Smooth projective varieties with a torus action of complexity 1 and Picard number 2

TL;DR

This paper classifies all smooth projective varieties with a complexity one torus action and Picard number at most two, including Fano varieties, revealing they are constructed via iterated cone methods from lower-dimensional smooth varieties.

Contribution

It provides an explicit classification of smooth varieties with a complexity one torus action and Picard number two, including a description of Fano varieties through cone constructions.

Findings

All such varieties are explicitly described.

Fano varieties are obtained via iterated cone constructions.

Classification holds in every dimension.

Abstract

We give an explicit description of all smooth varieties with a torus action of complexity one having Picard number at most two. As a consequence, we classify in every dimension the smooth (almost) Fano varieties with a torus action of complexity one having Picard number two. It turns out that all the Fano examples are obtained via an iterated generalized cone construction from a series of smooth varieties of dimension at most seven.