Proof. It is just the intersection number of the curve with the class of the $0$-section. Alternatively, use the cutting procedure from § 3.3 to get a tropical curve inside $\mathbb {R}^2$. The balancing condition ensures that the sum of the outgoing slopes of the unbounded ends add up to $0$. there are by assumption $d_2$ ends in the direction $(0,11)$. The ends on the right have slope $(u_i,v_i)$, with $\sum u_i= d_1$, and each one of them is associated to an end on the left with slope $(-u_i,\delta u_i-v_i)$. Thus, all add up to $(0,\delta \sum u_i)=(0,\delta d_1)$. The result follows.

Proof. Assume $h:\Gamma \to \mathbb {T} X$ of bidegree $(d_1,d_2)\neq (0,0)$ has no unbounded end. We thus have $d_2=0$ since it has no top ends, and $\delta =0$ since the curve has $\delta d_1$ lower ends and $d_1\neq 0$. Thus, $\mathbb {T} X$ is some $\mathbb {T} X_{0,\alpha }$.

Then, apply the cutting procedure for some admissible set $\mathcal {Q}'$ to get a curve inside $\mathbb {R}^2$. The pairs of ends have slopes $\pm v_e$ for some $v_e\in \mathbb {Z}_{>0}\times \mathbb {Z}$, and the marked points on it differ by some integer multiple of $(l,-\alpha )$. For each pair of ends, let $(x_e,y_e)$ be the marked point from $\mathcal {Q}'$ on the end directed by $-v_e$, and $(x_e+k_el,y_e-k_e\alpha )$ the point on the end directed by $v_e$, for some $k_e\in \mathbb {Z}$. Menelaus relation for $\Gamma '$ ensures that

\[ \sum_e \det(-v_e,(x_e,y_e))+\det(v_e,(x_e+k_el,y_e+k_e\alpha))=0. \]

Thus, $\det (\sum _e k_e v_e,(l,-\alpha ))=0$. As $\sum _e k_e v_e\in \mathbb {Z}_{>0}\times \mathbb {Z}$, we have that $(l,-\alpha )$ has rational slope and $\alpha \in \mathbb {Q} l$. In particular, $\mathbb {T} X_{0,\alpha }$ contains tropical elliptic curves going once around the cylinder direction: the superabundant loops of slope $(l,-\alpha )$ as in Example 3.18. These are parametrized by the coordinate of their intersection point with a fixed fiber.

To conclude, assume $\Gamma$ has at least one vertex. Consider a superabundant loop $\Upsilon$ of slope $(l,-\alpha )$ of highest coordinate that also passes through a vertex $V$ of $\Gamma$. It is possible since there are only a finite number of vertices in $\Gamma$. Not all the adjacent edges of $\Gamma$ to $V$ can be contained in the same line since $h$ is an immersion. But if some edge does not have slope $(l,-\alpha )$, by balancing condition, there is at least one edge that goes above $\Upsilon$, and as $\Gamma$ has no unbounded ends, it has to meet another vertex $W$. A superabundant loop passing through $W$ would be higher than $\Upsilon$, contradicting the assumption on the choice of $\Upsilon$. Thus, all the edges of $\Gamma$ are contained in $\Upsilon$. As $h$ is an immersion, $\Gamma$ has no vertices. It is thus of genus $1$ and is a superabundant loop.

Proof. Assume $h:\Gamma \to \mathbb {T} X$ is simple. Using the cutting procedure with an admissible set $\mathcal {Q}'$, we get a curve $h':\Gamma '\to \mathbb {R}^2$ with a pairing between the non-vertical ends. The deformations of $h:\Gamma \to \mathbb {T} X$ correspond to deformations of $h':\Gamma '\to \mathbb {R}^2$ that keep satisfying the gluing condition from the cutting procedure: paired ends coincide under the projection $\mathbb {R}^2\to \mathbb {T} X$. Recall that for an end with outing slope $u_e\in \mathbb {Z}^2$, its moment $\mu _e$ is $\det (u_e,P)$, where $P$ is any point on $e$, e.g. $q_e$ from the cutting process. The gluing condition amounts to find deformations of $h:\Gamma '\to \mathbb {R}^2$ such that for each pair $\{e,e'\}$ the quantity $\mu _e+\mu _{e'}$ remains constant.

The moments satisfy the Menelaus relation: $\sum _e\mu _e=0$. It is, in fact, the only relation between them: if we had another relation, we could use both relations to get a relation not involving all the moments. Then, let $e_0$ be one of the end not involved in the relation. For each other end $e\neq e_0$, as $\Gamma '$ is trivalent, it is possible to deform a path from $e_0$ to $e$. This changes the value of $\mu _e$ and $\mu _{e_0}$ but not for any other end, preventing a relation not involving all the ends to exist.

