arXiv subjects

Gianni Petrella

Publications and source records attributed to Gianni Petrella.

7 recordsLinked to original sources

arXiv2026

Benchmarks in Leipzig

Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the three-day workshop Benchmarks in Leipzig with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100~questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs and their predecessors, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive. In September 2026, we added a fourth stage in which the next generation of models attempted all 100 questions once more, after which only 1 question remains unsolved.

math.HO↗

arXiv2026

Ample stability implies rigidity for quiver moduli

We show that ample stability suffices to apply Teleman quantisation to the endomorphism bundle of the universal representation on a quiver moduli space, and thus in particular to deduce its infinitesimal rigidity. This answers a question left open in our earlier work with Brecan, Franzen, and Reineke on the sheaf cohomology of universal bundles. This sufficiency makes several recent results applicable in the widest possible range, subject only to ample stability, which for coprime dimension vectors guarantees that the Picard rank of the moduli space is maximal.

math.AG↗

arXiv2026

Brauer groups of resolved quiver moduli via gerbes

We show that the Brauer group of any resolution of singularities of the moduli space of semistable quiver representations is trivial. We do this by extending the quiver-curve dictionary, translating a proof of the analogous result by Biswas-Hogadi-Holla for moduli of vector bundles on a curve to the setting of moduli of quiver representations, giving an algebro-geometric proof. This gives a new proof of this triviality, first proved by Le Bruyn-Schofield, building on algebraic (resp. cohomological) vanishing results due to Saltman (resp. Colliot-Thélène-Sansuc). Reversing the logic, our approach gives a new algebro-geometric proof of these vanishing results.

math.AG↗

arXiv2025

Finding the walls for quiver moduli

We give an effective characterisation of the walls in the variation of geometric invariant theory problem associated to a quiver and a dimension vector.

math.AG↗

arXiv2025

The QuiverTools package for SageMath and Julia

We introduce QuiverTools, a new software package, available in both a SageMath and Julia version, to study quivers and their moduli spaces of representations. Its key features are the computation of general subdimension vectors, leading to canonical decompositions, and checking the existence of (semi)stable representations, as well as the enumeration of Harder-Narasimhan types and related calculations for Teleman quantization. Computations related to intersection theory on quiver moduli are also implemented.

math.AG↗

arXiv2025

Partial semiorthogonal decompositions for quiver moduli

We embed several copies of the derived category of a quiver and certain line bundles in the derived category of an associated moduli space of representations, giving the start of a semiorthogonal decomposition. This mirrors the semiorthogonal decompositions of moduli of vector bundles on curves. Our results are obtained with QuiverTools, an open-source package of tools for quiver representations, their moduli spaces and their geometrical properties.

math.AG↗

arXiv2023

Rigidity and Schofield's partial tilting conjecture for quiver moduli

We explain how Teleman quantization can be applied to moduli spaces of quiver representations to compute the higher cohomology of the endomorphism bundle of the universal bundle. We use this to prove Schofield's partial tilting conjecture, and to show that moduli spaces of quiver representations are (infinitesimally) rigid as varieties.

math.AG↗