Feuilleter
no. 2
Tropical Möbius strips and ruled surfaces
[Rubans de Möbius tropicaux et surfaces réglées]
Thomas Blomme1 ; Victoria Schleis2
1Université de Genève, 5-7 rue du Conseil Général, 1205 Genève, Switzerland
2Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 2, pp. 297-350

We consider the enumeration of tropical curves in Möbius strips for two different lattice structures and relate them to the enumeration of curves in two rational ruled surfaces over a complex elliptic curve. Using this correspondence, we prove regularity results such as the piecewise quasi-polynomiality of relative invariants and the quasi-modularity of their generating series.

On étudie dans ce papier l’énumération des courbes tropicales dans un ruban de Möbius pour deux structures affines entières distinctes. Cette dernière est reliée à l’énumération des courbes complexes dans certaines surfaces réglées sur une courbe elliptique. En utilisant cette correspondance, on montre des propriétés de quasi-polynomialité de quasi-modularité des séries génératrices des invariants obtenus.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1848
Classification : 14N10, 14T90, 05A15, 14H99
Keywords: Enumerative geometry, tropical refined invariants, relative invariants, floor diagrams

Thomas Blomme  1   ; Victoria Schleis  2

1 Université de Genève, 5-7 rue du Conseil Général, 1205 Genève, Switzerland
2 Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Thomas Blomme; Victoria Schleis. Tropical Möbius strips and ruled surfaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 2, pp. 297-350. doi: 10.5802/afst.1848
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[1] Dan Abramovich; Qile Chen Stable logarithmic maps to Deligne–Faltings pairs. II, Asian J. Math., Volume 18 (2014) no. 3, pp. 465-488 | DOI | Zbl | MR

[2] Dan Abramovich; Qile Chen; Mark Gross; Bernd Siebert Decomposition of degenerate Gromov–Witten invariants, Compos. Math., Volume 156 (2020) no. 10, pp. 2020-2075 | DOI | Zbl | MR

[3] Federico Ardila; Erwan Brugallé The double Gromov–Witten invariants of Hirzebruch surfaces are piecewise polynomial, Int. Math. Res. Not., Volume 2017 (2017) no. 2, pp. 614-641 | DOI | Zbl | MR

[4] Florian Block; Andreas Gathmann; Hannah Markwig Psi-floor diagrams and a Caporaso–Harris type recursion, Isr. J. Math., Volume 191 (2012), pp. 405-449 | DOI | Zbl | MR

[5] Florian Block; Lothar Göttsche Refined curve counting with tropical geometry, Compos. Math., Volume 152 (2016) no. 1, pp. 115-151 | DOI | Zbl | MR

[6] Thomas Blomme A Caporaso–Harris type Formula for relative refined invariants (2019) | arXiv | Zbl

[7] Thomas Blomme Floor diagrams and enumerative invariants of line bundles over an elliptic curve (2021) | arXiv | Zbl

[8] Thomas Blomme Refined count for rational tropical curves in arbitrary dimension, Math. Ann., Volume 382 (2022) no. 3-4, pp. 1199-1244 | DOI | Zbl | MR

[9] Thomas Blomme Tropical curves in abelian surfaces I: enumeration of curves passing through points (2022) | arXiv

[10] Thomas Blomme Tropical curves in abelian surfaces II: enumeration of curves in linear systems (2022) | arXiv | Zbl

[11] Thomas Blomme Tropical curves in abelian surfaces III: pearl diagrams and multiple cover formulas (2022) | arXiv | Zbl

[12] Janko Böhm; Christoph Goldner; Hannah Markwig Tropical mirror symmetry in dimension one (2018) | arXiv

[13] Janko Böhm; Christoph Goldner; Hannah Markwig Counts of (tropical) curves in E×ℙ 1 and Feynman integrals, Ann. Inst. Henri Poincaré D, Comb. Phys. Interact., Volume 9 (2022) no. 1, pp. 121-158 | DOI | Zbl | MR

[14] Pierrick Bousseau Tropical refined curve counting from higher genera and lambda classes, Invent. Math., Volume 215 (2019) no. 1, pp. 1-79 | DOI | Zbl | MR

[15] Erwan Brugallé Floor diagrams relative to a conic, and GW-W invariants of Del Pezzo surfaces, Adv. Math., Volume 279 (2015), pp. 438-500 | DOI | Zbl | MR

[16] Erwan Brugallé; Grigory Mikhalkin Enumeration of curves via floor diagrams, Comptes Rendus. Mathématique, Volume 345 (2007) no. 6, pp. 329-334 | DOI | Zbl | Numdam

[17] Erwan Brugallé; Grigory Mikhalkin Floor decompositions of tropical curves: the planar case (2008) | arXiv | Zbl

[18] Erwan Brugallé; Andrés Jaramillo Puentes Polynomiality properties of tropical refined invariants (2020) | arXiv | Zbl

[19] Erwan Brugallé; Kristin Shaw A bit of tropical geometry, Am. Math. Mon., Volume 121 (2014) no. 7, pp. 563-589 | DOI | Zbl | MR

[20] Renzo Cavalieri; Paul Johnson; Hannah Markwig; Dhruv Ranganathan Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space, Math. Proc. Camb. Philos. Soc., Volume 171 (2021) no. 1, pp. 165-205 | DOI | Zbl | MR

