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First steps in tropical intersection theory. (English) Zbl 1193.14074
The authors present the tropical counterpart of basic notions and constructions of intersection theory (cycles, Cartier divisors, intersection product, rational equivalence). This is a detailed and transparent elaboration (with some applications) of outlines given by G. Mikhalkin [Proceedings of ICM, Madrid, Spain, 2006. Volume II: Invited lectures. Zürich: EMS, 827–852 (2006; Zbl 1103.14034)], A. Gathmann, M. Kerber and H. Markwig [Compos. Math. 145, No. 1, 173–195 (2009; Zbl 1124.05049)], etc., for the intersection theory on tropical varieties in \(\mathbb{R}^n\) and on abstract tropical varieties. This context is more complicated than the tropical intersection theory in \(\mathbb{R}^n\) (also known as Minkowski weights), because the displacement rule does not help anymore in general: the simplest example is an attempt to count the self-intersection number of a classical line in a tropical plane.

MSC:
14T05 Tropical geometry (MSC2010)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
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References:
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