Toric models of smooth Fano threefolds

Abstract

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 $(X,D){\simeq }_{\mathrm{cbir}}({\mathbb{P}}^3,H_0+H_1+H_2+H_3)$, where the $H_i$'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.

23 pages. arXiv admin note: text overlap with arXiv:2309.16784; text overlap with arXiv:0806.2107 by other authors