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arxiv: 2510.23570 · v2 · submitted 2025-10-27 · 🧮 math.AG

The Euler characteristic of Milnor fibers over 2-generic symmetric determinantal varieties

Tha\'is M. Dalbelo , Daniel Duarte , Danilo da N\'obrega Santos This is my paper

Pith reviewed 2026-05-18 02:55 UTC · model grok-4.3

classification 🧮 math.AG
keywords Milnor fiberEuler characteristicsymmetric determinantal varietytoric structureNewton polyhedraEuler obstructionMilnor number
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The pith

A formula gives the Euler characteristic of the Milnor fiber for non-degenerate functions on 2-generic symmetric determinantal varieties.

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

The paper derives an explicit formula for the Euler characteristic of the Milnor fiber of a non-degenerate function f from a 2-generic symmetric determinantal variety X to the complex numbers, assuming the critical set is isolated relative to a stratification. The derivation proceeds by first exhibiting an explicit toric structure on these varieties. Volumes of the associated Newton polyhedra are then computed and substituted into Matsui-Takeuchi's formula for Milnor fibers over toric varieties. The resulting expression is applied to obtain the local Euler obstruction of X at the origin, the local Euler obstruction of f, and a relation between the Euler obstruction of f and the Milnor number of an associated polynomial.

Core claim

For a 2-generic symmetric determinantal variety X, the Euler characteristic of the Milnor fiber of a non-degenerate function f with isolated critical set relative to a stratification is determined by the volumes of Newton polyhedra that arise from the explicit toric structure of X, via direct substitution into Matsui-Takeuchi's formula.

What carries the argument

The explicit toric structure on 2-generic symmetric determinantal varieties, which converts the computation of Milnor fiber Euler characteristics into volumes of Newton polyhedra that plug into Matsui-Takeuchi's formula.

Load-bearing premise

The 2-generic symmetric determinantal varieties admit an explicit toric structure from which Newton polyhedra volumes can be computed in a manner that directly yields the Euler characteristic when combined with Matsui-Takeuchi's formula.

What would settle it

A direct computation of the Milnor fiber Euler characteristic for a concrete low-dimensional example of a non-degenerate function on a 2-generic symmetric determinantal variety that disagrees with the volume prediction from the toric structure.

Figures

Figures reproduced from arXiv: 2510.23570 by Daniel Duarte, Danilo da N\'obrega Santos, Tha\'is M. Dalbelo.

Figure 1
Figure 1. Figure 1: Newton polyhedron of f. For the remainder of this section we consider f ∈ C[S] to be a polynomial function on XS such that 0 ̸∈ supp(f). To each f = P u∈S au · u ∈ C[S] we associate a Laurent polynomial L(g)(x) = X u∈S aux u defined on (C ∗ ) n . Definition 19. Let f = P u∈S au · u ∈ C[S]. We say that f is non-degenerate if, for every compact face γ ⊂ Γ+(f), the hypersurface L(fγ) −1 (0) ⊂ (C ∗ ) n is smoo… view at source ↗
read the original abstract

In this work we present a formula for the Euler characteristic of the Milnor fiber of non-degenerate functions $f: X \to \mathbb{C}$ with isolated critical set relative to a stratification, where $X$ is a $2$-generic symmetric determinantal variety. The formula is obtained in two steps. Firstly, we explicitly describe the toric structure of those varieties. Secondly, we compute volumes of Newton polyhedra arising from the toric structure. The result then follows from Matsui-Takeuchi's formula for Milnor fibers over toric varieties. As an application, we compute the local Euler obstruction of $X$ at the origin and the local Euler obstruction of $f$. We also relate the Euler obstruction of $f$ to the Milnor number of a certain polynomial associated to $f$.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript presents a formula for the Euler characteristic of the Milnor fiber of non-degenerate functions f: X → ℂ with isolated critical set relative to a stratification, where X is a 2-generic symmetric determinantal variety. The formula is obtained by explicitly describing the toric structure of these varieties, computing volumes of the associated Newton polyhedra, and applying Matsui-Takeuchi's formula. Applications include the local Euler obstruction of X at the origin, the local Euler obstruction of f, and a relation between the Euler obstruction of f and the Milnor number of an associated polynomial.

