14 1. THE TROPICS We will call two marked parameterized tropical curves h : (Γ,x1,...,xk) → MR and h : (Γ , x1,...,xk) → MR equivalent if there is a homeomorphism ϕ : Γ → Γ with ϕ(Ex i ) = Ex i and h = h ◦ ϕ. We will define a marked tropical curve to be an equivalence class of parameterized marked tropical curves. The genus of h is b1(Γ). We wish to talk about the degree of a tropical curve, and to do so, we need to fix a fan Σ. In fact, for the moment, we will only make use of the set of one- dimensional cones in Σ, Σ[1]. Denote by TΣ the free abelian group generated by Σ[1]. For ρ ∈ Σ[1], denote by tρ ∈ TΣ the corresponding generator. We have a map r : TΣ → M tρ → mρ where mρ is the primitive generator of the ray ρ. Definition 1.12. A marked tropical curve h is in XΣ if for each E ∈ Γ∞[1] which is not a marked edge, h(E) is a translate of some ρ ∈ Σ[1]. If h is a curve in XΣ, the degree of h is Δ(h) ∈ TΣ defined by Δ(h) = ρ∈Σ[1] dρtρ where dρ is the number of edges E ∈ Γ∞ [1] with h(E) a translate of ρ, counted with weight. For Δ ∈ TΣ, Δ = ∑ ρ∈Σ[1] dρtρ, define |Δ| := ρ∈Σ[1] dρ. The following lemma is a straightforward application of the balancing condition, obtained by summing the balancing conditions over all vertices of Γ: Lemma 1.13. r(Δ(h)) = 0. Example 1.14. Let Σ be the fan for P2. This is the complete fan in MR = R2 whose one-dimensional rays are ρ0,ρ1,ρ2 generated by m0 = (−1, −1), m1 = (1, 0) and m2 = (0, 1) see Figure 14. The two-dimensional cones are σi,i+1, with indices taken modulo 3 and where σi,i+1 is generated by mi and mi+1. We shall see in Chapter 3 that this fan defines P2 as a toric variety (Example 3.2). In particular, we shall give meaning to the symbol “XΣ”, which is actually a variety, and in the case of this particular Σ, XΣ = P2. Then the examples of Figures 2, 3 and 4 are tropical curves in XΣ = P2. The degree of Figure 2 is tρ 0 + tρ 1 + tρ 2 , while the degree of Figures 3 and 4 is 2(tρ 0 + tρ 1 + tρ 2 ). In general, a tropical curve in P2 will be, by the above lemma, of degree d(tρ 0 + tρ 1 + tρ 2 ), in which case we say the curve is degree d in P2 (compare with Example 1.6). So in particular, Figure 2 is a tropical line, and all degree one curves in P2 are just translates of this example. Figures 3 and 4 are tropical conics. It is reasonable to ask what the relationship is between this new definition of tropical curve and the earlier notion of a tropical hypersurface in MR with dim MR = 2. In particular, one can ask whether or not h(Γ) is a tropical hypersurface in MR. Of course, to pose this question, one must first define weights on h(Γ), as h is in general not an embedding. Viewing h(Γ) as a one-dimensional polyhedral complex,

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