Jumps, folds and hypercomplex structures

We investigate the geometry of the Kodaira moduli space M of sections of π:Z→P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi :Z\rightarrow {\mathbb {P}}^1$$\end{document}, the normal bundle of which is allowed to jump from O(1)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(1)^{n}$$\end{document} to O(1)n-2m⊕O(2)m⊕Om\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(1)^{n-2m}\oplus {\mathcal {O}}(2)^{m}\oplus {\mathcal {O}}^{m}$$\end{document}. In particular, we identify the natural assumptions which guarantee that the Obata connection of the hypercomplex part of M extends to a logarithmic connection on M.