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

TL;DR

This paper constructs full strong exceptional collections of line bundles on certain smooth toric Deligne-Mumford stacks, advancing the understanding of their derived categories and aiming to prove a broader conjecture.

Contribution

It provides explicit constructions of exceptional collections on smooth toric Fano stacks with low Picard number and extends results to dimension two, supporting the conjecture for all such stacks.

Findings

Constructed exceptional collections for Picard number ≤ 2

Extended results to all dimension two cases

Progress towards proving the conjecture for all smooth toric nef-Fano stacks

Abstract

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.