On Smooth Projective Toric Fano 4-Folds

This paper describes work done to explore the topic of smooth projective toric Fano 4-folds that possess a full strong exceptional collection of line bundles. In previous work done over the summer of 2024, we found that 72/124 of the smooth toric Fano 4-folds possess a full strong exceptional collection of line bundles. We use data from the polymake database, which lists a classification of smooth reflexive lattice Fano polytopes in the first nine dimensions. In addition to getting the ratio of 72/124, we also wanted to check Bondal's numerical criterion to figure out why the other 4-folds did not possess a full strong exceptional collection of line bundles. We were able to find that the passing and failing of Bondal's numerical criterion coincided directly with the possession of a full strong exceptional collection of line bundles.