The Poincar\'e polynomial of the type B analogue of $\overline{\mathcal{M}}_{0,n+1}$

TL;DR

This paper derives formulas for the Poincaré polynomial of a type B moduli space of rational curves, extending known type A results, and explores its generating functions, gamma-positivity, and related differential equations.

Contribution

It provides the first formulas for the Poincaré polynomial of the type B analogue of the moduli space of rational curves with marked points, extending classical type A results.

Findings

Derived explicit formulas for the Poincaré polynomial in type B.

Established functional and differential equations for the generating function.

Proved gamma-positivity and related combinatorial properties.

Abstract

We establish formulas for the Poincar\'e polynomial of the type B analogue of the Deligne--Knudsen--Mumford moduli space of rational curves with $n$ marked points, providing type B counterparts to results by Keel, Manin, Getzler and Yuzvinsky. We establish functional and differential equations satisfied by the bivariate exponential generating function of these polynomials. We show how this generating function relates to the classical one in type A. We deduce the gamma-positivity of these polynomials via a quadratic recursion and discuss a type B analogue of a formula found by Aluffi, Marcolli and Nascimento in type A.