arXiv subjects

Søren Gammelgaard

Publications and source records attributed to Søren Gammelgaard.

7 recordsLinked to original sources

arXiv2026

Quiver Schemes for Nongeneric Stability and Cornering

We use the Le Bruyn--Procesi theorem to prove several results on quiver schemes for nongeneric stability conditions: We show that the stability-zero finite-type Nakajima quiver schemes are reduced points and give an example of a closely related nonreduced quiver scheme. In broader generality, we prove that adding modules supported on stability-zero vertices induces closed embeddings of quiver schemes, and show how in many cases the quiver scheme associated with the cornered algebra defines a limit to this system of embeddings. As an application, we show that there is an isomorphism between the underlying reduced schemes of certain equivariant Quot schemes and Nakajima quiver varieties.

math.AG↗

arXiv2025

On the unirationality of conic bundles with discriminant of degree eight

We study the unirationality of surface conic bundles $π\colon S\to\mathbb P^1$ over an arbitrary field $k$ with discriminant degree $d_S=8$, the first case beyond the del Pezzo range. We divide these surfaces in four families and produce explicit rational multisections via tangent constructions and Cremona transformations. Over $C_1$ fields we obtain Zariski dense loci of minimal, hence non $k$-rational, yet $k$-unirational conic bundles in each family; for one of the types we prove that the dense unirational locus is indeed Zariski open. Finally, we investigate the deformation theory of these conic bundles and how their unirationality behaves under specialization.

math.AG↗

arXiv2025

Noncommutative projective partial resolutions and quiver varieties

Let $Γ\in \mathrm{SL}_2(\mathbb{C})$ be a finite subgroup. We introduce a class of projective noncommutative surfaces $\mathbb{P}^2_I$, indexed by a set of irreducible $Γ$-representations. Extending the action of $Γ$ from $\mathbb{C}^2$ to $\mathbb{P}^2$, we show that these surfaces generalise both $[\mathbb{P}^2/Γ]$ and $\mathbb{P}^2/Γ$. We prove that isomorphism classes of framed torsion-free sheaves on any $\mathbb{P}^2_I$ carry a canonical bijection to the closed points of appropriate Nakajima quiver varieties. In particular, we provide geometric interpretations for a class of Nakajima quiver varieties using noncommutative geometry. Our results partially generalise several previous results on such quiver varieties.

math.AG↗

arXiv2024

Moduli spaces of framed sheaves on compactified Kleinian singularities

Consider a Kleinian singularity $ \mathbb{C}^2/Γ$, where $ Γ$ is a finite subgroup of $ SL_2(\mathbb{C}) $. In this paper, we construct moduli spaces of framed sheaves on a projective Deligne-Mumford stack compactifying the singularity, and we show that these moduli spaces are quasiprojective schemes.

math.AG↗

arXiv2023

Quiver Varieties and Framed Sheaves on Compactified Kleinian Singularities

Consider a Kleinian singularity $ \mathbb{C}^2/Γ$, where $ Γ$ is a finite subgroup of $ SL_2(\mathbb{C}) $. In this paper, we introduce a natural stack compactifying the singularity by adding a smooth stacky divisor, and we show that sets of framed sheaves on this stack satisfying certain additional criteria are closely related to a class of Nakajima quiver varieties. This partially extends our previous work on punctual Hilbert schemes of Kleinian singularities.

math.AG↗

arXiv2021

Quot schemes for Kleinian orbifolds

For a finite subgroup $Γ\subset {\mathrm{SL}}(2,\mathbb{C})$, we identify fine moduli spaces of certain cornered quiver algebras, defined in earlier work, with orbifold Quot schemes for the Kleinian orbifold $[\mathbb{C}^2/Γ]$. We also describe the reduced schemes underlying these Quot schemes as Nakajima quiver varieties for the framed McKay quiver of $Γ$, taken at specific non-generic stability parameters. These schemes are therefore irreducible, normal and admit symplectic resolutions. Our results generalise our previous work on the Hilbert scheme of points on $\mathbb{C}^2/Γ$; we present arguments that completely bypass the ADE classification.

math.AG↗

arXiv2021

Punctual Hilbert schemes for Kleinian singularities as quiver varieties

For a finite subgroup $Γ\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/Γ$ as a Nakajima quiver variety for the framed McKay quiver of $Γ$, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of stability parameter.

math.AG↗