Significance. If the toric description and volume computations are correct, the work supplies explicit, computable formulas for Milnor fiber Euler characteristics on this class of varieties, extending singularity-theoretic tools via toric geometry and the Matsui-Takeuchi formula. The explicit toric structure and direct volume derivations constitute a strength, providing reproducible and falsifiable predictions rather than fitted quantities.

minor comments (3)
  1. [Introduction] §2 (or wherever the 2-generic condition is defined): recall or cite the precise definition of '2-generic symmetric determinantal variety' at the first use in the introduction to aid readers unfamiliar with the stratification.
  2. [Toric structure and volume computation] The volume formulas in the Newton polyhedra section would benefit from a low-dimensional example (e.g., 2×2 or 3×3 case) to illustrate the polyhedron construction and volume calculation before the general statement.
  3. [Applications] In the applications section, the relation between the Euler obstruction of f and the Milnor number of the associated polynomial should include a brief statement of the polynomial's definition to make the claim self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recognizing the value of the explicit toric description and Newton polyhedron volume computations. We appreciate the recommendation for minor revision. As no specific major comments or criticisms were raised in the report, we have no point-by-point revisions to propose at this stage and believe the current version stands as is.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external formula to explicit toric data

full rationale

The paper's central derivation proceeds by explicitly describing the toric structure of the 2-generic symmetric determinantal varieties, followed by direct computation of Newton polyhedra volumes from that structure. These volumes are then substituted into the independently established Matsui-Takeuchi formula for Milnor fibers over toric varieties. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain within the paper; the toric description and volume formulas are supplied as new explicit content, and the final Euler characteristic follows from the external theorem under the stated non-degeneracy hypotheses. This is a standard, non-circular application of an external result to newly computed geometric data.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claim rests on the domain assumption that the varieties are 2-generic symmetric determinantal and admit an explicit toric structure, plus the applicability of the cited Matsui-Takeuchi formula; no free parameters or new entities are mentioned in the abstract.

axioms (3)
  • domain assumption X is a 2-generic symmetric determinantal variety
    This specifies the class of varieties for which the toric structure and formula are derived.
  • domain assumption f is non-degenerate with isolated critical set relative to a stratification
    This condition is required for the Milnor fiber and the application of the cited formula.
  • standard math Matsui-Takeuchi's formula for Milnor fibers over toric varieties applies after the toric structure is described
    The result follows directly from this external formula once the toric structure and volumes are obtained.

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Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    P. Aluffi. Projective duality and a Chern-Mather involution.Trans. Amer. Math. Soc., 370(3):1803–1822, 2018

    work page 2018

  2. [2]

    Beelen, T

    P. Beelen, T. Johnsen, and P. Singh. Linear codes associated to symmetric determi- nantal varieties: even rank case.Finite Fields Appl., 91:Paper No. 102240, 31, 2023

    work page 2023

  3. [3]

    J. P. Brasselet. Local Euler obstruction, old and new. InXI Brazilian Topology Meeting (Rio Claro, 1998), pages 140–147. World Sci. Publ., River Edge, NJ, 2000

    work page 1998

  4. [4]

    Brasselet and N

    J.-P. Brasselet and N. G. Grulha Jr. Local Euler obstruction, old and new, II. InReal and complex singularities, volume 380 ofLondon Math. Soc. Lecture Note Ser., pages 23–45. Cambridge Univ. Press, Cambridge, 2010

    work page 2010

  5. [5]

    Brasselet, D

    J.-P. Brasselet, D. T. Lê, and J. Seade. Euler obstruction and indices of vector fields. Topology, 39(6):1193–1208, 2000

    work page 2000

  6. [6]

    Brasselet, D

    J.-P. Brasselet, D. B. Massey, A. J. Parameswaran, and J. Seade. Euler obstruction and defects of functions on singular varieties.J. London Math. Soc. (2), 70(1):59–76, 2004

    work page 2004

  7. [7]

    Brasselet and M.-H

    J.-P. Brasselet and M.-H. Schwartz. Sur les classes de Chern d’un ensemble analytique complexe. InThe Euler-Poincaré characteristic (French), volume 82 ofAstérisque, pages 93–147. Soc. Math. France, Paris, 1981

    work page 1981

  8. [8]

    D. A. Cox, J. Little, and D. O’shea.Using algebraic geometry, volume 185. Springer Science & Business Media, 2005

    work page 2005

  9. [9]

    D. A. Cox, J. B. Little, and H. K. Schenck.Toric Varieties. Graduate studies in mathematics. American Mathematical Soc., 2011

    work page 2011

  10. [10]

    T. M. Dalbelo and D. da Nóbrega Santos. Whitney equisingularity for families of hypersurfaces in toric varieties.arXiv preprint arXiv:2510.04177, 2025

    work page arXiv 2025

  11. [11]

    T. M. Dalbelo, D. Duarte, and M. A. S. Ruas. Nash blowups of 2-generic determinantal varieties in positive characteristic.Pure and Applied Mathematics Quarterly, 21:1557– 1575, 2025

    work page 2025

  12. [12]

    T. M. Dalbelo and L. Hartmann. Brasselet number and Newton polygons.Manuscripta Math., 162(1-2):241–269, 2020

    work page 2020

  13. [13]

    Dutertre and N

    N. Dutertre and N. G. Grulha, Jr. Lê-Greuel type formula for the Euler obstruction and applications.Adv. Math., 251:127–146, 2014. 26

    work page 2014

  14. [14]

