Geometric invariant theory and Einstein–Weyl geometry

In this article, we give a survey of geometric invariant theory for Toric Varieties, and present an application to the Einstein–Weyl geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted projective space CP1,1,2CP1,1,2. We also find and classify all possible quotients.