Deformation of tropical Hirzebruch surfaces and enumerative geometry
Authors:
Erwan Brugallé and Hannah Markwig
Journal:
J. Algebraic Geom. 25 (2016), 633-702
DOI:
https://doi.org/10.1090/jag/671
Published electronically:
June 2, 2016
MathSciNet review:
3533183
Full-text PDF
Abstract |
References |
Additional Information
Abstract:
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
- 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, DOI https://doi.org/10.1090/conm/276/04512
- 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, DOI https://doi.org/10.1186/s40687-014-0019-0
- 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, DOI https://doi.org/10.2140/ant.2015.9.267
- D. Abramovich and C. Chen, Logarithmic stable maps to Deligne-Faltings pairs II. arXiv:1102.4531v2
- Lars Allermann and Johannes Rau, First steps in tropical intersection theory, Math. Z. 264 (2010), no. 3, 633–670. MR 2591823, DOI https://doi.org/10.1007/s00209-009-0483-1
- 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, DOI https://doi.org/10.1112/S0010437X13007409
- Benoît Bertrand, Erwan Brugallé, and Grigory Mikhalkin, Tropical open Hurwitz numbers, Rend. Semin. Mat. Univ. Padova 125 (2011), 157–171. MR 2866125, DOI https://doi.org/10.4171/RSMUP/125-10
- 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
- Pascale Harinck, Alain Plagne, and Claude Sabbah (eds.), 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. MR 1500296
- E. Brugallé and G. Mikhalkin, Floor decompositions of tropical curves in any dimension. In preparation, preliminary version available at the homepage http://www.math.jussieu.fr/$\sim$brugalle/articles/FDn/FDGeneral.pdf.
- E. Brugallé and G. Mikhalkin, Realizability of superabundant curves. In preparation.
- 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
- 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, DOI https://doi.org/10.4153/CJM-2014-014-4
- Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume (2000), 560–673. GAFA 2000 (Tel Aviv, 1999). MR 1826267, DOI https://doi.org/10.1007/978-3-0346-0425-3_4
- 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
- Andreas Gathmann, Tropical algebraic geometry, Jahresber. Deutsch. Math.-Verein. 108 (2006), no. 1, 3–32. MR 2219706
- 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, DOI https://doi.org/10.1007/s00208-007-0092-4
- 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, DOI https://doi.org/10.1515/CRELLE.2007.006
- Mark Gross and Bernd Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510. MR 3011419, DOI https://doi.org/10.1090/S0894-0347-2012-00757-7
- A. Gathmann, K. Schmitz, and A. Winstel, The realizability of curves in a tropical plane. arXiv:1307.5686
- Eleny-Nicoleta Ionel, GW invariants relative to normal crossing divisors, Adv. Math. 281 (2015), 40–141. MR 3366837, DOI https://doi.org/10.1016/j.aim.2015.04.027
- 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, DOI https://doi.org/10.4007/annals.2004.159.935
- Kunihiko Kodaira, Complex manifolds and deformation of complex structures, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Springer-Verlag, New York, 1986. Translated from the Japanese by Kazuo Akao; With an appendix by Daisuke Fujiwara. MR 815922
- Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247–258. MR 1338784, DOI https://doi.org/10.4310/MRL.1995.v2.n3.a2
- Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113
- 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, DOI https://doi.org/10.1007/s002220100146
- Hannah Markwig, Three tropical enumerative problems, Trends in mathematics, Universitätsdrucke Göttingen, Göttingen, 2008, pp. 69–96. MR 2906041
- G. Mikhalkin, Phase-tropical curves I. Realizability and enumeration. In preparation.
