Sign-reversing involutions in moduli spaces of curves
Vance Blankers, Maria Gillespie, Jake Levinson
TL;DR
The paper introduces sign-reversing involutions to evaluate alternating sums in graphically stable and multicolored moduli spaces of genus 0 curves. By two-stage SRIs, it converts ψ-class intersection numbers on $\overline{M}_{0,[r_1,\dots,r_m]}$ into fixed-point counts of decorated diagrams, yielding explicit positive formulas and a sharp nonvanishing criterion in terms of color-wise matchings. A parallel two-involution approach for $\overline{M}_{0,\Gamma}$ with two dominating vertices produces a tropical Euler-characteristic analogue tied to acyclic orientations of $\Gamma\setminus\{P,Q\}$. The results unify computations across Losev–Manin/heavy-light cases, compatibilities with color-merge reductions, and a border-strip conjecture, and demonstrate a versatile combinatorial toolkit for moduli of curves.
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 $ψ$ 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 $\overline{M}_{0,Γ}$ 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 $Γ\smallsetminus \{P, Q\}$.