Improved Bounds on Ultra-Log Concavity of the Grothendieck Class of M0,n

The class of the fine moduli space of stable n-pointed curves of genus zero, M0,n, in the Grothendieck ring of varieties encodes its Poincaré polynomial. Aluffi-Chen-Marcolli conjecture that the Grothendieck class of M0,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 rk\, H²ˡ(M0,n). As a consequence, we prove that in the range l ≤ n/10log n, the ultra-log-concavity inequality \[({rk\, H²⁽ˡ⁻¹⁾({M0,n})}{{n-3}{l-1}})^2 ≥ {rk\, H²⁽ˡ⁻²⁾({M0,n})rk\, H²ˡ({M0,n})}{{n-3}{l-2}{n-3}{l}} \] holds for n sufficiently large.