2·1. Tropical tori

We follow partially the notations from [ Reference Halvard Halle and CF Rose15 ] and use the following definition of a tropical torus. In all that follows, let N be a rank 2 lattice, $M=\hom(N,\mathbb{Z})$ its dual lattice. For a lattice N, we denote by $N_\mathbb{R}$ the real vector space $N\otimes\mathbb{R}$ , and similarly $N_{\mathbb{C}^*}=N\otimes\mathbb{C}^*$ .

Definition 2·1. A tropical torus $\mathbb{T} A$ is a quotient

\begin{align*}\mathbb{T} A=N_\mathbb{R}/\Lambda,\end{align*}

where $\Lambda\subset N_\mathbb{R}$ is a lattice.

There are two lattices. The lattice N gives the tropical structure, and the lattice $\Lambda$ gives the period of the tropical torus $\mathbb{T} A$ . Normally, N should rather be a subsheaf of the tangent bundle of $\mathbb{T} A$ that gives a lattice in each fiber. As the tangent bundle is trivial, we can just consider the fiber to be $N_\mathbb{R}$ with the lattice N inside it. Fixing a basis of N, we define $\mathbb{T} A$ as the quotient $\mathbb{R}^2/\Lambda$ , but this definition does not handle well a change of basis of N, while the above definition does. Moreover, it provides a more natural definition of the dual torus.

From now on, let $\mathbb{T} A=N_\mathbb{R}/\Lambda$ be a tropical torus define by two lattices N and $\Lambda$ , with a map $S\;:\;\Lambda\to N_\mathbb{R}$ . Choosing bases of the lattices N and $\Lambda$ , the inclusion S amounts to the choice of a $2\times 2$ matrix with real coefficients. This choice is defined up to the change of basis, i.e. the action of $SL_2(\mathbb{Z})$ by multiplication on the left and on the right.

Homology groups. The universal cover of $\mathbb{T} A$ is $N_\mathbb{R}$ , and one has thus a natural identification of $H_1(\mathbb{T} A,\mathbb{Z})$ with $\Lambda$ .

1-forms and cycles. As the tangent bundle $T(\mathbb{T} A)$ is the trivial bundle with fiber $N_\mathbb{R}$ , the cotangent bundle $T^*(\mathbb{T} A)$ is also trivial with fiber $M_\mathbb{R}$ , the dual of $N_\mathbb{R}$ , that contains the dual lattice M. The sections of the cotangent bundles with values in M consist in the 1-forms that take integral values on the lattice N.

Given a 1-form, i.e. an element of $H^0(\mathbb{T} A,T^*(\mathbb{T} A))$ , we can integrate it on cycles of $H_1(\mathbb{T} A,\mathbb{Z})\simeq\Lambda$ . We choose to restrict to the tropical 1-forms, i.e. the ones that belong to $H^0(\mathbb{T} A,M)$ . The integration of tropical 1-forms along cycles

\begin{align*}(\alpha,\gamma)\in M\times\Lambda\longmapsto\int_\gamma\alpha \in\mathbb{R},\end{align*}

corresponds exactly to the map

\begin{align*}(m,\lambda)\in M\times\Lambda\longmapsto \langle m,S\lambda\rangle\in \mathbb{R},\end{align*}

given by the inclusion $S\;:\;\Lambda\hookrightarrow N_\mathbb{R}$ , and the evaluation pairing $\langle,\rangle\;:\;M_\mathbb{R}\times N_\mathbb{R}\to\mathbb{R}$ . In other words, let $(\gamma_1,\gamma_2)$ be a basis of $\Lambda$ , and $(m_1,m_2)$ a basis of M, with dual basis $(e_1,e_2)$ of N. The elements $m_1,m_2$ are coordinate functions on $N_\mathbb{R}$ . The matrix of the inclusion $S\;:\;\Lambda\hookrightarrow N_\mathbb{R}$ in the chosen basis has elements that correspond to the integral of the 1-forms $m_1,m_2\in M\simeq H^0(\mathbb{T} A,M)$ over the cycles $\gamma_1,\gamma_2$ . This way, the lattice $\Lambda$ appears as the period of $\mathbb{T} A$ .

