Moduli of unstable bundles of HN-length two with fixed algebra of endomorphisms

TL;DR

This paper characterizes unstable bundles of HN-length two on complex curves, describing their endomorphism algebras, stratifying the moduli space, and analyzing topological properties, with distinctions from stable bundle moduli spaces.

Contribution

It provides necessary and sufficient conditions for endomorphism algebras of these bundles and constructs stratifications of their moduli spaces based on algebra dimension.

Findings

Stratification of moduli space by endomorphism algebra dimension

Conditions for unstable bundles to have specific endomorphism algebras

Topological properties depend on the curve's generality

Abstract

Let X be a smooth irreducible complex projective curve of genus g > 1. In this paper, we give necessary and sufficient conditions for an unstable bundle of HN-lenght 2 to have a particular algebra of endomorphisms. Then, fixing the dimension of the algebra of endomorphisms we obtain a stratification of the moduli scheme such that each strata is a coarse moduli space. A particular case of interest is when the unstable bundles are simple. In that case the moduli space is fine. Topological properties of those moduli spaces are described and will depend on the generality of the curve X. Such results differ from the corresponding results for the moduli space of stable bundles, where non-emptiness, dimension etc. are independent of the curve.