A Parabolic approach to Intersection Theory on singular moduli spaces of vector bundles

Moduli spaces of semistable vector bundles of fixed rank and degree on curves are fundamental objects in algebraic geometry. When rank and degree are coprime the stability and semistability conditions coincide and the corresponding moduli space is a smooth projective variety whose geometry, topology and intersection theory have been studied extensively over the past several decades. When we drop the coprimality assumption, singularities appear and this has historically posed significant challenges for computing intersection pairings. In this talk we present a recent work in collaboration with Olga Trapeznikova, which introduces a novel and more accessible methodology for tackling this problem. The core strategy involves realizing the intersection cohomology of the singular moduli space as a canonical subspace of the cohomology of a smooth moduli space of parabolic bundles. We will show how this parabolic approach enables the calculation of intersection pairings by leveraging the Hecke correspondence and the Jeffrey-Kirwan residue formula. By comparing this methodology to the standard Jeffrey-Kirwan-Kiem-Woolf blow-up construction, we will discuss how this frame work not only provides a computationally efficient alternative but also offers a clear geometric interpretation for arbitrary rank r.