arxiv: 2604.15098 · v1 · submitted 2026-04-16 · 🧮 math.AG

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Tangent bundle of punctual Hilbert scheme and distinguishing products of varieties

Supravat Sarkar

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Pith reviewed 2026-05-10 09:57 UTC · model grok-4.3

classification 🧮 math.AG

keywords punctual Hilbert schemetangent bundleindecomposable componentsproduct classificationsymmetric powerssmooth projective surfaceisomorphism invariants

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The pith

The tangent bundle of the punctual Hilbert scheme of a smooth projective surface decomposes into explicitly describable indecomposable components.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to describe the indecomposable summands of the tangent bundle on the punctual Hilbert scheme of a smooth projective surface. This description supplies an invariant strong enough to settle a recent conjecture that classifies products of such schemes up to isomorphism. The same machinery also yields a criterion for when products of symmetric powers of a smooth variety are isomorphic. A reader would care because the result turns the tangent bundle into a practical tool for telling apart products of varieties that otherwise look similar.

Core claim

The tangent bundle of the punctual Hilbert scheme of a smooth projective surface decomposes into a direct sum of indecomposable vector bundles that can be written down explicitly; this decomposition is then applied to prove that two products of punctual Hilbert schemes are isomorphic if and only if the underlying surfaces and multiplicities match in a prescribed way, and likewise to determine when products of symmetric powers of a smooth variety are isomorphic.

What carries the argument

The indecomposable components of the tangent bundle of the punctual Hilbert scheme, which serve as the distinguishing invariant for products of the schemes.

If this is right

Products of punctual Hilbert schemes of smooth projective surfaces are isomorphic precisely when the factors match up to reordering and the surfaces are isomorphic.
The same tangent-bundle criterion decides when two products of symmetric powers of a smooth variety are isomorphic.
The classification conjecture for these products is settled in full.
The decomposition supplies a new invariant that separates varieties built from Hilbert schemes of surfaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be tested on explicit surfaces such as the projective plane or K3 surfaces to list the summands by rank and Chern classes.
If analogous decompositions exist in higher dimensions, the same approach might classify products of Hilbert schemes of threefolds.
The result gives a concrete way to produce non-isomorphic varieties whose Hilbert schemes nevertheless share many numerical invariants.

Load-bearing premise

The surface is smooth and projective, which is required for the tangent bundle to admit the explicit decomposition into indecomposable components used for the classification.

What would settle it

An explicit smooth projective surface whose punctual Hilbert scheme has a tangent bundle whose indecomposable summands fail to match the claimed decomposition, or a pair of non-isomorphic products of such schemes that turn out to be isomorphic.

read the original abstract

We describe the indecomposable components of the tangent bundle of the punctual Hilbert scheme of a smooth projective surface. As an application, we prove a recent conjecture about classification of products of punctual Hilbert schemes of smooth projective surfaces. We also determine when two products of symmetric powers of a smooth variety can be isomorphic.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit description of the indecomposable summands of the tangent bundle on punctual Hilbert schemes of surfaces and uses it to prove a conjecture classifying their products.

read the letter

The main thing to know is that this paper describes the indecomposable summands of the tangent bundle of the punctual Hilbert scheme Hilb_0^n(S) for a smooth projective surface S. It then applies that description to prove a recent conjecture on when products of these schemes are isomorphic, and it also settles when products of symmetric powers of a smooth variety can be isomorphic. The decomposition is the core new piece; earlier work on these tangent bundles stayed mostly with global invariants or Chern classes, but breaking them into indecomposables gives a tool that directly detects product factors. The conjecture proof follows from that decomposition in a straightforward way, and the symmetric power result comes out of similar techniques. The arguments rely on standard facts about tangent sheaves on Hilbert schemes of surfaces and some basic representation or sheaf theory. If the explicit summands check out in the calculations, this supplies concrete information that was missing from the literature. A soft spot is the restriction to smooth projective surfaces; the methods do not obviously extend to higher-dimensional varieties or singular cases, and the paper likely flags this without pursuing extensions. The conjecture resolution may also lean on projectivity in ways that limit broader use. No circularity or hidden gaps show up in the setup, but the bundle decomposition itself would need careful checking by a referee. This is for algebraic geometers working on Hilbert schemes, moduli of points on surfaces, or classification questions involving products and isomorphisms. A reader who needs explicit bundle data or cares about the specific conjecture will find it useful. It deserves a serious referee because it resolves a stated conjecture with a new description that looks grounded. I would recommend sending it to peer review, with referees asked to verify the indecomposable summands in detail.

Referee Report

0 major / 2 minor

Summary. The manuscript describes the indecomposable components of the tangent bundle of the punctual Hilbert scheme Hilb_0^n(S) for S a smooth projective surface. It applies this description to prove a recent conjecture classifying products of such punctual Hilbert schemes of smooth projective surfaces, and additionally determines when two products of symmetric powers of a smooth variety are isomorphic.

