Sign-reversing involutions in moduli spaces of curves

TL;DR

This paper introduces combinatorial sign-reversing involutions to compute intersection products and topological invariants in moduli spaces of curves, providing explicit formulas and conditions.

Contribution

It presents novel sign-reversing involutions for calculating intersection numbers and topological characteristics in moduli spaces of curves, with explicit combinatorial formulas.

Findings

Explicit formula for psi class intersection products on genus zero multicolored spaces.

Necessary and sufficient condition for nonzero intersection products based on graph matchings.

Calculation of the tropical Euler characteristic for certain graphical moduli spaces.

Abstract

We use sign-reversing involutions to solve two computational problems that arise naturally in the geometry of moduli spaces of curves. In particular, we give an explicit combinatorial formula for arbitrary $\psi$ class intersection products on the genus zero multicolored spaces $\overline{M}_{0,[r_1,\ldots,r_m]}$ using a novel sign reversing involution on decorated diagrams. As an application, we give a necessary and sufficient condition for when these intersection products are nonzero in terms of matchings on graphs. We also calculate the analog of the tropical Euler characteristic for the graphical moduli spaces $\overline{M}_{0,\Gamma}$ for graphs with two dominant vertices $P, Q$, by constructing two new sign-reversing involutions to simplify the sum. We show that (up to sign) it is the number of acyclic orientations of $\Gamma \smallsetminus \{P, Q\}$.