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Realizability of matroid quotients

Alessio Borzì

Abstract

We characterize the realizability of a quotient of matroids, over an infinite field $K$, in terms of the realizability over $K$ of a single matroid associated to it, called the Higgs major. This result extends to realizability of flag matroids. Further, we provide some applications to the relative realizability problem for Bergman fans in tropical geometry.

Realizability of matroid quotients

Abstract

We characterize the realizability of a quotient of matroids, over an infinite field , in terms of the realizability over of a single matroid associated to it, called the Higgs major. This result extends to realizability of flag matroids. Further, we provide some applications to the relative realizability problem for Bergman fans in tropical geometry.
Paper Structure (13 sections, 20 theorems, 25 equations)

This paper contains 13 sections, 20 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Morphisms, strong maps and quotients
  5. Realizability
  6. Modular cuts and weak maps
  7. Factorizations and majors
  8. Realizability of quotients of matroids
  9. Applications and examples
  10. Linear case
  11. Tropical Fano schemes
  12. Nonlinear case
  13. Questions and future work

Key Result

Theorem A

Let $f:M_1 \twoheadrightarrow M_2$ be a quotient of matroids, and let $K$ be an infinite field. The following statements are equivalent:

Theorems & Definitions (51)

  • Theorem A: Theorem \ref{['thm: characterization of realizability of quotients']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.9
  • Lemma 2.10: kennedy1975majors
  • ...and 41 more