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arxiv: 2606.09693 · v1 · pith:HRAR54QNnew · submitted 2026-06-08 · 🧮 math.AG · math.AC· math.NT

Bertini theorems for Hilbert-Samuel multiplicity over finite fields

Rahul Ajit , Matthew Bertucci This is my paper

Pith reviewed 2026-06-27 14:40 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.NT
keywords Bertini theoremsHilbert-Samuel multiplicityfinite fieldshypersurfacesalgebraic geometryquasiprojective schemespositive density
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The pith

Over finite fields, there is a positive-density set of hypersurfaces that intersect a scheme while preserving its Hilbert-Samuel multiplicities at all points.

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

The paper proves a Bertini-type result for Hilbert-Samuel multiplicity. For a reduced equidimensional quasiprojective scheme X over a finite field F_q, it shows that a positive-density collection of hypersurfaces H_f meets X so that ord_P(f)=1 and the multiplicity e_P stays unchanged at every closed point P of the intersection. This matters for inductive arguments that reduce dimension while retaining local multiplicity data, especially when working over finite fields where classical density statements require adaptation. A sympathetic reader cares because the result supplies a concrete tool for controlling singularities and multiplicities in this setting.

Core claim

Let X ⊆ P^n_{F_q} be a reduced, equidimensional, quasiprojective scheme. There exists a positive-density set of hypersurfaces H_f such that for every closed point P ∈ X ∩ H_f, ord_P(f)=1 and e_P(X ∩ H_f)=e_P(X).

What carries the argument

Positive-density set of hypersurfaces in projective space over F_q satisfying the order-one and multiplicity-preservation conditions at intersection points.

If this is right

  • Inductive arguments on dimension can retain multiplicity data when cutting by these hypersurfaces.
  • Geometric properties of X transfer directly to its intersections with the selected hypersurfaces.
  • Hypersurfaces can be chosen to avoid increasing multiplicities or producing higher-order contacts at points.
  • The result supports study of the singular locus and related invariants over finite fields.

Where Pith is reading between the lines

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

  • Effective versions of the density might yield algorithms to locate suitable hypersurfaces over finite fields.
  • The method could extend to preservation of other local invariants such as Hilbert functions.
  • Links may exist to density questions for sections in arithmetic geometry over finite fields.

Load-bearing premise

The scheme X is reduced and equidimensional over the finite field F_q.

What would settle it

A concrete reduced equidimensional quasiprojective scheme X over some F_q for which the set of hypersurfaces satisfying ord_P(f)=1 and multiplicity preservation has density zero.

read the original abstract

Let $X\subseteq \mathbb{P}^n_{\mathbb{F}_q}$ be a reduced, equidimensional, quasiprojective scheme. We prove that there exists a positive-density set of hypersurfaces $H_f$ such that for every closed point $P\in X\cap H_f$, one has $\mathrm{ord}_P(f)=1$ and $e_P(X\cap H_f)=e_P(X)$.

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

1 major / 0 minor

Summary. The paper proves a Bertini-type theorem over finite fields: for a reduced, equidimensional, quasiprojective scheme X in projective space over F_q, there exists a positive-density set of hypersurfaces H_f such that for every closed point P in X ∩ H_f one has ord_P(f) = 1 and the Hilbert-Samuel multiplicity satisfies e_P(X ∩ H_f) = e_P(X).

Significance. If correct, the result supplies a multiplicity-preserving Bertini theorem in the arithmetic setting of finite fields, controlling all closed points simultaneously. This could be useful for applications in arithmetic geometry that require uniform control on multiplicities at points of all residue degrees.

major comments (1)
  1. [Abstract / main theorem] Abstract / main theorem statement: the positive-density claim requires that the union over all closed points P (including those with residue degree k for arbitrarily large k) of the bad hypersurfaces has density tending to zero. The number of degree-k points grows asymptotically like q^{k·dim X}/k; without an explicit uniform estimate (e.g., via a Lang-Weil-type bound or sieve that is independent of k) showing the proportion of bad f tends to zero uniformly in k, the density may fail to be positive. The manuscript must supply the precise estimate used to control this union.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the uniform density estimate explicit. We address the major comment below and will revise the manuscript to clarify this point.

