King's Conjecture and the Cox category

TL;DR

This paper proves King's Conjecture for a category constructed from derived categories of toric varieties associated with a Cox ring, extending classical ideas to a broader class of semiprojective toric varieties.

Contribution

It provides a new realization of King's Conjecture for a category built from all toric varieties linked to a Cox ring, generalizing Beilinson and Bondal's approaches.

Findings

Proves King's Conjecture in a new categorical context

Extends classical methods to all semiprojective toric varieties

Connects derived categories with Cox ring structures

Abstract

We state and prove a realization of King's Conjecture for a category glued from the derived categories of all of the toric varieties arising from a given Cox ring. Our perspective extends ideas of Beilinson and Bondal to all semiprojective toric varieties.