Frobenius morphisms and derived categories on two dimensional toric Deligne-Mumford stacks

TL;DR

This paper explores the use of Frobenius morphisms on two-dimensional toric Deligne-Mumford stacks, demonstrating that the push-forward of the structure sheaf generates their derived categories and identifying exceptional collections in examples.

Contribution

It introduces a generalized Frobenius endomorphism for two-dimensional toric DM stacks and shows it generates the derived category, with explicit exceptional collections in examples.

Findings

Push-forward of structure sheaf generates derived category

Identification of full strong exceptional collections

Application to two-dimensional toric DM orbifolds

Abstract

For a toric Deligne-Mumford (DM) stack, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism on a two-dimensional toric DM stack, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on the toric DM stack. We also choose a full strong exceptional collection from the set of direct summands of the push-forward in several examples of two dimensional toric DM orbifolds.