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Peer-reviewedCC BYSelecta Mathematica

Building sets, Chow rings, and their Hilbert series

  • Christopher Eur1
  • Luis Ferroni2
  • Jacob P. Matherne3
  • Roberto Pagaria4
  • Lorenzo Vecchi5
5 affiliations
  1. 1Carnegie Mellon University, Pittsburgh, PA, USA
  2. 2Università di Pisa, Pisa, Italy
  3. 3North Carolina State University, Raleigh, NC, USA
  4. 4Università di Bologna, Bologna, Italy
  5. 5KTH Royal Institute of Technology, Stockholm, Sweden
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Abstract

We establish formulas for the Hilbert series of the 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 ℳ̅0,n+1 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.

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Cite

BibTeXRIS
BibTeX
@article{eur2026building,
  title = {{Building sets, Chow rings, and their Hilbert series}},
  author = {Christopher Eur and Luis Ferroni and Jacob P. Matherne and Roberto Pagaria and Lorenzo Vecchi},
  journal = {Selecta Mathematica},
  year = {2026},
  month = aug,
  doi = {10.1007/s00029-026-01186-2},
  url = {https://doi.org/10.1007/s00029-026-01186-2},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Building sets, Chow rings, and their Hilbert series
AU  - Christopher Eur
AU  - Luis Ferroni
AU  - Jacob P. Matherne
AU  - Roberto Pagaria
AU  - Lorenzo Vecchi
PY  - 2026
DA  - 2026/08/11
JO  - Selecta Mathematica
DO  - 10.1007/s00029-026-01186-2
UR  - https://doi.org/10.1007/s00029-026-01186-2
AB  - We establish formulas for the Hilbert series of the 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 ℳ̅_0,n+1 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.
SN  - 1022-1824
SN  - 1420-9020
ER  -