Explicit formulas for the Grothendieck class of $\overline{\mathcal M}_{0,n}$

TL;DR

This paper derives explicit formulas for the Grothendieck class of the moduli space of genus 0 stable curves with n marked points, linking it to Betti numbers, Stirling and Bernoulli numbers, and solving related differential equations.

Contribution

It provides the first explicit formulas for the Grothendieck class of n, connecting it to combinatorial identities and Betti number recursions, extending previous theoretical work.

Findings

Explicit formulas for the Grothendieck class in terms of Stirling and Bernoulli numbers.

Verification of log-concavity conjecture for Betti number polynomials for many cases.

Representation of the generating function as a series of rational functions involving Lambert W-function.

Abstract

We obtain explicit expressions for the class in the Grothendieck group of varieties of the moduli space of genus 0 stable curves with n marked points. This information is equivalent to the Poincar\'e polynomial; it implies explicit expressions for the Betti numbers of the moduli space in terms of Stirling numbers or, alternatively, Bernoulli numbers. The expressions are obtained by solving a differential equation characterizing the generating function for the Grothendieck class as shown in work of Yuri Manin from the 1990s. This differential equation is equivalent to S. Keel's recursion for the Betti numbers of these moduli spaces. Our proof reduces the solution to two combinatorial identities which follow from applications of Lagrange series. We also study generating functions for the individual Betti numbers. In previous work it had been shown that these functions are determined by a set of polynomials with positive rational coefficients, which are conjecturally log-concave. We verify this conjecture for many infinite families of these polynomials, corresponding to the generating functions for the $2k$-Betti numbers of the moduli spaces for all $k\le 100$. Further, studying these polynomials allows us to prove that the generating function for the Grothendieck class of the moduli spaces may be written as a series of rational functions in the Lefschetz motive and the principal branch of the Lambert W-function. We include an interpretation of the main result in terms of Stirling matrices and a discussion of the Euler characteristic of the moduli space.