Blow-Ups of Toric Surfaces and the Mori Dream Space Property

Abstract

We study the question of whether the blow-ups of toric surfaces of Picard number one at the identity point of the torus are Mori dream spaces. For some of these toric surfaces, the question whether the blow-up is a Mori dream space is equivalent to countably many planar interpolation problems. We state a conjecture which generalizes a theorem of González and Karu. We give new examples of Mori dream spaces and not Mori dream spaces among these blow-ups.