Pith. sign in

REVIEW 4 major objections 5 minor 91 references

The Iarrobino scheme: a self-dual analogue of the Hilbert scheme of points

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The Iarrobino scheme is a self-dual Hilbert scheme of points, and over any smooth curve it is smooth of dimension $2d-1$.

desk verdict New moduli construction worth taking seriously; the main theorems are plausible but the printed proofs leave two load-bearing gaps that a revision should fill. read the letter →

arxiv 2508.21705 v1 pith:F7KC3M5F submitted 2025-08-29 math.AG math.AC

Joachim Jelisiejew This is my paper · ORCID
classification math.AGmath.AC MSC 14C0514D2214M2713H10
keywords IarrobinoschemeHilbertofpointsself-dualmodulesGorensteinschemescompletedquadricsQuotsymmetricdecompositionmodulispaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces the Iarrobino scheme, a self-dual analogue of the Hilbert scheme of points. For a scheme $X$, it parameterises flags of zero-dimensional subschemes together with symmetric isomorphisms between successive quotient modules. This compactifies the locus of oriented Gorenstein subschemes, where the extra data is a trivialisation of the dualising sheaf. The main result is that for a smooth curve $C$, the space $\operatorname{Iar}_d(C)$ is smooth, irreducible, of dimension $2d-1$, and the forgetful map $\tau_C$ is flat and projective with integral fibres of dimension $d-1$.

What carries the argument

The load-bearing object is the variety of completed quadrics $CQ(V)$, the closure of the space of full-rank quadrics inside a product of projective spaces of exterior powers. A point of $CQ(V)$ is a broken quadric: a sequence of symmetric maps $q_0: V^\vee\to V$, then $q_1: \ker q_0 \to \operatorname{coker} q_0$, and so on, ending with a full-rank quadric. The paper defines compatible and anticompatible bundles on $CQ(V)$, whose fibres are endomorphisms preserving the associated flag and acting symmetrically or antisymmetrically on each subquotient. The Iarrobino scheme is then the compatibility locus of the universal family over the Hilbert scheme with the compatible bundle, and the smoothness results for curves are proved using the smoothness of $CQ(V)$, Tyrrell's explicit affine patches, and an infinitesimal lifting argument.

What would settle it

For $C = \mathbb{A}^1$ and $E = O^{\oplus r}$, compute the complete local ring at a point of $\mathrm{CQuot}^d(E)$ corresponding to a split module with a rank-one first quadric; if its tangent space has dimension larger than $d(1+r)-1$, the infinitesimal lifting assertion in the proof of Theorem 4.14 is false.

Watch

Extended reading notes

Core claim

For every quasi-projective $k$-scheme $X$ with $\operatorname{char} k \neq 2$ and every integer $d$, the paper constructs a quasi-projective scheme $\operatorname{Iar}_d(X)$ with a projective morphism $\tau_X:\operatorname{Iar}_d(X)\to \mathrm{Hilb}_d(X)$. Its $k$-points are flags $X \supsetneq Z_0 \supsetneq \cdots$ of zero-dimensional subschemes of degree $d$, together with symmetric $O_X$-module isomorphisms $q_i: (I_{Z_{i+1}}/I_{Z_i})^\vee \to I_{Z_{i+1}}/I_{Z_i}$. This compactifies the oriented Gorenstein locus, where there is only one term and $q_0$ trivialises the dualising sheaf. The central structural result is that for a smooth connected curve $C$, the scheme $\operatorname{Iar}_d(C)$ is smooth and irreducible of dimension $2d-1$, the forgetful map $\tau_C$ is flat and projective with integral fibres of dimension $d-1$, and $(\tau_C)_*O_{\operatorname{Iar}_d(C)} = O_{\mathrm{Hilb}_d(C)}$.

Load-bearing premise

The smoothness proof for the self-dual Quot scheme over a curve assumes that every compatible self-duality structure on an infinitesimal deformation can be lifted to the next infinitesimal order once the underlying module is freely extended; if that lifting fails, the smoothness claim does not follow.

