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A proof of the Brian\c{c}on-Iarrobino Conjecture in three dimensions

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Pith's one-line read The paper proves that for every tetrahedral number $l = \binom{k+2}{3}$, the Hilbert scheme $\mathrm{Hilb}^{l}(\mathbb{A}^{3})$ is most singular at the point cut out by the $k$-th power of the maximal ideal, resolving the 1978…

desk verdict The main idea is likely right, but a load-bearing lemma is unproved; the paper needs a real referee and a repaired proof before it can be accepted. read the letter →

arxiv 2508.21717 v1 pith:Q2HJYXYG submitted 2025-08-29 math.AG math.ACmath.COmath.RT

Owen Mackenzie , Fatemeh Rezaee This is my paper · ORCID
classification math.AGmath.ACmath.COmath.RT MSC 14C0513D1013P10
keywords HilbertschemeofpointsBriançon–IarrobinoconjecturetangentspacedimensionBorel-fixedidealmonomialtetrahedralnumbermaximalsingularitycolength
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 Hilbert scheme $\mathrm{Hilb}^{l}(\mathbb{A}^{3})$ parameterizes ideals of colength $l$ in $\mathbb{C}[x,y,z]$ — geometrically, configurations of $l$ points with multiplicities. The paper proves a 1978 conjecture stating that when $l$ is a tetrahedral number, $l = \binom{k+2}{3}$, the largest possible dimension of the tangent space is attained at the ideal $(x,y,z)^{k}$. The proof refines a known upper bound by subtracting an explicit correction term, then shows the resulting function of the smallest pure exponent is strictly increasing. At the conjectured ideal the corrected upper bound equals the known tangent-space dimension, so the bound is sharp. The same argument gives a necessary condition for maximal singularity: the smallest pure exponent must be $k$.

What carries the argument

The argument works with the decomposition $I = \bigoplus_i x^{i} I_i$, where each $I_i$ is a monomial ideal in $\mathbb{C}[y,z]$; this is the filtration introduced in [37]. For two such ideals $J, J'$, the paper proves $\hom(J, S/J') = l + l' - \operatorname{Card}(B_n^{J,J'} \setminus A_n^{J,J'})$, where the subtracted term counts 'ghost vectors', connected components that lie in the upper half-plane but are unbounded and hence do not contribute to the tangent space. A computable lower bound for that term is $t(J, J')$, the number of monomials of $S/J'$ sitting at or above the height of $J'$. The new input is Corollary 2.2, which asserts $t(I_i, I_j) \geq \binom{j-i+1}{2}$ for the layers of any Borel-fixed ideal; summing these triangular lower bounds produces the term $2\binom{m_1+2}{4}$ that makes the earlier upper bound sharp.

What would settle it

Take the layers $I_i$ of every Borel-fixed ideal of small colength, compute $t(I_i, I_j)$ directly from the monomial sets, and check whether $t(I_i, I_j) \geq \binom{j-i+1}{2}$ holds in all cases; one pair with a smaller value, or one Borel-fixed ideal whose tangent-space dimension exceeds $\psi(m_1)$, would refute the central claim.

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Extended reading notes

Core claim

The central result is Theorem A: for any 0-dimensional Borel-fixed ideal $I = (x^{m_1}, y^{m_2}, z^{m_3}, \text{mixed generators})$ of colength $l = \binom{k+2}{3}+\Delta$ with $0 \leq \Delta \leq \binom{k+2}{2}-1$, the tangent-space dimension satisfies $T(I) \leq (2m_1+1)l - 2\binom{m_1+2}{4}$. Theorem B states that the right-hand side, viewed as a function $\psi(m_1)$, is strictly increasing. Since Lemma 2.3 forces $m_1 \leq k$ in the tetrahedral range, one gets $T(I) \leq \psi(k)$. A direct computation gives $\psi(k) = \binom{k+2}{2}\binom{k+1}{2} = T(m_k)$, where $m_k = (x,y,z)^{k}$. Hence the tangent space at $[m_k]$ is at least as large as at any other point, and the maximum singularity of $\mathrm{Hilb}^{l}(\mathbb{A}^{3})$ occurs at $[m_k]$.

