Exceptional Collections for Toric Fano Fivefolds

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.

💡 Research Summary

The paper investigates when smooth projective toric Fano fivefolds admit full strong exceptional collections of line bundles, using the minimal symmetric resolution of the diagonal constructed by Hanlon‑Hicks‑Lazarev (HHL). Building on the classical Beilinson resolution for projective space and the Bayer‑Popescu‑Sturmfels cellular resolution for unimodular toric varieties, the author previously introduced a non‑minimal virtual cellular resolution that works for any smooth projective toric variety. HHL later supplied a symmetric minimal resolution that applies to any toric subvariety of a smooth projective toric variety.

The central question is whether the set of line bundles appearing on one side of the HHL resolution forms a full strong exceptional collection (FSEC). To answer this, the author simultaneously checks three criteria: (1) agreement with the Bondal‑Thomsen collection, (2) satisfaction of Bondal’s numerical criterion (for each toric curve C, the intersection numbers a_i with the n‑1 toric divisors containing C must be > −1, and −1 may appear at most once), and (3) the unimodular condition of Bayer‑Popescu‑Sturmfels (all maximal minors of the ray matrix are 0 or ±1).

Using the Macaulay2 database of smooth toric Fano varieties, the author iterates over all 866 fivefolds (accessed via smoothFanoToricVariety(5,q)). For each variety X, the HHL resolution of the diagonal O_Δ is computed, yielding a locally free resolution expressed as box products of line bundles. The line bundles on the left‑hand side constitute a set E. A directed graph G is built whose vertices are the bundles in E and whose edges i→j indicate a non‑zero Hom⁰(L_i, L_j). The presence of any directed cycle of length > 1 would prevent a topological ordering, implying that E cannot be arranged into an exceptional sequence. The absence of such cycles guarantees an ordering; together with the vanishing of higher Ext groups (checked via Bondal’s criterion) this yields a full strong exceptional collection.

The computational pipeline runs on a high‑performance cluster at the Simons Laufer Mathematics Institute. For each X, the author computes the toric fan, the ray matrix B, checks unimodularity, evaluates Bondal’s numerical condition for every toric curve, constructs G, and tests for cycles using standard graph algorithms. The code and data are publicly released on GitHub (https://github.com/reggiea91/ADJOINT2025_Fivefolds.git), ensuring reproducibility.

Results: among the 866 smooth toric Fano fivefolds, 554 satisfy the unimodular condition, but only 300 satisfy the stricter combination of HHL‑induced line bundle collection, Bondal’s numerical criterion, and acyclicity of G. Thus, the HHL resolution yields a full strong exceptional collection for exactly 300 varieties, i.e., roughly 34.6 % of the total. This proportion is markedly lower than in dimension four, where 72 of 124 varieties (≈58 %) admit such collections, indicating that the existence of FSEC becomes rarer as dimension grows.

The paper also provides explicit positive and negative examples (indices 851 and 200, respectively), illustrating how the graph G either forms a directed acyclic graph (DAG) with no higher Exts, or contains cycles leading to non‑exceptionality.

Finally, the author discusses extensions to higher dimensions. The same methodology can be applied to the polymake database, which contains toric Fano varieties up to dimension nine, though the computational cost escalates dramatically. The work suggests that while the HHL resolution is a powerful tool, additional geometric or combinatorial constraints (such as Bondal’s criterion) are necessary to guarantee full strong exceptional collections in higher dimensions. The paper thus contributes a concrete quantitative benchmark (300/866) for fivefolds and lays out a reproducible computational framework for future investigations in toric geometry and derived categories.