On the classification of toric Fano 4-folds

Abstract

The biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝ d , the so-called Fano polyhedra, up to an isomorphism of the standard lattice % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqeeuuDJXwAKb% sr4rNCHbacfaGae8hjHi6aaWbaaSqabeaaieGacaGFKbaaaOGaeyOG% IWSae8xhHe6aaWbaaSqabeaacaGFKbaaaaaa!418C! $$\mathbb{Z}^d \subset \mathbb{R}^d$$ . In this paper, we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds somorphism.