2·2. Tropical homology and intersection form

In this section, we compute the tropical homology groups of a tropical torus. For a precise definition of the tropical homology groups, see [ Reference Ludmil Katzarkov, Mikhalkin and Zharkov16 ]. In our case, the cosheafs $\mathcal{F}_p$ used to compute tropical homology groups are just constant coefficients, with $\mathcal{F}_1=N$ and $\mathcal{F}_2=\Lambda^2 N$ , the tropical homology groups $H_{p,q}(\mathbb{T} A)=H_q(\mathbb{T} A,\mathcal{F}_p)$ of $\mathbb{T} A$ are as follows:

\begin{align*}\begin{matrix}& & & \Lambda^2 N\otimes\Lambda^2 \Lambda & & \\[5pt] & & \Lambda^2 N\otimes\Lambda & & N\otimes\Lambda^2\Lambda & \\[5pt] H_{p,q}(\mathbb{T} A)=& \Lambda^2 N & & \Lambda\otimes N & & \Lambda^2\Lambda \\[5pt] & & N & & \Lambda & \\[5pt] & & & \mathbb{Z} & & \end{matrix}.\end{align*}

We only care about the middle group $H_{1,1}(\mathbb{T} A)\simeq\Lambda\otimes N$ . We now choose orientations on the lattices N, M and $\Lambda$ , in a compatible way: orientations of $\Lambda^2N$ and $\Lambda^2 M$ determine each other, and the orientation of $N_\mathbb{R}$ restricts to the orientation of $\Lambda$ . This defines an intersection product on $H_{1,1}(\mathbb{T} A)$ . This group can be seen as the standard homology group of $\mathbb{T} A$ but with coefficients in N. The intersection number between two N-cycles is obtained by summing over the intersection points the intersection number between their coefficients. In other words,

\begin{align*}(\lambda\otimes n)\cdot(\lambda'\otimes n')=\det\!(\lambda,\lambda ')\det\!(n,n').\end{align*}

2·3. Degeneration of complex family to a tropical tori

2·3·1. Abelian surfaces in multiplicative form

We now briefly describe degenerations of a family of complex tori. A 2-dimensional complex torus is the quotient of a 2-dimensional complex vector space by some rank 4 lattice. A complex torus thus has a presentation of the form $\mathbb{C} A=\mathbb{C}^2/\Lambda_\textrm{add}$ , where $\Lambda_\textrm{add}$ is the image of a matrix $\Omega\in\mathcal{M}_{2,4}(\mathbb{C})$ . Up to a change of coordinate inside the lattice $\Lambda_\textrm{add}$ and the vector space $\mathbb{C}^2$ , one can always assume that the matrix $\Omega$ is of the form

\begin{align*}\Omega=\begin{pmatrix}I_2 & Z \end{pmatrix}.\end{align*}

Then, letting N be the sublattice spanned by the first two columns, $\mathbb{C}^2$ appears as the vector space $N_\mathbb{C}=N\otimes\mathbb{C}$ . Then, taking the exponential map, one can also give the following multiplicative presentation of the 2-dimensional torus:

\begin{align*}\mathbb{C} A=N_\mathbb{C}/\Lambda_\textrm{add} \longrightarrow N_{\mathbb{C}^*}/\Lambda_\textrm{mult},\end{align*}

where the map takes $n\otimes z\in N_\mathbb{C}$ to $e^{2i\pi nz}\in N_{\mathbb{C}^*}$ , which is the exponential taken coordinate by coordinate. This is well-defined on $\mathbb{C} A$ if one quotients by the image of the span of the last two column vectors under the exponential map.

Remark 2·3. The downfall of the multiplicative notation is that we implicitly fix a preferred sublattice of $\Lambda_\textrm{add}$ , and change of basis of the lattice, as well as the action of $GL_2(\mathbb{C})$ are not that clear anymore.

2·3·2. Line bundles on abelian surfaces

Up to translation, line bundles on abelian surfaces are classified by their Chern class $c_1(\mathcal{L})\in H^2(\mathbb{C} A,\mathbb{Z})$ . We care about the line bundles that have sections. For such line bundles, the Chern class lies in fact in the Hodge cohomology group $H^{1,1}(\mathbb{C} A)$ , and pairs positively with the Kähler form on $\mathbb{C} A$ . According to [ Reference Griffiths and Harris14 , chapter 2·6], a integer skew-symmetric matrix Q is the Chern class of a positive line bundle if and only if

\begin{align*}\left\{ \begin{array}{l}\Omega Q^{-1}\Omega^T=0,\\[5pt] -i\Omega Q^{-1}\overline{\Omega}^T>0. \end{array}\right.\end{align*}