Editorial extensions

If this is right

  • Over a smooth curve $C$, $\operatorname{Iar}_d(C)$ is smooth, irreducible of dimension $2d-1$, and $\tau_C$ is flat and projective with integral fibres of dimension $d-1$.
  • For a smooth threefold $X$, a point of $\operatorname{Iar}_d(X)$ whose intermediate subquotients are principal ideals detects smooth points of $\mathrm{Hilb}_d(X)$, giving a new route toward understanding smoothness of Hilbert schemes of points on threefolds.
  • The characteristic numbers of the algebra $k[x]/(f)$ are independent of $f$, because the relevant subvariety is a fibre of the flat map $\tau_{\mathbb{A}^1}$ over the Hilbert scheme.
  • The Iarrobino scheme provides a modular interpretation of the varieties $X_V$ attached to finite algebras as closures of the unbroken part of the fibre of $\tau$, explaining their lower semicontinuity.
  • The same construction yields self-dual analogues of the Quot scheme of points and of the stacks of coherent sheaves and finite algebras, so the formalism extends beyond Hilbert schemes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the compatibility-locus construction is as robust as it appears, the same smoothness argument might extend to other bases where completed quadrics are smooth, giving self-dual Quot schemes beyond the curve case without additional hypotheses.
  • The interpretation of Iarrobino's symmetric decomposition as a torus limit on $\operatorname{Iar}_d(\mathbb{A}^n)$ suggests that Bialynicki-Birula decompositions of the Iarrobino scheme could produce new numerical constraints on Hilbert functions of Gorenstein algebras.
  • A concrete testable direction is to compute intersection numbers on $\operatorname{Iar}_d(X)$ for $X$ of higher dimension, where the fibres of $\tau_X$ need not be irreducible; flatness would then be replaced by virtual structure, giving a different enumerative theory from the one on the usual Hilbert scheme.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces, for any quasi-projective k-scheme X (char k != 2) and d >= 0, a scheme Iard(X) that parameterizes flags X = Z0 ? Z1 ? ... of zero-dimensional subschemes of degree d together with symmetric isomorphisms (I_{Z_{i+1}}/I_{Z_i})^vee -> I_{Z_{i+1}}/I_{Z_i}, and a projective forgetful morphism tau: Iard(X) -> Hilbd(X). The construction is made through the variety of completed quadrics and new compatible and anticompatible vector bundles on it. The paper proves the scheme exists and is quasi-projective, gives a concrete description of its k-points, defines Quot, stack, and commuting-matrix analogues, and then studies the curve case: Theorem 4.14 claims CQuotd(E) is smooth and connected of dimension d(1 + rk E) - 1, and Theorem 4.18 claims Iard(C) is smooth, irreducible, of dimension 2d - 1, with flat projective integral fibres over Hilbd(C) satisfying (tau_C)_* O = O. Applications to deformation theory of Hilbert schemes, characteristic numbers of algebras, and enumerative geometry are discussed.

Significance. If the main theorems hold, this is a substantial new construction: it gives a natural compactification of the oriented Gorenstein locus, a self-dual analogue of the Hilbert and Quot schemes, and a new bridge between Hilbert schemes and the completed quadrics used in recent enumerative work. The explicit pointwise description of Iard(X) and the self-contained treatment of completed quadrics and of the (anti)compatible bundles are clear strengths. The paper also convincingly situates the construction relative to existing literature and to ongoing work. However, the central new geometric statements for curves are not fully proved as written: the smoothness and connectedness proof of Theorem 4.14 contains two unproved lifting/extension steps, and the irreducibility and integrality of the fibres in Theorem 4.18 are asserted rather than established. The significance of the paper is therefore conditional on repairing these arguments.

major comments (4)
  1. [Section 4.4, Theorem 4.14 (smoothness)] The infinitesimal lifting argument is plausible, but two steps are load-bearing and unproved. First, the sentence 'The element [eq•] is a map Spec(A) -> CQ(fM), -> CQ(fM')' assumes a natural morphism from the completed quadrics of a quotient module fM to the completed quadrics of a free extension fM'; no such inclusion is proved, and it is not automatic for completed quadrics on quotients. Second, the statement that the compatible bundle C is smooth over CQ(fM') does not by itself imply that a section Spec(A) -> C over CQ(fM')|Spec(A) extends to Spec(B) -> C, because the base map Spec(A) -> CQ(fM') must simultaneously be lifted to Spec(B) and the section must be lifted compatibly. These two points are exactly what is needed to conclude that the deformation extends, so Theorem 4.14 is not justified as written.
  2. [Section 4.4, Theorem 4.14 (connectedness)] The connectedness proof ends by reducing to torus-fixed quotients of the form (4.17) and then says 'connect any module of the form (4.17) ... to the module M0. This is an elementary construction, which we leave to the reader.' Example 4.3 similarly says 'A willing reader can do this now by hand' for the irreducibility of the compatible locus over k[x]/(x^d). These are not cosmetic exercises: without an explicit deformation argument one cannot conclude that the open locus V is dense or that there are no extra components, and the claim that CQuotd(E) is connected remains incomplete.
  3. [Section 4.4, Theorem 4.18] The assertion that every fibre tau_C^{-1}([Z]) is irreducible is never proved. The displayed argument only establishes the dimension bound and then deduces reducedness from the existence of a smooth unbroken point; a smooth point in one component of a possibly reducible fibre does not imply generic reducedness. The integrality of the fibres is used both for flatness via Miracle Flatness and for the Stein factorization conclusion (tau_C)_* O = O, so the statement of Theorem 4.18 is not supported by the proof as written.
  4. [Section 4.4.1, Proposition 4.21] The proof of independence of characteristic numbers uses flatness of tau_{A^1} via [Ful98, Section 10.2]. Since the flatness of tau_{A^1} is a consequence of Theorem 4.18, this application is conditional on the missing fibre-integrality proof. The paper should either prove the needed flatness directly for the universal family over Hilbd(A^1) or state explicitly that the result is conditional on Theorem 4.18 being completed.
minor comments (5)
  1. [Throughout, e.g. Abstract and Proposition 4.7] The word 'bijectively' appears throughout; this should be replaced by 'bijectively'.
  2. [Section 3.4, after Definition 3.6] The manuscript uses 'anticompactible' once; this is a typo for 'anticompatible'.
  3. [Lemma 3.18] The check that (3.17) is the limit of Lambda^bullet q(t) is left to the reader; a short justification would improve readability, though this is not load-bearing.
  4. [Proposition 3.26 and Example 3.31] The notation A^{d-1} is used both for affine space and for the base field; this is potentially confusing but harmless.
  5. [References] The reference [JPS25] is cited as arXiv:19587, which appears to be an incomplete identifier; the full arXiv number should be given.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Iarrobino scheme construction is self-contained; noted gaps are missing proofs, not circular reductions.

