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

The class of the fine moduli space of stable n-pointed curves of genus zero, M₀, ₍, in the Grothendieck ring of varieties encodes its Poincaré polynomial. Aluffi-Chen-Marcolli conjecture that the Grothendieck class of M₀, ₍ 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\, H^2l (M₀, ₍). As a consequence, we prove that in the range l n10 n, the ultra-log-concavity inequality \ (rk\, H^2 (l-1) ({M₀, ₍) }n-3{l-1}) ² rk\, H^2 (l-2) ({M₀, ₍) rk\, H^2l (M₀, ₍) }n-3{l-2n-3l} \ holds for n sufficiently large.