Explicit description of jumping phenomena on moduli spaces of parabolic connections and Hilbert schemes of points on surfaces

TL;DR

This paper provides explicit descriptions of moduli spaces of rank 2 parabolic connections and Higgs bundles on the projective line, revealing jumping phenomena and detailed global structures for specific cases.

Contribution

It offers explicit descriptions of moduli spaces, including apparent singularities and dual parameters, and details the global structure for the case n=5.

Findings

Explicit descriptions of Zariski open sets of moduli spaces

Global descriptions of moduli spaces for n=5

Analysis of jumping phenomena in moduli spaces

Abstract

In this paper, we investigate the apparent singularities and the dual parameters of rank 2 parabolic connections on $\mathbb{P}^1$ and rank 2 (parabolic) Higgs bundle on $\mathbb{P}^1$. Then we obtain explicit descriptions of Zariski open sets of the moduli space of the parabolic connections and the moduli space of the Higgs bundles. For $n=5$, we can give global descriptions of the moduli spaces in detail.