Grothendieck 类的对数凹性：$\overline{\mathcal M}_{0,n}$
Log concavity of the Grothendieck class of $\overline{\mathcal M}_{0,n}$
AI总结：
本文利用递归公式证明模空间 $\overline{\mathcal M}_{0,n}$ 的 Grothendieck 类满足渐近超对数凹性及 $\gamma$-正性，支持其实零点猜想，并推广了拟阵 Chow 环的相关猜想。
摘要：
利用模空间 $\overline{\mathcal M}_{0,n}$ 的 Grothendieck 类的已知递归公式，我们证明了它们作为 Lefschetz 类的多项式满足超对数凹性的渐近形式。我们还观察到这些多项式是 $\gamma$-正的。这两个性质，连同数值证据，支持了这些多项式仅具有实零点的猜想。该猜想可被视为 Ferroni-Schröter 和 Huh 关于拟阵 Chow 环的 Hilbert 级数的猜想的一个可能扩展的特例。我们通过研究由递归得到的微分方程来证明渐近超对数凹性，这些方程的解是 $\overline{\mathcal M}_{0,n}$ 的各个 Betti 数的生成函数。我们获得了这些生成函数的相当完整的描述，确定了它们的渐近行为；它们的主导项由 Lambert W 函数的系数控制。$\gamma$-正性质直接由递归得出，扩展了 Ferroni 等人证明拟阵 Chow 环的 Hilbert 级数的 $\gamma$-正性的论证。
英文摘要：
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.
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