full rationale

The central construction is not circular. Iard(X) and CQuotd(E) are defined as closed compatibility loci inside CQ(U), and the curve smoothness theorem is proved by an infinitesimal lifting argument whose inputs are the smoothness of completed quadrics (Tyrrell patches, [TK88]) and a reduction to A1 via Cohen's structure theorem. The author's self-citations ([JRS24], [JRS25], [Jel19]) appear as applications, motivation, or explicitly future work, and they are not load-bearing premises. Proposition 4.21 reproves a result of [JPS25] independently rather than importing it, and Theorem 1.7 is presented as an application of [JRS24], not as an input to the construction. The genuine weaknesses are omitted arguments, not circularity: Theorem 4.18 asserts irreducibility of fibres without supplying a proof, and Theorem 4.14 defers a connecting deformation with 'This is an elementary construction, which we leave to the reader'; Example 4.3 similarly says 'A willing reader can do this now by hand.' These are correctness risks, but they do not exhibit a claim reducing to its own inputs by construction. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the author's prior work to force a choice, and no known result is merely relabelled. The derivation chain is therefore independent of its outputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The central construction rests on the standard theory of completed quadrics and Hilbert schemes; no numerical parameters are fitted to data. The genuinely new objects (Iard, CQuot, compatible and anticompatible bundles) are given explicit moduli interpretations and proven geometric properties, so they carry independent evidence within the paper.

assumptions (5)
  • domain assumption The variety of completed quadrics CQ(V) is smooth projective and represents the functor of projectively d-complete symmetric forms, with duality isomorphism CQ(V) -> CQ(V^vee).
    Used throughout Sections 3 and 4 to define Iard(X); the paper recalls the construction but relies on [TK88] for smoothness, functor of points, and duality (Propositions 3.15, 3.33, Theorem 3.36).
  • standard math For a zero-dimensional Gorenstein scheme over a field, the dualizing sheaf is trivializable, and the Buchsbaum-Eisenbud structure theorem gives smoothness of the Gorenstein locus for dimension at most 3.
    Used in Lemma 4.12 and Section 1.2.1; quoted from [BE77, KMR98] without proof.
  • standard math Cohen's structure theorem: complete local rings of finite k-schemes on a smooth curve are products of power series rings.
    Used in the reduction to A1 in the proof of Theorem 4.14, specifically in equation (4.15).
  • standard math Bialynicki-Birula decomposition and Gm-limits behave as stated, including existence and uniqueness of limits for projective maps.
    Used in Section 4.5 for torus limits on Hilbert and Iarrobino schemes; citations [BB73, JS19].
  • domain assumption The base field k has characteristic different from 2 throughout.
    Stated in Section 2; used for symmetric and antisymmetric decomposition, inverses of quadrics, and trace pairing arguments.
invented entities (3)
  • Iarrobino scheme Iard(X) independent evidence
    purpose: Fine moduli space of oriented Gorenstein zero-dimensional subschemes with self-dual filtration, compactifying the Gorenstein locus.
    The paper proves it exists as a quasi-projective scheme with a projective map to Hilbd(X) (Theorem 1.1) and proves smoothness and flatness for curves (Theorem 4.18), providing independent checks of the proposed moduli interpretation.
  • Self-dual Quot scheme CQuotd(E) independent evidence
    purpose: Quot scheme analogue parameterizing quotients equipped with compatible broken quadrics.
    Proved smooth and connected over a smooth curve (Theorem 4.14), giving an independent handle beyond the definition.
  • Compatible and anticompatible bundles C and aC on CQ(V) independent evidence
    purpose: Record endomorphisms of V preserving a broken quadric, symmetric or antisymmetric on subquotients; used to define compatibility conditions.
    Constructed explicitly as kernels of representation-theoretic maps with fibre descriptions (Propositions 3.13, 3.26, 3.30); these are checkable algebraic data.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Iarrobino scheme: a self-dual analogue of the Hilbert scheme of points." pith.science (2026). https://pith.science/paper/F7KC3M5F