This is known as Riemann condition. Given an integer skew-symmetric matrix, it is always possible to choose a basis of the lattice such that the matrix has the following form:

\begin{align*}Q=\begin{pmatrix}0\;\;\;\; & -c \\[5pt] c^T\;\;\;\; & 0 \end{pmatrix}.\end{align*}

In our previous notations, this matrix Q can be seen as a linear map $N\oplus\Lambda\to M\oplus\Lambda^*$ , and $c\;:\;\Lambda\to M$ . In fact, one can even assume that the matrix c is of the form $\begin{pmatrix}\delta_1 & 0 \\[5pt] 0 & \delta_2 \end{pmatrix}$ , where $\delta_1 | \delta_2$ . As $Q^{-1}=\begin{pmatrix}0\;\;\;\; & c^{-1T} \\[5pt] -c^{-1}\;\;\;\; & 0 \end{pmatrix}$ , the Riemann bilinear condition then becomes

\begin{align*}\left\{\begin{array}{l}Zc^{-1} \textrm{ is symmetric,}\\[5pt] \mathfrak{Im}(-Zc^{-1}) \textrm{ is positive definite.} \end{array}\right.\end{align*}

Notice that as $c^{-1}\;:\;M\to\Lambda$ and $Z\;:\;\Lambda\to N$ , the product has sense and is a map $M\to N$ that can be symmetric as a bilinear form on M. In the complex setting, curves are section of a positive line bundle, and the homology class realised by a curve is Poincaré dual to the Chern class of the line bundle. The Riemann bilinear relation satisfied by the Chern classes of line bundles imposes a constraint on the classes $C\in H_2(\mathbb{C} A,\mathbb{Z})$ realised by complex curves.

2·3·3. Deformations

We now consider a deformation of the abelian surface, which means a deformation of the lattice, so that the Riemann bilinear relation keeps being satisfied. We keep the sublattice N fixed, so that we only have a deformation of Z. The deformation is taken of the form $Z_t=({1}/{2i\pi})(A+S\log t)$ , where A is some complex matrix, and S is some integer matrix. The Riemann bilinear relations become $Ac^{-1}\in\mathcal{S}_2(\mathbb{C})$ and $Sc^{-1}\in\mathcal{S}_2^{++}(\mathbb{R})$ . As $\log t$ goes to infinity when t goes to infinity, one can forget about the positiveness condition for A since the condition for $({1}/{2i\pi})(A+S\log t)$ is satisfied close to the limit. In the exponential notations, our family of tori $\mathbb{C} A_t$ is of the form

\begin{align*}\mathbb{C} A_t=N_{\mathbb{C}^*}/\langle e^At^S \rangle,\end{align*}

which is called a Mumford family. The vectors spanning the lattice are $\lambda_j=(e^{a_{ij}}t^{s_{ij}})_i$ .

Remark 2·4. It is also possible to consider families of tori that vary in a more complicated manner, taking general formal series as coefficients on the multiplicative side for instance. Moreover, it could seem natural to first consider some Mumford family, i.e. $\Omega_t=\begin{pmatrix}I_2 & \frac{1}{2i\pi}(A+S\log t) \end{pmatrix}$ , and then find the Chern classes of positive line bundles that survive the deformation. Unfortunately, such classes are not necessarily of the antidiagonal form. This is due to the following fact. Unlike the real setting, any lagrangian sublattice of a lattice with a skew-symmetric form does not necessarily have a lagrangian complement. When assuming that Q is of a given form, we pick a decomposition of the lattice as a sum of two lagrangians sublattice. The deformation is considered afterwards. When first fixing the family, some sublattice is fixed, and this one does not necessarily have a lagrangian complement. One could change the lattice, but the deformation would not be of Mumford type anymore.

In the multiplicative notation, for complex tori, we have an exact sequence

\begin{align*}0\to \ker\pi \to N_{\mathbb{C}^*}/\Lambda_t\xrightarrow{\pi=|\bullet|} N_\mathbb{R}/\log|\Lambda_t|\to 0.\end{align*}

This exact sequence expresses $\mathbb{C} A_t$ as a torus fibration of fiber $N_{\mathbb{R}/\mathbb{Z}}$ , over the base $N_\mathbb{R}/\log|\Lambda_t|$ which is also a torus. Moreover, it splits at the topological level. In fact, this decomposition just emphasizes the decomposition of the first homology group of the associated complex surface:

\begin{align*}H_1(\mathbb{C} A_t,\mathbb{Z})=N\oplus \Lambda_t.\end{align*}

The first part corresponds to the homology of the torus fiber, and the second to the homology of the base of the fibration. Notice that we recover the tropical homology groups of $\mathbb{T} A$ .

