On the anticanonical complex

TL;DR

This paper extends the concept of the anticanonical complex from toric Fano varieties to higher complexity torus actions, enabling new classifications of three-dimensional canonical Fano varieties with specific automorphism groups.

Contribution

It generalizes the anticanonical complex to higher complexity varieties and applies this to classify certain three-dimensional Fano varieties.

Findings

Extended the anticanonical complex to higher complexity varieties

Classified three-dimensional canonical Fano intrinsic quadrics with specific automorphism groups

Demonstrated the utility of the generalized complex in variety classification

Abstract

The anticanonical complex has been introduced as a natural generalization of the toric Fano polytope and so far has been succesfully used for the study of varieties with a torus action of complexity one. In the present article we enlarge the area of application of the anticanonical complex to varieties with a torus action of higher complexity, for example, general arrangement varieties. As an application of our techniques we classify the three-dimensional canonical Fano intrinsic quadrics with automorphism group having a maximal torus of dimension one.