@misc{pith2026250821705,
  author       = {Pith},
  title        = {Pith review of: The Iarrobino scheme: a self-dual analogue of the Hilbert scheme of points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F7KC3M5F}},
  note         = {Machine review of arXiv:2508.21705}
}
abstract

For a fixed quasi-projective scheme $X$ we introduce a self-dual analogue of ${\mathrm{Hilb}}_d(X)$ which we call the Iarrobino scheme of $X$. It is a fine moduli space of oriented Gorenstein zero-dimensional subschemes of $X$ together with some additional data (a self-dual filtration) which is vacuous over a big open set but non-trivial over the compactification. Via the link between Hilbert schemes and varieties of commuting matrices, Iarrobino schemes correspond to commuting symmetric matrices. We provide also self-dual analogues of the Quot scheme of points and of the stacks of coherent sheaves and finite algebras. A crucial role in the construction is played by the variety of completed quadrics. We prove that the resulting analogues of Hilbert and Quot schemes are smooth for $X$ a smooth curve and that they have very rich geometry. We give applications, in particular to deformation theory of (usual) Hilbert schemes of points on threefolds, and to enumerative geometry \`a la June Huh.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

91 extracted references · 55 canonical work pages

  1. [1]

    Hodge theory for combinatorial geometries

    Karim Adiprasito, June Huh, and Eric Katz. Hodge theory for combinatorial geometries. Ann. of Math. (2) , 188(2):381--452, 2018

    2018

  2. [2]

    Wonderful varieties with a view towards Poisson geometry

    Ana B a libanu. Wonderful varieties with a view towards P oisson geometry. arXiv:2307.06663 , 2023

    work page Pith review arXiv 2023

  3. [3]

    Bia ynicki-Birula

    A. Bia ynicki-Birula. Some theorems on actions of algebraic groups. Ann. of Math. (2) , 98:480--497, 1973

    1973

  4. [4]

    Secant varieties to high degree V eronese reembeddings, catalecticant matrices and smoothable G orenstein schemes

    Weronika Buczy \'n ska and Jaros aw Buczy \'n ski. Secant varieties to high degree V eronese reembeddings, catalecticant matrices and smoothable G orenstein schemes. J. Algebraic Geom. , 23:63--90, 2014

    2014

  5. [5]

    Buchsbaum and David Eisenbud

    David A. Buchsbaum and David Eisenbud. Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3 . Amer. J. Math. , 99(3):447--485, 1977

    1977

  6. [6]

    Donaldson- T homas type invariants via microlocal geometry

    Kai Behrend. Donaldson- T homas type invariants via microlocal geometry. Ann. of Math. (2) , 170(3):1307--1338, 2009

    2009

  7. [7]

    The punctual H ilbert scheme: an introduction

    Jos\' e Bertin. The punctual H ilbert scheme: an introduction. In Geometric methods in representation theory. I , volume 24 of S\' e min. Congr. , pages 1--102. Soc. Math. France, Paris, 2012

    2012

  8. [8]

    Symmetric obstruction theories and H ilbert schemes of points on threefolds

    Kai Behrend and Barbara Fantechi. Symmetric obstruction theories and H ilbert schemes of points on threefolds. Algebra Number Theory , 2(3):313--345, 2008

    2008

Show all 91 references
  1. [9]

    On the motive of Q uot schemes of zero-dimensional quotients on a curve

    Massimo Bagnarol, Barbara Fantechi, and Fabio Perroni. On the motive of Q uot schemes of zero-dimensional quotients on a curve. New York J. Math. , 26:138--148, 2020

    2020

  2. [10]

    On the cactus rank of cubics forms

    Alessandra Bernardi and Kristian Ranestad. On the cactus rank of cubics forms. J. Symbolic Comput. , 50:291--297, 2013

    2013

  3. [11]

    Log homogeneous varieties

    Michel Brion. Log homogeneous varieties. In Proceedings of the XVI th L atin A merican A lgebra C olloquium ( S panish) , Bibl. Rev. Mat. Iberoamericana, pages 1--39. Rev. Mat. Iberoamericana, Madrid, 2007

    2007

  4. [12]

    Irreducibility of the G orenstein loci of H ilbert schemes via ray families

    Gianfranco Casnati, Joachim Jelisiejew, and Roberto Notari. Irreducibility of the G orenstein loci of H ilbert schemes via ray families. Algebra Number Theory , 9(7):1525--1570, 2015

    2015

  5. [13]

