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Log concavity of the Grothendieck class of $\overline{\mathcal M}_{0,n}$
Algebraic Geometry · Submitted 5 Feb 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | result 0.76 | 0.95 | 0.84 | asserted 1.00 |
| s1 | We also observe that these polynomials are $\gamma$-positive. | result 0.99 | 0.92 | 0.08 | asserted 0.98 |
| s2 | Both properties, along with numerical evidence, support the conjecture that these polynomials only have real zeros. | interpretation 0.99 | 0.65 | 0.16 | hedged possibility 0.76 |
| s3 | 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. | interpretation 0.94 | 0.25 | 0.05 | hedged possibility 1.00 |
| s4 | 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 $\overline{\mathcal M}_{0,n}$. | method 0.97 | 0.96 | 0.19 | asserted 1.00 |
| s5 | 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. | result 0.84 | 0.96 | 0.08 | asserted 1.00 |
| s6 | 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. | method 0.51 | 0.86 | 0.06 | asserted 1.00 |
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Original 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.
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References
How the literature uses it
2 arXiv papers cite this in 16 sentences
Papers citing it · 13 sentences without a confident label
Builds on it 1 paper
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In [ref]–[ref] we study the finer structure of the formulas obtained in the first part of the paper, building upon work carried out in [ref] .
Explicit formulas for the Grothendieck class of $\overline{\mathcal M}_{0,n}$ · Introduction · 0.85
Uses its data or values 1 paper
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The following list of the first several polynomials $p^(k)_m$ is reproduced from [ref] .
Explicit formulas for the Grothendieck class of $\overline{\mathcal M}_{0,n}$ · The polynomials $p^(k)_m$ and $Γ_mj$, and proof of Theorems [ref] and [ref] · 1.00
Cites it as background 1 paper
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In [ref] , Aluffi-Chen-Marcolli proved the asymptotic ultra-log-concavity for the Betti numbers of $_0,n$.
Recursive algorithm and log-concavity of representations on the cohomology of $\overline{\mathcal M}_{0,n}$ · Introduction · 0.84
What it cites · 9 sentences without a confident label
Cites as background 1 paper
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For a survey on $γ$-positivity, we refer the reader to [ref] ; we follow the terminology in [ref] .
Hilbert-Poincar\'e series of matroid Chow rings and intersection cohomology · $_0,n$ is $γ$-positive · 0.98
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