To conclude, according to [Reference MikhalkinMik05], $h':\Gamma '\to \mathbb {R}^2$ varies in a space of dimension $|\mu _0|+|\mu _\infty |+2|\mathcal {Q}'|+g(\Gamma ')-1$. Thanks to Proposition 4.2, there is at least one vertical unbounded end. The gluing condition thus impose $|\mathcal {Q}'|$ independent conditions. As $g(\Gamma )=g(\Gamma ')+|\mathcal {Q}'|$, we get the result for simple combinatorial types.

We now assume that $h$ is still an immersion but $\Gamma$ is not trivalent anymore. Furthermore, we can assume that no vertex can be split into two vertices adding an edge of slope $0$: if it were the case, we could split the vertex $V$ into $V_1$ and $V_2$ adding an artificial bounded edge of slope $0$ and then delete it. The vertices $V_1$ and $V_2$ are still vertices of the new curve after the splitting, or might become marked points, if they were not at least trivalent. We would get into one of the following situations.

• – If the edge was disconnecting, we now have two simple tropical curves $h_j:\Gamma _j\to \mathbb {T} X$ and genus $g_j$ of smaller degree with tangency profile $(\mu _0^j,\mu _\infty ^j)$, for $j=1,2$. By induction on the degree, we can assume they vary in a space of respective dimension at most ${|\mu _0^j|+|\mu _\infty ^j|+g_j-1}$. Deformations of $\Gamma$ are obtained by taking deformations of $\Gamma _1$ and $\Gamma _2$ mapping $V_1$ and $V_2$ to the same image. Thus, as $g=g_1+g_2$, $\Gamma$ varies in a space of dimension at most

  \[ |\mu_0|+|\mu_\infty|+g-2<|\mu_0|+|\mu_\infty|+g-1. \]

• – If the edge is not disconnecting, we get a new parametrization of the image of $\Gamma$ by a simple tropical curve $\widetilde {h}:\widetilde {\Gamma }\to \mathbb {T} X$ which is of genus $g-1$. Deformations of $\Gamma$ are the deformations of $\widetilde {\Gamma }$ that map $V_1$ and $V_2$ to the same image. By induction, we can still assume that it varies in a space of dimension at most

  \[ |\mu_0|+|\mu_\infty|+(g-1)-1<|\mu_0|+|\mu_\infty|+g-2. \]

Therefore, we can assume no vertex can be split. Then, it is possible to find a one-parameter family deforming $\Gamma$ into a nearby simple tropical curve: for any non-trivalent vertex, choose a loop or a string between two unbounded ends and deform it, and continue for each remaining non-trivalent vertex. Thanks to the assumption on vertices, no cycle or string can arrive at a vertex and leave with the same slope, so each deformation deletes at least one non-trivalent vertex. If the dimension of the deformation space of $h:\Gamma \to \mathbb {T} X$ was at least $|\mu _0|+|\mu _\infty |+g-1$, this nearby simple tropical curve would vary in a space of dimension at least one more (using the deformation provided by the one parameter family), which contradicts the dimension count for simple tropical curves.

Proof. We consider the moduli space $\mathcal {M}_{g,n}(\mathbb {T} X,(d_1,d_2))$, with $n=\delta d_1+2d_2+g-1$ marked points. The enumerative problem consists in finding the antecedent of a generic $\mathcal {P}\in \mathbb {T} X^n$ by the evaluation map. The moduli space is the union of a finite number of polyhedra corresponding to the possible combinatorial types. We want to prove that only the linear maps associated to simple combinatorial types are surjective.

If a combinatorial type does not correspond to an immersion, then, as in the planar case, it is possible to reparametrize the image by an immersion which is of strictly smaller genus or fewer ends. This is done by gluing the parallel adjacent edges, and deleting the contracted edges. Thus, we restrict to the immersed case.

If the combinatorial type corresponds to an immersion, we have seen that the dimension is as expected for a trivalent curve, and strictly less for a non-trivalent one, unless we are in presence of superabundant loops, but the assumption ensures that there are none. In particular, for the immersions of smaller genus or with fewer ends obtained from non-immersed curves, the dimension is also strictly less. Then the evaluation map cannot be surjective for these combinatorial types. The result follows.

Proof. For the curves which are not superabundant, the proof of this correspondence theorem follows exactly the same steps as other versions of the correspondence theorem that take place inside toric surfaces. This follows from the fact that it is possible to degenerate the surfaces $\mathbb {C} X$ into a union of toric surfaces glued over their toric divisors, and endow it with a smooth log structure. We explain the main steps and refer the reader to [Reference Nishinou and SiebertNS06] for more details.

