Exceptional Collections for Toric Fano Fourfolds

Beilinson first gave a resolution of the diagonal for $\mathbb{P}^n$. Generalizing this, a modification of the cellular resolution of the diagonal given by Bayer-Popescu- Sturmfels gives a (non-minimal, in general) virtual resolution of the diagonal for smooth projective toric varieties and toric Deligne-Mumford stacks which are a global quotient of a smooth projective variety by a finite abelian group. In the past year, Hanlon-Hicks-Lazarev gave in particular a symmetric, minimal resolution of the diagonal for smooth projective toric varieties. We give implications for exceptional collections on smooth projective toric Fano varieties in dimension 4. We find that for 72 out of 124 smooth projective toric Fano 4-folds, the Hanlon-Hicks-Lazarev resolution of the diagonal yields a full strong exceptional collection of line bundles, which coincides exactly with satisfying a numerical criterion due to Bondal.