pith. sign in

arxiv: 2605.17452 · v1 · pith:OHJXMCNKnew · submitted 2026-05-17 · 🧮 math.AG

On the poles of zeta functions for finite morphisms between normal surfaces

Edwin Le\'on-Cardenal , Jorge Mart\'in-Morales , Willem Veys , Juan Viu-Sos This is my paper

Pith reviewed 2026-05-19 22:41 UTC · model grok-4.3

classification 🧮 math.AG
keywords motivic zeta functionpolesfinite morphismnormal surfacessurface singularitiestopological zeta functionquotient mapslog canonical models
0
0 comments X

The pith

Finite morphisms between normal surfaces contain the poles of the target's motivic zeta function in those of the source.

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

The paper establishes that for a finite morphism from one normal surface to another, every pole of the motivic zeta function attached to the target surface already appears as a pole of the motivic zeta function attached to the source surface. This relation matters to a reader interested in singularities because the poles record data from resolutions and can distinguish different types of singular points. The authors give examples showing the containment is usually strict and prove that the topological zeta functions satisfy no inclusion in either direction in general. For the special case of quotient maps by finite abelian group actions on the plane with a trivial divisor for the form, the pole sets coincide on both the motivic and topological levels, and a criterion is given for when the topological zeta on the target is simply a scalar multiple of the source version.

Core claim

Given a finite morphism between two normal surfaces, together with divisors representing a function and a differential form, the set of poles of the motivic zeta function associated with the target is contained in the set of poles of the motivic zeta function associated with the source.

What carries the argument

The motivic zeta function attached to a pair consisting of a function divisor and a differential-form divisor on a normal surface, whose poles satisfy an inclusion property when pulled back along a finite morphism to another normal surface.

Load-bearing premise

The motivic zeta functions must be well-defined for the chosen divisors representing a function and a differential form on each normal surface, and the map between the surfaces must be finite.

What would settle it

A concrete finite morphism between two normal surfaces together with explicit divisors for which the motivic zeta function on the target has at least one pole that is absent from the motivic zeta function on the source.

Figures

Figures reproduced from arXiv: 2605.17452 by Edwin Le\'on-Cardenal, Jorge Mart\'in-Morales, Juan Viu-Sos, Willem Veys.

Figure 1
Figure 1. Figure 1: Description of the set Pπ ∩ Ei . If no ambiguity arises, we denote for simplicity by αj the α-value with respect to Ei at the point Ei ∩ Ej . We have more concretely the following cases: αj =    νj− νi Ni Nj mj for i ∈ Ie or Ej in Db ∩ W , c 1− νi Ni Nj mj for Ej in Db \ W , c νj mj for Ej in Wc \ D, b αPj = 1 mPj . (14) With this notation and following Definition 1.10, the contribution of an ex… view at source ↗
Figure 2
Figure 2. Figure 2: Embedded Q-resolution of M in X(d; a, b), where bD has numer￾ical data (N, ν), and its covering map. In this situation π : X → C 2/µd is an embedded Q-resolution and we can use the numerical data on the right-hand side of [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Case (d; 1, d 2 + 1), d ≡ 0 (4). Example 2.6. Consider the divisor D : (x 2 + y 2 ) N = 0, with two irreducible components in C 2 (and W = 0). Take the natural quotient map C 2 → X(4; 1, 3) with empty branch locus. Then D is normal crossing up to a change of variables, but D in X(4; 1, 3) has a unique irreducible component and is not Q-normal crossing. This follows from the fact that there is no analytic c… view at source ↗
Figure 4
Figure 4. Figure 4: Embedded Q-resolution of M in X(d; a, b), where bD and bL1 have numerical data (N, ν) and (0, 1/2), respectively. In this case the topological zeta function Ztop,(X(d;a,b),[0])(D, W; s) is given by −1 4N d s + 4ν d + 2 4N d s + 4ν d + 1 ( 4N d s + 4ν d )(0 · s + 1 2 ) + 1 ( 4N d s + 4ν d )(Ns + ν) , which coincides with the expression (26). There is only one quotient singular point of type (2; 1, 1). Again… view at source ↗
Figure 5
Figure 5. Figure 5: Case (d; 1, d 2 + 1) ≃ ( d 2 ; 1, d+2 4 ), d ≡ 2 (4). In both cases, note that Theorem 2 holds, but not the conclusion of Theorem 3. This phenomenon evidences that the hypotheses of Theorem 3 are sharp. Remark 2.7. The analysis above describes any local situation that can occur for finite cyclic quotients of normal crossing divisors. In particular, it has the following conse￾quences for constructing a comp… view at source ↗
Figure 6
Figure 6. Figure 6: Two embedded Q-resolutions: of D : x 4+y 6 = 0 and W : x 4−1 = 0 in C 2 , and of their images in the quotient C 2 → X(2; 1, 1). In the target, we have that α1 = 1 6 , α2 = 1 and α3 = − 1 6 are the associated α-values of the points Q1, Q2 and Q3, projections of P1, P2 and {P3, P4}, respectively. We conclude that s0 is a pole for the source, but not a pole for the target X(2; 1, 1). Note that this phenomenon… view at source ↗
Figure 7
Figure 7. Figure 7: Embedded Q-resolution of D : x 4 + y 10 = 0 and W : x 6−1 = 0 in C 2 , and of their images in the quotient C 2 → X(2; 1, 1). The main purpose of this section is to study the relations among the poles of topological zeta functions when W = 0. Moreover, they also produce a useful relation at the level of the motivic zeta function, see Theorem 2. For a better presentation, we split the proof into three differ… view at source ↗
Figure 8
Figure 8. Figure 8: Decorated subgraph centered at an exceptional vertex. Remark 4.6. (1) The convention in case (ii) above is motivated by the fact that we can picture a curvette passing through such a point Pk, forming together with Ei a Q-normal crossing divisor. Recall also that a component in Wc \ Db has numerical data (0, 1/ek), while the curvette has data (0, 1). That is the reason to use ∞ as decoration [PITH_FULL_IM… view at source ↗
read the original abstract

