Toric models of smooth Fano threefolds

TL;DR

This paper proves that most smooth Fano threefolds that are rational can be modeled as toric varieties, showing they have a simple combinatorial structure and zero birational complexity.

Contribution

It establishes that a general rational smooth Fano threefold admits a toric model, linking rationality and smoothness to toric geometry.

Findings

Existence of a boundary divisor making the threefold toric

General rational smooth Fano threefolds have birational complexity zero

Conditions of rationality, generality, and smoothness are necessary

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 $(X,D)\simeq_{\rm cbir} (\mathbb{P}^3,H_0+H_1+H_2+H_3)$, 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.