Einstein-Weyl spaces and (1,n)-curves in the quadric surface (Q1076351)

Following the ideas of Hitchin in the twistorial approach to 3- dimensional Einstein-Weyl geometry we construct a series of complex surfaces containing rational curves with self-intersection number 2. These mini twistor spaces are obtained by taking an n-fold covering of a neighbourhood of a (1,n)-curve in the quadric \({\mathbb{C}}{\mathbb{P}}_ 1\times {\mathbb{C}}{\mathbb{P}}_ 1\) branched along the curve. We describe the corresponding Einstein-Weyl geometry on the parameter space of curves.