    On the G orenstein locus of some punctual H ilbert schemes

    Gianfranco Casnati and Roberto Notari. On the G orenstein locus of some punctual H ilbert schemes. J. Pure Appl. Algebra , 213(11):2055--2074, 2009

    2009

  6. [14]

    On the irreducibility and the singularities of the G orenstein locus of the punctual H ilbert scheme of degree 10

    Gianfranco Casnati and Roberto Notari. On the irreducibility and the singularities of the G orenstein locus of the punctual H ilbert scheme of degree 10. J. Pure Appl. Algebra , 215(6):1243--1254, 2011

    2011

  7. [15]

    De Concini and C

    C. De Concini and C. Procesi. Complete symmetric varieties. In Invariant theory ( M ontecatini, 1982) , volume 996 of Lecture Notes in Math. , pages 1--44. Springer, Berlin, 1983

    1982

  8. [16]

    De Concini and T

    C. De Concini and T. A. Springer. Compactification of symmetric varieties. volume 4, pages 273--300. 1999. Dedicated to the memory of Claude Chevalley

    1999

  9. [17]

    A pplications of intersection theory: from maximum likelihood to chromatic polynomials

    Rodica Dinu, Mateusz Micha ek, and Tim Seynnaeve. A pplications of intersection theory: from maximum likelihood to chromatic polynomials. To appear in Einstein Semester Proceedings ”Varieties, Polyhedra, Computation”, EMS Series of Congress Reports, arXiv:2111.02057 , 2021

    2021 arXiv

  10. [18]

    3264 and all that---a second course in algebraic geometry

    David Eisenbud and Joe Harris. 3264 and all that---a second course in algebraic geometry . Cambridge University Press, Cambridge, 2016

    2016

  11. [19]

    Commutative algebra , volume 150 of Graduate Texts in Mathematics

    David Eisenbud. Commutative algebra , volume 150 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1995. With a view toward algebraic geometry

    1995

  12. [20]

    Joan Elias and Maria E. Rossi. Isomorphism classes of short G orenstein local rings via M acaulay's inverse system. Trans. Amer. Math. Soc. , 364(9):4589--4604, 2012

    2012

  13. [21]

    Elias and M

    J. Elias and M. E. Rossi. A constructive approach to one-dimensional G orenstein k -algebras. Trans. Amer. Math. Soc. , 374(7):4953--4971, 2021

    2021

  14. [22]

    On the homology of the H ilbert scheme of points in the plane

    Geir Ellingsrud and Stein Arild Str mme. On the homology of the H ilbert scheme of points in the plane. Invent. Math. , 87(2):343--352, 1987

    1987

  15. [23]

    Recovering the good component of the H ilbert scheme

    Torsten Ekedahl and Roy Skjelnes. Recovering the good component of the H ilbert scheme. Ann. of Math. (2) , 179(3):805--841, 2014

    2014

  16. [24]

    Essence of independence: H odge theory of matroids since J une H uh

    Christopher Eur. Essence of independence: H odge theory of matroids since J une H uh. Bull. Amer. Math. Soc. (N.S.) , 61(1):73--102, 2024

    2024

  17. [25]

    Gelfand- T setlin algebras and cohomology rings of L aumon spaces

    Boris Feigin, Michael Finkelberg, Igor Frenkel, and Leonid Rybnikov. Gelfand- T setlin algebras and cohomology rings of L aumon spaces. Selecta Math. (N.S.) , 17(2):337--361, 2011

    2011

  18. [26]

    Kleiman, Nitin Nitsure, and Angelo Vistoli

    Barbara Fantechi, Lothar G \"o ttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli. Fundamental algebraic geometry , volume 123 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2005. Grothendieck's FGA explained

    2005

  19. [27]

    Algebraic families on an algebraic surface

    John Fogarty. Algebraic families on an algebraic surface. Amer. J. Math , 90:511--521, 1968

    1968

  20. [28]

    I rrational components of the H ilbert scheme of points

    Gavril Farkas, Rahul Pandharipande, and Alessio Sammartano. I rrational components of the H ilbert scheme of points. arXiv:2405.11997 , 2025

    2025 arXiv

  21. [29]

    B. L. Feigin and A. I. Tsymbaliuk. Equivariant K -theory of H ilbert schemes via shuffle algebra. Kyoto J. Math. , 51(4):831--854, 2011

    2011

  22. [30]

    Intersection theory , volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete

    William Fulton. Intersection theory , volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] . Springer-Verlag, Berlin, ...

