  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 Gonzalez and Karu. We give new
examples and non-examples of Mori Dream Spaces among these blow-ups.
