Improved Bounds on Ultra-Log Concavity of the Grothendieck Class of M_0,n

2025/09/15 by Eduardo Mendes Nascimento, Nascimento, Eduardo
Mathematics · #Algebraic Geometry and Number Theory #advanced mathematical theories #Advanced Topology and Set Theory

paper · pdf · doi:10.48550/arxiv.2509.11805

Abstract

The class of the fine moduli space of stable n-pointed curves of genus zero, M_0,n, in the Grothendieck ring of varieties encodes its Poincaré polynomial. Aluffi-Chen-Marcolli conjecture that the Grothendieck class of 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 rk H^2l(M_0,n). As a consequence, we prove that in the range l ≤ (n)/(10log n), the ultra-log-concavity inequality (\fracrk H^2(l-1)(M_0,n)\binomn-3l-1)² ≥ \fracrk H^2(l-2)(M_0,n)rk H^2l(M_0,n)\binomn-3l-2\binomn-3l holds for n sufficiently large.

Citations

• Building sets, Chow rings, and their Hilbert series
• Operads and moduli spaces of genus 0 Riemann surfaces
• Generating Functions in Algebraic Geometry and Sums over Trees

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