- 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, DOI https://doi.org/10.1007/0-306-48658-X_6
- Grigory Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology 43 (2004), no. 5, 1035–1065. MR 2079993, DOI https://doi.org/10.1016/j.top.2003.11.006
- Grigory Mikhalkin, Enumerative tropical algebraic geometry in $\Bbb R^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377. MR 2137980, DOI https://doi.org/10.1090/S0894-0347-05-00477-7
- Grigory Mikhalkin, Tropical geometry and its applications, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 827–852. MR 2275625
- 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, DOI https://doi.org/10.17323/1609-4514-2007-7-3-507-531
- B. Parker, Gromov-Witten invariants of exploded manifolds. arXiv:1102.0158
- 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, DOI https://doi.org/10.1090/conm/377/06998
- Kristin M. Shaw, A tropical intersection product in matroidal fans, SIAM J. Discrete Math. 27 (2013), no. 1, 459–491. MR 3032930, DOI https://doi.org/10.1137/110850141
- 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, DOI https://doi.org/10.1007/978-0-8176-8277-4_18
- Bernd Sturmfels and Jenia Tevelev, Elimination theory for tropical varieties, Math. Res. Lett. 15 (2008), no. 3, 543–562. MR 2407231, DOI https://doi.org/10.4310/MRL.2008.v15.n3.a14
- Ravi Vakil, Counting curves on rational surfaces, Manuscripta Math. 102 (2000), no. 1, 53–84. MR 1771228, DOI https://doi.org/10.1007/s002291020053
- Magnus Dehli Vigeland, Smooth tropical surfaces with infinitely many tropical lines, Ark. Mat. 48 (2010), no. 1, 177–206. MR 2594592, DOI https://doi.org/10.1007/s11512-009-0116-2
References
- 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), DOI https://doi.org/10.1090/conm/276/04512
- 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, DOI https://doi.org/10.1186/s40687-014-0019-0
- 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, DOI https://doi.org/10.2140/ant.2015.9.267
- D. Abramovich and C. Chen, Logarithmic stable maps to Deligne-Faltings pairs II. arXiv:1102.4531v2
- Lars Allermann and Johannes Rau, First steps in tropical intersection theory, Math. Z. 264 (2010), no. 3, 633–670. MR 2591823 (2011e:14110), DOI https://doi.org/10.1007/s00209-009-0483-1
- 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, DOI https://doi.org/10.1112/S0010437X13007409
- Benoît Bertrand, Erwan Brugallé, and Grigory Mikhalkin, Tropical open Hurwitz numbers, Rend. Semin. Mat. Univ. Padova 125 (2011), 157–171. MR 2866125, DOI https://doi.org/10.4171/RSMUP/125-10
- 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)
- 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)
- E. Brugallé and G. Mikhalkin, Floor decompositions of tropical curves in any dimension. In preparation, preliminary version available at the homepage http://www.math.jussieu.fr/$\sim$brugalle/articles/FDn/FDGeneral.pdf.
- E. Brugallé and G. Mikhalkin, Realizability of superabundant curves. In preparation.
- 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)
- 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, DOI https://doi.org/10.4153/CJM-2014-014-4
- 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), DOI https://doi.org/10.1007/978-3-0346-0425-3_4
- 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)
- Andreas Gathmann, Tropical algebraic geometry, Jahresber. Deutsch. Math.-Verein. 108 (2006), no. 1, 3–32. MR 2219706 (2007e:14088)
- 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), DOI https://doi.org/10.1007/s00208-007-0092-4
- 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), DOI https://doi.org/10.1515/CRELLE.2007.006
- Mark Gross and Bernd Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510. MR 3011419, DOI https://doi.org/10.1090/S0894-0347-2012-00757-7
- A. Gathmann, K. Schmitz, and A. Winstel, The realizability of curves in a tropical plane. arXiv:1307.5686
- Eleny-Nicoleta Ionel, GW invariants relative to normal crossing divisors, Adv. Math. 281 (2015), 40–141. MR 3366837, DOI https://doi.org/10.1016/j.aim.2015.04.027
- 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), DOI https://doi.org/10.4007/annals.2004.159.935
- 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)
- Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247–258. MR 1338784 (96f:58062), DOI https://doi.org/10.4310/MRL.1995.v2.n3.a2
- Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113 (2004k:14096)
- 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), DOI https://doi.org/10.1007/s002220100146
- Hannah Markwig, Three tropical enumerative problems, Trends in mathematics, Universitätsdrucke Göttingen, Göttingen, 2008, pp. 69–96. MR 2906041
- G. Mikhalkin, Phase-tropical curves I. Realizability and enumeration. In preparation.
- 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), DOI https://doi.org/10.1007/0-306-48658-X_6
- Grigory Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersurfaces, Topology 43 (2004), no. 5, 1035–1065. MR 2079993 (2005i:14055), DOI https://doi.org/10.1016/j.top.2003.11.006
- Grigory Mikhalkin, Enumerative tropical algebraic geometry in $\mathbb {R}^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377. MR 2137980 (2006b:14097), DOI https://doi.org/10.1090/S0894-0347-05-00477-7
- 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)
- 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)
- B. Parker, Gromov-Witten invariants of exploded manifolds. arXiv:1102.0158
- 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), DOI https://doi.org/10.1090/conm/377/06998
- Kristin M. Shaw, A tropical intersection product in matroidal fans, SIAM J. Discrete Math. 27 (2013), no. 1, 459–491. MR 3032930, DOI https://doi.org/10.1137/110850141
- 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, DOI https://doi.org/10.1007/978-0-8176-8277-4_18
- Bernd Sturmfels and Jenia Tevelev, Elimination theory for tropical varieties, Math. Res. Lett. 15 (2008), no. 3, 543–562. MR 2407231 (2009f:14124), DOI https://doi.org/10.4310/MRL.2008.v15.n3.a14
- Ravi Vakil, Counting curves on rational surfaces, Manuscripta Math. 102 (2000), no. 1, 53–84. MR 1771228 (2001h:14069), DOI https://doi.org/10.1007/s002291020053
- Magnus Dehli Vigeland, Smooth tropical surfaces with infinitely many tropical lines, Ark. Mat. 48 (2010), no. 1, 177–206. MR 2594592 (2011e:14112), DOI https://doi.org/10.1007/s11512-009-0116-2
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
Email:
erwan.brugalle@math.cnrs.fr
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
MR Author ID:
741165
Email:
hannah@math.uni-tuebingen.de
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.