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

Luis Ferroni, Roberto Pagaria, Lorenzo Vecchi

Abstract

We establish formulas for the Poincaré 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.

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

Abstract

We establish formulas for the Poincaré polynomial of the type B analogue of the Deligne--Knudsen--Mumford moduli space of rational curves with 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.
Paper Structure (10 sections, 25 theorems, 78 equations)

This paper contains 10 sections, 25 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. The lattice of signed partitions
  3. A toolkit on generating functions
  4. Type B analogues for the Poincaré polynomial of Mbar0n+1
  5. Formulas relating A and B
  6. A functional equation for B
  7. A differential equation for B
  8. A quadratic recursion
  9. Compositional formulas for B
  10. Iterative formulas for type B

Key Result

Theorem 1.1

The generating functions in type $A$ and $B$ are related by Solving for $A$ the last equation gives the equivalent form:

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • ...and 33 more