17 - Moduli Spaces of Torsion Free Sheaves on Nodal Curves and Generalisations – I

Summary

The contents of this paper are essentially the same as in the unpublished note [13], but for some more comments on ramified H-bundles, H a semisimple algebraic group. The work of Ivan Kausz [4] is closely related to [13].

Introduction

Let X be an irreducible projective curve whose only singularities are nodes, say only one at x₀ ∈ X. We take the base field as the field ℂ of complex numbers. The moduli spaces of (semi-stable) torsion free sheaves on X provide a good generalisation of the moduli spaces of vector bundles on smooth projective curves. They provide compactifications of the moduli spaces of (semi-stable) vector bundles on X and have good specialisation properties i.e. when a smooth projective curve specialises to X, these objects specialise well. An interesting question is to generalise the moduli spaces of torsion free sheaves in the context of reductive or semi-simple algebraic groups H i.e. to have moduli spaces on X which are compactifications of the moduli spaces of semi-stable principal H-bundles (in the sense of A. Ramanathan [8]). There is some progress now due to the works of A. Schmitt and Usha Bhosle ([2],[11]).