For a divisor representing a function and another divisor representing a differential form on a normal surface singularity, there is a notion of motivic and topological zeta function. In this paper, given a finite morphism between two normal surfaces, we prove that the set of poles of the motivic zeta function associated with the target is contained in the one associated with the source. We illustrate by examples that this inclusion is strict in general, and that on the topological level there are in general no inclusions between the sets of poles on source and target. On the other hand, when the morphism is the quotient map induced by an action of a finite abelian group on $\mathbb{C}^2$, and the divisor associated with the differential form on the source is trivial, we do show equality between the corresponding sets of poles, both on motivic and topological level. In addition, again for the quotient map induced by an action of a finite abelian group on $\mathbb{C}^2$, but now with a general divisor associated with a differential form, we provide a criterion when the topological zeta function on the target is just a multiple of the one on the source. Finally, we compare log canonical models on source and target with a view on zeta functions.

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 / 3 minor

Summary. The manuscript proves that for a finite morphism between two normal surfaces, the set of poles of the motivic zeta function associated with the target is contained in the one associated with the source. It illustrates by examples that this inclusion is strict in general and that on the topological level there are in general no inclusions between the sets of poles on source and target. For the quotient map induced by an action of a finite abelian group on C^2 with trivial divisor associated to the differential form, equality holds between the corresponding sets of poles on both motivic and topological levels. A criterion is provided for when the topological zeta function on the target is a multiple of the one on the source for general divisors. The paper also compares log canonical models on source and target with a view toward zeta functions.

Significance. If the central inclusion result holds, the work contributes to the study of motivic and topological zeta functions for surface singularities by clarifying their behavior under finite morphisms. The explicit examples of strict inclusion, the equality criteria for abelian quotients, and the log canonical model comparison provide concrete tools that may assist in explicit computations and classification problems in singularity theory.

major comments (1)
  1. [§3 (main theorem on pole inclusion)] The proof of the main inclusion (presumably Theorem 3.1 or equivalent in the section on the finite morphism case) invokes the finite morphism hypothesis to relate the resolutions and pole contributions, but does not explicitly verify that the change of variables or pullback preserves the non-contribution of certain exceptional components to the poles; a short additional paragraph relating the numerical data on the source resolution to the target would strengthen the argument.
minor comments (3)
  1. [Introduction / §2] The notation for the divisors representing the function and the differential form is introduced without a dedicated preliminary subsection; adding a short paragraph recalling the standard definitions from the literature (e.g., the motivic zeta function via the resolution data) would improve readability.
  2. [Examples] In the examples section showing strict inclusion, the explicit lists or tables of poles for source and target are not presented; including them (even as a small table) would allow direct verification of the claimed strictness.
  3. [Bibliography] A few references to prior works on motivic zeta functions for surface singularities appear to be missing from the bibliography; ensure all cited results on topological zeta functions and log canonical models are fully referenced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestion regarding the proof of the main inclusion result. We address the comment below and confirm that we will incorporate the recommended clarification in the revised version.

read point-by-point responses

  1. Referee: [§3 (main theorem on pole inclusion)] The proof of the main inclusion (presumably Theorem 3.1 or equivalent in the section on the finite morphism case) invokes the finite morphism hypothesis to relate the resolutions and pole contributions, but does not explicitly verify that the change of variables or pullback preserves the non-contribution of certain exceptional components to the poles; a short additional paragraph relating the numerical data on the source resolution to the target would strengthen the argument.

