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Tropical images of intersection points. (English) Zbl 1331.14059
Let $$K$$ be an algebraically closed non-Archimedean field with a nontrivial valuation $$K^*\to\mathbb R$$. The main example is the field of Puiseux series with complex coefficients. Consider two algebraic curves $$X$$ and $$Y$$ in the algebraic torus $$(K^*)^2$$. Assume that $$X$$ and $$Y$$ intersect in only finitely many points. The coordinatewise valuation map applied to the intersection points (with multiplicity) yields a divisor $$D$$ inside the intersection $$\mathrm{Trop}(X)\cap \mathrm{Trop}(Y)$$ of the tropicalizations of the two curves.
The main result of the paper under review is that $$D$$ is linearly equivalent on $$\mathrm{Trop}(X)$$ to the stable intersection divisor $$E$$ of $$\mathrm{Trop}(X)$$ and $$\mathrm{Trop}(Y)$$. I.e., $$D-E$$ is the divisor associated to a suitable tropical rational function on $$\mathrm{Trop}(X)$$. Two proofs are given, one based on tropical modifications and one on Berkovich analytic spaces, although the latter proof requires an additional assumption on $$X$$.
The author conjectures that his result is the only restriction on $$D$$ (in a suitable sense). Detailed examples are given as evidence supporting this conjecture.

##### MSC:
 14T05 Tropical geometry (MSC2010) 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14H50 Plane and space curves
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##### References:
 [1] Baker, M; Norine, S, Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math., 215, 766-788, (2007) · Zbl 1124.05049 [2] Baker, M., Payne, S., Rabinoff, J.: Nonarchimedean geometry, tropicalization, and metrics on curves (2012). Preprint: http://arxiv.org/abs/1104.0320 · Zbl 06609386 [3] Brugallé, EA; López de Medrano, LM, Inflection points of real and tropical plane curves, J. Singul., 4, 74-103, (2012) · Zbl 1292.14042 [4] Gathmann, A; Kerber, M, A Riemann-Roch theorem in tropical geometry, Mathematische Zeitschrift, 259, 217-230, (2008) · Zbl 1187.14066 [5] Gubler, W.: A guide to tropicalizations. In: Algebraic and combinatorial aspects of tropical geometry. Contemp. Math., , vol. 589, pp. 125-189. Amer. Math. Soc., Providence (2013) · Zbl 1318.14061 [6] Hasse, C; Musiker, G; Yu, J, Linear systems on tropical curves, Mathematische Zeitschrift, 270, 1111-1140, (2012) · Zbl 1408.14201 [7] Luo, Y.: Tropical Convexity and Canonical Projections (2013). Preprint: http://arxiv.org/abs/1304.7963 · Zbl 1124.05049 [8] Maclagan, D., Sturmfels, B.: Introduction to tropical geometry (2014). Preprint: http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook.html · Zbl 1321.14048 [9] Mikhalkin, G.: Tropical Geometry and its applications. In: International Congress of Mathematicians, vol. II, pp. 827-852. Eur. Math. Soc., Zürich (2006) · Zbl 1103.14034 [10] Osserman, B; Payne, S, Lifting tropical intersections, Doc. Math., 18, 121-175, (2013) · Zbl 1308.14069 [11] Osserman, B.; Rabinoff J.: Lifting non-proper tropical intersections. Contemp. Math. (2014, to appear) · Zbl 1320.14078
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