Obstructed bundles of rank two on a quintic surface

TL;DR

This paper investigates the structure of the moduli space of stable rank two bundles on a very general quintic surface, revealing complex geometric features such as non-reduced components and unexpectedly smooth components of higher dimension.

Contribution

It provides a detailed analysis of obstructed points in the moduli space using spectral coverings, establishing sharp bounds for when the moduli space behaves well.

Findings

Identification of generically non-reduced components

Existence of generically smooth components of higher than expected dimension

Sharp bounds for the 'good' behavior of the moduli space

Abstract

In this note we consider the moduli space of stable bundles of rank two on a very general quintic surface. We study the potentially obstructed points of the moduli space via the spectral covering of a twisted endomorphism. This analysis leads to generically non-reduced components of the moduli space, and components which are generically smooth of more than the expected dimension. We obtain a sharp bound asked for by O'Grady saying when the moduli space is good.