Classifying weak Fano toric varieties of Picard rank $3$

TL;DR

This paper develops a systematic computational approach to classify all smooth weak Fano toric varieties with Picard rank 3 across dimensions, explicitly detailing classifications in three and four dimensions.

Contribution

It introduces a method using Macaulay2 for classification and provides explicit classifications for dimensions 3 and 4.

Findings

28 isomorphism classes of weak Fano toric threefolds

114 isomorphism classes of weak Fano toric fourfolds

Systematic classification method for Picard rank 3 varieties

Abstract

We provide a systematic method to classify all smooth weak Fano toric varieties of Picard rank $3$ in any dimension using Macaulay2, and describe the classification explicitly in dimensions $3$ and $4$. There are $28$ and $114$ isomorphism classes of rank $3$ weak Fano toric threefolds and fourfolds, respectively.