NCCRs of cones over del Pezzo surfaces

Abstract: Non-commutative crepant resolutions (NCCRs) are non-commutative versions of classical crepant resolutions in algebraic geometry. For 3-dimensional terminal Gorenstein singularities Iyama and Wemyss proved that all NCCRs are connected by mutations, which may be viewed as a non-commutative analogue of Kawamata's result that all crepant resolutions are connected by flops. In this paper we prove the corresponding result for a class of canonical Gorenstein singularities which are not terminal, namely anticanonical cones over del Pezzo surfaces. More precisely, we first obtain a classification of NCCRs of anticanonical del Pezzo cones, showing that every NCCR arises from a geometric helix on the corresponding del Pezzo surface. We then prove that all such geometric helices are connected to each other by mutations, up to simple operations which include tensoring by line bundles and shifts. A crucial ingredient in our proofs is the polygons that can be associated to exceptional collections on del Pezzo surfaces following the works of Hille and Perling. We obtain some interesting observations about these polygons which may be of independent interest.

• The paper’s main contribution is the complete classification of NCCRs for cones over del Pezzo surfaces via tilting bundles derived from geometric helices.
• It demonstrates that all NCCRs are connected by sequences of Iyama–Wemyss mutations, aligning these with DWZ quiver mutations for derived equivalences.
• The study integrates algebraic geometry and toric combinatorics using HP polygons to explicitly classify Gram matrices and minimal exceptional collections.

Non-Commutative Crepant Resolutions of Cones over del Pezzo Surfaces

Introduction and Motivations

The paper "NCCRs of cones over del Pezzo surfaces" (2604.11319) addresses the structure and classification of non-commutative crepant resolutions (NCCRs) for cones over del Pezzo surfaces. NCCRs fundamentally generalize the traditional notion of crepant resolutions of singularities for Gorenstein rings and are of central interest in non-commutative algebraic geometry, singularity theory, and closely tied to derived categories and mirror symmetry.

For three-dimensional terminal Gorenstein singularities, it is established (Iyama–Wemyss) that all NCCRs are interconnected via sequences of mutations, analogizing the connectivity of commutative crepant resolutions by flops (Kawamata). This work extends the hinge of that connectivity principle to the canonical Gorenstein setting, specifically to cones over smooth del Pezzo surfaces, whose vertex singularity is canonical but not terminal.

Main Results: Classification and Mutation Connectivity of NCCRs

Classification Theorem

Let $X$ be a smooth del Pezzo surface. Form the affine cone $Z = \mathrm{Spec}(R_X)$, with $$R_X = \bigoplus_{k \geq 0} \Gamma(X, \omega_X^{-k})$$, and let $\widehat{R_X}$ denote its completion at the singularity. The primary result is a complete classification:

• Every NCCR of $\widehat{R_X}$ is Morita equivalent to the completion of the endomorphism ring of a tilting bundle naturally arising from a geometric helix (generated by a very strong exceptional collection) on $X$.
• Such tilting bundles are constructed by pulling back direct sums of objects from very strong exceptional collections on $X$ to $Y = \mathrm{Tot}(\omega_X)$.

Thus, the set of NCCRs (up to Morita equivalence and regrading) is parametrized by geometric helices of $X$, modulo elementary operations (line bundle twists, derived shifts, orthogonal block reorderings, and rotations).

Mutation Fan Theorem

The connectivity result extends the Iyama–Wemyss theorem:

• All NCCRs of $\widehat{R_X}$ (equivalently, all rolled-up helix algebras of geometric helices) are related by sequences of mutations.

Formally, every NCCR can be reached from any other by a sequence of Iyama–Wemyss mutations, which coincide in this context with Derksen–Weyman–Zelevinsky (DWZ) quiver mutations. These encompass both combinatorial and geometric operations and induce derived equivalences among the corresponding NCCR algebras.

Mutations and Geometric Helices

The authors establish that all geometric helices on del Pezzo surfaces are interconnected via quiver mutations, possibly composed with line bundle tensoring, simultaneous shifts, and other benign operations. Crucially, a complete classification of minimal exceptional collections is achieved—those not further reducible by quiver mutation with respect to decreasing the sum of vector bundle ranks.

Polygons and Toric Data

A notable technical tool is the Hille–Perling (HP) polygon, associating to every exceptional collection on a del Pezzo surface a certain convex lattice polygon, encoding both algebraic and geometric data. The paper:

• Shows that the HP polygon structure is tightly linked to duality in the exceptional collection, and that its combinatorial properties determine the shape of the associated quiver and its mutation behavior.
• Proves a version of Herzog's conjecture on quiver shapes, under the absence of parallel edges in the polygon, constraining possible quiver types for rolled-up helix algebras.

Quiver mutations thereof correspond to explicit combinatorial and geometric transformations of the polygon, and minimality of the collection matches with a geometric "forbidden region" condition for the polygon's origin.

Numerical and Computational Aspects

Strong numerical results are established:

• For every del Pezzo surface, the possible Gram matrices (Euler characteristics of pairs in the collection), block types, and minimal polygons are completely classified.
• The transitions and equivalence relations between these minimal objects are effectively determined by explicit computation. This is achieved by a combination of geometric bounding inequalities, lattice theory, and computer-aided enumeration (with code provided).

Theoretical Implications

These results have deep implications for the structure of non-commutative resolutions in algebraic geometry:

• The explicit, mutation-theoretic connectivity of NCCRs in the context of canonical (non-terminal) Gorenstein singularities broadens the scope of known geometric–algebraic symmetries.
• The classification implies that, despite apparent non-uniqueness, the landscape of non-commutative desingularizations above these cones is under tight combinatorial and geometric control, dictated by the exceptional geometry of the del Pezzo surface.
• The translation to toric and polygonal language opens avenues for studying more general relationships between quivers, potentials, and (non-)commutative algebraic geometry in higher dimensions or with less restrictive singularities.

Future Directions

Key directions suggested include:

• Generalization to other Fano varieties or higher-dimensional analogs, leveraging the interplay of exceptional collections, helices, and NCCRs.
• Investigating further geometric constraints on possible quiver types and block structures for more general classes of singularities.
• Deepened study of how the polygonal combinatorics and mutation patterns encountered here manifest in the context of mirror symmetry, cluster algebras, or symplectic geometry.

Conclusion

The authors provide a comprehensive, intricate, and highly technical analysis of non-commutative crepant resolutions for cones over del Pezzo surfaces, establishing both a detailed classification and a mutation-theoretic completeness for NCCRs in this context. The elucidation of the connection between algebraic mutations, categorical helices, and toric polygon combinatorics is both structurally robust and computationally explicit, setting a benchmark for future investigations in non-commutative and derived algebraic geometry.