16 1. THE TROPICS Finally, we note that f is convex, i.e., given by a tropical polynomial. In addition, clearly h(Γ) = V (f). We are ready to talk about moduli spaces of such curves. For this, we need to talk about the combinatorial type of a marked tropical curve h : (Γ,x1,...,xn) MR. This is the data of the labelled graph (Γ,x1,...,xn), the weight function w, along with, for each flag (V, E) of Γ, the primitive tangent vector m(V,E) M to h(E) pointing away from h(V ). A combinatorial equivalence class is the set of all tropical curves of the same combinatorial type. We denote by [h] the combinatorial equivalence class of a curve h. Definition 1.16. For g, k 0, Σ a fan in MR, Δ with r(Δ) = 0, denote by Mg,k(Σ, Δ) the set of tropical curves in of genus g, degree Δ and with k marked points. If [h] is a combinatorial equivalence class of curves of genus g with k marked points of degree Δ in XΣ, we denote by M[h] g,k (Σ, Δ) Mg,k(Σ, Δ) the set of all curves of combinatorial equivalence class [h]. Proposition 1.17. Mg,k(Σ, Δ) = M[h] g,k (Σ, Δ), where the disjoint union is over all combinatorial equivalence classes of curves of degree Δ and genus g with k marked points. For a given combinatorial equivalence class [h] of a curve h : (Γ,x1,...,xk) MR, M [h] g,k (Σ, Δ) is the interior of a polyhedron of dimension e + k + (3 dim MR)(g 1) ov(Γ), where ov(Γ) = V ∈Γ[0] ( Valency(V ) 3 ) is the overvalence of Γ and e is the number of non-compact unmarked edges of Γ. Proof. First note the topological Euler characteristic χ(Γ) = 1 g satisfies χ(Γ) = [0] [1] = #Γ[0] #(Γ[1] \ Γ∞).[1] On the other hand, 3(#Γ[0]) + ov(Γ) = V ∈Γ[0] Valency(V ) = 2(#Γ[1] \ Γ[1]) + #Γ[1], from which we conclude that # of compact edges of Γ = # non-compact edges of Γ + 3g 3 ov(Γ) = e + k + 3g 3 ov(Γ). Now to describe all possible tropical curves with the given topological type, we choose a reference vertex V Γ[0], and we need to choose h(V ) MR and affine lengths1 E of each bounded edge h(E). However, these lengths cannot be chosen independently. Indeed, suppose we have a cycle E1,...,Em of edges in Γ, ∂Ei = 1 The affine length of a line segment of rational slope in MR with endpoints m1, m2 is the number R 0 such that m1 m2 = mprim for some primitive mprim M.
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