Asymptotic computations of tropical refined invariants in genus $0$ and $1$

Thomas Blomme; Gurvan Mével

doi:10.5802/jep.288

Asymptotic computations of tropical refined invariants in genus

0

and

1

[Calculs asymptotiques des invariants tropicaux raffinés en genre

0

et

1

]

Thomas Blomme ¹ ; Gurvan Mével ²

¹ Université de Neuchâtel, rue Émile Argan 11, Neuchâtel 2000, Suisse
² CNRS & Nantes Université, UMR 6629 Laboratoire de Mathématiques Jean Leray, 2 rue de la Houssinière, F-44322 Nantes Cedex 3, France

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 185-234.

Résumé
Abstract

Block et Göttsche ont introduit une multiplicité polynomiale pour compter les courbes tropicales. Itenberg et Mikhalkin ont montré que cette multiplicité donnait lieu à des invariants de comptage, appelés invariant tropicaux raffinés. Récemment, Brugallé et Jaramillo-Puentes ont étudié les propriétés polynomiales des coefficients de ces invariants, et montré qu’à genre fixé ils coïncident asymptotiquement avec des polynômes en la classe d’homologie des courbes que l’on regarde. On appelle invariant raffiné asymptotique la série génératrice de ces polynômes. En genre $0$ , elle a été calculée par le second auteur dans le cas $h$ -transverse. Dans cet article, on donne une nouvelle démonstration de la formule pour l’invariant raffiné asymptotique en genre $0$ , en utilisant une variante de la méthode des diagrammes en étages. Cette technique nous permet également de calculer l’invariant asymptotique en genre $1$ . Le résultat exhibe de surprenantes propriétés de régularité, liées à la série génératrice des nombres de partitions et à des formes quasi-modulaires.

Block and Göttsche introduced a Laurent polynomial multiplicity to count tropical curves. Itenberg and Mikhalkin then showed that this multiplicity leads to invariant counts called tropical refined invariants. Recently, Brugallé and Jaramillo-Puentes studied the polynomiality properties of the coefficients of these invariants and showed that for fixed genus $g$ , the coefficients ultimately coincide with polynomials in the homology class of the curves that we consider. We call the generating series of these polynomials asymptotic refined invariant. In genus $0$ , the asymptotic refined invariant has been computed by the second author in the $h$ -transverse case. In this paper, we give a new proof of the formula for the asymptotic refined invariant for $g = 0$ using variations on the floor diagram algorithm. This technique also enables us to compute the asymptotic refined invariant for $g = 1$ . The result exhibits surprising regularity properties related to the generating series of partition numbers and quasi-modular forms.

Reçu le : 2024-04-04
Accepté le : 2024-12-26
Publié le : 2025-01-17

DOI : 10.5802/jep.288

Classification : 14T15, 14T90, 05A15, 14N10
Keywords: Tropical refined invariants, floor diagrams, generating series, asymptotic behavior
Mots-clés : Invariants tropicaux raffinés, diagrammes en étages, séries génératrices, comportement asymptotique

Affiliations des auteurs :

Thomas Blomme ¹ ; Gurvan Mével ²

¹ Université de Neuchâtel, rue Émile Argan 11, Neuchâtel 2000, Suisse
² CNRS & Nantes Université, UMR 6629 Laboratoire de Mathématiques Jean Leray, 2 rue de la Houssinière, F-44322 Nantes Cedex 3, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2025__12__185_0,
     author = {Thomas Blomme and Gurvan M\'evel},
     title = {Asymptotic computations of tropical~refined~invariants in genus~$0$ and $1$},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {185--234},
     publisher = {\'Ecole polytechnique},
     volume = {12},
     year = {2025},
     doi = {10.5802/jep.288},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.288/}
}

TY  - JOUR
AU  - Thomas Blomme
AU  - Gurvan Mével
TI  - Asymptotic computations of tropical refined invariants in genus $0$ and $1$
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2025
SP  - 185
EP  - 234
VL  - 12
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.288/
DO  - 10.5802/jep.288
LA  - en
ID  - JEP_2025__12__185_0
ER  -

%0 Journal Article
%A Thomas Blomme
%A Gurvan Mével
%T Asymptotic computations of tropical refined invariants in genus $0$ and $1$
%J Journal de l’École polytechnique — Mathématiques
%D 2025
%P 185-234
%V 12
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.288/
%R 10.5802/jep.288
%G en
%F JEP_2025__12__185_0

Thomas Blomme; Gurvan Mével. Asymptotic computations of tropical refined invariants in genus $0$ and $1$. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 185-234. doi : 10.5802/jep.288. https://jep.centre-mersenne.org/articles/10.5802/jep.288/

Bibliographie
Cité par

[BG16a] F. Block & L. Göttsche - “Fock spaces and refined Severi degrees”, Internat. Math. Res. Notices 2016 (2016) no. 21, p. 6553-6580 | DOI | MR | Zbl

