Torsors on semistable curves and degenerations

In this paper, we answer two long-standing questions on the classification of G-torsors on curves for an almost simple, simply connected algebraic group G over the field of complex numbers. The first is the construction of a flat degeneration of the moduli of G-torsors on smooth projective curves when the smooth curve degenerates to an irreducible nodal curve and the second one is to give an intrinsic definition of (semi)stability for a G-torsor on an irreducible projective 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.