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 birational model where introduce apparent singular points and bundle has fixed Birkhoff–Grothendieck decomposition. The essential poles parabolic structures. first one only depends formal type of points. latter determines connection (accessory parameters). As consequence, an open set corresponding moduli space is canonically identified with some Hilbert scheme explicit blow-up Hirzebruch surface. This generalizes previous results obtained by Szabó to case. Our work more generally related ideas descriptions Oblezin, Dubrovin–Mazzocco, Saito–Szabó in logarithmic After version this appeared, Komyo used our compute isomonodromic Hamiltonian systems Garnier systems.