[BG16b] F. Block & L. Göttsche - “Refined curve counting with tropical geometry”, Compositio Math. 152 (2016) no. 1, p. 115-151 | DOI | MR | Zbl

[BJP22] E. Brugallé & A. Jaramillo Puentes - “Polynomiality properties of tropical refined invariants”, Comb. Theory 2 (2022), article ID 1, 53 pages | DOI | MR | Zbl

[Blo19] T. Blomme - “A Caporaso-Harris type Formula for relative refined invariants”, 2019 | arXiv

[BM07] E. Brugallé & G. Mikhalkin - “Enumeration of curves via floor diagrams”, Comptes Rendus Mathématique 345 (2007) no. 6, p. 329-334 | DOI | Numdam | Zbl

[BM09] E. Brugallé & G. Mikhalkin - “Floor decompositions of tropical curves: the planar case”, in Proc. Geometry-Topology Conference (Gökova, 2008), Gökova Geometry/Topology Conference (GGT), Gökova, 2009, p. 64-90 | arXiv | MR | Zbl

[Bou19] P. Bousseau - “Tropical refined curve counting from higher genera and lambda classes”, Invent. Math. 215 (2019) no. 1, p. 1-79 | DOI | MR | Zbl

[CH98] L. Caporaso & J. Harris - “Counting plane curves of any genus”, Invent. Math. 131 (1998) no. 2, p. 345-392 | DOI | MR | Zbl

[DFI95] P. Di Francesco & C. Itzykson - “Quantum Intersection Rings”, The moduli space of curves, Birkhäuser Boston, Boston, MA (1995), p. 81-148 | DOI | Zbl

[Ful93] W. Fulton - Introduction to toric varieties, Princeton University Press, Princeton, NJ, 1993 no. 131 | DOI | MR

[GM07] A. Gathmann & H. Markwig - “The Caporaso–Harris formula and plane relative Gromov-Witten invariants in tropical geometry”, Math. Ann. 338 (2007) no. 4, p. 845-868 | DOI | MR | Zbl

[GS14] L. Göttsche & V. Shende - “Refined curve counting on complex surfaces”, Geom. Topol. 18 (2014) no. 4, p. 2245-2307 | DOI | MR | Zbl

[GS19] L. Göttsche & F. Schroeter - “Refined broccoli invariants”, J. Algebraic Geom. 28 (2019) no. 1, p. 1-41 | DOI | MR | Zbl

[Göt98] L. Göttsche - “A conjectural generating function for numbers of curves on surfaces”, Comm. Math. Phys. 196 (1998) no. 3, p. 523-533 | MR | Zbl

[IM13] I. Itenberg & G. Mikhalkin - “On Block–Göttsche multiplicities for planar tropical curves”, Internat. Math. Res. Notices 2013 (2013) no. 23, p. 5289-5320 | DOI | MR | Zbl

[IS23] I. Itenberg & E. Shustin - “Real enumerative invariants relative to the anti-canonical divisor and their refinement”, 2023 | arXiv

[KM94] M. Kontsevich & Y. Manin - “Gromov-Witten classes, quantum cohomology, and enumerative geometry”, Comm. Math. Phys. 164 (1994) no. 3, p. 525-562 | DOI | MR | Zbl

[KST11] M. Kool, V. Shende & R. P. Thomas - “A short proof of the Göttsche conjecture”, Geom. Topol. 15 (2011) no. 1, p. 397-406 | DOI | MR | Zbl

[Mik05] G. Mikhalkin - “Enumerative tropical algebraic geometry in $ℝ^{2}$ ”, J. Amer. Math. Soc. 18 (2005) no. 2, p. 313-377 | DOI | MR | Zbl

[Mik17] G. Mikhalkin - “Quantum indices and refined enumeration of real plane curves”, Acta Math. 219 (2017) no. 1, p. 135-180 | DOI | MR | Zbl

[Mév23] G. Mével - “Universal polynomials for tropical refined invariants in genus 0”, 2023 | arXiv

[Mév24a] G. Mével - “Combinatorial Göttsche-Schroeter invariants in any genus”, 2024 | arXiv

[Mév24b] G. Mével - Propriétés asymptotiques des invariants tropicaux raffinés, Ph. D. Thesis, Nantes Université, 2024

[NPS18] J. Nicaise, S. Payne & F. Schroeter - “Tropical refined curve counting via motivic integration”, Geom. Topol. 22 (2018) no. 6, p. 3175-3234 | DOI | MR | Zbl

[SS24] E. Shustin & U. Sinichkin - “Refined tropical invariants and characteristic numbers”, 2024 | arXiv

[Tze12] Y.-j. Tzeng - “A proof of the Göttsche-Yau-Zaslow formula”, J. Differential Geom. 90 (2012), p. 439-472 | MR | Zbl

Cité par Sources :