Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Deformation of tropical Hirzebruch surfaces and enumerative geometry

Authors: Erwan Brugallé and Hannah Markwig
Journal: J. Algebraic Geom. 25 (2016), 633-702
Published electronically: June 2, 2016
MathSciNet review: 3533183
Full-text PDF

Abstract | References | Additional Information


We illustrate the use of tropical methods by generalizing a formula due to Abramovich and Bertram, extended later by Vakil. Namely, we exhibit relations between enumerative invariants of the Hirzebruch surfaces $ \Sigma _n$ and $ \Sigma _{n+2}$, obtained by deforming the first surface to the latter.

Our strategy involves a tropical counterpart of deformations of Hirzebruch surfaces and tropical enumerative geometry on a tropical surface in three-space.

References [Enhancements On Off] (What's this?)

  • [AB01] Dan Abramovich and Aaron Bertram, The formula $ 12=10+2\times 1$ and its generalizations: counting rational curves on $ \mathbf {F}_2$, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), Contemp. Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001, pp. 83-88. MR 1837110 (2002f:14071),
  • [ABBRa] Omid Amini, Matthew Baker, Erwan Brugallé, and Joseph Rabinoff, Lifting harmonic morphisms I: Metrized complexes and Berkovich skeleta, Res. Math. Sci. 2 (2015), Art. 7, 67. MR 3375652,
  • [ABBRb] Omid Amini, Matthew Baker, Erwan Brugallé, and Joseph Rabinoff, Lifting harmonic morphisms II: Tropical curves and metrized complexes, Algebra Number Theory 9 (2015), no. 2, 267-315. MR 3320845,
  • [AC] D. Abramovich and C. Chen,
    Logarithmic stable maps to Deligne-Faltings pairs II.
  • [AR10] Lars Allermann and Johannes Rau, First steps in tropical intersection theory, Math. Z. 264 (2010), no. 3, 633-670. MR 2591823 (2011e:14110),
  • [BBM] Benoît Bertrand, Erwan Brugallé, and Grigory Mikhalkin, Genus 0 characteristic numbers of the tropical projective plane, Compos. Math. 150 (2014), no. 1, 46-104. MR 3164359,
  • [BBM11] Benoît Bertrand, Erwan Brugallé, and Grigory Mikhalkin, Tropical open Hurwitz numbers, Rend. Semin. Mat. Univ. Padova 125 (2011), 157-171. MR 2866125,
  • [Bea83] Arnaud Beauville, Complex algebraic surfaces, London Mathematical Society Lecture Note Series, vol. 68, Cambridge University Press, Cambridge, 1983. Translated from the French by R. Barlow, N. I. Shepherd-Barron and M. Reid. MR 732439 (85a:14024)
  • [BIT08] Géométrie tropicale, Éditions de l'École Polytechnique, Palaiseau, 2008 (French). Papers from the Mathematical Days X-UPS held at the École Polytechnique, Palaiseau, May 14-15, 2008; Edited by Pascale Harinck, Alain Plagne and Claude Sabbah. MR 1500296 (2010a:14003)
  • [BMa] E. Brugallé and G. Mikhalkin,
    Floor decompositions of tropical curves in any dimension.
    In preparation, preliminary version available at the homepage$ \sim $brugalle/articles/FDn/FDGeneral.pdf.
  • [BMb] E. Brugallé and G. Mikhalkin,
    Realizability of superabundant curves.
    In preparation.
  • [BM08] Erwan Brugallé and Grigory Mikhalkin, Floor decompositions of tropical curves: the planar case, Proceedings of Gökova Geometry-Topology Conference 2008, Gökova Geometry/Topology Conference (GGT), Gökova, 2009, pp. 64-90. MR 2500574 (2011e:14111)
  • [BS] Erwan Brugallé and Kristin Shaw, Obstructions to approximating tropical curves in surfaces via intersection theory, Canad. J. Math. 67 (2015), no. 3, 527-572. MR 3339531,
  • [EGH00] Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. Special Volume (2000), 560-673. MR 1826267 (2002e:53136),
  • [Ful84] William Fulton, Introduction to intersection theory in algebraic geometry, CBMS Regional Conference Series in Mathematics, vol. 54, published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1984. MR 735435 (85j:14008)
  • [Gat06] Andreas Gathmann, Tropical algebraic geometry, Jahresber. Deutsch. Math.-Verein. 108 (2006), no. 1, 3-32. MR 2219706 (2007e:14088)
  • [GM07a] Andreas Gathmann and Hannah Markwig, The Caporaso-Harris formula and plane relative Gromov-Witten invariants in tropical geometry, Math. Ann. 338 (2007), no. 4, 845-868. MR 2317753 (2008e:14075),