The union of the images of all tropical curves passing through $\mathcal {P}$ induces a polyhedral subdivision of the strip $[0;l]\times \mathbb {R}$. Up to adding fibers to this union, we can assume that for each vertical edge in the subdivision, the whole vertical line containing it is in the subdivision. Using the $\mathbb {Z}$ action generated by the map $(x,y)\mapsto (x+l,y+\delta x)$, it extends periodically to a polyhedral subdivision $\Xi _1$ of $\mathbb {R}^2$. If $\delta =0$, the action is rather generated by $(x,y)\mapsto (x+l,y-\alpha )$. For any value of $(d_1,d_2)$, we can assume that $\alpha,l\in \mathbb {Q}$ but such that there are no superabundant loop of bidegree $(k,0)$ for any $k\leqslant d_1$. By periodically extending the subdivision of $[0;l]$, we also get a subdivision of $\mathbb {R}$. We consider the cones over both subdivisions.

• – The cone over the subdivision in $\mathbb {R}$ is a fan in $\mathbb {R}^2$ whose rays are generated by $(1,p_i)$, where $p_i$ is a point of the above subdivision of $\mathbb {R}$. It is denoted by $\Sigma$, and is depicted in Figure 7. The fan $\Sigma$ defines an almost toric variety $\mathcal {T}$: it is a partial compactification of $(\mathbb {C}^*)^2$ by a chain of $\mathbb {C} P^1$, glued along their toric boundary. The ‘almost toric’ is because the variety is not constructed from a finite fan. This non-finite fan projects to $\mathbb {R}_+$, giving a map from $\mathcal {T}$ to $\mathbb {C}$. The variety $\mathcal {T}$ is then fibered over $\mathbb {C}$, the special fiber being the chain of $\mathbb {C} P^1$.

  

  Figure 7. The fan $\Sigma$ defining the variety $\mathcal {T}$.

  The action of $\mathbb {Z}$ on $(\mathbb {C}^*)^2$ generated by the map $(t,z)\mapsto (t,a t^l z)$ descends to an action on the fan. The action on $\mathbb {R}^2$ is given by the following matrix:

  \[ \begin{pmatrix} 1 & 0 \\ l & 1 \\ \end{pmatrix}, \]
  i.e. the shear transformation of size $l$ parallel to the horizontal direction. Taking the quotient by the action, the quotient of $\mathcal {T}$ is a family of elliptic curves $\mathbb {C} E_t=\mathbb {C}^*/\langle a t^l\rangle$ that degenerates to the central fiber $\mathbb {C} E_0$, which is a chain of $\mathbb {C} P^1$ glued along their toric boundary. The difference is that the chain makes a cycle.

• – Similarly, we consider the cone over $\Xi _1$ gives a fan inside $\mathbb {R}^3$. A cone of $\Xi$ is generated by $(1,\sigma )$, where $\sigma$ is some cell of $\Xi _1$. The fan $\Xi$ defines an almost toric variety $\mathcal {T}'$ of dimension three. It is a partial compactification of $(\mathbb {C}^*)^3$ whose boundary consists of toric surfaces glued along their toric divisors.

  The fan $\Xi _1$ is mapped to $\Sigma$ by $(t,x,y)\mapsto (t,x)$. We consider the $\mathbb {Z}$-action on $(\mathbb {C}^*)^3$ generated by the map

  \[ (t,z,w)\longmapsto(t,a t^l z,z^\delta w). \]
  This action is compatible with the previous $\mathbb {Z}$-action on $(\mathbb {C}^*)^2$, and the projection $(t,z,w)\mapsto t$. The quotient of $\mathcal {T}'$ by the $\mathbb {Z}$-action is then a family $\mathcal {X}_t$ that admits a factorization $\mathcal {X}_t\to \mathbb {C} E_t$. This maps realizes $\mathcal {X}_t$ as a line bundle of degree $\delta$ over the elliptic curve $\mathbb {C} E_t$.

  The central fiber $\mathcal {X}_0$ is a union of toric surfaces glued along their toric boundary.

The previous description extends the setting of a degeneration of toric surfaces to a more general setting. It is possible to be more general by considering the quotient of $(\mathbb {C}^*)^3$ by a map $(t,z,w)\mapsto (t,a t^l z,b t^a z^\delta w)$, but up to a change of basis, we can assume that the map is of the assumed form. In the case of a degree $0$ line bundle, it is of the form

\[ (t,z,w)\mapsto(t,a t^l z,b t^{\alpha} w). \]

We have obtained a family of elliptic curves $\mathbb {C} E_t$ that degenerates to a union of $\mathbb {C} P^1$ glued over their toric divisors, and a degeneration of the degree $\delta$ line bundles over $\mathbb {C} E_t$ to a union of toric varieties glued over their toric divisors. We can now apply the method from [Reference Nishinou and SiebertNS06] to get a correspondence theorem. We refer the reader to it for more details. We summarize the main steps.

