Moduli space of irregular rank two parabolic bundles over the Riemann sphere and its compactification

TL;DR

This paper explores the structure and compactification of moduli spaces of rank 2 parabolic bundles over the Riemann sphere, introducing refined bundles and analyzing their geometric properties.

Contribution

It introduces refined parabolic bundles for better compactification and studies their stability, elementary transformations, and moduli spaces related to weak del Pezzo surfaces.

Findings

Criteria for unramified irregular singular parabolic connections

Construction of compactified moduli spaces using refined bundles

Description of moduli spaces related to weak del Pezzo surfaces

Abstract

In this paper, we study rank 2 (quasi) parabolic bundles over the Riemann sphere with an effective divisor and these moduli spaces. First we consider a criterium when a parabolic bundle admits a unramified irregular singular parabolic connection. Second, to give a good compactification of the moduli space of semistable parabolic bundles, we introduce a generalization of parabolic bundles, which is called refined parabolic bundles. Third, we discuss a stability condition of refined parabolic bundles and define elementary transformations of the refined parabolic bundles. Finally, we describe the moduli spaces of refined parabolic bundles when the dimensions of the moduli spaces are two. These are related to geometry of some weak del Pezzo surfaces.