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arXiv · 2504.16776

Building sets, Chow rings, and their Hilbert series

Christopher Eur·Luis Ferroni·Jacob P. Matherne·Roberto Pagaria·Lorenzo Vecchi
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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 Poincar\'e 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--Schr\"oter for matroids with maximal building sets, and of Aluffi--Chen--Marcolli for braid matroids with minimal building sets.

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Christopher Eur, Luis Ferroni, Jacob P. Matherne, Roberto Pagaria, Lorenzo Vecchi. 2025-04-23. Building sets, Chow rings, and their Hilbert series. https://doi.org/10.1007/s00029-026-01186-2

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