We prove that a general rational smooth Fano threefold admits a toric model. More precisely, for a general rational smooth Fano threefold X, we show the existence of a boundary divisor D for which (X,D)cbir (P³,H₀+H₁+H₂+H₃), where the Hᵢ's are the coordinate hyperplanes. In particular, a general rational smooth Fano threefold has birational complexity zero. We argue that the three conditions: rationality, generality, and smoothness are indeed necessary for the theorem.
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Loginov et al. (2024) studied this question.
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