Exceptional Collections for Toric Fano Fivefolds

TL;DR

This paper investigates the existence of exceptional collections of line bundles on smooth projective toric Fano fivefolds, providing the first count of cases where such collections are full and strong.

Contribution

It extends the resolution of the diagonal to toric Fano fivefolds and identifies 300 cases where this yields a full strong exceptional collection.

Findings

Counted 300 out of 866 toric Fano 5-folds with full strong exceptional collections.

Extended diagonal resolutions to toric Deligne-Mumford stacks.

Connected diagonal resolutions to the existence of exceptional collections.

Abstract

Resolutions of the diagonal of toric varieties has been an active area of study since Beilinson's celebrated resolution of the diagonal for $\PP^n$ and the disproof of King's conjecture. The author generalized a cellular resolution of the diagonal given by Bayer-Popescu-Sturmfels to yield a virtual resolution of the diagonal for smooth projective toric varieties, which extends to toric Deligne-Mumford stacks which are a global quotient of a smooth projective variety by a finite abelian group. Moreover, a celebrated result of Hanlon-Hicks-Lazarev gives a symmetric, minimal resolution of the diagonal for smooth projective toric varieties. This work studies when smooth projective toric Fano varieties in dimension 5 yield exceptional collections of line bundles using a resolution of the diagonal. We give the first known count of 300 out of 866 smooth projective toric Fano 5-folds for which the Hanlon-Hicks-Lazarev resolution of the diagonal yields a full strong exceptional collection of line bundles.