Generalized intersection pairings on moduli spaces of vector bundles over a curve

TL;DR

This paper introduces a generalized intersection pairing for Artin stacks with proper moduli spaces, compares various existing constructions for moduli of vector bundles over curves, and provides explicit low-rank computations.

Contribution

It establishes wall crossing formulas linking different intersection pairings and offers a unified framework for understanding their relationships.

Findings

Wall crossing formulas relate different intersection pairings.

Explicit computations are provided for low-rank cases.

The framework applies to moduli stacks of semistable bundles over curves.

Abstract

We introduce the notion of a generalized intersection pairing for an Artin stack with a proper good moduli space and nonempty stable part. For the moduli stack of semistable bundles over a smooth projective curve, there are four known constructions by partial desingularization, parabolic bundles, stable pairs and wall crossing. In this paper, we compare all these generalized intersection pairings by establishing wall crossing formulas between them. Explicit computations for low rank cases are included.