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Enumerative tropical algebraic geometry in \(\mathbb{R}^2\). (English) Zbl 1092.14068
Tropical geometry is a relatively new mathematical domain which has multiple and deep connections with various branches of mathematics. It can be seen as an algebraic geometry over the \((\max,+)\)-semifield \(\mathbb{R}_{\text{trop}}\), the semifield of real numbers equipped with maximum for addition and addition for multiplication. The main result of the paper establishes an important correspondence between the complex algebraic world and the tropical one. This correspondence gives a possibility to enumerate curves of a given genus which pass through given generic points on a toric surface.
The paper contains an introduction to tropical geometry (this introduction is mainly oriented towards enumerative questions in tropical geometry), the statement and the proof of the correspondence theorem, as well as a combinatorial algorithm which is designed for the tropical curve count and gives rise to important applications of the correspondence theorem in enumerative geometry (in particular, it produces a new formula for enumeration of algebraic curves of arbitrary genus on toric surfaces; this formula expressing the number of curves in terms of certain lattice paths in lattice polygons was announced in [G. Mikhalkin, C. R., Math., Acad. Sci. Paris 336, No. 8, 629–634 (2003; Zbl 1027.14026)]).

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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