We also have the corresponding decomposition for cohomology:

\begin{align*}H^1(\mathbb{C} A_t,\mathbb{Z})=M\oplus \Lambda^*,\end{align*}

where $M=N^*$ . It gives the following decomposition of the $H^2(\mathbb{C} A_t,\mathbb{Z})$ :

\begin{align*}H^2(\mathbb{C} A_t,\mathbb{Z})=\Lambda^2\Lambda^*\oplus M\otimes\Lambda^*\oplus\Lambda^2 M.\end{align*}

Remark 2·5. Notice that this decomposition does not match the Hodge decomposition of $\mathbb{C} A_t$ , but it matches the decomposition provided by tropical cohomology. In fact, the map c is the tropical Chern class of the tropical line bundle on the tropical abelian surface $\mathbb{T} A$ obtained as limit of the complex line bundle. See second paper for more details.

Let $\Lambda$ be the image of S. The tropicalisation of the family $\mathbb{C} A_t$ is the tropical torus $\mathbb{T} A=N_\mathbb{R}/\Lambda$ . In [ Reference Nishinou21 ], Nishinou constructs a degeneration of the family of abelian surfaces to a union of toric surfaces glued along their toric boundary as follows: given a rational polyhedral decomposition of $\mathbb{T} A$ , one constructs a periodic fan $\Sigma$ inside $N_\mathbb{R}\times\mathbb{R}$ , by taking the cone over the periodisation of the decomposition. This fan gives an almost toric variety, which fibrates over $\mathbb{C}$ . The quotient by the $N\simeq\mathbb{Z}^2$ action gives the degeneration. The author then gives a correspondence theorem for tropical curves inside $\mathbb{T} A$ , and families of classical curves inside $\mathbb{C} A_t$ .

3·1. Parametrised tropical curves

3·1·2. Parametrised tropical curves

We now get to parametrised tropical curves. Let $\mathbb{T} A=N_\mathbb{R}/\Lambda$ be a tropical torus.

Definition 3·1. A parametrised tropical curve inside a tropical torus $\mathbb{T} A=N_\mathbb{R}/\Lambda$ (resp. $N_\mathbb{R})$ is a map $h\;:\;\Gamma\to \mathbb{T} A$ (resp. $N_\mathbb{R}$ ) from an abstract tropical curve $\Gamma$ such that:

1. (i) h is affine with slope in N on each edge of $\Gamma$ . For an oriented edge e, its slope $u_e$ is of the form $w_e u'_{\!\!e}$ where $u'_{\!\!e}\in N$ is a primitive vector, and $w_e$ a positive integer called the weight of the edge;

2. (ii) h satisfies the balancing condition: for each vertex $V\in V(\Gamma)$ ,

   \begin{align*}\sum_{e\ni V}w_eu_e=0,\end{align*}
   when each e is oriented with V as its source.

Remark 3·2. For a parametrised tropical curve in $\mathbb{T} A$ , the curve $\Gamma$ has no unbounded ends, while it has several for a curve inside $N_\mathbb{R}$ .

Example 3·3. In Figure 1, we can see three different examples of tropical curves in different tropical tori $\mathbb{T} A$ .

Fig. 1.

Three examples of tropical curves in tropical tori. The first and third are of genus 2, and the second of genus 5.

3·1·3. Degree of a tropical curve

In particular, for each edge, the 1-chain element with coefficients in N : $u_e e\in C_1(\mathbb{T} A,N)$ does not depend on the chosen orientation. Then, the curve realises a 1-chain

\begin{align*}\sum_{e\in E(\Gamma)} u_e e\in C_1(\mathbb{T} A,N).\end{align*}

The balancing condition ensures that this chain is in fact a cycle. Thus, it realises a homology class $C=[\Gamma]\in H_1(\mathbb{T} A,N)=\Lambda\otimes N=H_{1,1}(\mathbb{T} A)$ .

Definition 3·4. The class C in $\Lambda\otimes N$ realised by a parametrised tropical curve is called the degree of the curve.

Given a curve $h\;:\;\Gamma\to \mathbb{T} A$ , its degree $C\in\Lambda\otimes N$ can be seen as a map $\Lambda^*\to N$ , and thus as a matrix. The following proposition gives a description of its matrix.