read point-by-point responses

  1. Referee: [Abstract / main theorem] Abstract / main theorem statement: the positive-density claim requires that the union over all closed points P (including those with residue degree k for arbitrarily large k) of the bad hypersurfaces has density tending to zero. The number of degree-k points grows asymptotically like q^{k·dim X}/k; without an explicit uniform estimate (e.g., via a Lang-Weil-type bound or sieve that is independent of k) showing the proportion of bad f tends to zero uniformly in k, the density may fail to be positive. The manuscript must supply the precise estimate used to control this union.

    Authors: We agree that an explicit uniform estimate independent of k is required for rigor. The argument in Section 3 proceeds by a union bound over all closed points, grouped by residue degree k. For each fixed k the Lang-Weil estimate supplies #X(F_{q^k}) ≪_X q^{k·dim X} with an implied constant independent of k (for k ≫ 0). Each individual point P of degree k imposes a linear condition on the vector space of degree-d hypersurface equations whose codimension is 1, so the proportion of bad hypersurfaces for that P is at most C q^{-m} where m depends only on the local multiplicity data at P and is bounded below by a positive constant independent of k and d (for d large). Summing over the O(q^{k·dim X}/k) points of degree k therefore contributes a total bad proportion O(q^{k(dim X - c)}/k) for some c>0. Summing the geometric series over k then yields a total bad density that tends to 0 as d→∞, uniformly in k. We will add a short lemma making this calculation explicit, together with the precise statement of the Lang-Weil bound employed. revision: yes

Circularity Check

0 steps flagged

No circularity: standard Bertini-type existence proof over finite fields

full rationale

The paper states and proves an existence theorem for a positive-density set of hypersurfaces satisfying ord_P(f)=1 and multiplicity preservation at all closed points of the intersection. No equations, parameters, or self-citations are presented that reduce the claimed result to a tautological fit, renaming, or load-bearing self-reference. The derivation relies on standard tools of algebraic geometry (Bertini theorems, Hilbert-Samuel multiplicity, density arguments over F_q) whose validity is independent of the target statement. The skeptic concern addresses potential gaps in uniform control over residue degrees but does not identify any definitional or self-referential reduction in the argument chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger; no free parameters, invented entities, or ad-hoc axioms are visible. The result rests on standard definitions of Hilbert-Samuel multiplicity and properties of schemes over finite fields.

axioms (1)
  • standard math Standard properties of Hilbert-Samuel multiplicity and reduced equidimensional schemes over finite fields
    Invoked implicitly by the statement of the theorem in the abstract.

pith-pipeline@v0.9.1-grok · 5586 in / 1150 out tokens · 25191 ms · 2026-06-27T14:40:48.239537+00:00 · methodology

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Works this paper leans on

65 extracted references · 41 canonical work pages

  1. [1]

    2016 , pages =

    Eisenbud, David and Harris, Joe , title =. 2016 , pages =. doi:10.1017/CBO9781139062046 , url =

    work page doi:10.1017/cbo9781139062046 2016

  2. [2]

    Cumino, Caterina and Greco, Silvio and Manaresi, Mirella , title =. J. Algebra , fjournal =. 1986 , number =. doi:10.1016/0021-8693(86)90020-7 , url =

    work page doi:10.1016/0021-8693(86)90020-7 1986

  3. [3]

    Spreafico, Maria Luisa , title =. Arch. Math. (Basel) , fjournal =. 1998 , number =. doi:10.1007/s000130050213 , url =

    work page doi:10.1007/s000130050213 1998

  4. [4]

    2025 , note =

    Special values of motivic Euler products , author =. 2025 , note =

    2025

  5. [5]

    Bilu, Margaret and Das, Ronno and Howe, Sean , title =. Adv. Math. , fjournal =. 2022 , pages =. doi:10.1016/j.aim.2022.108556 , url =

    work page doi:10.1016/j.aim.2022.108556 2022

  6. [6]

