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

Fano threefolds with 2-torus action - a picture book