• – Endow the special fiber $\mathcal {X}_0$, which is a union of toric surfaces glued along their toric divisors, with a log-structure obtained from the toric construction.

• – Find the prelog curves in the special fiber $\mathcal {X}_0$, i.e. the reducible nodal curves that matches the point constraints. These curves are the limit curves of the families of curves inside $\mathcal {X}_t$ that match the constraints.

• – Given a prelog curve, find the possible log-structures on it so that they can be deformed to the general fiber.

The count of the solutions yields the expected multiplicity $m^\mathbb {C}_\Gamma =\prod m_V$. It is obtained as follows.

• – The complement of the marked points on a curve $\Gamma$ passing through $\mathcal {P}$ is a union of trees, each one possessing a unique unbounded end. Indeed, there are no cycles, otherwise the curve would vary in a one parameter family. Analogously, there are no components with at least two unbounded ends. Thus, there is exactly one unbounded end in each component.

• – When looking for the prelog curves, their search can be done by pruning the trees in the forest from the previous point. Each vertex provides ${m_V}/{w_{V1}w_{V2}}$ solutions, where $w_{VA}$ and $w_{V2}$ are the weights of the adjacent edges not in direction of the unbounded end of the tree.

• – The upgrading from prelog curves to log curves gives a $w$ factor for each edge of each tree. In particular, edges which are not marked contribute $w$, and edges marked $w^2$.

The product yields the expected multiplicity. The case of relative invariants is handled similarly. The difference is that we now may have tangencies with the toric divisors corresponding to the ends of the tropical curve and not only bounded edges. The count of prelog curves is the same, but we do not multiply anymore by $w$ when enhancing the prelog curves to log curves, yielding the expected multiplicity.

Proof. The proof reduces to the proof of invariance in [Reference Itenberg and MikhalkinIM13]. We consider a generic path $\mathcal {P}_s$ between two generic choices of point configuration $\mathcal {P}_0$ and $\mathcal {P}_1$. Given the assumption, there are no superabundant curves that can appear. Moreover, assume only one of the points moves: $p_s\in \mathcal {P}_s$.

We know that locally, a deformation of $p_s$ induces a deformation of the solutions, which have the same multiplicity since the latter is constant on the combinatorial types. We know that the complement of the marked points in any of the curves solution to the problem is a union of trees that each possess a unique unbounded end. Otherwise it would be possible to deform the curve in a one parameter family, contradicting the generic choice of the configuration. As the marked point $p_s\in \mathcal {P}_s$ moves, it is possible to deform any curve passing through $\mathcal {P}_s$. We have two possibilities:

• – both sides of the edge that $p_s$ splits belong to the same tree;

• – they belong to different components of $\Gamma \backslash h^{-1}(\mathcal {P}_s)$.

In the first case, only the unique path between both sides of the cut edge move. This is a cycle in $\Gamma$. Meanwhile in the second case, it is the path from the marked points to both unbounded ends in each tree that move. According to [Reference Itenberg and MikhalkinIM13], which uses Proposition 3.9 in [Reference Gathmann and MarkwigGM07b], the only possibilities that can occur when moving a point are:

1. (a) the curve degenerates to a curve that has a quadrivalent vertex;

2. (b) the curve degenerates to a curve with a vertex that coincides with a marked point;

3. (c) the curve degenerates to a curve that has two quadrivalent vertices related by a pair of parallel edges.

In each case, the invariance of counts with refined multiplicity comes from [Reference Itenberg and MikhalkinIM13].

Proof. Let $e$ be a bounded edge of $\Gamma$ with slope $(u,v)$. Assume $u\neq 0$. The intersection index with a fiber $F$ gives intersection number $|u|$, and it cannot be more than the intersection number between $\Gamma$ and $F$. Thus, one has $|u|\leqslant \Gamma \cdot F=d_1$. Assuming $v\neq 0$, we can similarly intersect with a section and get this time $|v|\leqslant \Gamma \cdot E_0=\delta d_1+d_2$.

Proof. Let $e$ be a bounded edge of $\Gamma$. If the length of $e$ is too big, it rolls several times around the cylinder and provides too many intersection points with a fiber $F$. In fact, it provides at least $\lfloor {l(e)}/{l} \rfloor$ intersection points, which each contribute $|u|$ to the intersection index. Then, one gets that $\lfloor {l(e)}/{l} \rfloor |u|\leqslant d_1$. Hence, as $|u|\geqslant 1$, it suffices to take $M=l(d_1+1)$.