# The locus of the representation of logarithmic connections by Fuchsian equations

## TL;DR

This paper investigates how logarithmic connections on rank 3 vector bundles over the Riemann sphere can be represented by third-order Fuchsian equations, identifying coordinates and constructing parts of the moduli space.

## Contribution

It provides a detailed analysis of the moduli space of rank 3 logarithmic connections and constructs a portion of this space via blow-up techniques.

## Key findings

- Coordinates on the moduli space are explicitly found.
- A non-trivial part of the moduli space is constructed.
- The case of rank 3 vector bundles is thoroughly analyzed.

## Abstract

The generic element of the moduli space of logarithmic connections with parabolic points on holomorphic vector bundle over the Riemann sphere can be represented by a Fuchsian equation with some singularities and some apparent singularities. We analyze the case of rank $3$ vector bundle which leads to third order Fuchsian equation. We find coordinates on an open subset of the moduli space and we construct a non-trivial part of the moduli space by blowing up along a variety in a special case.