Exceptional Collections for Toric Fano Fourfolds

TL;DR

This work analyzes when the Hanlon-Hicks-Lazarev diagonal resolution yields a full strong exceptional collection of line bundles on smooth projective toric Fano fourfolds by computing Hom-spaces via $H^*(X,\mathcal{O}(E-D))$ and testing Bondal's numerical criterion. Using the polymake 4D Fano polytope database and diagonal-resolutions, it confirms a positive result for $72$ of $124$ fourfolds, exactly when Bondal's criterion holds, and provides explicit positive and negative examples (e.g., polytopes F.4D.0085 and F.4D.0000). The authors supply complete data and code for replication and show that the failure or success is detected by the presence of cycles in the Hom-quiver and by the intersection behavior along toric curves. These findings reinforce the link between diagonal resolutions, tilting theory, and Homological Mirror Symmetry in toric Fano geometry, and they motivate a dimension-dependent conjecture on the prevalence of such collections. They also publish data to enable broader exploration and future generalizations.

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.

Figures (1)

• Figure 1: Quiver showing nonzero $Hom^0(E_i, E_j)$ for $i\neq j$ in $\mathcal{E}$. Indexing on vertices from $0$ to $11=|\mathcal{E}|-1$.

Theorems & Definitions (2)

• Theorem 1.1
• Conjecture 6.1