Duke Mathematical Journal

Equations of tropical varieties

Jeffrey Giansiracusa and Noah Giansiracusa

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We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as T=(R{},max,+) by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-Archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of T-points this reduces to Kajiwara–Payne’s extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of T-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.

Article information

Duke Math. J., Volume 165, Number 18 (2016), 3379-3433.

Received: 13 June 2014
Revised: 7 January 2016
First available in Project Euclid: 22 August 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14T05: Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]
Secondary: 14A20: Generalizations (algebraic spaces, stacks)

tropical geometry tropical scheme Hilbert polynomial tropicalization max-plus algebra bend relations


Giansiracusa, Jeffrey; Giansiracusa, Noah. Equations of tropical varieties. Duke Math. J. 165 (2016), no. 18, 3379--3433. doi:10.1215/00127094-3645544. https://projecteuclid.org/euclid.dmj/1471873079

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  • [1] D. Abramovich, L. Caporaso, and S. Payne, The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), 765–809.
  • [2] D. Abramovich, Q. Chen, S. Marcus, and J. Wise, Boundedness of the space of stable logarithmic maps, to appear in J. Eur. Math. Soc. (JEMS), preprint, arXiv:1408.0869v2 [math.AG].
  • [3] D. Abramovich and J. Wise, Invariance in logarithmic Gromov-Witten theory, preprint, arXiv:1306.1222v2 [math.AG].
  • [4] D. Alessandrini and M. Nesci, On the tropicalization of the Hilbert scheme, Collect. Math. 64 (2013), 39–59.
  • [5] F. Ardila and F. Block, Universal polynomials for Severi degrees of toric surfaces, Adv. Math. 237 (2013), 165–193.
  • [6] M. Baker, Specialization of linear systems from curves to graphs, with an appendix by B. Conrad, Algebra Number Theory 2 (2008), 613–653.
  • [7] S. D. Banerjee, Tropical geometry over higher dimensional local fields, J. Reine Angew. Math. 698 (2015), 71–87.
  • [8] V. G. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Math. Surveys Monogr. 33, Amer. Math. Soc., Providence, 1990.
  • [9] L. Caporaso and F. Viviani, Torelli theorem for graphs and tropical curves, Duke Math. J. 153 (2010), 129–171.
  • [10] A. Connes and C. Consani, Schemes over $\mathbb{F}_{1}$ and zeta functions, Compos. Math. 146 (2010), 1383–1415.
  • [11] F. Cools, J. Draisma, S. Payne, and E. Robeva, A tropical proof of the Brill-Noether theorem, Adv. Math. 230 (2012), 759–776.
  • [12] D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17–50.
  • [13] A. Deitmar, $\mathbb{F}_{1}$-schemes and toric varieties, Beitr. Algebra Geom. 49 (2008), 517–525.
  • [14] A. Dickenstein, E. M. Feichtner, and B. Sturmfels, Tropical discriminants, J. Amer. Math. Soc. 20 (2007), 1111–1133.
  • [15] N. Durov, New approach to Arakelov geometry, preprint, arXiv:0704.2030v1 [math.AG].
  • [16] A. Fink, J. Giansiracusa, and N. Giansiracusa, Tropical hypersurfaces and valuated matroids, in preparation.
  • [17] J. Flores and C. Weibel, Picard groups and class groups of monoid schemes, J. Algebra 415 (2014), 247–263.
  • [18] S. Fomin and G. Mikhalkin, Labeled floor diagrams for plane curves, J. Eur. Math. Soc. (JEMS) 12 (2010), 1453–1496.
  • [19] S. Fomin and A. Zelevinsky, Cluster algebras, I: Foundations, J. Amer. Math. Soc. 15 (2002), 497–529.
  • [20] B. Frenk, Tropical varieties, maps and gossip, Ph.D. dissertation, Eindhoven University of Technology, Eindhoven, Netherlands, 2013.
  • [21] A. Gathmann and H. Markwig, Kontsevich’s formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008), 537–560.
  • [22] A. A. Gerasimov and D. R. Lebedev, Representation theory over tropical semifield and Langlands duality, Comm. Math. Phys. 320 (2013), 301–346.
  • [23] J. Giansiracusa and N. Giansiracusa, The universal tropicalization and the Berkovich analytification, preprint, arXiv:1410.4348v2 [math.AG].
  • [24] A. Gibney and D. Maclagan, Lower and upper bounds for nef cones, Int. Math. Res. Not. IMRN 2012, no. 14, 3224–3255.
