# Generalization of a conjecture of Mumford

## TL;DR

This paper extends Mumford's conjecture on relations in the cohomology of moduli spaces from smooth to irreducible nodal curves, showing how relations degenerate and computing related invariants.

## Contribution

It generalizes Mumford's relations to nodal curves and demonstrates their degeneration from smooth cases, also computing the Hodge-Poincare polynomial.

## Key findings

- Relations arise as degenerations of Mumford relations in nodal curves
- Computed the Hodge-Poincare polynomial for the moduli space on nodal curves
- Established a link between smooth and nodal curve moduli space relations

## Abstract

A conjecture of Mumford predicts a complete set of relations between the generators of the cohomology ring of the moduli space of rank 2 semi-stable sheaves with fixed odd degree determinant on a smooth, projective curve of genus at least 2. The conjecture was proven by Kirwan. In this article, we generalize the conjecture to the case when the underlying curve is irreducible, nodal. In fact, we show that these relations (in the nodal curve case) arise naturally as degeneration of the Mumford relations shown by Kirwan in the smooth curve case. As a byproduct, we compute the Hodge-Poincare polynomial of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed determinant on an irreducible, nodal curve.