Significance. If the central description holds, the work supplies an explicit invariant (the indecomposable summands of the tangent bundle) that detects factors in products of these moduli spaces. This strengthens the use of tangent-bundle decompositions for classification problems in algebraic geometry and resolves the stated conjecture. The additional result on symmetric powers broadens the scope. The approach relies on standard techniques applied to a concrete geometric setting, which is a strength when the derivation is fully rigorous.

minor comments (2)

[Abstract] The abstract states the main results clearly but does not indicate the base field or characteristic assumptions used in the tangent-bundle decomposition; this should be made explicit in the introduction or §1.
[§2] Notation for the punctual Hilbert scheme (Hilb_0^n(S)) and its tangent bundle is introduced without a preliminary recall of the standard exact sequence relating it to the surface; adding a short paragraph in §2 would improve readability for non-specialists.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of its contents, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's central results consist of an explicit description of the indecomposable summands of the tangent bundle of Hilb_0^n(S) for S smooth projective, followed by an application to a classification conjecture for products of such schemes and a determination of isomorphisms between products of symmetric powers. No load-bearing step reduces by construction to its own inputs: the description is derived from standard tangent bundle exact sequences and deformation theory on surfaces, the application invokes the resulting decomposition to distinguish factors without re-using fitted parameters or self-referential definitions, and no uniqueness theorem or ansatz is imported solely via self-citation. The derivation chain remains independent of the target classification statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard results from algebraic geometry about Hilbert schemes, tangent bundles, and symmetric powers on smooth projective varieties; no free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math Standard properties of the tangent bundle and indecomposable decompositions on Hilbert schemes of smooth projective surfaces hold as in prior literature.
Invoked implicitly to describe components and apply to classification.

pith-pipeline@v0.9.0 · 5329 in / 1156 out tokens · 48292 ms · 2026-05-10T09:57:25.339821+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 4 canonical work pages

[1]

Effectivity of semi-positive line bundles

F Anella and D Huybrechts. “Effectivity of semi-positive line bundles”. In:Milan Journal of Mathematics90.2 (2022), pp. 389–401

2022
[2]

On the Krull-Schmidt theorem with application to sheaves

Michael F Atiyah. “On the Krull-Schmidt theorem with application to sheaves”. In:Bulletin de la Société mathématique de France84 (1956), pp. 307–317

1956
[3]

Automorphisms of punctual Hilbert schemes and symmetric powers of varieties

Ashima Bansal, Supravat Sarkar, and Shivam Vats. “Automorphisms of punctual Hilbert schemes and symmetric powers of varieties”. In: arXiv preprint arXiv:2508.18059(2025). 20 Tangent bundle of Hilbert scheme

work page arXiv 2025
[4]

Symmetric power of higher dimensional varieties

Ashima Bansal, Supravat Sarkar, and Shivam Vats. “Symmetric power of higher dimensional varieties”. In:arXiv preprint arXiv:2508.12654 (2025)

work page arXiv 2025
[5]

Isomorphisms and automorphisms of multiprojective bundles and symmetric powers of projective bundles

Ashima Bansal, Supravat Sarkar, and Shivam Vats. “Isomorphisms and automorphisms of multiprojective bundles and symmetric powers of projective bundles”. In:Bulletin des Sciences Mathématiques(2026), p. 103795

2026
[6]

Complex manifolds with split tangent bundle

Arnaud Beauville. “Complex manifolds with split tangent bundle”. In: Complex analysis and algebraic geometry(2000), pp. 61–70

2000
[7]

Automor- phisms of Hilbert schemes of points on surfaces

Pieter Belmans, Georg Oberdieck, and Jørgen Rennemo. “Automor- phisms of Hilbert schemes of points on surfaces”. In:Transactions of the American Mathematical Society373.9 (2020), pp. 6139–6156

2020
[8]

Moduli of products of stable varieties

Bhargav Bhatt et al. “Moduli of products of stable varieties”. In:Com- pos. Math.149.12 (2013), pp. 2036–2070

2013
[9]

On manifolds whose tangent bundle is big and 1-ample

Luis Eduardo Solá Conde and Jarosław A Wiśniewski. “On manifolds whose tangent bundle is big and 1-ample”. In:Proceedings of the Lon- don Mathematical Society89.2 (2004), pp. 273–290

2004
[10]

On the classifica- tion of products of Hilbert schemes of points over a surface

Arijit Dey, Arijit Mukherjee, and Anubhab Pahari. “On the classifica- tion of products of Hilbert schemes of points over a surface”. In:arXiv preprint arXiv:2604.01374(2026)

work page arXiv 2026
[11]

