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arxiv: 2408.15931 · v2 · submitted 2024-08-28 · 🧮 math.AG · math.AC

On δ-sequences and surfaces at infinity

C. Galindo , F. Monserrat , C.-J. Moreno-\'Avila , J.-J. Moyano-Fern\'andez This is my paper

Pith reviewed 2026-05-23 22:10 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords δ-sequencessemigroup at infinityone place at infinitycurvessurfaces at infinityresolution of singularitiesdual graph
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The pith

Different δ-sequences can generate the same semigroup at infinity while each encodes distinct geometric data on curves with one place at infinity.

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

The paper develops explicit constructions for δ-sequences that generate the semigroup at infinity of a curve with only one place at infinity. These sequences carry concrete geometric information, including the dual graph arising from resolution of the singularity at infinity. Multiple distinct δ-sequences can produce identical semigroups, so the work identifies families of such sequences for a fixed semigroup. This identification makes it possible to isolate which geometric features are determined by the semigroup alone. The same construction problem is posed for the δ-semigroups attached to surfaces at infinity.

Core claim

In most cases the semigroup at infinity S of a curve C with only one place at infinity is generated by a δ-sequence. This sequence provides geometrical information on C such as the dual graph of the resolution of the singularity of C at infinity. Since different δ-sequences can generate the same semigroup, the paper shows how to construct δ-sequences and how to obtain different families that generate the same semigroup S, allowing study of the geometrical content encoded by S. An analogous problem arises when considering surfaces at infinity and their δ-semigroups.

What carries the argument

A δ-sequence, which generates the semigroup at infinity S and determines the dual graph of the resolution at infinity.

If this is right

  • Explicit constructions exist for δ-sequences attached to given curves with one place at infinity.
  • Multiple families of δ-sequences can be produced for any fixed semigroup S.
  • The dual graph of the resolution at infinity is recoverable from any generating δ-sequence.
  • The same construction technique applies to δ-semigroups of surfaces at infinity.

Where Pith is reading between the lines

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

  • The families may distinguish geometric realizations of the same semigroup that are not visible from S alone.
  • The approach could extend to higher-dimensional varieties whose semigroups at infinity are defined analogously.
  • Comparison of families for a fixed S might yield new numerical invariants of the curve beyond the semigroup.

Load-bearing premise

The semigroup at infinity of a curve with one place at infinity is generated by a δ-sequence in most cases.

What would settle it

An explicit curve with one place at infinity whose semigroup at infinity admits no generating δ-sequence.

read the original abstract

In most cases the semigroup at infinity $S$ of a curve $C$ with only one place at infinity is generated by a $\delta$-sequence. This sequence provides geometrical information on $C$ such as the dual graph of the resolution of the singularity of $C$ at infinity. Since different $\delta$-sequences can generate the same semigroup, it is an interesting problem to know the geometrical behaviour of curves $C$ sharing the same semigroup $S$. An analogous problem arises in a more general context when considering surfaces at infinity and their $\delta$-semigroups. We show how to construct $\delta$-sequences, and how to obtain different families that generate the same semigroup $S$, allowing us to study the geometrical content encoded by $S$.

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 / 0 minor

Summary. The paper claims that in most cases the semigroup at infinity S of a curve C with only one place at infinity is generated by a δ-sequence. This sequence provides geometrical information on C such as the dual graph of the resolution of the singularity of C at infinity. Since different δ-sequences can generate the same semigroup, the authors consider the geometrical behaviour of curves C sharing the same S. An analogous problem is posed for surfaces at infinity and their δ-semigroups. The contribution consists of showing how to construct δ-sequences and how to obtain different families that generate the same semigroup S, allowing study of the geometrical content encoded by S.

Significance. If the constructions are explicit and correct, the work could provide concrete tools for extracting and comparing geometrical data (such as dual graphs) from a fixed semigroup S, which may aid classification and analysis of singularities at infinity for both curves and surfaces.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting the potential utility of the constructions if they are explicit and correct. The recommendation is listed as uncertain, but no specific major comments or points of criticism are provided in the report. We therefore have no individual comments to address point by point. The constructions of δ-sequences and the families generating identical semigroups are presented explicitly in Sections 3 and 4 of the manuscript, with concrete examples for both curves and surfaces.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states the generation of S by a δ-sequence as established background ('In most cases the semigroup at infinity S ... is generated by a δ-sequence') and focuses on explicit constructions of sequences and families sharing S to study geometrical content such as dual graphs. No derivation, equation, or central claim reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the work is framed as constructive and independent of any internal fitting or renaming of prior results within the paper itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, which supplies no concrete free parameters, background axioms, or newly postulated entities.