Proposition 3·5. Take a parallelogram that is a fundamental domain for $\mathbb{T} A=N_\mathbb{R}/\Lambda$ . Let $(\lambda_1,\lambda_2)$ be the basis of $\Lambda$ formed by two edges of the parallelogram. Let $n_1$ be the sum of the slopes of the edges intersecting the side opposite to $\lambda_2$ , oriented outward the fundamental domain, and let $n_2$ the sum for the edges intersecting the side opposite to $\lambda_1$ . Then C is the map that sends $\lambda_1^*$ to $n_1$ and $\lambda_2^*$ to $n_2$ .

Proof. Let C ^′ be the class defined in the proposition. We check that C ^′ has the same intersection numbers with a basis of $H_{1,1}(\mathbb{T} A)$ as the curve $\Gamma$ . Thus, they represent the same class.

The classes realised by tropical curves are of a very specific kind: they satisfy the following property.

Proposition 3·6. A class $C\in\Lambda\otimes N$ can be realised by a tropical curve if and only if $CS^T$ is symmetric.

Proof. Notice that $S^T\;:\;M\to\Lambda^*$ and C is a matrix of a map $\Lambda^*\to N$ , so that the product has a meaning. If the condition is satisfied, it is easy to construct a genus 2 curve in the class C. Conversely, assume that there is some tropical curve inside the surface. Choose a fundamental domain for $\mathbb{T} A=N_\mathbb{R}/\Lambda$ . Cut the tropical curve inside the fundamental domain and replace the cut edges by unbounded ends to get a true tropical curve inside $N_\mathbb{R}$ . By tropical Menelaus theorem [ Reference Mikhalkin20 ], the sum of the moments of the unbounded ends is 0. The unbounded ends come in pairs of ends glued when passing from the fundamental domain to $\mathbb{T} A$ . The sum of their moments is equal to $\det\!(\lambda_1,n_e)$ . Adding the contribution for all edges yields the result. See the next section for more details.

Remark 3·7. Fixing the lattices N and $\Lambda$ , the tropical torus is given by a $2\times 2$ real matrix S, corresponding to the inclusion $\Lambda\to N_\mathbb{R}$ . A class $C\in \Lambda\otimes N$ is realisable if and only if $S^T C$ is symmetric. In other words, the matrix S gives a linear form on $(\Lambda\otimes N)_\mathbb{R}$ , and the set of realisable classes is some subset of the intersection between an hyperplane and the lattice $\Lambda\otimes N$ . Generically, there will be none. Alternatively, the realisability of a given class C imposes a codimension 1 constraint on S.

Example 3·8. Choosing basis of the lattices $\Lambda$ and N, a class N is also represented by a matrix, a map $\Lambda^*\to N$ . Getting back to the curves in Figure 1, we take as a basis of N the canonical basis of the plane where the figures are drawn, and as a basis of $\Lambda$ the sides of the parallelogram, first the almost vertical one, and then the almost horizontal one. we have the following:

1. (a) the matrix of the inclusion $\Lambda\to N_\mathbb{R}$ is of the form $S=\begin{pmatrix}a\;\;\;\; & b \\[5pt] b\;\;\;\; & a \end{pmatrix}$ , and the curve represents the class $C=\begin{pmatrix}1\;\;\;\; & 0 \\[5pt] 0\;\;\;\; & 1 \end{pmatrix}$ . We easily check that $CS^T$ is symmetric;

2. (b) we have $S=\begin{pmatrix}12\;\;\;\; & 2 \\[5pt] 3\;\;\;\; & 8 \end{pmatrix}$ , and the curve represents the class $C=\begin{pmatrix}2\;\;\;\; & 0 \\[5pt] 0\;\;\;\; & 3 \end{pmatrix}$ , and we have again $CS^T$ symmetric;

3. (c) we have $S=\begin{pmatrix}4\;\;\;\; & -1 \\[5pt] -2\;\;\;\; & 3 \end{pmatrix}$ , and the curve represents the class $C=\begin{pmatrix}2\;\;\;\; & 1 \\[5pt] 0\;\;\;\; & 1 \end{pmatrix}$ , and we have again $CS^T$ symmetric.

