Log concavity of the Grothendieck class of $\overline{\mathcal M}_{0,n}$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909098710138880 |
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| author | Aluffi, Paolo Chen, Stephanie Marcolli, Matilde |
| author_facet | Aluffi, Paolo Chen, Stephanie Marcolli, Matilde |
| contents | 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 $γ$-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-Schröter 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 $γ$-positivity property follows directly from the recursion, extending the argument of Ferroni et al. proving $γ$-positivity for the Hilbert series of the Chow ring of matroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02646 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Log concavity of the Grothendieck class of $\overline{\mathcal M}_{0,n}$ Aluffi, Paolo Chen, Stephanie Marcolli, Matilde Algebraic Geometry 14C15, 14H10, 13D40, 05A15, 30C15 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 $γ$-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-Schröter 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 $γ$-positivity property follows directly from the recursion, extending the argument of Ferroni et al. proving $γ$-positivity for the Hilbert series of the Chow ring of matroids. |
| title | Log concavity of the Grothendieck class of $\overline{\mathcal M}_{0,n}$ |
| topic | Algebraic Geometry 14C15, 14H10, 13D40, 05A15, 30C15 |
| url | https://arxiv.org/abs/2402.02646 |