Multigraded Factorial Rings and Fano varieties with torus action

Understand

In a first result, we describe all finitely generated factorial algebras over an algebraically closed field of characteristic zero that come with an effective multigrading of complexity one by means of generators and relations.

• This enables us to construct systematically varieties with free divisor class group and a complexity one torus action via their Cox rings.
• For the Fano varieties of this type that have a free divisor class group of rank one, we provide explicit bounds for the number of possible deformation types depending on the dimension and the index of the Picard group in the divisor class group.
• As a consequence, one can produce classification lists for fixed dimension and Picard index.