A Gieseker type degeneration of moduli stacks of vector bundles on curves

TL;DR

This paper constructs a regular, well-behaved degeneration model of the moduli stack of vector bundles on a specific type of stable curve, aiming to facilitate recursive calculations of cohomological invariants.

Contribution

It introduces a Gieseker type degeneration of the moduli stack of vector bundles on curves with a double point, with desirable geometric properties.

Findings

The model is regular and has a normal crossing divisor as its special fibre.

The normalization of the special fibre is a locally trivial KGl-bundle.

Potential to derive recursion formulas for cohomological invariants.

Abstract

Given a generically smooth stable curve over a discrete valuation ring such that its special fibre is irreducible with one double point, we construct a moduli stack over that descrete valuation ring which is a model for the moduli stack of vector bundles over the generic fibre of the curve. The model has the following nice properties: 1. It is regular. 2. Its special fibre is a normal crossing divisor. 3. The normalization of its special fibre is a locally trivial KGl-bundle over the moduli stack of vector bundles over the normalization of the special fibre of the curve. Here KGl is a canonical compactification of the general linear group. Our motivation is that such a model may help to give recursion formulae for such cohomological invariants of the moduli stack of vector bundles on smooth curves which depend only on the genus of the curve.