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Balázs Szendrői

Publications and source records attributed to Balázs Szendrői.

9 recordsLinked to original sources
arXiv2026

Slicing up multigraded linear series

Multigraded linear series generalize the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We investigate the collection of the natural cornering morphisms into elementary bigraded linear series obtained from direct summands of the original globally generated vector bundle. Our main result is a condition on the injectivity of the product morphism. We apply our result in two examples: modules over the reconstruction algebra and equivariant Hilbert and Quot schemes of quotient stacks.

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arXiv2026

The projective coinvariant algebra, Young invariants and bigraded coordinate rings of Segre embeddings

This paper studies a flat degeneration P_n of the classical coinvariant algebra R_n, a bigraded Artinian Gorenstein algebra that arises from the coordinate ring of the Segre embedding of the n-fold self-product of the projective line. The Frobenius character of P_n is computed by a natural bigraded refinement of the classical Lusztig--Stanley formula for the character of the coinvariant algebra. Young invariants in P_n get related to coordinate rings of general Segre embeddings of products of projective spaces; their bigraded Hilbert polynomials get expressed in terms of major-descent generating functions of words in multisets. Relations to the diagonal coinvariant algebra, cohomological interpretations including quantum cohomology, and Garsia-Stanton-style bases are also explored.

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arXiv2025

Euler characteristics of affine ADE Nakajima quiver varieties via collapsing fibers

We prove a universal substitution formula that compares generating series of Euler characteristics of Nakajima quiver varieties associated with affine ADE diagrams at generic and at certain nongeneric stability conditions via a study of collapsing fibres in the associated variation of GIT map, unifying and generalising earlier results of the last two authors with Némethi and of Nakajima. As a special case, we compute generating series of Euler characteristics of noncommutative Quot schemes of Kleinian orbifolds. In type A and rank 1, we give a second, combinatorial proof of our substitution formula, using torus localisation and partition enumeration. This gives a combinatorial model of the fibers of the variation of GIT map, and also leads to relations between our results and the representation theory of the affine and finite Lie algebras in type A.

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arXiv2024

Kleinian singularities: some geometry, combinatorics and representation theory

We review the relationship between discrete groups of symmetries of Euclidean three-space, constructions in algebraic geometry around Kleinian singularities including versions of Hilbert and Quot schemes, and their relationship to finite-dimensional and affine Lie algebras via the McKay correspondence. We focus on combinatorial aspects, such as the enumeration of certain types of partition-like objects, reviewing in particular a recently developed root-of-unity-substitution calculus. While the most complete results are in type A, we also develop aspects of the theory in type D, and end with some questions about the exceptional type E cases.

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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.

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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.

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arXiv2018

Euler characteristics of Hilbert schemes of points on simple surface singularities

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corresponding affine Lie algebra; we conjecture that the same result holds also in type E. Our results are consistent with known results in type A, and are new for type D.

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arXiv2015

Euler characteristics of Hilbert schemes of points on surfaces with simple singularities

This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite subgroup of SL(2, C), we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type A singularities. We announce a proof of our conjecture for singularities of type D. The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding (extended) affine Lie algebra; we discuss possible representation-theoretic consequences of this fact. Our results, respectively conjectures, imply the modularity of the generating function for surfaces with type A and type D, respectively arbitrary, simple singularities, confirming predictions of S-duality.

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