1.4. AFFINE MANIFOLDS WITH SINGULARITIES 19 the aﬃne line containing h(Ei), violating the fact that there is only one such curve. So a curve of type [h] passing through general points P1,...,P|Δ|−1 is simple, as desired. This result allows us to count tropical curves passing through a general set of points. However, as Mikhalkin showed, to get a meaningful result these must be counted with a suitable multiplicity, which we now define. Definition 1.21. Let h : Γ → MR be a simple tropical curve, with dim MR = 2. We define for V ∈ Γ[0] with adjacent edges E1, E2 and E3, MultV (h) = wΓ(E1)wΓ(E2)|m(V,E 1 ) ∧ m(V,E 2 ) | = wΓ(E2)wΓ(E3)|m(V,E 2 ) ∧ m(V,E 3 ) | = wΓ(E3)wΓ(E1)|m(V,E 3 ) ∧ m(V,E 1 ) | if none of E1,E2,E3 are marked, and otherwise MultV (h) = 1. Here for m1,m2 ∈ M, we identify 2 M with Z so that |m1 ∧ m2| makes sense. The equalities follow from the balancing condition. We then define the (Mikhalkin) multiplicity of h to be Mult(h) = V ∈Γ[0] MultV (h). Finally, for a given fan Σ and degree Δ, we write N 0,trop Δ,Σ = h Mult(h) where the sum is over all h ∈ M0,|Δ|−1(Σ, Δ) passing through |Δ|−1 general points in MR. While the generality of these points guarantees that the sum makes sense, it is not obvious that N 0,trop Δ,Σ doesn’t depend on the choice of these points. This will be shown later, twice, once in Chapter 4 and once in Chapter 5. In fact, the same definition can be made for curves of genus g. Indeed, one can show that, for a choice of |Δ|+g −1 general points in MR, there are a finite number of simple genus g curves passing through these points (see [80], Proposition 2.23). Using the same definition of multiplicity, one obtains numbers NΔ,trop.g, Σ If dim MR 2, there are similar definitions for counting formulas for genus zero curves: see [86] for precise statements. However, because of superabundant families of g 0 curves, there are more serious issues in higher genus. Mikhalkin’s main result in [80] relates the numbers N g,trop Δ,Σ , which are of course purely combinatorial, to counts of holomorphic curves in the toric variety XΣ. Stat- ing this result rather imprecisely here, he shows that the numbers N g,trop Δ,Σ coincide with the numbers N g,hol Δ,Σ of holomorphic curves of genus g in XΣ passing through |Δ| + g − 1 points in general position. The fact that this count can be computed in this purely combinatorial fashion was the first significant result in tropical geometry. We shall give a proof of this result for genus zero in Chapter 4. 1.4. Aﬃne manifolds with singularities We shall now discuss possible generalizations of the discussion of tropical curves. The question we want to pose here is: all our curves have lived in MR, a real aﬃne space. Are there interesting choices for more general target manifolds?

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