. Moduli of $n$ weighted points in a flag up to translation and scaling
  10. The Losev-Manin compactification and the polypermutohedral variety
  11. Pullback of building sets and nontrivial toric fibrations
  12. Proof of Theorem \ref{['GFM: thm:weightedFM_forFlags']}
  13. Pullback of building sets and refinements
  14. Generalized Fulton-MacPherson compactification and polystellahedral varieties
  15. Generalized Fulton-MacPherson compactification
  16. ...and 3 more sections

Key Result

Theorem 1.1

Consider an $n$-tuple $\mathbf{a} = (a_1, \dots, a_n) \in \mathbb{Z}_{>0}^n$ with $a_{i+1} \leq a_{i}$, and a weight vector $\mathbf{w} \in \mathcal{D}^{\operatorname{T}}_n$ (Definition pp: twa weights). There exists a smooth, normal crossings, geometric compactification $T^{\mathbf{a}}_{\mathbf{w}}

Theorems & Definitions (113)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: cf. ferroni2022matroids
  • Definition 2.4: cf. eur2024intersection
  • Definition 2.5
  • Proposition 2.6: schrijver2003combinatorial
  • Example 2.7: Permutohedra and stellahedra as polymatroids
  • Definition 2.8: eur2024intersection
  • ...and 103 more