    1998

  23. [31]

    Giovenzana, L

    F. Giovenzana, L. Giovenzana, M. Graffeo, and P. Lella. Unexpected but recurrent phenomena for Q uot and H ilbert schemes of points. Rend. Semin. Mat. Univ. Politec. Torino , 82(1):145--170, 2024

    2024

  24. [32]

    Gustavsen, Dan Laksov, and Roy Mikael Skjelnes

    Trond S. Gustavsen, Dan Laksov, and Roy Mikael Skjelnes. An elementary, explicit, proof of the existence of H ilbert schemes of points. J. Pure Appl. Algebra , 210(3):705--720, 2007

    2007

  25. [33]

    Refined knot invariants and H ilbert schemes

    Eugene Gorsky and Andrei Negu t . Refined knot invariants and H ilbert schemes. J. Math. Pures Appl. (9) , 104(3):403--435, 2015

    2015

  26. [34]

    Flag H ilbert schemes, colored projectors and K hovanov- R ozansky homology

    Eugene Gorsky, Andrei Negu t , and Jacob Rasmussen. Flag H ilbert schemes, colored projectors and K hovanov- R ozansky homology. Adv. Math. , 378:Paper No. 107542, 115, 2021

    2021

  27. [35]

    Hilbert schemes of zero-dimensional subschemes of smooth varieties , volume 1572 of Lecture Notes in Mathematics

    Lothar G \"o ttsche. Hilbert schemes of zero-dimensional subschemes of smooth varieties , volume 1572 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1994

    1994

  28. [36]

    Hilbert schemes, polygraphs and the M acdonald positivity conjecture

    Mark Haiman. Hilbert schemes, polygraphs and the M acdonald positivity conjecture. J. Amer. Math. Soc. , 14(4):941--1006 (electronic), 2001

    2001

  29. [37]

    Notes on M acdonald polynomials and the geometry of H ilbert schemes

    Mark Haiman. Notes on M acdonald polynomials and the geometry of H ilbert schemes. In Symmetric functions 2001: surveys of developments and perspectives , volume 74 of NATO Sci. Ser. II Math. Phys. Chem. , pages 1--64. Kluwer Acad. Publ., Dordrecht, 2002

    2001

  30. [38]

    Vanishing theorems and character formulas for the H ilbert scheme of points in the plane

    Mark Haiman. Vanishing theorems and character formulas for the H ilbert scheme of points in the plane. Invent. Math. , 149(2):371--407, 2002

    2002

  31. [39]

    Log-concavity of characteristic polynomials and the B ergman fan of matroids

    June Huh and Eric Katz. Log-concavity of characteristic polynomials and the B ergman fan of matroids. Math. Ann. , 354(3):1103--1116, 2012

    2012

  32. [40]

    The geometry of moduli spaces of sheaves

    Daniel Huybrechts and Manfred Lehn. The geometry of moduli spaces of sheaves . Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 2010

    2010

  33. [41]

    Cohomology of large semiprojective hyperk \"a hler varieties

    Tam \'a s Hausel and Fernando Rodriguez Villegas. Cohomology of large semiprojective hyperk \"a hler varieties. Ast\'erisque , (370):113--156, 2015

    2015

  34. [42]

    Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

    June Huh. Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs. J. Amer. Math. Soc. , 25(3):907--927, 2012

    2012

  35. [43]

    Tropical geometry of matroids

    June Huh. Tropical geometry of matroids. In Current developments in mathematics 2016 , pages 1--46. Int. Press, Somerville, MA, 2018

    2016

  36. [44]

    Huibregtse

    Mark E. Huibregtse. An elementary construction of the multigraded H ilbert scheme of points. Pacific J. Math. , 223(2):269--315, 2006

    2006

  37. [45]

    Associated graded algebra of a G orenstein A rtin algebra

    Anthony Iarrobino. Associated graded algebra of a G orenstein A rtin algebra. Mem. Amer. Math. Soc. , 107(514):viii+115, 1994

    1994

  38. [46]

    Symmetric decomposition of the associated graded algebra of an A rtinian G orenstein algebra

    Anthony Iarrobino and Pedro Macias Marques. Symmetric decomposition of the associated graded algebra of an A rtinian G orenstein algebra. J. Pure Appl. Algebra , 225(3):Paper No. 106496, 49, 2021

    2021

  39. [47]

    Connected sums of graded A rtinian G orenstein algebras and L efschetz properties

    Anthony Iarrobino, Chris McDaniel, and Alexandra Seceleanu. Connected sums of graded A rtinian G orenstein algebras and L efschetz properties. J. Pure Appl. Algebra , 226(1):Paper No. 106787, 52, 2022

    2022

  40. [48]

    Classifying local A rtinian G orenstein algebras

    Joachim Jelisiejew. Classifying local A rtinian G orenstein algebras. Collect. Math. , 68(1):101--127, 2017

    2017

  41. [49]

    Elementary components of H ilbert schemes of points

    Joachim Jelisiejew. Elementary components of H ilbert schemes of points. Journal of the L ondon M athematical S ociety , 100(1):249--272, 2019

    2019

  42. [50]

    Pathologies on the H ilbert scheme of points

    Joachim Jelisiejew. Pathologies on the H ilbert scheme of points. Invent. Math. , 220(2):581--610, 2020

