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Building sets, Chow rings, and their Hilbert series

Christopher Eur, Luis Ferroni, Jacob P. Matherne, Roberto Pagaria, Lorenzo Vecchi

TL;DR

This work develops comprehensive Hilbert-series formulas for the Feichtner–Yuzvinsky Chow rings D(𝕄,𝔾) of arbitrary building sets on polymatroids, unifying recursive, chain-sum, and nested-set approaches. It introduces the 𝔾-reduced characteristic polynomial and demonstrates incidence-algebra inverses that yield two practical recursions for 𝐻_{𝕄}^{𝔾}(x), plus efficient nonrecursive formulas that leverage spanning 𝔾-nested sets. The authors apply these results to braid matroids with the minimal building set, deriving new closed forms for the Poincaré polynomials of 𝑀̄_{0,n+1} and recovering known formulas by Manin, Getzler, Keel, and AMN, with additional generating-function perspectives. They also construct examples showing that, unlike special cases, the Hilbert-series coefficients can be non-log-concave, highlighting the limits of log-concavity and real-rootedness conjectures in the general building-set setting, and they discuss geometric interpretations via sequential blow-ups (wonderful compactifications) and Koszulity considerations. These results deepen the interplay between combinatorics, algebraic geometry, and moduli spaces, providing versatile tools for computing and analyzing Chow rings beyond maximal or minimal building-set scenarios.

Abstract

We establish formulas for the Hilbert series of the Feichtner--Yuzvinsky Chow ring of a polymatroid using arbitrary building sets. For braid matroids and minimal building sets, our results produce new formulas for the Poincaré polynomial of the moduli space $\overline{\mathcal{M}}_{0,n+1}$ of pointed stable rational curves, and recover several previous results by Keel, Getzler, Manin, and Aluffi--Marcolli--Nascimento. We also use our methods to produce examples of matroids and building sets for which the corresponding Chow ring has Hilbert series with non-log-concave coefficients. This contrasts with the real-rootedness and log-concavity conjectures of Ferroni--Schröter for matroids with maximal building sets, and of Aluffi--Chen--Marcolli for braid matroids with minimal building sets.

Building sets, Chow rings, and their Hilbert series

TL;DR

This work develops comprehensive Hilbert-series formulas for the Feichtner–Yuzvinsky Chow rings D(𝕄,𝔾) of arbitrary building sets on polymatroids, unifying recursive, chain-sum, and nested-set approaches. It introduces the 𝔾-reduced characteristic polynomial and demonstrates incidence-algebra inverses that yield two practical recursions for 𝐻_{𝕄}^{𝔾}(x), plus efficient nonrecursive formulas that leverage spanning 𝔾-nested sets. The authors apply these results to braid matroids with the minimal building set, deriving new closed forms for the Poincaré polynomials of 𝑀̄_{0,n+1} and recovering known formulas by Manin, Getzler, Keel, and AMN, with additional generating-function perspectives. They also construct examples showing that, unlike special cases, the Hilbert-series coefficients can be non-log-concave, highlighting the limits of log-concavity and real-rootedness conjectures in the general building-set setting, and they discuss geometric interpretations via sequential blow-ups (wonderful compactifications) and Koszulity considerations. These results deepen the interplay between combinatorics, algebraic geometry, and moduli spaces, providing versatile tools for computing and analyzing Chow rings beyond maximal or minimal building-set scenarios.

Abstract

We establish formulas for the Hilbert series of the Feichtner--Yuzvinsky Chow ring of a polymatroid using arbitrary building sets. For braid matroids and minimal building sets, our results produce new formulas for the Poincaré polynomial of the moduli space of pointed stable rational curves, and recover several previous results by Keel, Getzler, Manin, and Aluffi--Marcolli--Nascimento. We also use our methods to produce examples of matroids and building sets for which the corresponding Chow ring has Hilbert series with non-log-concave coefficients. This contrasts with the real-rootedness and log-concavity conjectures of Ferroni--Schröter for matroids with maximal building sets, and of Aluffi--Chen--Marcolli for braid matroids with minimal building sets.

Paper Structure

This paper contains 19 sections, 18 theorems, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Building sets
  5. Nested sets and Chow rings
  6. Hilbert series for arbitrary building sets
  7. G-reduced characteristic polynomials
  8. Recursions for Hilbert series of Chow rings
  9. Explicit formulas for the Hilbert series
  10. A more efficient non-recursive formula
  11. The geometry of the formulas
  12. Formulas for M0n+1.
  13. A formula in terms of partitions
  14. A formula by Aluffi--Marcolli--Nascimento
  15. Generating functions and recursions
  16. ...and 4 more sections

Key Result

Theorem 1.1

For every nonempty polymatroid $\mathsf{M}$ and every building set $\mathcal{G}$, the following two recursions hold: and

Figures (2)

  • Figure 1: Hierarchy of properties for palindromic polynomials
  • Figure 2: The lattice of flats of a polymatroid $\mathsf{M}$

Theorems & Definitions (49)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: aluffi-marcolli-nascimento
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 39 more