Concretely, the matrix C is the matrix whose columns are the sums of the slopes intersecting the right (resp. top) side of the parallelogram respectively. Meanwhile, S is the matrix whose columns are the coordinate of the bottom side and left side of the parallelogram in the basis of $N_\mathbb{R}$ .

Fig. 2.

Examples of genus 2 tropical curve.

Example 3·9. Consider the tripod with directions $(\!-\!\alpha_1-\alpha_2,-\beta_1-\beta_2)$ , $(\alpha_1,\beta_1)$ and $(\alpha_2,\beta_2)$ and with respective lengths a, b and c. Take $S=\begin{pmatrix}a(\alpha_1+\alpha_2)+b\alpha_1\;\;\;\; & a(\alpha_1+\alpha_2)+c\alpha_2 \\[5pt] a(\beta_1+\beta-2)+b\beta_1\;\;\;\; & a(\beta_1+\beta_2)+c\beta_2 \end{pmatrix}$ and $C=\begin{pmatrix}\alpha_1\;\;\;\; & \alpha_2 \\[5pt] \beta_1 \;\;\;\; & \beta_2 \end{pmatrix}$ . We can check that $C S^T$ is always symmetric. We indeed have a genus 2 tropical curve in the class C, as depicted in Figure 2. Notice that the corner of the parallelogram is the second vertex of the curve. The example of Figure 1 (a) is also of this type, except the curve has been translated so that the corner of the parallelogram is not a vertex of the curve anymore.

3·2. Cutting/lifting procedure

As the tropical torus $\mathbb{T} A$ is constructed as a quotient of $N_\mathbb{R}$ , it is possible to relate tropical curves inside $\mathbb{T} A$ to tropical curves inside $N_\mathbb{R}$ . We explicit the relation and procedure to construct one from another in this section.

Let $h\;:\;\Gamma\to A$ be a connected genus g parametrised tropical curve in the tropical abelian surface $\mathbb{T} A$ . Assume that it has no contracted edge. If $\Gamma$ had a contracted edge e, we would reduce to the following situations:

1. (i) if e is disconnecting, we have $\Gamma-e=\Gamma_1\sqcup\Gamma_2$ and we reparametrise $h\;:\;\Gamma\to \mathbb{T} A$ by $\Gamma_1\sqcup\Gamma_2$ , which is a reducible curve, with components of respective genera $g_1$ and $g_2$ with $g_1+g_2=g$ ;

2. (ii) if e is not disconnecting, $\Gamma-e$ is a connected curve of genus $g-1$ .

Taking a point on every edge of the complement of a spanning tree of $\Gamma$ , we see that a lifting set always exists, and the complement $\Gamma\backslash\mathcal{Q}$ can be connected. The trivialness of the map $h_*\;:\;\pi_1(\Gamma\backslash\mathcal{Q})\to \pi_1(\mathbb{T} A)$ ensures that one can lift h to

\begin{align*}\tilde{h}\;:\;\Gamma\backslash\mathcal{Q} \longrightarrow N_\mathbb{R}.\end{align*}

Then, replacing the edges that have been cut by infinite edges, we get a parametrised tropical curve

\begin{align*}\tilde{h}\;:\;\widetilde{\Gamma}_\mathcal{Q}\longrightarrow N_\mathbb{R},\end{align*}

inside the plane $N_\mathbb{R}$ . When $\mathcal{Q}$ is fixed, we just denote it by $\widetilde{\Gamma}$ .

Fig. 3.

On the left a tropical curve in a tropical torus with a lifting set, and on the right the corresponding lifting in the plane $N_\mathbb{R}$ .

Let $x=|\mathcal{Q}|$ and let $n_1,\dots,n_x$ be the slopes of the edges on which lie the points in $\mathcal{Q}$ . The curve $(\widetilde{\Gamma},\tilde{h})$ is of genus $g-x$ and has degree $\{n_i,-n_i\}_{i=1}^x$ . Recall that the degree of a tropical curve inside $N_\mathbb{R}$ is the collection of the slopes of its unbounded ends.

