1.3. PARAMETERIZED TROPICAL CURVES 17 {Vi−1,Vi} with Vm = V0. We of course have by definition of m(V i−1 ,Ei) that Vi = Vi−1 + Ei m(V i−1 ,Ei) . Thus V0 = Vm = V0 + ∑m i=1 Ei m(V i−1 ,Ei) , or m i=1 Ei m(V i−1 ,Ei) = 0 in MR. So, for each cycle, we obtain the above linear equation, which imposes dim MR linear conditions on the E ’s. Thus, given that there exists a tropical curve of the given combinatorial type, the set of all curves of this combinatorial type is MR × (Re+k+3g−3−ov(Γ) 0 L) where L Re+k+3g−3−ov(Γ) is a linear subspace of codimension g · dim MR and hence the whole cell is of dimension e + k + (3 dim MR)(g 1) ov(Γ). Remark 1.18. One should view the case where all vertices of Γ are trivalent as a generic situation. However, there are tropical curves of genus g 1 in MR for dim MR 3 which are not trivalent and cannot be viewed as limits of trivalent curves. Of course, for g = 0, equality always holds for the dimension, but for g 1 equality need not hold. A curve of a given combinatorial type is said to be superabundant if the moduli space of curves of that type is larger than e + k + (3 dim MR)(g 1) ov(Γ). Otherwise a curve is called regular. Superabundant curves cause a great deal of difficulty for tropical geometry, and we shall handle this by restricting further to plane curves, i.e., dim MR = 2. Furthermore, as we shall only need the genus zero case for our discussion, we shall often restrict our attention to this case also. Restricting to the case that dim MR = 2, we define Definition 1.19. A marked tropical curve h : Γ MR for dim MR = 2 is simple if (1) Γ is trivalent (2) h is injective on the set of vertices and there are no disjoint edges E1, E2 with a common vertex V for which h|E 1 and h|E 2 are non-constant and h(E1) h(E2) (3) Each unbounded unmarked edge of Γ has weight one. By our discussion above, simple curves move in a family of dimension at least |Δ| + k + g 1, as now e = |Δ|. However, one can show that simple curves in dimension two are always regular, see [80], Proposition 2.21, so in fact simple curves move in (|Δ| + k + g 1)-dimensional families. We know this for g = 0 already, but since we shall not be focussing on higher genus curves, we omit a proof of this fact. Lemma 1.20. Fix Σ a fan in MR, dim MR = 2, and a degree Δ TΣ. Let P1,...,P|Δ|−1 MR be general points.2 Then there are a finite number of marked 2 By general, we mean that (P1, . . . , P|Δ|−1) M |Δ|−1 R lies in some dense open subset of M |Δ|−1 R .
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