18 1. THE TROPICS genus zero tropical curves h : (Γ,x1,...,x|Δ|−1) → MR in XΣ with h(xi) = Pi for all i. Furthermore, these curves are simple, and there is at most one such curve of any given combinatorial type. Proof. First note there are only a finite number of combinatorial types of curves of degree Δ in XΣ. Indeed, given a curve h : Γ → MR, we know by Proposi- tion 1.15 that h(Γ) is in fact a tropical hypersurface in MR. The degree Δ in fact determines the Newton polytope (up to translation) of a defining equation for h(Γ). Furthermore, specifying a regular subdivision of the Newton polytope is equivalent to specifying the combinatorial type of h(Γ). For each possible combinatorial type of h(Γ), there are only a finite number of ways of parameterizing such a curve. Since there are a finite number of lattice subdivisions of the Newton polytope, this implies there are only a finite number of combinatorial types. So in fact we can prove the result just by fixing one combinatorial type of curve, [h]. This gives the description as in the proof of Proposition 1.17, M[h] 0,|Δ|−1 (Σ, Δ) ∼ MR × Re+|Δ|−4−ov(Γ), 0 obtained after choosing a reference vertex V ∈ Γ[0]. We have an evaluation map ev : M[h] 0,|Δ|−1 (Σ, Δ) → (MR)|Δ|−1 sending h : (Γ,x1,...,x|Δ|−1) → MR to ev(h) = (h(x1),...,h(x|Δ|−1)). Note that in fact ev is an aﬃne linear map. Indeed, to compute h(xi) given h corresponding to a point in MR × Re+|Δ|−4−ov(Γ), 0 let E1,...,En be the sequence of edges traversed from the reference vertex V to the vertex adjacent to Ex i , with ∂Ei = {Vi−1,Vi}, V0 = V . Then h(xi) = h(V ) + n i=1 Ei m(V i−1 ,Ei) where Ei is the aﬃne length of Ei. This shows that h(xi) depends aﬃne linearly on h(V ) and the length of the edges. Thus, unless dim M[h] 0,|Δ|−1 (Σ, Δ) ≥ dim((MR)|Δ|−1), there is no curve of com- binatorial type [h] through general (P1,...,P|Δ|−1) ∈ (MR)|Δ|−1. This inequality of dimensions only holds if e + |Δ| − 2 − ov(Γ) ≥ 2(|Δ| − 1), or e − ov(Γ) ≥ |Δ|. Since e ≤ |Δ| and ov(Γ) ≥ 0, strict inequality never holds and equality only holds if e = |Δ| and ov(Γ) = 0, i.e., all unbounded edges of Γ are weight 1 and Γ is trivalent. If this equality holds, then either the image of ev is codimension ≥ 1, in which case again there are no curves of combinatorial type [h] through general (P1,...,P|Δ|−1), or else ev is a local isomorphism and then there is at most one curve of combinatorial type [h] passing through any P1,...,P|Δ|−1 ∈ MR. Finally, it is easy to see that the general curve in M [h] 0,|Δ|−1 (Σ, Δ) is injective on the set of vertices. Also, if there is a vertex V with attached edges E1, E2 with h|E 1 , h|E 2 non-constant and h(E1) ⊆ h(E2), then we are free to move h(V ) along

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