Singular principal bundles on reducible nodal curves

TL;DR

This paper explores the construction of moduli spaces for semistable principal bundles on reducible nodal curves, revealing their asymptotic behavior and proposing a framework for degenerations in algebraic geometry.

Contribution

It introduces a notion of semistability for principal bundles on reducible nodal curves and connects moduli spaces of pseudo bundles to singular principal bundles for large stability parameters.

Findings

Moduli spaces of pseudo bundles become moduli spaces of singular principal bundles for large δ.

A new notion of semistability for principal bundles on reducible nodal curves is proposed.

Construction of a universal moduli space of semistable singular principal bundles.

Abstract

Studying degenerations of moduli spaces of semistable principal bundles on smooth curves leads to the problem of constructing and studying moduli spaces on singular curves. In this note, we will see that the moduli spaces of $\delta$-semistable pseudo bundles on a nodal curve constructed by the first author become, for large values of $\delta$, the moduli spaces for semistable singular principal bundles. The latter are reasonable candidates for degenerations and a potential basis of further developments as on irreducible nodal curves. In particular, we find a notion of semistability for principal bundles on reducible nodal curves. The understanding of the asymptotic behavior of $\delta$-semistability rests on a lemma from geometric invariant theory. The results will allow the construction of a universal moduli space of semistable singular principal bundles relative to the moduli space $\overline{\mathcal M}_g$ of stable curves of genus $g$.