Load-bearing premise

The load-bearing premise is the unproved Corollary 2.2: for the two-variable ideals $I_i$ in the decomposition of a Borel-fixed ideal, the count $t(I_i, I_j)$ is always at least the triangular number $\binom{j-i+1}{2}$; if this inequality fails for some ideal, the paper's upper bound on $T(I)$ does not follow.

Editorial extensions

If this is right

  • For each tetrahedral $l = \binom{k+2}{3}$, the point $[m_k]$ has tangent-space dimension exactly $\binom{k+2}{2}\binom{k+1}{2}$, and no point of $\mathrm{Hilb}^{l}(\mathbb{A}^{3})$ has a larger tangent space.
  • Any maximally singular point of $\mathrm{Hilb}^{l}(\mathbb{A}^{3})$ at a tetrahedral $l$ must have smallest pure exponent $m_1 = k$, confirming the necessary condition proposed in [40].
  • The upper bound $T(I) \leq (2m_1+1)l - 2\binom{m_1+2}{4}$ is valid for every Borel-fixed ideal whose colength lies in the interval $[\binom{k+2}{3}, \binom{k+3}{3}-1]$, giving a uniform estimate in that range.
  • Because the upper bound is increasing in $m_1$ and equals the lower bound at $m_k$, the maximal singularity problem for tetrahedral $l$ is settled exactly, not up to a multiplicative constant.

Reading between the lines

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

  • A reader who wants to verify the paper will find that Corollary 2.2 is the only step stated without proof; a direct combinatorial proof of $t(I_i, I_j) \geq \binom{j-i+1}{2}$ would make the argument self-contained, and a counterexample would destroy the theorem.
  • The same ghost-vector correction is a natural template for higher dimensions: for $\mathrm{Hilb}^{l}(\mathbb{A}^{N})$ with $l = \binom{k+N-1}{N}$, the expected extremal ideal is the power $(x_1,\dots,x_N)^{k}$, and the correction terms should be higher binomial coefficients, a direction the paper does not pursue.
  • The monotonicity of $\psi$ suggests that just above a tetrahedral number the maximizer should still have smallest pure exponent $k$; the announced sequel addresses certain non-tetrahedral $l$, but the mechanism here is already visible in the inequality $T(I) \leq \psi(m_1)$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proposes a proof of the 1978 Briançon-Iarrobino Conjecture for the Hilbert scheme Hilb^l(A^3) when l is a tetrahedral number. The strategy is to refine the Ramkumar-Sammartano upper bound for the tangent space by introducing a correction term, obtaining T(I) ≤ (2m1+1)l − 2 binom(m1+2,4) for every Borel-fixed ideal of colength l. The authors then prove that the function ψ(m1) defined by this bound is strictly increasing, and they combine this with the known value T(m_k) = ψ(k) to conclude that the maximum singularity occurs at [m_k]. The paper also derives a conjectural necessary condition for maximal singularity. The main technical ingredients are a decomposition I = ⊕ x^i I_i, a bound hom(J,S/J') ≤ l + l' − t(J,J'), and a purported lower bound t(I_i,I_j) ≥ binom(j-i+1,2).

Significance. If the proof is completed, the result resolves a major conjecture in the study of Hilbert schemes of points, open since 1978, and it supplies a clean explanation of why the maximum singularity occurs at the tetrahedral ideal m_k. The overall strategy is attractive, and the arithmetic identity ψ(k) = T(m_k) checks out. The paper is clearly organized and the examples in Section 1 are helpful. However, the central new combinatorial inequality, Corollary 2.2, is stated without proof and is used essentially in the proof of Theorem A; this is a load-bearing gap in the current version.

