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Cox rings generalize the homogeneous coordinate rings of projective spaces to varieties with finitely generated divisor class groups. In this talk, we study negative curves on two families of blowups of toric surfaces of Picard number one and classify all varieties in the families according to whether their Cox rings are finitely generated. The main tools are a geometric interpretation of the finite generation of the Cox rings for these varieties and positive characteristic methods. This talk is based on joint work with Javier Gonzalez and Kalle Karu.