# Torsors on semistable curves and degenerations

## TL;DR

This paper defines (semi)stability for $G$-torsors on irreducible nodal curves and constructs a flat degeneration of their moduli space during curve degeneration, using advanced algebraic geometry tools.

## Contribution

It provides an intrinsic stability definition and a degeneration construction for $G$-torsors on singular curves, extending classical theories.

## Key findings

- Intrinsic stability for $G$-torsors on nodal curves established.
- Constructed flat degeneration of moduli space during curve degeneration.
- Applied generalized Bruhat-Tits group schemes and McKay correspondence.

## Abstract

In this paper we answer two long-standing questions in the classification of $G$-torsors on curves for an almost simple, simply connected algebraic group $G$ over the field of complex numbers. The first question is to give an intrinsic definition of (semi)stability for a $G$-torsor on an {\em irreducible nodal curve} and the second one is the construction of a flat degeneration of the moduli space of semistable $G$-torsors when the smooth curve degenerates to an irreducible nodal curve. A generalization of the classical Bruhat-Tits group schemes to two-dimensional regular local rings and an application of the geometric formulation of the McKay correspondence provide the key tools.