A guide to tropical modifications

This paper surveys tropical modifications, which have already become folklore in tropical geometry. Tropical modifications are used in tropical intersection theory and in a study of singularities. They admit interpretations in various contexts such as hyperbolic geometry, Berkovich spaces, and non-standard analysis. Our main goal is to mention different points of view, to give references, and to demonstrate the abilities of tropical modifications. We assume that the reader had already met "tropical modifications" somewhere and wants to understand them better. There are novelties here: a new obstruction (Theorem 2.29) for realizability of non-transversal intersections is found and a tropical version of Weil's reciprocity law (Theorem 2.10) is proven via tropical Menelaus Theorem. A generalization of tropical momentum is given in Section 2.6

Tropical fans and the moduli spaces of tropical curves

We give a rigorous definition of tropical fans (the "local building blocks for tropical varieties") and their morphisms. For such a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point - a statement that can be viewed as the beginning of a tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some R^r) together with the evaluation and forgetful morphisms. Using our results this gives new, easy, and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any R^r) through given points are independent of the points.Comment: minor corrections to match published versio