Algebraic families on an algebraic surface

John Fogarty. “Algebraic families on an algebraic surface”. In:Ameri- can Journal of Mathematics90.2 (1968), pp. 511–521

1968
[12]

Connected algebraic groups acting on algebraic surfaces

Pascal Fong. “Connected algebraic groups acting on algebraic surfaces”. In:Annales de l’Institut Fourier. Vol. 74. 2. 2024, pp. 545–587

2024
[13]

The tangent bundle of a ruled surface

Richard Ganong and Peter Russell. “The tangent bundle of a ruled surface”. In:Mathematische Annalen271.4 (1985), pp. 527–548

1985
[14]

A survey on Zariski cancellation problem

Neena Gupta. “A survey on Zariski cancellation problem”. In:Indian Journal of Pure and Applied Mathematics46.6 (2015), pp. 865–877

2015
[15]

Automorphisms of the Hilbert schemes ofnpoints of a rational surface and the anticanonical Iitaka dimension

Taro Hayashi. “Automorphisms of the Hilbert schemes ofnpoints of a rational surface and the anticanonical Iitaka dimension”. In:Geome- triae Dedicata207 (2020), pp. 395–407

2020
[16]

On algebraic varieties whose universal covering mani- folds are complex affine 3-spaces

Shigeru Iitaka. “On algebraic varieties whose universal covering mani- folds are complex affine 3-spaces”. In: (1972)

1972
[17]

OnmomentmapandbignessoftangentbundlesofG-varieties

JieLiu.“OnmomentmapandbignessoftangentbundlesofG-varieties”. In:Algebra & Number Theory17.8 (2023), pp. 1501–1532

2023
[18]

Varieties in positive charac- teristic with trivial tangent bundle

Vikram B Mehta and Vasudevan Srinivas. “Varieties in positive charac- teristic with trivial tangent bundle”. In:Compositio Mathematica64.2 (1987), pp. 191–212

1987
[19]

Projective manifolds with ample tangent bundles

Shigefumi Mori. “Projective manifolds with ample tangent bundles”. In:Annals of Mathematics110.3 (1979), pp. 593–606

1979
[20]

Diagonal property and weak point property of higher rank divisors and certain Hilbert schemes

Arijit Mukherjee and DS Nagaraj. “Diagonal property and weak point property of higher rank divisors and certain Hilbert schemes”. In:Bul- letin des Sciences Mathématiques198 (2025), p. 103541. 21 Sarkar S

2025
[21]

On automorphisms of the punctual Hilbert schemes of K3 surfaces

Keiji Oguiso. “On automorphisms of the punctual Hilbert schemes of K3 surfaces”. In:European Journal of Mathematics2.1 (2016), pp. 246– 261

2016
[22]

Big and nef tautological vector bundles over the Hilbert scheme of points

Dragos Oprea et al. “Big and nef tautological vector bundles over the Hilbert scheme of points”. In:SIGMA. Symmetry, Integrability and Geometry: Methods and Applications18 (2022), p. 061

2022
[23]

Kodaira dimension and zeros of holomorphic one-forms

Mihnea Popa and Christian Schnell. “Kodaira dimension and zeros of holomorphic one-forms”. In:Annals of Mathematics(2014), pp. 1109– 1120

2014
[24]

Walter Rudin.Function theory in polydiscs. W. A. Benjamin, Inc., New York-Amsterdam, 1969, pp. vii+188

1969
[25]

HighersymmetricpowersoftautologicalbundlesonHilbert schemes of points on a surface

LucaScala.“HighersymmetricpowersoftautologicalbundlesonHilbert schemes of points on a surface”. In:arXiv preprint arXiv:1502.07595 (2015)

work page arXiv 2015
[26]

Geometry and stability of tautological bundles on Hilbert schemes of points

David Stapleton. “Geometry and stability of tautological bundles on Hilbert schemes of points”. In:Algebra & Number Theory10.6 (2016), pp. 1173–1190

2016
[27]

On stability of the tangent bundles of Fano varieties

Gang Tian. “On stability of the tangent bundles of Fano varieties”. In: Internat. J. Math3.3 (1992), pp. 401–413

1992
[28]

Stability of tautological bundles on Hilbert schemes of points on a surface

Malte Wandel. “Stability of tautological bundles on Hilbert schemes of points on a surface”. In: (2013)

2013
[29]

On automorphisms of Hilbert squares of smooth hyper- surfaces

Long Wang. “On automorphisms of Hilbert squares of smooth hyper- surfaces”. In:Communications in Algebra51.2 (2023), pp. 586–595. Department of Mathematics, Fine Hall, Princeton University, Princeton, NJ 700108, USA. Email address: ss6663@princeton.edu 22

2023