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

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 bundle has a fixed Birkhoff–Grothendieck decomposition. The essential poles and the apparent poles provide two parabolic structures. The first one only depends on the formal type of the singular points. The latter one determines 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 previous results obtained by Szabó to the irregular case. Our work is more generally related to ideas and descriptions of Oblezin, Dubrovin–Mazzocco, and Saito–Szabó in the logarithmic case. After the first version of this work appeared, Komyo used our normal form to compute isomonodromic Hamiltonian systems for irregular Garnier systems.