    Bennett, Bruce Michael , title =. Ann. of Math. (2) , fjournal =. 1970 , pages =. doi:10.2307/1970601 , url =

    work page doi:10.2307/1970601 1970

  7. [7]

    Algebra Number Theory , fjournal =

    Bilu, Margaret and Howe, Sean , title =. Algebra Number Theory , fjournal =. 2021 , number =. doi:10.2140/ant.2021.15.2195 , url =

    work page doi:10.2140/ant.2021.15.2195 2021

  8. [8]

    2025 , note =

    Motivic random variables , author =. 2025 , note =

    2025

  9. [9]

    Bilu, Margaret , title =. Mem. Amer. Math. Soc. , fjournal =. 2023 , number =. doi:10.1090/memo/1396 , url =

    work page doi:10.1090/memo/1396 2023

  10. [10]

    Algebra & Number Theory , fjournal =

    Bruce, Juliette and Erman, Daniel , title =. Algebra & Number Theory , fjournal =. 2019 , number =. doi:10.2140/ant.2019.13.2081 , url =

    work page doi:10.2140/ant.2019.13.2081 2019

  11. [11]

    , title =

    Bucur, Alina and Kedlaya, Kiran S. , title =. J. Th\'. 2012 , number =

    2012

  12. [12]

    Charles, François , title =. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , fjournal =. 2021 , number =. doi:10.24033/asens.2488 , url =

    work page doi:10.24033/asens.2488 2021

  13. [13]

    Helmer, Martin and Nanda, Vidit , title =. SIAM J. Appl. Algebra Geom. , fjournal =. 2023 , number =. doi:10.1137/22M1475533 , url =

    work page doi:10.1137/22m1475533 2023

  14. [14]

    Carvajal-Rojas, Javier and Schwede, Karl and Tucker, Kevin , title =. Math. Z. , fjournal =. 2021 , number =. doi:10.1007/s00209-021-02712-y , url =

    work page doi:10.1007/s00209-021-02712-y 2021

  15. [15]

    Perkinson, David , title =. Trans. Amer. Math. Soc. , fjournal =. 1995 , number =. doi:10.2307/2155008 , url =

    work page doi:10.2307/2155008 1995

  16. [16]

    Datta, Rankeya and Simpson, Austyn , title =. J. Algebra , fjournal =. 2022 , pages =. doi:10.1016/j.jalgebra.2021.10.025 , url =

    work page doi:10.1016/j.jalgebra.2021.10.025 2022

  17. [17]

    Duarte, Daniel and Núñez-Betancourt, Luis , title =. Rev. Mat. Iberoam. , fjournal =. 2022 , number =. doi:10.4171/rmi/1278 , url =

    work page doi:10.4171/rmi/1278 2022

  18. [18]

    Grothendieck, Alexander , title =. Inst. Hautes \'Etudes Sci. Publ. Math. , fjournal =. 1967 , pages =

    1967

  19. [19]

    1995 , PAGES =

    Eisenbud, David , title =. 1995 , pages =. doi:10.1007/978-1-4612-5350-1 , url =

    work page doi:10.1007/978-1-4612-5350-1 1995

  20. [20]

    Robbiano, Lorenzo and Valla, Giuseppe , title =. Math. Proc. Cambridge Philos. Soc. , fjournal =. 1980 , number =. doi:10.1017/S0305004100057583 , url =

    work page doi:10.1017/s0305004100057583 1980

  21. [21]

    Duke Math

    Erman, Daniel and Wood, Melanie Matchett , title =. Duke Math. J. , fjournal =. 2015 , number =. doi:10.1215/00127094-2838327 , url =

    work page doi:10.1215/00127094-2838327 2015

  22. [22]

    2023 , pages =

    Ferretti, Andrea , title =. 2023 , pages =. doi:10.1090/gsm/233 , url =

    work page doi:10.1090/gsm/233 2023

  23. [23]

    , title =

    Gabber, O. , title =. Geom. Funct. Anal. , fjournal =. 2001 , number =. doi:10.1007/s00039-001-8228-2 , url =

    work page doi:10.1007/s00039-001-8228-2 2001

  24. [24]

