arXiv subjects
Publications and source records attributed to Mattia Morbello.
We extend the previous compactifications of the fifth Painlevé foliation studying the behaviour of the leaves for the time parameter $t$ being equal to $0$ or $\infty$. In particular, we find, for each boundary component, a first integral of the Hamiltonian vector field, providing an explicit description of the foliation.
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 $\mathfrak{Con}^V_Θ$ that extends the Okamoto compactification.