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.

Keywords:

• Mori dream spaces
• toric varieties
• interpolation

Acknowledgement

The author is very grateful to Ana-Maria Castravet for introducing this topic and the guidance throughout the research. The author would like to thank Kalle Karu and Xiaolei Zhao for valuable correspondences and suggestions. The author thanks José González for help with completing the proof of Proposition 5.1.