Log concavity of the Grothendieck class of $\overline{\mathcal M}_{0,n}$

Abstract

Using a known recursive formula for the Grothendieck classes of the moduli spaces $\overline{\mathcal M}_{0,n}$, we prove that they satisfy an asymptotic form of ultra-log-concavity as polynomials in the Lefschetz class. We also observe that these polynomials are $\gamma$-positive. Both properties, along with numerical evidence, support the conjecture that these polynomials only have real zeros. This conjecture may be viewed as a particular case of a possible extension of a conjecture of Ferroni-Schr\"oter and Huh on Hilbert series of Chow rings of matroids. We prove asymptotic ultra-log-concavity by studying differential equations obtained from the recursion, whose solutions are the generating functions of the individual betti numbers of $\overline{\mathcal M}_{0,n}$. We obtain a rather complete description of these generating functions, determining their asymptotic behavior; their dominant term is controlled by the coefficients of the Lambert W function. The $\gamma$-positivity property follows directly from the recursion, extending the argument of Ferroni et al. proving $\gamma$-positivity for the Hilbert series of the Chow ring of matroids.