Degenerations of Godeaux surfaces and exceptional vector bundles

TL;DR

This paper explores the relationship between boundary components of the moduli space of Godeaux surfaces and the construction of exceptional rank 2 vector bundles, enhancing understanding of surface degenerations and vector bundle stability.

Contribution

It explicitly constructs stable exceptional vector bundles on Godeaux surfaces and describes boundary components of their moduli space, linking degenerations to vector bundle theory.

Findings

Explicit construction of stable rank 2 exceptional vector bundles.

Description of boundary components of Godeaux surface moduli space.

Establishment of correspondence between boundary components and vector bundles.

Abstract

A recent construction of Hacking relates the classification of stable vector bundles on a surface of general type with $p_g = 0$ and the boundary of the moduli space of deformations of the surface. In the present paper we analyze this relation for Godeaux surfaces. We provide a description of certain boundary components of the moduli space of Godeaux surfaces. Also we explicitly construct certain exceptional vector bundles of rank 2 on Godeaux surfaces, stable with respect to the canonical class, and examine the correspondence between the boundary components and such exceptional vector bundles.