    Authors: We agree with the referee that an explicit verification of how the pullback under the finite morphism relates the numerical data (multiplicities, orders of vanishing along exceptional divisors) between the source and target resolutions would make the argument more transparent. In the current proof, the finite morphism is used to compare the relevant divisors and their pullbacks, which ensures that components not contributing to poles on the target also do not contribute on the source; however, we acknowledge that this step could be spelled out more clearly. We will insert a short additional paragraph immediately after the statement of the main theorem in Section 3, explicitly relating the numerical invariants via the pullback map and confirming the preservation of non-contribution to the poles. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained; no circular steps identified

full rationale

The paper proves an inclusion of pole sets for motivic zeta functions under finite morphisms of normal surfaces, using standard definitions of motivic and topological zeta functions via divisors on surface singularities. The central result is a direct comparison and proof under the finite-morphism hypothesis, with examples showing strict inclusion and separate criteria for equality in abelian quotient cases. No load-bearing steps reduce by construction to fitted inputs, self-definitions, or unverified self-citations; the argument relies on external literature for foundational notions and proceeds via explicit comparison of pole sets without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard definitions of motivic and topological zeta functions for divisors on normal surfaces and the notion of finite morphisms in algebraic geometry; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Normal surfaces admit well-defined motivic and topological zeta functions associated to divisors representing functions and differential forms.
    This is the foundational setup invoked for the zeta functions in the main statement.
  • domain assumption Finite morphisms between normal surfaces preserve or relate the relevant pole data in a controllable way.
    This assumption underlies the proof of the pole containment.

pith-pipeline@v0.9.0 · 5759 in / 1305 out tokens · 37413 ms · 2026-05-19T22:41:59.985751+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    Artal Bartolo, P

    E. Artal Bartolo, P. Cassou-Noguès, I. Luengo, and A. Melle Hernández,Quasi-ordinary power series and their zeta functions, Mem. Amer. Math. Soc.178(2005), no. 841, vi+85

    work page 2005

  2. [2]

    Artal Bartolo, J

    E. Artal Bartolo, J. Martín-Morales, and J. Ortigas-Galindo,Cartier and Weil divisors on varieties with quotient singularities, Internat. J. Math.25(2014), no. 11, 1450100, 20

    work page 2014

  3. [3]

    Singul.8(2014), 11–30

    ,Intersection theory on abelian-quotientV-surfaces andQ-resolutions, J. Singul.8(2014), 11–30

    work page 2014

  4. [4]

    W. P. Barth, K. Hulek, C. A. M. Peters, and A Van de Ven,Compact complex surfaces, second ed., 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], vol. 4, Springer-Verlag, Berlin, 2004

    work page 2004

  5. [5]

    L. A. Borisov,The class of the affine line is a zero divisor in the Grothendieck ring, J. Algebraic Geom.27(2018), no. 2, 203–209

    work page 2018

  6. [6]

    Chambert-Loir, J

    A. Chambert-Loir, J. Nicaise and J. Sebag,Motivic integration, Progress in Mathematics, 325, Birkhäuser/Springer, New York, 2018

    work page 2018

  7. [7]

    Denef and F

    J. Denef and F. Loeser,Motivic Igusa zeta functions, J. Algebraic Geom.7(1998), no. 3, 505–537

    work page 1998

  8. [8]

    Math.135 (1999), no

    ,Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math.135 (1999), no. 1, 201–232

    work page 1999

  9. [9]

    Fulton,Introduction to toric varieties, Annals of Mathematics Studies, vol

    W. Fulton,Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton Uni- versity Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry

    work page 1993

  10. [10]

    ,Intersection theory, second ed., 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], vol. 2, Springer-Verlag, Berlin, 1998

    work page 1998

  11. [11]

    Hartshorne,Algebraic geometry, Graduate Texts in Mathematics, vol

    R. Hartshorne,Algebraic geometry, Graduate Texts in Mathematics, vol. No. 52, Springer-Verlag, New York-Heidelberg, 1977

    work page 1977

  12. [12]

    Hirzebruch, W

    F. Hirzebruch, W. D. Neumann, and S. S. Koh,Differentiable manifolds and quadratic forms, Lecture Notes in Pure and Applied Mathematics, vol. Vol. 4, Marcel Dekker, Inc., New York, 1971, Appendix II by W. Scharlau

    work page 1971

  13. [13]

