Moduli Spaces of Unstable Objects: Sheaves of Harder-Narasimhan Length 2

TL;DR

This paper develops a method using Non-Reductive Geometric Invariant Theory to construct moduli spaces for unstable objects, specifically $ au$-stable sheaves of Harder-Narasimhan length 2, extending prior results for vector bundles.

Contribution

It introduces a new approach to moduli of unstable objects via Non-Reductive GIT and constructs moduli spaces for $ au$-stable sheaves of HN length 2.

Findings

Constructed moduli spaces for $ au$-stable sheaves of HN length 2.

Extended previous results from vector bundles on curves to more general sheaves.

Provided a self-contained method for quotienting unstable strata in GIT.

Abstract

Given a moduli problem posed using Geometric Invariant Theory, one can use Non-Reductive Geometric Invariant Theory to quotient unstable HKKN strata and construct 'moduli spaces of unstable objects', extending the usual moduli classifications. After giving a self-contained account of how to do this, we apply this method to construct moduli spaces for certain unstable coherent sheaves of HN length 2 on a projective scheme, which we call $\tau$-stable sheaves. This extends a previous result of Brambila-Paz and Mata-Guti\'{e}rrez for rank two vector bundles on a curve.