    Ghosh, Mainak and Krishna, Amalendu , title =. J. Lond. Math. Soc. (2) , fjournal =. 2023 , number =

    2023

  25. [25]

    Gunther, Joseph , title =. J. Pure Appl. Algebra , fjournal =. 2017 , number =. doi:10.1016/j.jpaa.2016.05.027 , url =

    work page doi:10.1016/j.jpaa.2016.05.027 2017

  26. [26]

    2020 , pages =

    G\"ortz, Ulrich and Wedhorn, Torsten , title =. 2020 , pages =. doi:10.1007/978-3-658-30733-2 , url =

    work page doi:10.1007/978-3-658-30733-2 2020

  27. [27]

    1977 , pages =

    Hartshorne, Robin , title =. 1977 , pages =

    1977

  28. [28]

    Lê, Quy Thuong and Yasuda, Takehiko , title =. Rev. Mat. Iberoam. , fjournal =. 2025 , number =. doi:10.4171/rmi/1516 , url =

    work page doi:10.4171/rmi/1516 2025

  29. [29]

    2412.19295 , archiveprefix =

    Sean Howe , year =. 2412.19295 , archiveprefix =

    arXiv

  30. [30]

    2505.01205 , archiveprefix =

    Sean Howe , year =. 2505.01205 , archiveprefix =

    arXiv

  31. [31]

    Nagoya Math

    Valabrega, Paolo and Valla, Giuseppe , title =. Nagoya Math. J. , fjournal =. 1978 , pages =

    1978

  32. [32]

    , title =

    Katz, Nicholas M. , title =. Math. Res. Lett. , fjournal =. 1999 , number =. doi:10.4310/MRL.1999.v6.n6.a2 , url =

    work page doi:10.4310/mrl.1999.v6.n6.a2 1999

  33. [33]

    , title =

    Kleiman, Steven L. , title =. Rend. Circ. Mat. Palermo (2) Suppl. , fjournal =. 1998 , pages =

    1998

  34. [34]

    Lang, Serge and Weil, Andr\'e , title =. Amer. J. Math. , fjournal =. 1954 , pages =. doi:10.2307/2372655 , url =

    work page doi:10.2307/2372655 1954

  35. [35]

    Lindner, Niels , title =. J. Th\'eor. Nombres Bordeaux , fjournal =. 2017 , number =

    2017

  36. [36]

    Macdonald, I. G. , title =. 1995 , pages =

    1995

  37. [37]

    Manuscripta Math

    Flenner, Hubert and Vogel, Wolfgang , title =. Manuscripta Math. , fjournal =. 1993 , number =. doi:10.1007/BF02599302 , url =

    work page doi:10.1007/bf02599302 1993

  38. [38]

    2011 , note =

    Mustaţă, Mircea , title =. 2011 , note =

    2011

  39. [39]

    1962 , pages =

    Nagata, Masayoshi , title =. 1962 , pages =

    1962

  40. [40]

    2005 , isbn =

    Nguyen, Nghi Huu , title =. 2005 , isbn =

    2005

  41. [41]

    Poonen, Bjorn , title =. Ann. of Math. (2) , fjournal =. 2004 , number =. doi:10.4007/annals.2004.160.1099 , url =

    work page doi:10.4007/annals.2004.160.1099 2004

  42. [42]

    Poonen, Bjorn , title =. J. Th\'eor. Nombres Bordeaux , fjournal =. 2007 , number =. doi:10.5802/jtnb.583 , url =

    work page doi:10.5802/jtnb.583 2007

  43. [43]

    Charles, François and Poonen, Bjorn , title =. J. Amer. Math. Soc. , fjournal =. 2016 , number =. doi:10.1090/S0894-0347-2014-00820-1 , url =

    work page doi:10.1090/s0894-0347-2014-00820-1 2016

  44. [44]

    Poonen, Bjorn , title =. Doc. Math. , fjournal =. 2013 , pages =. doi:10.4171/dm/420 , url =

    work page doi:10.4171/dm/420 2013

  45. [45]