  • [GM07b] Andreas Gathmann and Hannah Markwig, The numbers of tropical plane curves through points in general position, J. Reine Angew. Math. 602 (2007), 155-177. MR 2300455 (2008a:14073),
  • [GS13] Mark Gross and Bernd Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451-510. MR 3011419,
  • [GSW] A. Gathmann, K. Schmitz, and A. Winstel,
    The realizability of curves in a tropical plane.
  • [Ion] Eleny-Nicoleta Ionel, GW invariants relative to normal crossing divisors, Adv. Math. 281 (2015), 40-141. MR 3366837,
  • [IP04] Eleny-Nicoleta Ionel and Thomas H. Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2) 159 (2004), no. 3, 935-1025. MR 2113018 (2006b:53110),
  • [Kod86] Kunihiko Kodaira, Complex manifolds and deformation of complex structures, with an appendix by Daisuke Fujiwara, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Springer-Verlag, New York, 1986. Translated from the Japanese by Kazuo Akao. MR 815922 (87d:32040)
  • [Ler95] Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247-258. MR 1338784 (96f:58062),
  • [Li02] Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199-293. MR 1938113 (2004k:14096)
  • [LR01] An-Min Li and Yongbin Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151-218. MR 1839289 (2002g:53158),
  • [Mar08] Hannah Markwig, Three tropical enumerative problems, Trends in mathematics, Universitätsdrucke Göttingen, Göttingen, 2008, pp. 69-96. MR 2906041
  • [Mik] G. Mikhalkin,
    Phase-tropical curves I. Realizability and enumeration.
    In preparation.
  • [Mik04a] Grigory Mikhalkin, Amoebas of algebraic varieties and tropical geometry, Different faces of geometry, Int. Math. Ser. (N. Y.), vol. 3, Kluwer/Plenum, New York, 2004, pp. 257-300. MR 2102998 (2005m:14110),
  • [Mik04b] Grigory Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology 43 (2004), no. 5, 1035-1065. MR 2079993 (2005i:14055),
  • [Mik05] Grigory Mikhalkin, Enumerative tropical algebraic geometry in $ \mathbb{R}^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313-377. MR 2137980 (2006b:14097),
  • [Mik06] Grigory Mikhalkin, Tropical geometry and its applications, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 827-852. MR 2275625 (2008c:14077)
  • [MO07] Grigory Mikhalkin and Andrei Okounkov, Geometry of planar log-fronts, Mosc. Math. J. 7 (2007), no. 3, 507-531, 575 (English, with English and Russian summaries). MR 2343146 (2008g:14110)
  • [Par] B. Parker,
    Gromov-Witten invariants of exploded manifolds.
  • [RGST05] Jürgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald, First steps in tropical geometry, Idempotent mathematics and mathematical physics, Contemp. Math., vol. 377, Amer. Math. Soc., Providence, RI, 2005, pp. 289-317. MR 2149011 (2006d:14073),
  • [Sha13] Kristin M. Shaw, A tropical intersection product in matroidal fans, SIAM J. Discrete Math. 27 (2013), no. 1, 459-491. MR 3032930,
  • [Shu12] Eugenii Shustin, Tropical and algebraic curves with multiple points, Perspectives in analysis, geometry, and topology, Progr. Math., vol. 296, Birkhäuser/Springer, New York, 2012, pp. 431-464. MR 2884046,
  • [ST08] Bernd Sturmfels and Jenia Tevelev, Elimination theory for tropical varieties, Math. Res. Lett. 15 (2008), no. 3, 543-562. MR 2407231 (2009f:14124),
  • [Vak00] Ravi Vakil, Counting curves on rational surfaces, Manuscripta Math. 102 (2000), no. 1, 53-84. MR 1771228 (2001h:14069),
  • [Vig09] Magnus Dehli Vigeland, Smooth tropical surfaces with infinitely many tropical lines, Ark. Mat. 48 (2010), no. 1, 177-206. MR 2594592 (2011e:14112),

Additional Information

Erwan Brugallé
Affiliation: Université Pierre et Marie Curie, Paris 6, 4 place Jussieu, 75 005 Paris, France – and – CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France

Hannah Markwig
Affiliation: Universität des Saarlandes, Fachrichtung Mathematik, Postfach 151150, 66041 Saarbrücken, Germany
Address at time of publication: Eberhard Karls Universität Tübingen, Arbeitsbereich Geometrie, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Received by editor(s): July 11, 2013
Received by editor(s) in revised form: May 28, 2014
Published electronically: June 2, 2016
Article copyright: © Copyright 2016 University Press, Inc.

American Mathematical Society