Exceptional Collections of Line Bundles for Smooth Toric Fano Surfaces and Threefolds

The cellular resolution of the diagonal given by Bayer-Popescu-Sturmfels for unimodular projective toric varieties yields a full, strong exceptional collection of line bundles on unimodular projective toric surfaces. The Hanlon-Hicks-Lazarev resolution of the diagonal yields a full, strong exceptional collection of line bundles for 16 of the 18 smooth toric Fano threefolds.