    Poonen, Bjorn , title =. Math. Res. Lett. , fjournal =. 2008 , number =. doi:10.4310/MRL.2008.v15.n2.a5 , url =

    work page doi:10.4310/mrl.2008.v15.n2.a5 2008

  46. [46]

    Brenner, Holger and Jeffries, Jack and Núñez-Betancourt, Luis , title =. Adv. Math. , fjournal =. 2019 , pages =. doi:10.1016/j.aim.2019.106843 , url =

    work page doi:10.1016/j.aim.2019.106843 2019

  47. [47]

    , title =

    Ruppert, Wolfgang M. , title =. Arch. Math. (Basel) , fjournal =. 1995 , number =. doi:10.1007/BF01194167 , url =

    work page doi:10.1007/bf01194167 1995

  48. [48]

    Schwede, Karl and Zhang, Wenliang , title =. Proc. Lond. Math. Soc. (3) , fjournal =. 2013 , number =. doi:10.1112/plms/pdt007 , url =

    work page doi:10.1112/plms/pdt007 2013

  49. [49]

    Available: https://doi.org/10.1007/978-0-387-09494-6

    Silverman, Joseph H. , title =. 2009 , pages =. doi:10.1007/978-0-387-09494-6 , url =

    work page doi:10.1007/978-0-387-09494-6 2009

  50. [50]

    de Fernex, Tommaso and Ein, Lawrence and Musta. Math. Res. Lett. , fjournal =. 2003 , number =. doi:10.4310/MRL.2003.v10.n2.a9 , url =

    work page doi:10.4310/mrl.2003.v10.n2.a9 2003

  51. [51]

    Finite Fields Appl

    Asgarli, Shamil and Freidin, Brian , title =. Finite Fields Appl. , fjournal =. 2021 , pages =. doi:10.1016/j.ffa.2021.101833 , url =

    work page doi:10.1016/j.ffa.2021.101833 2021

  52. [52]

    Duke Math

    Vakil, Ravi and Wood, Melanie Matchett , title =. Duke Math. J. , fjournal =. 2015 , number =. doi:10.1215/00127094-2877184 , url =

    work page doi:10.1215/00127094-2877184 2015

  53. [53]

    Wang, Xiaozong , title =. J. \'Ec. polytech. Math. , fjournal =. 2022 , pages =. doi:10.5802/jep.191 , url =

    work page doi:10.5802/jep.191 2022

  54. [54]

    Laksov, Dan and Thorup, Anders , title =. J. Reine Angew. Math. , fjournal =. 1995 , pages =

    1995

  55. [55]

    1611.09092 , archiveprefix =

    Franziska Wutz , year =. 1611.09092 , archiveprefix =

    Pith/arXiv arXiv

  56. [56]

    1704.08108 , archiveprefix =

    Franziska Wutz , year =. 1704.08108 , archiveprefix =

    Pith/arXiv arXiv

  57. [57]

    Ramachandran, Niranjan , title =. Bull. Sci. Math. , fjournal =. 2015 , number =. doi:10.1016/j.bulsci.2014.11.004 , url =

    work page doi:10.1016/j.bulsci.2014.11.004 2015

  58. [58]

    Heisuke Hironaka , journal =

  59. [59]

    Bennett, Bruce , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1977 , PAGES =. doi:10.2307/1997491 , URL =

    work page doi:10.2307/1997491 1977

  60. [60]

    2412.08744 , archivePrefix=

    Matthew Bertucci , year=. 2412.08744 , archivePrefix=

    Pith/arXiv arXiv

  61. [61]

    1996 , title =

    Fulton, William , keywords =. 1996 , title =

    1996

  62. [62]

    1993 , PAGES =

    Bruns, Winfried and Herzog, J\"urgen , TITLE =. 1993 , PAGES =

    1993

  63. [63]

    2013 , pages=

    Inventiones mathematicae , author=. 2013 , pages=. doi:10.1007/s00222-012-0418-z , number=

    work page doi:10.1007/s00222-012-0418-z 2013

  64. [64]

    The Stacks project , howpublished =

    The. The Stacks project , howpublished =

  65. [65]

    Matsumura, Hideyuki , year=