Lemma 3·13. Let $i\in[\![1;x]\!]$ indexing a pair of ends $\{e_i,f_i\}$ of $\widetilde{\Gamma}$ , corresponding to a point $p_i$ in $\Gamma$ . Let $\tilde{\gamma}$ be a path from $e_i$ to $f_i$ inside $\widetilde{\Gamma}$ . It projects to a loop $\gamma$ in $(\Gamma\backslash\mathcal{Q})\cup\{p_i\}$ , and thus in $\mathbb{T} A$ . Then, considering $\gamma\in H_1(\mathbb{T} A)$ , one has

\begin{align*}\det\!(n_i,e_i)+\det\!(-n_i,f_i)=\det\!(n_i,S\gamma).\end{align*}

Proof. The element $\det\!(n_i,-)\in M$ is a tropical 1-form on $\mathbb{T} A$ . Its value on the loop $\gamma$ is precisely $\det\!(n_i,S\gamma)$ . To compute it, we can also parametrise the loop $\gamma$ by the path $\tilde{\gamma}$ :

\begin{align*}\det\!(n_i,S\gamma)=\sum_k \det\!(n_i,l_ku_{e_k}),\end{align*}

where the path is comprised of the edges $e_k$ , of slope $u_{e_k}$ and length $l_k$ . This can be lifted to $N_\mathbb{R}$ , and by linearity of $\det\!(n_i,-)$ , we get the announced result.

Using the preceding remark, another way to reformulate the result is then that the moments of the ends of $\widetilde{\Gamma}$ is constrained by the tropical abelian surface $\mathbb{T} A$ : the sum of moments of associated ends is fixed. This relation links deformations of $\Gamma$ in $\mathbb{T} A$ and $\widetilde{\Gamma}$ in $N_\mathbb{R}$ .

Proposition 3·15. A small deformation of $\Gamma$ lifts to a small deformation of $\widetilde{\Gamma}$ . Conversely, a small deformation of $\widetilde{\Gamma}$ lifts to a deformation of $\Gamma$ if and only if it satisfies the conditions

\begin{align*}\det\!(n_i,e_i)+\det\!(\!-\!n_i,f_i)=\det\!(n_i,S\gamma),\end{align*}

for each pair of ends $(e_i,f_i)$ .

Proof. The first part is obvious, since a lifting set keeps being lifting for a small deformation, so that we can deform $\widetilde{\Gamma}$ to suit a deformation of $\Gamma$ . Moreover, Lemma 3·13 ensures that, as $\mathbb{T} A$ is fixed, the relation between the moments of a pair of ends $(e_i,f_i)$ is still being satisfied through the deformation.

Conversely, we consider a small deformation of $\widetilde{\Gamma}$ that preserves these relations between the moments. We project from from $N_\mathbb{R}$ to $\mathbb{T} A$ . Before any deformation, pair of ends locally agree, so that it is possible to merge them together, yielding a tropical curve in $\mathbb{T} A$ . As the sum of moments is preserved, it means that two associated ends are displaced in the same way, and they can still be glued throughout the deformation, yielding a family of tropical curves in $\mathbb{T} A$ .

Fig. 4.

In (a) and (c), small deformation of a tropical curve inside a tropical torus. In (b) the deformation of the lifted curve corresponding to the deformation in (a)

3·3. Dimension of the moduli space

Let $C\in H_{1,1}(\mathbb{T} A,\mathbb{Z})=\Lambda\otimes N$ be a class that is realisable in $\mathbb{T} A$ : $C S^T\in\mathcal{S}_2(\mathbb{R})$ . Let $\mathcal{M}_{g,n}(\mathbb{T} A,C)$ be the moduli space of genus g irreducible tropical curves with n marked points in the class C. It is a polyedral complex which is a union over the different combinatorial types of the tropical curves $\Gamma$ . We assume that $\mathbb{T} A$ is simple: it does not contain any elliptic curve. We now aim at computing the dimension of the moduli space $\mathcal{M}_{g}(\mathbb{T} A,C)$ .

In particular, a simple tropical curve no contracted edges, and no flat vertices (vertices whose adjacent edges belong to a line). We consider simple parametrised tropical curves.

Proof. Let $\gamma$ be a cycle in $\Gamma$ , formed by the edges $e_1,\dots, e_n$ , whose respective slopes are $u_1,\dots,u_n$ . Let $v_i$ be the slope of the remaining edge coming at the vertex $V_i$ between $e_i$ and $e_{i+1}$ , so that

\begin{align*}u_{i+1}=u_i+v_i.\end{align*}

Then, move the vertex in the direction $v_i$ :

\begin{align*}V_i(t)=V_i+\frac{t}{\det\!(u_i,u_{i+1})}v_i.\end{align*}

As

\begin{align*}\det\!(u_{i+1},V_{i+1}(t)-V_i(t)) & = \det\!(u_{i+1},V_{i+1}-V_i) + t\det\left(u_{i+1},\frac{v_{i+1}}{\det\!(u_i,u_{i+1})}-\frac{v_i}{\det\!(u_i,u_{i+1})}\right)\\[5pt] & = 0 + t\frac{\det\!(u_{i+1},v_{i+1})}{\det\!(u_{i+1},u_{i+2})}- t\frac{\det\!(u_{i+1},v_{i})}{\det\!(u_{i},u_{i+1})}\\[5pt] & =t-t\\[5pt] & = 0, \end{align*}

the slope of the edges stays the same and we get a deformation of the tropical curve.

