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.

我们为黎曼球面上的二阶不规则连接提供了一个独特的范式。事实上，我们提供了一个双有理模型，我们在其中引入了明显的奇异点，并且束具有固定的 Birkhoff-Grothendieck 分解。本质极点和视在极点提供了两种抛物线结构。第一个只取决于奇异点的形式类型。后者决定连接（附件参数）。因此，对应的连接模空间的开集被规范地标识为在某些 Hirzebruch 表面的显式爆炸上的某些 Hilbert 方案的点的开集。这将 Szabó 先前获得的结果推广到不规则情况。我们的工作更普遍地与 Oblezin、Dubrovin-Mazzocco、和 Saito-Szabó 在对数情况下。在这项工作的第一个版本出现后，Komyo 使用我们的范式来计算不规则卡尼尔系统的等单哈密顿系统。