major comments (2)
  1. [Section 2, Corollary 2.2] The inequality binom(j-i+1,2) ≤ t(I_i,I_j) is asserted without proof, and it is exactly the term that makes the final bound sharp. In the proof of Theorem A, after applying Corollary 1.15, the displayed chain passes through ≤ (2m1+1)l − Σ_{0≤i<j≤m1-1} binom(j-i+1,2) − binom(m1+2,4). This step is legitimate only if Corollary 2.2 holds. The text offers no derivation: Lemma 2.1 concerns shifts α with α_x<0 and minimal generators γ, whereas t(I_i,I_j) is defined from the yz-slices and the height of I_j, and the bridge between these statements is not exhibited. Example 1.17 verifies equality for the extremal ideal m_k only, and Example 1.16 exhibits strict inequality in another case; neither establishes the general lower bound. Since this is the load-bearing novelty of the paper, the proof of Theorem A is incomplete as written. Please supply a complete proof of Corollary 2.2.
  2. [Section 1, Lemma 1.5, Claim 2] The well-definedness of g is not established. The proof defines g: B^{J,J'}_n → N^2 \ \tilde J by (U,α) ↦ (γ_y, −α_z−1) and must show (γ_y, −α_z−1) ∉ \tilde J. The text says that because U lies in the upper half-plane, γ_z+α_z+1>0, hence γ_z > −α_z−1, 'and so f(U,α)/∈ J'. This conclusion does not follow: \tilde J is upward closed, so the inequality γ_z > −α_z−1 gives no information about whether the smaller-height point (γ_y, −α_z−1) lies in \tilde J. One needs an argument using the maximality of γ_y (and minimality of γ_z among ties) to rule out (γ_y, −α_z−1) ∈ \tilde J; no such argument is supplied. Since the bijection B^{J,J'}_n ≅ N^2 \ \tilde J underlies the exact count in Corollary 1.9 and hence the bound in Lemma 1.5, this gap also affects the proof of Theorem A.
minor comments (4)
  1. [Section 1, proof of Lemma 1.5] In the proof of Claim 2, the displayed statement 'and so f(U,α)/∈ J' should read 'and so g(U,α) ∉ \tilde J'; the function being defined is g, not f.
  2. [Section 2, proof of Theorem A] The proof of the Claim is very terse: from (2.1) and (2.2) it asserts that the relevant components form a single bounded connected component and a single unbounded component, but the details of this implication are not spelled out. Please expand this step.
  3. [Section 2, proof of Theorem B] The derivative computation is correct, but the positivity step would be clearer if written as follows: for fixed k and ∆ ≥ 0, the displayed expression is decreasing in m1, so its minimum occurs at m1 = k, where it equals 3k^2 + 5k + 1 + 12∆ > 0.
  4. [Section 1, Definition 1.1] The phrase 'xy−z-octant' is nonstandard; consider replacing it with 'the first octant of the (x,y,z) lattice' or similar wording.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the upper bound is proved from combinatorial lemmas, and the sharpness check T(m_k)=ψ(k) uses external classical computations.

full rationale

Walked the claimed derivation chain. Theorem A is an upper bound proved by decomposing I = ⊕ x^i I_i, expanding T(I) over pairs Hom(I_i, S/I_j), subtracting zero-vector counts via the Claim and identity (2.3), and applying Corollary 1.15 together with the asserted lower bound t(I_i,I_j) ≥ binom(j-i+1,2). The latter is labelled Corollary 2.2 but no proof is displayed; this is a missing-support correctness gap, not circularity, because t is defined independently from the set {β ∈ N^2 \ tilde I_0 : β_z ≥ h}, and the asserted inequality is not identical to the theorem's conclusion. The sharp comparison T(m_k)=ψ(k) is obtained from [5, Proposition III.4] or [40, Corollary 1.9]; [5] is an external classical computation, and the formula is not fitted from the data being predicted. The self-citation [40, Lemma 1.7] enters only in the short argument for Corollary 2, not in the proof of the Briançon-Iarrobino Conjecture; even if that argument is incomplete, the main bound does not reduce to a self-citation. No parameter is fitted and then renamed a prediction: ψ is an explicit function of l and m_1, and Theorem B's monotonicity is proven from Lemma 2.3. Thus no circular step is exhibited, and the correct circularity finding is score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the proof is combinatorial. The main external inputs are the Borel-fixed reduction, the value of T(m_k), and the strong stability property. The paper's new combinatorial notions, such as ghost vectors and the function t, are definitions rather than postulated entities requiring independent evidence.

