Applications of tropical geometry to enumerative geometry

Abstract

The main purpose of this chapter is to present several applications of tropical geometry in enumerative geometry. The idea to use tropical curves in enumerative questions, and in particular in classical questions of enumeration of algebraic curves (satisfying some constraints) in algebraic varieties was suggested by M. Kontsevich. This idea was realized by G. Mikhalkin [38, 40] who established an appropriate correspondence theorem between the complex algebraic world and the tropical one. This correspondence allows one to calculate Gromov-Witten type invariants of toric surfaces, namely, to enumerate certain nodal complex curves of a given genus which pass through given points in a general position in a toric surface. Roughly speaking, Mikhalkin’s theorem affirms that the number of complex curves in question is equal to the number of their tropical analogs passing through given points in a general position in ℝ² and counted with multiplicities. In addition, [38] suggests a combinatorial algorithm for an enumeration of the required tropical curves. An extension of Mikhalkin’s correspondence theorem to the case of rational curves in toric varieties was proposed by T. Nishinou and B. Siebert in [45].