  • [25] J. S. Golan, Semirings and Their Applications, Kluwer, Dordrecht, 1999.
  • [26] M. Gross, Mirror symmetry for $\mathbb{P}^{2}$ and tropical geometry, Adv. Math. 224 (2010), 169–245.
  • [27] M. Gross, Tropical Geometry and Mirror Symmetry, CBMS Reg. Conf. Ser. Math. 114, Amer. Math. Soc., Providence, 2011.
  • [28] M. Gross, R. Pandharipande, and B. Siebert, The tropical vertex, Duke Math. J. 153 (2010), 297–362.
  • [29] W. Gubler, The Bogomolov conjecture for totally degenerate abelian varieties, Invent. Math. 169 (2007), 377–400.
  • [30] P. Hacking, S. Keel, and J. Tevelev, Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces, Invent. Math. 178 (2009), 173–227.
  • [31] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • [32] U. Hebisch and H. J. Weinert, Semirings: Algebraic Theory and Applications in Computer Science, Ser. Algebra 5, World Scientific, River Edge, N.J., 1998.
  • [33] R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects Math. E30, Vieweg, Braunschweig, 1996.
  • [34] T. Kajiwara, “Tropical toric geometry” in Toric Topology, Contemp. Math. 460, Amer. Math. Soc., Providence, 2008, 197–207.
  • [35] K. Kato, “Logarithmic structures of Fontaine-Illusie” in Algebraic Analysis, Geometry, and Number Theory (Baltimore, 1988), Johns Hopkins Univ. Press, Baltimore, 1989, 191–224.
  • [36] E. Katz, A tropical toolkit, Expo. Math. 27 (2009), 1–36.
  • [37] J. López Peña and O. Lorscheid, “Mapping $\mathbb{F}_{1}$-land: An overview of geometries over the field with one element” in Noncommutative Geometry, Arithmetic, and Related Topics, Johns Hopkins Univ. Press, Baltimore, 2011, 241–265.
  • [38] O. Lorscheid, The geometry of blueprints, I: Algebraic background and scheme theory, Adv. Math. 229 (2012), 1804–1846.
  • [39] D. Maclagan and F. Rincón, Tropical schemes, tropical cycles, and valuated matroids, preprint, arXiv:1401.4654v1 [math.AG].
  • [40] D. Maclagan and F. Rincón, Tropical ideals, in preparation.
  • [41] D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Grad. Stud. Math. 161, Amer. Math. Soc., Providence, 2015.
  • [42] A. Macpherson, Skeleta in non-Archimedean and tropical geometry, preprint, arXiv:1311.0502v2 [math.AG].
  • [43] C. Manon, Dissimilarity maps on trees and the representation theory of ${\mathrm{SL}}_{m}(\mathbb{C})$, J. Algebraic Combin. 33 (2011), 199–213.
  • [44] G. Mikhalkin, Enumerative tropical algebraic geometry in $\mathbb{R}^{2}$, J. Amer. Math. Soc. 18 (2005), 313–377.
  • [45] G. Mikhalkin, “Tropical geometry and its applications” in International Congress of Mathematicians, II, Eur. Math. Soc., Zürich, 2006, 827–852.
  • [46] G. Mikhalkin and I. Zharkov, “Tropical curves, their Jacobians and theta functions” in Curves and Abelian Varieties, Contemp. Math. 465, Amer. Math. Soc., Providence, 2008, 203–230.
  • [47] M. C. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), 747–791.
  • [48] B. Osserman and S. Payne, Lifting tropical intersections, Doc. Math. 18 (2013), 121–175.
  • [49] L. Pachter and B. Sturmfels, Tropical geometry of statistical models, Proc. Natl. Acad. Sci. USA 101 (2004), 16132–16137.
  • [50] S. Payne, Analytification is the limit of all tropicalizations, Math. Res. Lett. 16 (2009), 543–556.
  • [51] Q. Ren, S. V. Sam, and B. Sturmfels, Tropicalization of classical moduli spaces, Math. Comput. Sci. 8 (2014), 119–145.
  • [52] J. Richter-Gebert, B. Sturmfels, and T. Theobald, “First steps in tropical geometry” in Idempotent Mathematics and Mathematical Physics, Contemp. Math. 377, Amer. Math. Soc., Providence, 2005, 289–317.
  • [53] D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (2008), 1527–1558.
  • [54] D. Speyer and B. Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004), 389–411.
  • [55] J. Tevelev, Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007), 1087–1104.
  • [56] B. Toën and M. Vaquié, Au-dessous de ${\mathrm{Spec}}\mathbb{Z}$, J. K-Theory 3 (2009), 437–500.
  • [57] M. Ulirsch, Functorial tropicalization of logarithmic schemes: The case of constant coefficients, preprint, arXiv:1310.6269v2 [math.AG].