Building sets, Chow rings, and their Hilbert series

Abstract

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 $$\overline{\mathcal {M}}_{0,n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>M</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> 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.