A Compactification Over $\overline{M}_g$ Of The Universal Moduli Space of Slope-Semistable Vector Bundles

TL;DR

This paper constructs a projective moduli space of slope-semistable torsion-free sheaves over stable curves using GIT, providing a compactification that generalizes previous rank 1 results.

Contribution

It introduces a new GIT-based construction of a compact moduli space for pairs (C,E) with higher rank sheaves, extending Caporaso's rank 1 compactification.

Findings

The moduli space is projective and parametrizes slope-semistable sheaves.

The construction is functorial and compatible with existing compactifications in rank 1.

Basic properties of the moduli space are established.

Abstract

A projective moduli space of pairs (C,E) where E is a slope- semistable torsion free sheaf of uniform rank on a Deligne- Mumford stable curve C is constructed via G.I.T. There is a natural SL x SL action on the relative Quot scheme over the universal curve of the Hilbert scheme of pluricanonical, genus g curves. The G.I.T. quotient of this product action yields a functorial, compact solution to the moduli problem of pairs (C,E). Basic properties of the moduli space are studied. An alternative approach to the moduli problem of pairs has been suggested by D. Gieseker and I Morrison and completed by L. Caporaso in the rank 1 case. It is shown the contruction given here is isomorphic to Caporaso's compactification in the rank 1 case.