1.3. PARAMETERIZED TROPICAL CURVES 13 1.3. Parameterized tropical curves We shall now use the discussion of the balancing condition in §1.1 to define tropical curves in a more abstract setting. In theory, similar definitions could be given for tropical varieties of higher dimension, but we will not do so here. Let Γ be a connected graph with no bivalent vertices. Such a graph can be viewed in two different ways. First, it can be viewed as a purely combinatorial object, i.e., a set Γ [0] of vertices and a set Γ [1] of edges consisting of unordered pairs of elements of Γ [0] , indicating the endpoints of an edge. We can also view Γ as the topological realization of the graph, i.e., a topological space which is the union of line segments corresponding to the edges. We shall confuse these two viewpoints at will, hopefully without any confusion. Let Γ∞ [0] be the set of univalent vertices of Γ, and write Γ = Γ \ Γ∞. [0] Let Γ[0], Γ[1] denote the set of vertices and edges of Γ. Here we are thinking of Γ and Γ as topological spaces, so Γ now has some non-compact edges. Let Γ∞ [1] be the set of non-compact edges of Γ. A flag of Γ is a pair (V, E) with V Γ[0] and E Γ[1] with V E. In addition, all graphs will be weighted graphs, i.e., Γ comes along with a weight function w : Γ [1] N = {0, 1, 2,...}. We will often consider marked graphs, (Γ,x1,...,xk), where Γ is as above and x1,...,xk are labels assigned to non-compact edges of weight 0, i.e., we are given an inclusion {x1,...,xk} Γ∞[1] xi Ex i with w(Ex i ) = 0. We will use the convention in this book which is not actually quite standard in the tropical literature that w(E) = 0 unless E = Ex i for some xi. We can now define a marked parameterized tropical curve in MR, where as usual, M = Zn, MR = M ⊗Z R and N = HomZ(M, Z). Definition 1.11. A marked parameterized tropical curve h : (Γ,x1,...,xk) MR is a continuous map h satisfying the following two properties: (1) If E Γ[1] and w(E) = 0, then h|E is constant otherwise, h|E is a proper embedding of E into a line of rational slope in MR. (2) The balancing condition. Let V Γ[0], and let E1,...,E Γ[1] be the edges adjacent to V . Let mi M be a primitive tangent vector to h(Ei) pointing away from h(V ). Then i=1 w(Ei)mi = 0. If h : (Γ,x1,...,xn) MR is a marked parameterized tropical curve, we write h(xi) for h(Ex i ).
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