assumptions (4)
  • domain assumption For every l there exists a Borel-fixed ideal of colength l that attains the maximal tangent space dimension (Remark 0.4).
    Used to restrict Theorem A to Borel-fixed ideals. This is standard via generic initial ideals, but it is not proved in the paper.
  • domain assumption The tangent space at [I] has dimension hom(I,R/I), and T(m_k) = binom(k+2,2) binom(k+1,2) from [5, Proposition III.4] or [40, Corollary 1.9].
    External computation used to equate the upper bound with the actual tangent space dimension at m_k.
  • standard math Borel-fixed ideals satisfy the strong stability rule used in Lemma 2.1: if a minimal generator gamma and alpha with alpha_x < 0 satisfy gamma + alpha in tilde I, then gamma - e_x in tilde I.
    Basic property of Borel-fixed ideals in characteristic 0, invoked in the proof of Lemma 2.1.
  • standard math All work is over a characteristic 0 field, and C can be replaced by any characteristic 0 field (Remark 0.1).
    Needed for the Borel-fixed and strong stability theory used throughout.

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Pith. "Pith review of A proof of the Brian\c{c}on-Iarrobino Conjecture in three dimensions." pith.science (2026). https://pith.science/paper/Q2HJYXYG

@misc{pith2026250821717,
  author       = {Pith},
  title        = {Pith review of: A proof of the Brian\ccon-Iarrobino Conjecture in three dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q2HJYXYG}},
  note         = {Machine review of arXiv:2508.21717}
}
abstract

We resolve the 1978 Brian\c{c}on-Iarrobino Conjecture regarding the maximum singularity of $\mathcal{H}=\mathrm{Hilb}^{l}(\mathbb{A}^3)$, where $l$ is a tetrahedral number, by refining the work of Ramkumar-Sammartano in \cite{Ramkumar-Sammartano}. This also immediately implies the conjectural necessary condition for a point of $\mathcal{H}$ to have the maximal singularity, suggested by the second-named author in \cite{Rezaee-23-Conjectures}. In a sequel to this article, \cite{Mackenzie-Rezaee2}, we prove a generalized version of this conjecture for certain non-tetrahedral $l$, via proving the conjectural necessary condition.

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Works this paper leans on

43 extracted references · 37 canonical work pages

  1. [40]

    Rezaee,Conjectural criteria for the most singular points of the Hilbert schemes of points , Experimental Mathematics, Online first

    F . Rezaee,Conjectural criteria for the most singular points of the Hilbert schemes of points , Experimental Mathematics, Online first. https://doi.org/10.1080/10586458.2024.2400181. arXiv:2312.04520 (2024), 1–16

    arXiv 2024

  2. [5]

    Briançon and A

    J. Briançon and A. Iarrobino, Dimension of the punctual Hilbert scheme, J. Algebra 55 (1978), 536–544

    work page 1978

  3. [37]

    Ramkumar and A

    R. Ramkumar and A. Sammartano, On the tangent space to the Hilbert scheme of points in P3, Transactions of the American Mathematical Society 375 (2022), 6179–6203

    work page 2022

  4. [1]

    Arcara, A

    D. Arcara, A. Bertram, I. Coskun, and J. Huizenga, The minimal model program for the Hilbert scheme of points on P2 and Bridgeland stability, Adv. Math. 235 (2013), no. 3, 580–626

    work page 2013

  5. [2]

    Ascott, F

    A. Ascott, F . Rezaee, and Z. Zhou, A conjectural generalization of the Briançon-Iarrobino conjecture , arXiv:2506.17704 (2025)

    work page arXiv 2025

  6. [3]

    Behrend, J

    K. Behrend, J. Bryan, and B. Szendr˝oi, Motivic degree zero Donaldson–Thomas invariants, Invent. Math. 192 (2013), no. 1, 111–160

    work page 2013

  7. [4]

    Behrend and B

    K. Behrend and B. Fantechi,Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra Number Theory 2 (2008), 313–345

    work page 2008

  8. [6]

    Cartwright, D

    D. Cartwright, D. Erman, M. Velasco, and B. Viray,Hilbert schemes of 8 points, Algebra Number Theory 3 (2009), no. 7, 763–795

    work page 2009

Show all 43 references
  1. [7]