[21] Qile Chen Stable logarithmic maps to Deligne–Faltings pairs I, Ann. Math. (2), Volume 180 (2014) no. 2, pp. 455-521 | DOI | Zbl | MR

[22] Daniel Corey; Hannah Markwig; Dhruv Ranganathan Counting tropical curves in ℙ 1 ×ℙ 1 : computation & polynomiality properties (2022) | arXiv | Zbl

[23] Andreas Gathmann; Hannah Markwig The Caporaso–Harris formula and plane relative Gromov–Witten invariants in tropical geometry, Math. Ann., Volume 338 (2007) no. 4, pp. 845-868 | DOI | Zbl | MR

[24] Andreas Gathmann; Hannah Markwig The numbers of tropical plane curves through points in general position, J. Reine Angew. Math., Volume 2007 (2007) no. 602, pp. 155-177 | DOI | Zbl | MR

[25] Andreas Gathmann; Hannah Markwig Kontsevich’s formula and the WDVV equations in tropical geometry, Adv. Math., Volume 217 (2008) no. 2, pp. 537-560 | DOI | Zbl | MR

[26] Lothar Göttsche; Franziska Schrœter Refined broccoli invariants, J. Algebr. Geom., Volume 28 (2019) no. 1, pp. 1-41 | DOI | Zbl | MR

[27] Lothar Göttsche; Vivek Shende Refined curve counting on complex surfaces, Geom. Topol., Volume 18 (2014) no. 4, pp. 2245-2307 | DOI | Zbl | MR

[28] Ian P. Goulden; David M. Jackson; Ravi Vakil Towards the geometry of double Hurwitz numbers, Adv. Math., Volume 198 (2005) no. 1, pp. 43-92 | DOI | Zbl | MR

[29] Phillip Griffiths; Joseph Harris Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, 1994 | Zbl | DOI | MR

[30] Mark Gross; Bernd Siebert Logarithmic Gromov–Witten invariants, J. Am. Math. Soc., Volume 26 (2013) no. 2, pp. 451-510 | DOI | Zbl

[31] Robin Hartshorne Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1983 | Zbl | MR

[32] Ilia Itenberg; Ludmil Katzarkov; Grigory Mikhalkin; Ilia Zharkov Tropical homology, Math. Ann., Volume 374 (2019) no. 1-2, pp. 963-1006 | DOI | Zbl | MR

[33] Ilia Itenberg; Grigory Mikhalkin On Block–Göttsche multiplicities for planar tropical curves, Int. Math. Res. Not., Volume 2013 (2013) no. 23, pp. 5289-5320 | DOI | Zbl | MR

[34] Bumsig Kim; Hyenho Lho; Helge Ruddat The degeneration formula for stable log maps, Manuscr. Math., Volume 170 (2023) no. 1-2, pp. 63-107 | DOI | Zbl | MR

[35] Martijn Kool; Vivek Shende; Richard P. Thomas A short proof of the Göttsche conjecture, Geom. Topol., Volume 15 (2011) no. 1, pp. 397-406 | DOI | Zbl | MR

[36] Jun Li A degeneration formula of GW-invariants, J. Differ. Geom., Volume 60 (2002) no. 2, pp. 199-293 | DOI | Zbl | MR

[37] Travis Mandel; Helge Ruddat Descendant log Gromov–Witten invariants for toric varieties and tropical curves, Trans. Am. Math. Soc., Volume 373 (2020) no. 2, pp. 1109-1152 | DOI | Zbl | MR

[38] Grigory Mikhalkin Enumerative tropical algebraic geometry in ℝ 2 , J. Am. Math. Soc., Volume 18 (2005) no. 2, pp. 313-377 | DOI | Zbl | MR

[39] Grigory Mikhalkin Quantum indices and refined enumeration of real plane curves, Acta Math., Volume 219 (2017) no. 1, pp. 135-180 | DOI | Zbl | MR

[40] Takeo Nishinou Realization of tropical curves in abelian surfaces (2020) | arXiv | Zbl

[41] Takeo Nishinou; Bernd Siebert Toric degenerations of toric varieties and tropical curves, Duke Math. J., Volume 135 (2006) no. 1, pp. 1-51 | DOI | Zbl | MR

[42] Eugenii Shustin A tropical approach to enumerative geometry, St. Petersbg. Math. J., Volume 17 (2006) no. 2, pp. 343-375 | DOI | Zbl

[43] Eugenii Shustin A tropical calculation of the Welschinger invariants of real toric del Pezzo surfaces, J. Algebr. Geom., Volume 15 (2006) no. 2, pp. 285-322 | DOI | Zbl | MR

[44] Eugenii Shustin On refined count of rational tropical curves (2018) | arXiv | Zbl

[45] Bernd Sturmfels On vector partition functions, J. Comb. Theory, Ser. A, Volume 72 (1995) no. 2, pp. 302-309 | DOI | Zbl | MR

[46] Ilya Tyomkin Enumeration of rational curves with cross-ratio constraints, Adv. Math., Volume 305 (2017), pp. 1356-1383 | DOI | Zbl | MR

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