On the conjecture of King for smooth toric Deligne–Mumford stacks

We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne–Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne–Mumford stacks.