pith-pipeline@v0.9.0 · 5672 in / 947 out tokens · 29266 ms · 2026-05-23T22:10:14.156189+00:00 · methodology

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    Abhyankar

    S. Abhyankar. Lectures on expansion techniques in Algeb raic Geometry. T ata Institute of Fundamen- tal Research Lectures on Mathematics and Physics, 57, 1977. Tata Institute of Fundamental Research, Bombay

    work page 1977

  2. [2]

    S. S. Abhyankar and T. T. Moh. Newton-Puiseux expansion a nd generalized Tschirnhausen transfor- mation. I, II. J. Reine Angew. Math., 260:47–83; ibid. 261 (1973), 29–54, 1973

    work page 1973

  3. [3]

    Assi and P .-A

    A. Assi and P .-A. Garc´ ıa-S´ anchez. Algorithms for curves with one place at infinity. J. Symbolic Com- put., 74:475–492, 2016

    work page 2016

  4. [4]

    Bertin and P

    J. Bertin and P . Carbonne. Semi-groupes d’entiers et app lication aux branches. J. Algebra, 49(1):81– 95, 1977

    work page 1977

  5. [5]

    Campillo

    A. Campillo. Algebroid curves in positive characteristic , volume 613 of Lect. Notes Math. Springer, Berlin, 1980

    work page 1980

  6. [6]

    Campillo, F

    A. Campillo, F. Delgado, and S.-M. Gusein-Zade. Equivar iant Poincar´ e series and topology of valua- tions. Doc. Math., 21:271–286, 2016

    work page 2016

  7. [7]

    Campillo, F

    A. Campillo, F. Delgado, and S.-M. Gusein-Zade. On the to pological type of a set of plane valuations with symmetries. Math. Nachr ., 290(13):1925–1938, 2017

    work page 1925

  8. [8]

    Campillo, F

    A. Campillo, F. Delgado, and S.-M. Gusein-Zade. On real a nalogues of the Poincar´ e series.Bull. Lond. Math. Soc., 56(1):449–459, 2024

    work page 2024

  9. [9]

    Campillo, O

    A. Campillo, O. Piltant, and A. Reguera. Curves and divis ors on surfaces associated to plane curves with one place at infinity. Proc. London Math. Soc. , 84:559–580, 2002

    work page 2002

  10. [10]

    Casas-Alvero

    E. Casas-Alvero. Singularities of plane curves , volume 276 of London Math. Soc. Lect. Notes Ser . Cambridge Univ. Press, 2000

    work page 2000

  11. [11]

    S. D. Cutkosky, L. Ein, and R. Lazarsfeld. Positivity an d complexity of ideal sheaves. Math. Ann. , 321(2):213–234, 2001

    work page 2001

  12. [12]

    C. Delorme. Sous-mono¨ ıdes d’intersection compl` ete de N. Ann. Sci. ´Ecole Norm. Sup. (4) , 9(1):145– 154, 1976

    work page 1976

  13. [13]

    Fujimoto and M

    M. Fujimoto and M. Suzuki. Construction of affine plane c urves with one place at infinity. Osaka J. Math., 39(4):1005–1027, 2002

    work page 2002

  14. [14]

    Galindo and F

    C. Galindo and F. Monserrat. δ -sequences and evaluation codes defined by plane valuations at infinity. Proc. Lond. Math. Soc. (3) , 98(3):714–740, 2009

    work page 2009

  15. [15]

    Galindo and F

    C. Galindo and F. Monserrat. The Abhyankar-Moh Theorem for plane valuations at infinity.J. Algebra, 374:181–194, 2013

    work page 2013

  16. [16]

    Galindo and F

    C. Galindo and F. Monserrat. The cone of curves and the Co x ring of rational surfaces given by divisorial valuations. Adv. Math., 290:1040–1061, 2016

    work page 2016

  17. [17]

    Galindo, F

    C. Galindo, F. Monserrat, and C.-J. Moreno- ´Avila. Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces. Rev. Mat. Complut., 33:349–372, 2020

    work page 2020

  18. [18]

    Galindo, F

    C. Galindo, F. Monserrat, and C.-J. Moreno- ´Avila. Discrete equivalence of non-positive at infinity plane valuations. Results Math., 76(146), 2021

    work page 2021

  19. [19]

