Improved Bounds on Ultra-Log Concavity of the Grothendieck Class of $\overline{\mathcal{M}_{0,n}}$

Abstract

The class of the fine moduli space of stable $n$-pointed curves of genus zero, $\overline{\mathcal{M}_{0,n}}$, in the Grothendieck ring of varieties encodes its Poincar\'e polynomial. Aluffi-Chen-Marcolli conjecture that the Grothendieck class of $\overline{\mathcal{M}_{0,n}}$ is real-rooted (and hence ultra-log-concave), and they proved an asymptotic ultra-log-concavity result for these polynomials. We build upon their work, by providing effectively computable bounds for the error term in their asymptotic formula for $\mathrm{rk}\, H^{2l}(\overline{\mathcal{M}_{0,n}})$. As a consequence, we prove that in the range $l \le \frac{n}{10\log n}$, the ultra-log-concavity inequality \[\left(\frac{\mathrm{rk}\, H^{2(l-1)}(\overline{\mathcal{M}_{0,n}})}{\binom{n-3}{l-1}}\right)^2 \ge \frac{\mathrm{rk}\, H^{2(l-2)}(\overline{\mathcal{M}_{0,n}})\mathrm{rk}\, H^{2l}(\overline{\mathcal{M}_{0,n}})}{\binom{n-3}{l-2}\binom{n-3}{l}} \] holds for $n$ sufficiently large.