# On strong exceptional collections of line bundles of maximal length on Fano toric Deligne-Mumford stacks

## TL;DR

This paper investigates strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks with low Picard rank, proving they generate the entire derived category when of maximal length.

## Contribution

It establishes that such collections of line bundles of maximal length generate the derived category on these stacks, extending understanding of their categorical structure.

## Key findings

- Strong exceptional collections of line bundles generate the derived category.
- Maximal length collections correspond to the rank of K-theory.
- Results apply to stacks with Picard rank at most two.

## Abstract

We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $\mathbb{P}_{\mathbf{\Sigma}}$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of $\mathbb{P}_{\mathbf{\Sigma}}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $\mathbb{P}_{\mathbf{\Sigma}}$.