Intersection numbers of Chern classes of tautological line bundles on the moduli spaces of flexible polygons

TL;DR

This paper investigates the intersection theory of tautological line bundles on moduli spaces of flexible polygons, revealing geometric interpretations of Chern class intersections as counts of triangular configurations.

Contribution

It introduces a novel study of Chern classes and intersection numbers of tautological bundles on polygon moduli spaces, linking algebraic topology with geometric configurations.

Findings

Computed intersection numbers of Chern classes

Interpreted intersection numbers as counts of triangular configurations

Extended understanding of moduli spaces of flexible polygons

Abstract

Given a flexible $n$-gon with generic side lengths, the moduli space of its configurations in $\mathbb{R}^2$ as well as in $\mathbb{R}^3$ is a smooth manifold. It is equipped with $n$ \textit{tautological} line bundles whose definition is motivated by M. Kontsevich's tautological bundles over $\mathcal{M}_{0,n}$. We study their Euler classes, first Chern classes and intersection numbers, that is, top monomials in Chern (Euler) classes. The latter are interpreted geometrically as the signed numbers of some \textit{triangular configurations} of the flexible polygon.