Bubbling of rank two bundles over surfaces

TL;DR

This paper investigates the algebraic analogues of singularity formation in rank two holomorphic vector bundles over surfaces, introducing fertile families with bubbles and analyzing their properties and implications.

Contribution

It defines fertile families with bubbles, characterizes them via discriminants, and constructs examples demonstrating complex singularity behaviors in vector bundle families.

Findings

Existence of fertile families with bubbles established.

Characterization of these families using discriminants.

Examples showing negative answers to certain general questions.

Abstract

In this paper, motivated by the singularity formation of ASD connections in gauge theory, we study an algebraic analogue of the singularity formation of families of rank two holomorphic vector bundles over surfaces. For this, we define a notion of fertile families bearing bubbles and give a characterization of it using the related discriminant. Then we study families that locally form the singularity of the type $\mathcal{O}\oplus \mathcal{I}$ where $\mathcal{I}$ is an ideal sheaf defining points with multiplicities. We prove the existence of fertile families bearing bubbles by using elementary modifications of the original family. As applications, we study bubble trees for a few families that form singularities of low multiplicities and use examples to give negative answers to some plausible general questions.