[Paper Review] Moduli stacks and moduli schemes for rank 2 unstable bundles

L. Brambila‐Paz, Osbaldo Mata+1 more|arXiv (Cornell University)|Nov 12, 2009

Algebraic Geometry and Number Theory7 references5 citations

TL;DR

This paper constructs algebraic stacks and moduli schemes for rank 2 unstable vector bundles on a smooth projective curve over a field, parameterized by Harder-Narasimhan type and endomorphism algebra dimension. It establishes a geometric framework for classifying such bundles via their numerical invariants, providing a foundational structure for moduli theory in the unstable case.

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.

Motivation & Objective

• To develop a geometric classification framework for rank 2 unstable vector bundles on smooth projective curves over a field.
• To identify and fix numerical invariants—specifically Harder-Narasimhan type and dimension of the endomorphism algebra—for constructing moduli spaces.
• To construct algebraic stacks and moduli schemes that parameterize indecomposable unstable bundles with these fixed invariants.
• To provide a systematic algebraic-geometric approach to moduli problems in the unstable setting, extending existing results for stable bundles.

Proposed method

• The authors analyze the endomorphism algebra of indecomposable unstable vector bundles of rank 2 and degree d over a smooth projective curve X.
• They fix the Harder-Narasimhan type and the dimension of the endomorphism algebra as numerical invariants to stratify the moduli problem.
• Using algebraic geometry techniques, they construct algebraic stacks that parameterize bundles with these invariants.
• They further construct moduli schemes as coarse moduli spaces for the same class of bundles, ensuring representability under the fixed invariants.
• The construction relies on the geometric properties of the curve X and the structure of unstable bundles in the category of vector bundles.
• The framework is built on foundational results in vector bundle theory and algebraic stacks, particularly focusing on the unstable case.

Experimental results

Research questions

• RQ1 How can one construct moduli spaces for rank 2 unstable vector bundles when stability conditions do not apply?
• RQ2 What numerical invariants are sufficient to parameterize families of indecomposable unstable bundles?
• RQ3 Can algebraic stacks and moduli schemes be constructed for such bundles using Harder-Narasimhan type and endomorphism algebra dimension?
• RQ4 What is the role of the endomorphism algebra in classifying unstable bundles of rank 2?
• RQ5 How do the geometric and algebraic structures of the curve influence the moduli construction?

Key findings

• The paper successfully constructs algebraic stacks that classify rank 2 unstable vector bundles with fixed Harder-Narasimhan type and endomorphism algebra dimension.
• It establishes the existence of moduli schemes as coarse moduli spaces for the same class of bundles, providing a scheme-theoretic parameter space.
• The endomorphism algebra of indecomposable unstable bundles of rank 2 is shown to be a key invariant in their classification.
• The construction is valid for any geometrically irreducible smooth projective curve over a field, ensuring broad applicability.
• The framework provides a systematic approach to moduli problems in the unstable regime, previously less explored than the stable case.