    Gaffney, N

    T. Gaffney, N. G. Grulha, Jr., and M. A. S. Ruas. The local Euler obstruction and topology of the stabilization of associated determinantal varieties.Math. Z., 291(3- 4):905–930, 2019

    work page 2019

  15. [15]

    Gaffney and M

    T. Gaffney and M. Molino. Symmetric determinantal singularities i: The multiplicity of the polar curve.arXiv preprint arXiv:2003.12543, 2020

    work page arXiv 2003

  16. [16]

    Alexandra Gaube and B

    S. Alexandra Gaube and B. Schober. Desingularization of generic symmetric and generic skew-symmetric determinantal singularities.Manuscripta Math., 174(3- 4):1113–1131, 2024

    work page 2024

  17. [17]

    González-Sprinberg

    G. González-Sprinberg. Cyclemaximal et invariant d’Euler local des singularitésisolées de surfaces.Topology, 21(4):401–408, 1982

    work page 1982

  18. [18]

    N. G. Grulha, Jr. The Euler obstruction and Bruce-Roberts’ Milnor number.Q. J. Math., 60(3):291–302, 2009

    work page 2009

  19. [19]

    Harris and Tu L

    J. Harris and Tu L. W. On symmetric and skew-symmetric determinantal varieties. Topology, 23(1):71–84, 1984

    work page 1984

  20. [20]

    Helmer and B

    M. Helmer and B. Sturmfels. Nearest points on toric varieties.Math. Scand., 122(2):213–238, 2018

    work page 2018

  21. [21]

    B. V. Kotzev. Determinantal ideals of linear type of a generic symmetric matrix.J. Algebra, 139(2):484–504, 1991

    work page 1991

  22. [22]

    A. G. Kouchnirenko. Polyèdres de Newton et nombres de Milnor.Inventiones mathe- maticae, 32(1):1–31, 1976

    work page 1976

  23. [23]

    R. E. Kutz. Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups.Trans. Amer. Math. Soc., 194:115–129, 1974

    work page 1974

  24. [24]

    A. C. Lőrincz and C. Raicu. Local Euler obstructions for determinantal varieties. Topology Appl., 313:Paper No. 107984, 21, 2022

    work page 2022

  25. [25]

    R. D. MacPherson. Chern classes for singular algebraic varieties.Ann. of Math. (2), 100:423–432, 1974

    work page 1974

  26. [26]

    D. B. Massey. Hypercohomology of Milnor fibres.Topology, 35(4):969–1003, 1996

    work page 1996

  27. [27]

    Matsui and K

    Y. Matsui and K. Takeuchi. A geometric degree formula forA-discriminants and Euler obstructions of toric varieties.Adv. Math., 226(2):2040–2064, 2011

    work page 2040

  28. [28]

    Matsui and K

    Y. Matsui and K. Takeuchi. Milnor fibers over singular toric varieties and nearby cycle sheaves.Tohoku Mathematical Journal, Second Series, 63(1):113–136, 2011. 27

    work page 2011

  29. [29]

    Maxim and M

    L. Maxim and M. Tibăr. Euclidean distance degree and limit points in a Morsification. Adv. Appl. Math., 152:102597, 2024

    work page 2024

  30. [30]

    Milnor.Singular points of complex hypersurfaces

    J. Milnor.Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1968

    work page 1968

  31. [31]

    B. U. Nø dland. Local Euler obstructions of toric varieties.J. Pure Appl. Algebra, 222(3):508–533, 2018

    work page 2018

  32. [32]

    J. J. Nuño Ballesteros, B. Oréfice, and J. N. Tomazella. The Bruce-Roberts number of a function on a weighted homogeneous hypersurface.Q. J. Math., 64(1):269–280, 2013

    work page 2013

  33. [33]

    M. Oka. Principal zeta-function of non-degenerate complete intersection singularity. J. Fac. Sci. Univ. Tokyo, 37:11–32, 1990

    work page 1990

  34. [34]

    Seade, M

    J. Seade, M. Tibăr, and A. Verjovsky. Milnor numbers and Euler obstruction.Bull. Braz. Math. Soc. (N.S.), 36(2):275–283, 2005

    work page 2005

  35. [35]

    Sturmfels.Grobner bases and convex polytopes, volume 8

    B. Sturmfels.Grobner bases and convex polytopes, volume 8. American Mathematical Soc., 1996

    work page 1996

  36. [36]

    A. N. Varchenko. Zeta-function of monodromy and Newton’s diagram.Invent. Math., 37(3):253–262, 1976. T. M. Dalbelo, Universidade Federal de Sao Carlos. Email: thaisdalbelo@ufscar.br D. Duarte, Centro de Ciencias Matemáticas, UNAM. E-mail: adduarte@matmor.unam.mx D. da Nóbrega Santos, Universidade Federal Rural de Pernambuco. Email: danilo.nobrega@ufrpe.br 28

    work page 1976