Moduli stacks and moduli schemes for rank 2 unstable bundles

TL;DR

This paper constructs algebraic stacks and moduli schemes for rank 2 unstable vector bundles over a smooth projective curve, focusing on their endomorphism algebras and stability types.

Contribution

It introduces a new framework for classifying unstable rank 2 bundles via algebraic stacks and moduli schemes based on their endomorphism structures and stability invariants.

Findings

Describes the algebra of endomorphisms for indecomposable unstable bundles

Constructs moduli stacks and schemes for fixed invariants

Provides a classification framework for unstable bundles

Abstract

Let X be a geometrically irreducible smooth projective curve over a field k. We describe the algebra of endomorphisms of indecomposable unstable vector bundles over X of rank 2 and degree d. Fixing some numerical invariants, namely the Harder-Narasimhan type and the dimension of the algebra of endomorphisms, we construct algebraic stacks and moduli schemes for such bundles.