Moduli space of connections on rational irregular curves

TL;DR

This paper constructs a compactified moduli space of rank-two irregular connections on the Riemann sphere, extending previous compactifications by incorporating irregular stable nodal curves and an extra complex parameter.

Contribution

It explicitly compactifies the moduli space of irregular connections by introducing irregular stable nodal curves and analyzing the behavior of an additional complex parameter.

Findings

Constructed a compactification of the moduli space of irregular connections.

Introduced irregular stable nodal curves to describe boundary components.

Obtained a three-dimensional quasi-projective variety extending the Okamoto compactification.

Abstract

We study the compactification of the moduli space of a certain class of rank-two irregular connections on the Riemann sphere, presenting one double pole and two simple poles. To construct the compactification explicitly, we identify a class of such irregular connections with the data of a rational irregular curve together with an extra complex parameter. As a first step, we compactify the moduli space of rational irregular curves using a technique inspired by the Kapranov's compactification of the spaces $\mathcal{M}_{0,n}$. We then introduce the notion of irregular stable nodal curve to describe the curves lying on the boundary components, in the spirit of the work of Deligne and Mumford. Second, we study the behaviour of the extra complex parameter to complete the compactification, obtaining a three dimensional quasi-projective variety $\mathfrak{Con}^V_\Theta$ that extends the Okamoto compactification.