# Normal forms for rank two linear irregular differential equations and moduli spaces

## TL;DR

This paper establishes a unique normal form for rank two irregular connections on the Riemann sphere, linking moduli spaces to Hilbert schemes of points on blown-up Hirzebruch surfaces, generalizing previous logarithmic case results.

## Contribution

It introduces a birational model with apparent singular points and fixed bundle decomposition, providing a new geometric description of the moduli space for irregular connections.

## Key findings

- Normal form for rank two irregular connections on the Riemann sphere.
- Identification of moduli space with Hilbert scheme of points on blown-up Hirzebruch surface.
- Generalization of previous logarithmic case descriptions to irregular case.

## Abstract

We provide a unique normal form for rank two irregular connections on the Riemann sphere.In fact, we provide a birational model where we introduce apparent singular points and where the bundlehas a fixed Birkhoff-Grothendieck decomposition. The essential poles and the apparent poles provide twoparabolic structures. The first one only depend on the formal type of the singular points. The latter one determine the connection (accessory parameters). As a consequence, an open set of the corresponding moduli space of connections is canonically identified with an open set of some Hilbert scheme of points on the explicit blow-up of some Hirzebruch surface. This generalizes to the irregular case a description dueto Oblezin, and Saito-Szabo in the logarithmic case. This approach is also very close to the work of Dubrovin-Mazzocco with the cyclic vector.