    Douvropoulos, J

    T . Douvropoulos, J. Jelisiejew, B. I. U. Nødland, and Z. Teitler,The Hilbert scheme of 11 points inA3 is irreducible, Combinatorial algebraic geometry, Fields Inst. Commun., vol. 80, Fields Inst. Res. Math. Sci., Toronto, ON, 2017, pp. 321–352

    2017

  2. [8]

    Farkas, R

    G. Farkas, R. Pandharipande, and A. Sammartano, Irrational components of the Hilbert scheme of points , arXiv:2405.11997 (2024)

    2024 arXiv

  3. [9]

    Fogarty, Algebraic families on an algebraic surface, Amer

    J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 60 (1968), 511–521

    1968

  4. [10]

    Giovenzana, L

    F . Giovenzana, L. Giovenzana, M. Graffeo, and P . Lella,A counterexample to the parity conjecture, Algebraic geometry 12 (2025), no. 2, 173–188

    2025

  5. [11]

    Gorsky and A

    E. Gorsky and A. Negu¸ t,Refined knot invariants and Hilbert schemes , Journal de Mathématiques Pures et Appliquées 104 (2015), no. 3, 403–435

    2015

  6. [12]

    D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/

  7. [13]

    Grojnowski, Instantons and affine algebras I: The Hilbert scheme and vertex operators, Math

    I. Grojnowski, Instantons and affine algebras I: The Hilbert scheme and vertex operators, Math. Res. Letters 3 (1996), no. 275

    1996

  8. [14]

    Gross and F

    M. Gross and F . Rezaee,Bridgeland wall-crossing for Hilbn (P2) from scattering diagram, in preparation (2025)

    2025

  9. [15]

    Gross and F

    M. Gross and F . Rezaee,Decomposition of stability scattering diagram for P2 and its applications, to appear (2025)

    2025

  10. [16]

    Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique iv, Séminaire Bourbaki 6 (1995), no

    A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique iv, Séminaire Bourbaki 6 (1995), no. 221, 249–276

    1995

  11. [17]

    Haiman, t, q-Catalan numbers and the Hilbert scheme, Discrete Mathematics 193 (1998), no

    M. Haiman, t, q-Catalan numbers and the Hilbert scheme, Discrete Mathematics 193 (1998), no. 1, 201–224

    1998

  12. [18]

    Haiman, Hilbert schemes, polygraphs and the MacDonald positivity conjecture, Journal of the American Mathematical Society 14 (2001), no

    M. Haiman, Hilbert schemes, polygraphs and the MacDonald positivity conjecture, Journal of the American Mathematical Society 14 (2001), no. 4, 941–1006

    2001

  13. [19]

    Hartshorne, Connectedness of the hilbert scheme, Publ

    R. Hartshorne, Connectedness of the hilbert scheme, Publ. Math. Inst. Hautes Études Sci. 29 (1966), 5–48

    1966

  14. [20]

    A. A. Henni and M. Jardim, Commuting matrices and the Hilbert scheme of points on affine spaces, Adv. Geom. 18 (2018), no. 4, 467–482

    2018

  15. [21]

    Hu, On singular Hilbert schemes of points: local structures and tautological sheaves, arXiv:2101.05236 (2021)

    X. Hu, On singular Hilbert schemes of points: local structures and tautological sheaves, arXiv:2101.05236 (2021)

    2021 arXiv

  16. [22]

    Huizenga, Effective divisors on the hilbert scheme of points in the plane and interpolation for stable bundles, J

    J. Huizenga, Effective divisors on the hilbert scheme of points in the plane and interpolation for stable bundles, J. Algebraic Geom. 25 (2016), 19–75

    2016

  17. [23]

    J. W . Huizenga,Restrictions of Steiner bundles and divisors on the Hilbert scheme of points in the plane (PhD thesis), Harvard University (2012)

    2012

  18. [24]

    Iarrobino, Reducibility of the families of 0-dimensional schemes on a variety, Invent

    A. Iarrobino, Reducibility of the families of 0-dimensional schemes on a variety, Invent. Math. 15 (1972), 72–77

    1972

  19. [25]

