On Terminal Fano Varieties with a Torus Action of Complexity One

Abstract

In toric geometry, Fano varieties correspond to certain lattice polytopes, whose lattice points determine the type of singularities of the variety. In this spirit we approach the larger family of Fano varieties with a torus action of complexity one. After an introductory chapter in the language of Cox rings, the world of Fano varieties of complexity one is tackled with the goal of finding explicit classification for certain subfamilies. We associate to any such variety the anticanonical complex, a polyhedral complex that detects the type of singularities. This enables the classification of terminal Fano threefolds of complexity one with Picard number one and those that do not allow any divisorial contraction. Moreover the smooth (almost) Fano varieties of complexity one having Picard number two and arbitrary dimension are also classified.