    2020

  43. [51]

    Open problems in deformations of A rtinian algebras, H ilbert schemes and around

    Joachim Jelisiejew. Open problems in deformations of A rtinian algebras, H ilbert schemes and around. In Deformation of A rtinian algebras and J ordan type , volume 805 of Contemp. Math. , pages 3--25. Amer. Math. Soc., [Providence], RI, [2024] 2024

    2024

  44. [52]

    Masuti, and M

    Joachim Jelisiejew, Shreedevi K. Masuti, and M. E. Rossi. On the H ilbert function of A rtinian local complete intersections of codimension three. J. Pure Appl. Algebra , 227(7):Paper No. 107326, 21, 2023

    2023

  45. [53]

    Characteristic numbers of algebras

    Jakub Jagie a, Pawe Pielasa, and Anatoli Shatsila. Characteristic numbers of algebras. arXiv:19587 , 2025

    2025

  46. [54]

    The H ilbert scheme of points on a threefold, I

    Joachim Jelisiejew, Ritvik Ramkumar, and Alessio Sammartano. The H ilbert scheme of points on a threefold, I . arXiv.2409.17009 , 2024

    2024 arXiv

  47. [55]

    Schmiermann

    Joachim Jelisiejew, Andrea Ricolfi, and Reinier F. Schmiermann. Enumerative geometry of the I arrobino scheme. In preparation , 2025

    2025

  48. [56]

    B ia ynicki- B irula decomposition for reductive groups

    Joachim Jelisiejew and ukasz Sienkiewicz. B ia ynicki- B irula decomposition for reductive groups. Journal de Mathématiques Pures et Appliquées , 131:290 -- 325, 2019

    2019

  49. [57]

    Components and singularities of Q uot schemes and varieties of commuting matrices

    Joachim Jelisiejew and Klemen S ivic. Components and singularities of Q uot schemes and varieties of commuting matrices. J. Reine Angew. Math. , 788:129--187, 2022

    2022

  50. [58]

    Steven L. Kleiman. Toward a numerical theory of ampleness. Ann. of Math. (2) , 84:293--344, 1966

    1966

  51. [59]

    Steven L. Kleiman. Chasles's enumerative theory of conics: a historical introduction. In Studies in algebraic geometry , volume 20 of MAA Stud. Math. , pages 117--138. Math. Assoc. America, Washington, DC, 1980

    1980

  52. [60]

    Kleppe and Rosa M

    Jan O. Kleppe and Rosa M. Mir \'o -Roig. The dimension of the H ilbert scheme of G orenstein codimension 3 subschemes. J. Pure Appl. Algebra , 127(1):73--82, 1998

    1998

  53. [61]

    Proof of magnificent conjecture

    Martijn Kool and J rgen Vold Rennemo. Proof of magnificent conjecture. 2025. https://arxiv.org/abs/2507.02852

    2025

  54. [62]

    Gorenstein modules of finite length

    Michael Kunte. Gorenstein modules of finite length. Math. Nachr. , 284(7):899--919, 2011

    2011

  55. [63]

    Kapranov and E

    M. Kapranov and E. Vasserot. Kleinian singularities, derived categories and H all algebras. Math. Ann. , 316(3):565--576, 2000

    2000

  56. [64]

    N akajima’s C reation O perators and the K irwan M ap

    Jakub Koncki and Magdalena Zielenkiewicz. N akajima’s C reation O perators and the K irwan M ap. Transformation Groups , 2025

    2025

  57. [65]

    Completed quadrics and linear maps

    Dan Laksov. Completed quadrics and linear maps. In Algebraic geometry, B owdoin, 1985 ( B runswick, M aine, 1985) , volume 46, Part 2 of Proc. Sympos. Pure Math. , pages 371--387. Amer. Math. Soc., Providence, RI, 1987

    1985

  58. [66]

    On the birational geometry of spaces of complete forms I : collineations and quadrics

    Alex Massarenti. On the birational geometry of spaces of complete forms I : collineations and quadrics. Proc. Lond. Math. Soc. (3) , 121(6):1579--1618, 2020

    2020

  59. [67]

    Enumerative geometry meets statistics, combinatorics, and topology

    Mateusz Micha ek. Enumerative geometry meets statistics, combinatorics, and topology. Notices Amer. Math. Soc. , 70(4):588--597, 2023

    2023

  60. [68]

    Complete quadrics: S chubert calculus for G aussian models and semidefinite programming

    Laurent Manivel, Mateusz Micha ek, Leonid Monin, Tim Seynnaeve, and Martin Vodi c ka. Complete quadrics: S chubert calculus for G aussian models and semidefinite programming. J. Eur. Math. Soc. (JEMS) , 26(8):3091--3135, 2024

    2024

  61. [69]