    Jelisiejew, Pathologies on the Hilbert scheme of points, Invent

    J. Jelisiejew, Pathologies on the Hilbert scheme of points, Invent. Math. 220 (2020), no. 2, 581–610. A PROOF OF THE 1978 BRIANÇON-IARROBINO CONJECTURE IN THREE DIMENSIONS 13

    2020

  20. [26]

    Jelisiejew, M

    J. Jelisiejew, M. Kool, and R. F . Schmiermann,Behrend’s function is not constant on Hilbn (C3), To appear in Geometry and Topology, arXiv:2311.05408 (2023)

    2023

  21. [27]

    Jelisiejew, R

    J. Jelisiejew, R. Ramkumar, and A. Sammartano,The Hilbert scheme of points on a threefold, I, arXiv:2409.17009 (2024)

    2024 arXiv

  22. [28]

    Katz, The desingularization of Hilb4(P3) and its Betti numbers, Zero-dimensional schemes (Ravello, 1992) (1994), 231–242

    S. Katz, The desingularization of Hilb4(P3) and its Betti numbers, Zero-dimensional schemes (Ravello, 1992) (1994), 231–242

    1994

  23. [29]

    Li and X

    C. Li and X. Zhao, The MMP for deformations of Hilbert schemes of points on the projective plane, Algebraic Geometry 5 (2018), no. 3, 328–358

    2018

  24. [30]

    Li and X

    C. Li and X. Zhao, Birational models of moduli spaces of coherent sheaves on the projective plane, Geometry and Topology 23 (2019), 347–426

    2019

  25. [31]

    Mackenzie and F

    O. Mackenzie and F . Rezaee,Maximal singularities of the Hilbert scheme of a non-tetrahedral number of points in three dimensions, in preparation (2025)

    2025

  26. [32]

    Maulik, N

    D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande,Gromov–Witten theory and Donaldson–Thomas theory, I, Compos. Math. 142 (2006), no. 5, 1263–1285

    2006

  27. [33]

    Miller and B

    E. Miller and B. Sturmfels, Combinatorial commutative algebra, vol. 227, Graduate Texts in Mathematics, Springer-Verlag, New York, 2005

    2005

  28. [34]

    Nesterov, Hilbert schemes of points and Fulton-MacPherson compactifications, arXiv:2501.08269 (2025)

    D. Nesterov, Hilbert schemes of points and Fulton-MacPherson compactifications, arXiv:2501.08269 (2025)

    2025 arXiv

  29. [35]

    Pandharipande, A tour of the geometry of points in affine space, DMV Jahrestagung lecture (2022)

    R. Pandharipande, A tour of the geometry of points in affine space, DMV Jahrestagung lecture (2022)

    2022

  30. [36]

    Pandharipande and R

    R. Pandharipande and R. P . Thomas,Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), no. 2, 407–447

    2009

  31. [38]

    Ramkumar and A

    R. Ramkumar and A. Sammartano, Rational singularities of nested Hilbert schemes , International Mathematics Research Notices (2024), no. 2, 1061–1122

    2024

  32. [39]

    Ramkumar and A

    R. Ramkumar and A. Sammartano, On the parity conjecture for Hilbert schemes of points on threefolds, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze XXVI (2025), no. 3

    2025

  33. [41]

    A. T . Ricolfi, A sign that used to annoy me, and still does, J. Geom. Phys. 125 (2024)

    2024

  34. [42]

    Sturmfels, Four Counterexamples in Combinatorial Algebraic Geometry , Journal of Algebra 230 (2000), 282–294

    B. Sturmfels, Four Counterexamples in Combinatorial Algebraic Geometry , Journal of Algebra 230 (2000), 282–294

    2000

  35. [43]

    Vakil, Murphy’s law in algebraic geometry: badly-behaved deformation spaces , Invent

    R. Vakil, Murphy’s law in algebraic geometry: badly-behaved deformation spaces , Invent. Math. 164 (2006), no. 3, 569–590. CENTRE FOR MATHEMATICAL SCIENCES , UNIVERSITY OF CAMBRIDGE , WILBERFORCE ROAD, CB3 0WA, CAMBRIDGE , UNITED KINGDOM Email address: om379@cam.ac.uk Email ad...

    2006

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