On terminal Fano 3-folds with 2-torus action

Abstract

We classify the terminal Fano threefolds of Picard number one that come with an effective action of a two-torus. Our approach applies also to higher dimensions and generalizes the correspondence between toric Fano varieties and lattice polytopes: to any Fano variety with a complete intersection Cox ring we associate its "anti canonical complex", which is a certain polyhedral complex living in the lattice of one parameter groups of an ambient toric variety. For resolutions constructed via the tropical variety, the lattice points inside the anticanonical complex control the discrepancies. This leads, for example, to simple characterizations of terminality and canonicity.

missing items in the classification list added, other corrections

References in corpus (1)

• On G-Fano threefolds

Cited by in corpus (11)

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• Non-complete rational T-varieties of complexity one
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• On the anticanonical complex
• Iteration of Cox rings of klt singularities
• The anticanonical complex for non-degenerate toric complete intersections
• Classifying Fano Complexity-One $T$-Varieties via Divisorial Polytopes
• A software package to compute automorphisms of graded algebras
• On Gorenstein Fano Threefolds with an Action of a Two-Dimensional Torus
• Del Pezzo surfaces of Picard number one admitting a torus action
• On 3-folds having a holomorphic torus action with 6 fixed points