Jumps, folds, and hypercomplex structures

We investigate the geometry of the Kodaira moduli space $M$ of sections of $\pi :Z\to {\mathbb{P}}^1$, the normal bundle of which is allowed to jump from $\mathcal{O}(1{)}^n$ to ${\mathcal O}(1)^{n-2m}\oplus {\mathcal O}(2)^{m}\oplus {\mathcal O}^{m}$. 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 $M$.