    Wi \'s niewski

    Mateusz Micha ek, Leonid Monin, and Jaros aw A. Wi \'s niewski. Maximum likelihood degree, complete quadrics, and C ^* -action. SIAM J. Appl. Algebra Geom. , 5(1):60--85, 2021

    2021

  62. [70]

    The cohomology of the Q uot scheme on a smooth curve as a Y angian representation

    Alina Marian and Andrei Negu t . The cohomology of the Q uot scheme on a smooth curve as a Y angian representation. arXiv:2307.13671 , 2023

    2023

  63. [71]

    Maulik, N

    D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande. Gromov- W itten theory and D onaldson- T homas theory. I . Compos. Math. , 142(5):1263--1285, 2006

    2006

  64. [72]

    Maulik, N

    D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande. Gromov- W itten theory and D onaldson- T homas theory. II . Compos. Math. , 142(5):1286--1304, 2006

    2006

  65. [73]

    Combinatorial commutative algebra , volume 227 of Graduate Texts in Mathematics

    Ezra Miller and Bernd Sturmfels. Combinatorial commutative algebra , volume 227 of Graduate Texts in Mathematics . Springer-Verlag, New York, 2005

    2005

  66. [74]

    Lectures on H ilbert schemes of points on surfaces , volume 18 of University Lecture Series

    Hiraku Nakajima. Lectures on H ilbert schemes of points on surfaces , volume 18 of University Lecture Series . American Mathematical Society, Providence, RI, 1999

    1999

  67. [75]

    Quiver varieties and finite-dimensional representations of quantum affine algebras

    Hiraku Nakajima. Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc. , 14(1):145--238, 2001

    2001

  68. [76]

    More lectures on H ilbert schemes of points on surfaces

    Hiraku Nakajima. More lectures on H ilbert schemes of points on surfaces. In Development of moduli theory--- K yoto 2013 , volume 69 of Adv. Stud. Pure Math. , pages 173--205. Math. Soc. Japan, [Tokyo], 2016

    2013

  69. [77]

    Magnificent four

    Nikita Nekrasov. Magnificent four. Ann. Inst. Henri Poincar\'e D , 7(4):505--534, 2020

    2020

  70. [78]

    Algebraic spaces and stacks , volume 62 of American Mathematical Society Colloquium Publications

    Martin Olsson. Algebraic spaces and stacks , volume 62 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, 2016

    2016

  71. [79]

    Okounkov and R

    A. Okounkov and R. Pandharipande. Quantum cohomology of the H ilbert scheme of points in the plane. Invent. Math. , 179(3):523--557, 2010

    2010

  72. [80]

    Jeongseok Oh and Richard P. Thomas. Counting sheaves on C alabi- Y au 4-folds, I . Duke Math. J. , 172(7):1333--1409, 2023

    2023

  73. [81]

    Lectures on wonderful varieties

    Guido Pezzini. Lectures on wonderful varieties. Acta Math. Sin. (Engl. Ser.) , 34(3):417--438, 2018

    2018

  74. [82]

    Andrea T. Ricolfi. On the motive of the Q uot scheme of finite quotients of a locally free sheaf. J. Math. Pures Appl. (9) , 144:50--68, 2020

    2020

  75. [83]

    Andrea T. Ricolfi. An Invitation to Modern Enumerative Geometry . Springer International Publishing, 2022

    2022

  76. [84]

    Varieties of sums of powers

    Kristian Ranestad and Frank-Olaf Schreyer. Varieties of sums of powers. J. Reine Angew. Math. , 525:147--181, 2000

    2000

  77. [85]

    J. G. Semple. On complete quadrics. J. London Math. Soc. , 23:258--267, 1948

    1948

  78. [86]

    Matthew Satriano and Andrew P. Staal. Small elementary components of H ilbert schemes of points. Forum Math. Sigma , 11:Paper No. e45, 36, 2023

    2023

  79. [87]

    http://math.columbia.edu/algebraic_geometry/stacks-git, 2025

    S tacks P roject. http://math.columbia.edu/algebraic_geometry/stacks-git, 2025

    2025

  80. [88]

    Complete collineations revisited

    Michael Thaddeus. Complete collineations revisited. Math. Ann. , 315(3):469--495, 1999

    1999

  81. [89]

    Complete bilinear forms

    Anders Thorup and Steven Kleiman. Complete bilinear forms. In Algebraic geometry ( S undance, UT , 1986) , volume 1311 of Lecture Notes in Math. , pages 253--320. Springer, Berlin, 1988

    1986

  82. [90]

    J. A. Tyrrell. Complete quadrics and collineations in S_n . Mathematika , 3:69--79, 1956

    1956

  83. [91]

    Iarrobino's decomposition for self-dual modules

    Maciej Wojtala . Iarrobino's decomposition for self-dual modules. arXiv:2405.13829, 2024

    2024 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.