  We study blowups of weighted projective planes at a general point, and more
generally blowups of toric surfaces of Picard number one. Based on the positive
characteristic methods of Kurano and Nishida, we give a general method for
constructing examples of Mori Dream Spaces and non-Mori Dream Spaces among such
blowups. Compared to previous constructions, this method uses the geometric
properties of the varieties and applies to a number of cases. We use it to
fully classify the examples coming from two families of negative curves.