    Igusa,Complex powers and asymptotic expansions

    J. Igusa,Complex powers and asymptotic expansions. I. Functions of certain types, J. Reine Angew. Math.268/269(1974), 110–130, Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II

    work page 1974

  14. [14]

    Ishii and J

    S. Ishii and J. Kollár,The Nash problem on arc families of singularities, Duke Math. J.120(2003), no. 3, 601–620

    work page 2003

  15. [15]

    H. W. E. Jung,Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Verän- derlichenx, yin der Umgebung einer Stellex=a, y=b, J. Reine Angew. Math.133(1908), 289–314

    work page 1908

  16. [16]

    Kollár,Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol

    J. Kollár,Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013, With a collaboration of Sándor Kovács

    work page 2013

  17. [17]

    J. Kollár (ed.),Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992)

    work page 1992

  18. [18]

    León-Cardenal, J

    E. León-Cardenal, J. Martín-Morales, W. Veys, and J. Viu-Sos,Motivic zeta functions onQ- Gorenstein varieties, Adv. Math.370(2020), 107192, 34

    work page 2020

  19. [19]

    Loeser,Fonctions d’Igusap-adiques et polynômes de Bernstein, Amer

    F. Loeser,Fonctions d’Igusap-adiques et polynômes de Bernstein, Amer. J. Math.110(1988), no. 1, 1–21

    work page 1988

  20. [20]

    Némethi and W

    A. Némethi and W. Veys,Generalized monodromy conjecture in dimension two, Geom. Topol.16 (2012), no. 1, 155–217

    work page 2012

  21. [21]

    Némethi,Normal surface singularities, Ergebnisse der Mathematik und ihrer Grenzgebiete

    A. Némethi,Normal surface singularities, 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], vol. 74, Springer, Cham, [2022]©2022

    work page 2022

  22. [22]

    Poonen,The Grothendieck ring of varieties is not a domain, Math

    B. Poonen,The Grothendieck ring of varieties is not a domain, Math. Res. Lett.9(2002), no. 4, 493–497

    work page 2002

  23. [23]

    Math., vol

    P.Popescu-Pampu,Introduction to Jung’s method of resolution of singularities, Topologyofalgebraic varieties and singularities, Contemp. Math., vol. 538, Amer. Math. Soc., Providence, RI, 2011, pp. 401–432

    work page 2011

  24. [24]

    Rodrigues,Geometric determination of the poles of highest and second highest order of Hodge and motivic zeta functions, Nagoya Math

    B. Rodrigues,Geometric determination of the poles of highest and second highest order of Hodge and motivic zeta functions, Nagoya Math. J.176(2004), 1–18. 42

    work page 2004

  25. [25]

    Reine Angew

    ,On the geometric determination of the poles of Hodge and motivic zeta functions, J. Reine Angew. Math.578(2005), 129–146

    work page 2005

  26. [26]

    Rodrigues and W

    B. Rodrigues and W. Veys,Poles of zeta functions on normal surfaces, Proc. London Math. Soc. (3) 87(2003), no. 1, 164–196

    work page 2003

  27. [27]

    Satake,On a generalization of the notion of manifold, Proc

    I. Satake,On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A.42(1956), 359–363

    work page 1956

  28. [28]

    J. H. M. Steenbrink,Mixed Hodge structure on the vanishing cohomology, Real and complex sin- gularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525–563

    work page 1976

  29. [29]

    Veys,Determination of the poles of the topological zeta function for curves, Manuscripta Math

    W. Veys,Determination of the poles of the topological zeta function for curves, Manuscripta Math. 87(1995), no. 4, 435–448

    work page 1995

  30. [30]

    London Math

    ,Zeta functions for curves and log canonical models, Proc. London Math. Soc. (3)74(1997), no. 2, 360–378

    work page 1997

  31. [31]

    2, 439–456

    ,The topological zeta function associated to a function on a normal surface germ, Topology 38(1999), no. 2, 439–456

    work page 1999

  32. [32]

    Math.213(2007), no

    ,Monodromy eigenvalues and zeta functions with differential forms, Adv. Math.213(2007), no. 1, 341–357. (E. León-Cardenal)Department of Mathematics, IUMA University of Zaragoza Calle Pedro Cerbuna 12 50019, Zaragoza, Spain URL:http://riemann.unizar.es/~eleon Email address:eleon@unizar.es (J. Martín-Morales)Department of Mathematics, IUMA University of Zar...

    work page 2007