Exceptional Collections for Toric Fano Fourfolds

TL;DR

This paper investigates the structure of exceptional collections on smooth projective toric Fano fourfolds, showing that a significant subset admits full strong exceptional collections of line bundles derived from recent diagonal resolutions.

Contribution

It demonstrates that for 72 out of 124 toric Fano 4-folds, the Hanlon-Hicks-Lazarev diagonal resolution produces a full strong exceptional collection, linking geometric resolutions to algebraic properties.

Findings

72 of 124 toric Fano 4-folds admit full strong exceptional collections

The collections coincide with Bondal's numerical criterion

Implications for the structure of derived categories of toric varieties

Abstract

Beilinson first gave a resolution of the diagonal for $\mathbb{P}^n$. Generalizing this, a modification of the cellular resolution of the diagonal given by Bayer-Popescu- Sturmfels gives a (non-minimal, in general) virtual resolution of the diagonal for smooth projective toric varieties and toric Deligne-Mumford stacks which are a global quotient of a smooth projective variety by a finite abelian group. In the past year, Hanlon-Hicks-Lazarev gave in particular a symmetric, minimal resolution of the diagonal for smooth projective toric varieties. We give implications for exceptional collections on smooth projective toric Fano varieties in dimension 4. We find that for 72 out of 124 smooth projective toric Fano 4-folds, the Hanlon-Hicks-Lazarev resolution of the diagonal yields a full strong exceptional collection of line bundles, which coincides exactly with satisfying a numerical criterion due to Bondal.