    Galindo, F

    C. Galindo, F. Monserrat, and C-J. Moreno- ´Avila. Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruc h surfaces. Quaest. Math. , 46(11):2367– 2401, 2023

    work page 2023

  20. [20]

    Galindo, F

    C. Galindo, F. Monserrat, and J. J. Moyano-Fern´ andez. Minimal plane valuations. J. Alg. Geom. , 27:751–783, 2018

    work page 2018

  21. [21]

    Galindo, F

    C. Galindo, F. Monserrat, J. J. Moyano-Fern´ andez, and M. Nickel. Newton-Okounkov bodies of ex- ceptional curve valuations. Rev. Mat. Iberoam., 36(7):2147–2182, 2020. 26 C. GALINDO, F. MONSERRA T, C.-J. MORENO- ´AVILA, AND J.-J. MOY ANO-FERN´ANDEZ

    work page 2020

  22. [22]

    Kaveh and A

    K. Kaveh and A. Khovanskii. Newton-Okounkov bodies, se migroups of integral points, graded alge- bras and intersection theory. Ann. Math., 176:925–978, 2012

    work page 2012

  23. [23]

    Lazarsfeld

    R. Lazarsfeld. Positivity in algebraic geometry I. Classical setting: lin e bundles and linear series , volume 48. Springer-V erlag, Berlin, 2004

    work page 2004

  24. [24]

    Lazarsfeld and M

    R. Lazarsfeld and M. Mustat ¸˘ a. Convex bodies associated to linear series. Ann. Sci. ´Ec. Norm. Sup. , 42:783–835, 2009

    work page 2009

  25. [25]

    T. T. Moh. On analytic irreducibility at ∞ of a pencil of curves. Proc. Amer . Math. Soc., 44:22–24, 1974

    work page 1974

  26. [26]

    Okounkov

    A. Okounkov. Why would multiplicities be log-concave? In The orbit method in geometry and physics (Marseille, 2000) , volume 213 of Progr . Math., pages 329–347. Birkh¨ auser Boston, Boston, MA, 2003

    work page 2000

  27. [27]

    H. Pinkham. Courbes planes ayant une seule place a l’infi ni. Seminaire sur les singularities des sur- faces (Demazure-Pinkham-T eissier), Course donn ´e au Centre de Math. de l’Ecole Polytechnique , 1977–1978

    work page 1977

  28. [28]

    Reguera-L´ opez

    A.-J. Reguera-L´ opez. Semigroups and clusters at infinity. In Algebraic geometry and singularities (La R´abida, 1991), volume 134 of Progr . Math., pages 339–374. Birkh¨ auser, Basel, 1996

    work page 1991

  29. [29]

    J. C. Rosales and P . A. Garc´ ıa-S´ anchez.Numerical semigroups, volume 20. Springer New Y ork, 2009

    work page 2009

  30. [30]

    A. Sathaye. On planar curves. Amer . J. Math., 99(5):1105–1135, 1977

    work page 1977

  31. [31]

    Sathaye and J

    A. Sathaye and J. Stenerson. Plane polynomial curves. I n Algebraic geometry and its applications (West Lafayette, IN, 1990) , pages 121–142. Springer, New Y ork, 1994

    work page 1990

  32. [32]

    Seidenberg

    A. Seidenberg. V aluation ideals in polynomial rings. Trans. Amer . Math. Soc., 57:387–425, 1945

    work page 1945

  33. [33]

    C. M. Shor. On free numerical semigroups and the constru ction of minimal telescopic sequences. J. Integer Seq., 22(2):Art. 19.2.4, 29, 2019

    work page 2019

  34. [34]

    Spivakovsky

    M. Spivakovsky. V aluations in function fields of surfac es. Amer . J. Math., 112:107–156, 1990

    work page 1990

  35. [35]

    M. Suzuki. Affine plane curves with one place at infinity. Ann. Inst. F ourier (Grenoble), 49(2):375– 404, 1999

    work page 1999

  36. [36]

    O. Zariski. The moduli problem for plane branches , volume 39 of Univ. Lecture Series . Am. Math. Soc., Providence, RI, 2006. With an appendix by B. Teissier, Translated from the 1973. Current address : C. Galindo, C.-J. Moreno- ´Avila and J.-J. Moyano-Fern ´andez : Departamento de Matem´ aticas & Instituto Universitario de Matem´ aticas y Aplicaciones de...

    work page 2006