BorisovHua conjecture on exceptional collections for nef-Fano toric DM stacks

A smooth nef-Fano toric DM stack is a smooth toric DeligneMumford stack whose anticanonical divisor is nef. A full strong exceptional collection of line bundles is a strong exceptional collection of line bundles that generates the bounded derived category.

BorisovHua conjecture. Every smooth nef-Fano toric DM stack possesses a full strong exceptional collection of line bundles.

Borisov and Hua proved this in the case of Fano stacks with Picard number or dimension at most two, and the paper disproves the conjecture through toric Fano varieties with Picard number three. Hence the conjecture is refuted.