Fano Threefolds with 2-Torus Action

Following the work of Altmann and Hausen we give a combinatorial description for smooth Fano threefolds admitting a 2-torus action. We show that a whole variety of properties and invariants can be read off from this description. As an application we prove and disprove the existence of Kähler-Einstein metrics for some of these Fano threefolds, calculate their Cox rings and some of their toric canonical degenerations.

2010 Mathematics Subject Classification: Primary 14L30; Secondary 14J45, 32Q20

Keywords and Phrases: torus action, moment polytope, Fano variety, Kahler-Einstein metric, Cox ring

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