Key points are not available for this paper at this time.
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) ₂₁₈ₑ (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.
Loginov et al. (Fri,) studied this question.