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Factorization structures, cones, and polytopes

Roland Púček

TL;DR

The paper develops a comprehensive framework around factorization structures, unifying discrete toric geometry with differential-geometric constructions. It builds a complete structure theory (curves, complexification, degree, quotients) and classifies 2D and Segre-Veronese types, while deriving generalized Gale conditions and Vandermonde identities to describe faces of compatible cones and polytopes. It further shows how quotients preserve the factorization structure and provides explicit constructions of compatible cones and Delzant rational polytopes, enabling practical computation of extremal toric Kähler and Sasaki geometries. This framework offers a robust, computable route to generate explicit canonical geometries and Delzant polytopes, bridging discrete and differential geometry and enabling systematic exploration of toric extremal metrics.

Abstract

Factorization structures occur in toric differential and discrete geometry, and can be viewed in multiple ways, e.g., as objects determining substantial classes of explicit toric Sasaki and Kähler geometries, as special coordinates on such, or as an apex generalisation of cyclic polytopes featuring a generalised Gale's evenness condition. This article presents a comprehensive study of factorization structures. It establishes their structure theory and introduces their use in the geometry of cones and polytopes. The article explains the construction of polytopes and cones compatible with a given factorization structure, and exemplifies it for product Segre-Veronese and Veronese factorization structures, where the latter case includes cyclic polytopes. Further, it derives the generalised Gale's evenness condition for compatible cones, polytopes and their duals, and explicitly describes faces of these. Factorization structures naturally provide generalised Vandermonde identities, which relate normals of any compatible polytope, and which are used for Veronese factorization structure to find examples of Delzant and rational Delzant compatible polytopes. The article offers a myriad of factorization structure examples, which are later characterised to be precisely factorization structures with decomposable curves, and raises the question if these encompass all factorization structures, i.e., the existence of an indecomposable factorization curve.

Factorization structures, cones, and polytopes

TL;DR

The paper develops a comprehensive framework around factorization structures, unifying discrete toric geometry with differential-geometric constructions. It builds a complete structure theory (curves, complexification, degree, quotients) and classifies 2D and Segre-Veronese types, while deriving generalized Gale conditions and Vandermonde identities to describe faces of compatible cones and polytopes. It further shows how quotients preserve the factorization structure and provides explicit constructions of compatible cones and Delzant rational polytopes, enabling practical computation of extremal toric Kähler and Sasaki geometries. This framework offers a robust, computable route to generate explicit canonical geometries and Delzant polytopes, bridging discrete and differential geometry and enabling systematic exploration of toric extremal metrics.

Abstract

Factorization structures occur in toric differential and discrete geometry, and can be viewed in multiple ways, e.g., as objects determining substantial classes of explicit toric Sasaki and Kähler geometries, as special coordinates on such, or as an apex generalisation of cyclic polytopes featuring a generalised Gale's evenness condition. This article presents a comprehensive study of factorization structures. It establishes their structure theory and introduces their use in the geometry of cones and polytopes. The article explains the construction of polytopes and cones compatible with a given factorization structure, and exemplifies it for product Segre-Veronese and Veronese factorization structures, where the latter case includes cyclic polytopes. Further, it derives the generalised Gale's evenness condition for compatible cones, polytopes and their duals, and explicitly describes faces of these. Factorization structures naturally provide generalised Vandermonde identities, which relate normals of any compatible polytope, and which are used for Veronese factorization structure to find examples of Delzant and rational Delzant compatible polytopes. The article offers a myriad of factorization structure examples, which are later characterised to be precisely factorization structures with decomposable curves, and raises the question if these encompass all factorization structures, i.e., the existence of an indecomposable factorization curve.
Paper Structure (18 sections, 37 theorems, 165 equations, 3 figures)

This paper contains 18 sections, 37 theorems, 165 equations, 3 figures.

Table of Contents

  1. Factorization structures
  2. Factorization structures of dimension 2
  3. Segre-Veronese factorization structure
  4. Products and decomposable elements
  5. Motivation for definition of factorization structures rooted in discrete geometry
  6. Factorization structures in differential geometry
  7. Structure theory
  8. Factorization curves
  9. Complexification
  10. Degree
  11. Decomposability
  12. Quotient factorization structure
  13. Decomposable curves and Segre-Veronese factorization structures
  14. Characterisation of decomposable factorization structures
  15. Compatible cones and polytopes
  16. ...and 3 more sections

Key Result

Lemma 1.1.1

An inclusion $\varphi:\mathfrak{h}\to V_1^*\otimes V_2^*$ of a 3-dimensional vector space into the tensor product of two 2-dimensional vector spaces is a 2-dimensional factorization structure.

Figures (3)

  • Figure 1: cyclic polytope with 5 vertices on the curve $t\mapsto(t,t^2,t^3)$
  • Figure 2:
  • Figure 3:

Theorems & Definitions (115)

  • Definition 1.0.1
  • Remark 1.0.2
  • Remark 1.0.3
  • Lemma 1.1.1
  • proof
  • Definition 1.2.1
  • Remark 1.2.2
  • Example 1.2.3
  • Example 1.2.4
  • Example 1.2.5
  • ...and 105 more