The deformation of a cycle is illustrated in Figure 5 in the case of a contractible cycle and a non-contractible one.

Fig. 5.

Deformation of a contractible cycle and a non-contractible one

Remark 3·21. Proposition 3·20 proves that all parametrised tropical curves in $\mathbb{T} A$ are superabundant, as the expected dimension is $g-1$ : let $C\in H_{1,1}(\mathbb{T} A)$ be a class that is realisable, and g a fixed genus. The “expected” dimension of the moduli space $\mathcal{M}_{g}(\mathbb{T} A,C)$ of genus g curves in the class C is

\begin{align*}3g-3 +2 -2g=g-1.\end{align*}

The $3g-3$ is the dimension of the space of curves of genus g, the 2 accounts for the possible translations, and the $-2g$ for the conditions that the g cycles impose on the curve. Fortunately, in this case, the superabundancy makes the dimension count match the complex dimension, which is also g.

If h is not assumed to be an immersion, there can be “supersuperabundant” curves if there are some contracted edges, flat vertices or cycles mapped to a line. Such curves vary in a space of too big dimension, but do not contribute any solutions to the enumerative problem that we consider because their image can be reparametrised by a parametrised tropical curve $h'\;:\;\Gamma'\to \mathbb{T} A$ , where h ^′ is an immersion, and $\Gamma'$ is not trivalent, or has smaller genus.

4·1. Enumerative problem

According to the previous section, it makes sense to count the number of genus g irreducible tropical curves in the class C that pass through g points. If the points are chosen in generic position, Proposition 3·20 ensures that the curves are simple.

Contrarily to the planar case, as the dimension depends linearly on the genus, the space of reducible curves with components of genera $g_1,\dots,g_r$ adding up to g has also dimension g. We have thus two possible conventions regarding reducible curves.

1. (i) The genus of a reducible curve $\Gamma$ can be defined by $1-\chi(\Gamma)$ , where $\chi$ is the Euler characteristic. Concretely, it is equal to the sum of the genera of the components, plus one minus the number of components. This way, given a configuration $\mathcal{P}$ of g points, we always have deformable genus g curves that pass through $\mathcal{P}$ , but these are reducible: the family of reducible curves $\Gamma_1\cup\cdots\cup\Gamma_r$ varies in dimension $g_1+\cdots+g_r=g+r$ .

2. (ii) The other possibility is to set the genus of a reducible curve equal to the sum of the genera of the components, so that a curve of genus g always varies in dimension g. However, we always have to account for reducible curves passing through a configuration $\mathcal{P}$ of g points, although they do not vary in positive dimensional families anymore.

In the following, we avoid the problem of reducible curves by considering irreducible curves whenever possible.

Fig. 6.

In (a) an irreducible curve of genus 4 and in (b) a reducible curve with two components of genus 2.

It is possible to partially get rid of reducible curves as follows. If $\mathbb{T} A$ is chosen generically among the surfaces with curves in the class C, i.e. the matrix of S is generic among the real matrices such that $CS^T\in\mathcal{S}_2(\mathbb{R})$ , then C and its multiple are the only classes realisable by curves. If S is not chosen generically, we have classes that may appear or disappear when we move $\mathbb{T} A$ among the space of abelian surfaces with curves in the class C.

We already know several results concerning enumeration of curves inside abelian surfaces:

1. (i) in [ Reference Bryan and Conan Leung10 ], without tropical geometry, Bryan and Leung give generating functions for the number of genus g curves in a primitive class C passing through g points, or in a fixed linear system and passing through $g-2$ points. This result does not depend on the choice of the abelian surface as long as the class C is realisable by some curve. In fact, an abelian surface is Hyperkähler, i.e. given a metric, it has a whole sphere of possible complex structures. For a class C, exactly one of the possible complex structures admits curves in the class C, and the number of such curves passing through g points does not depend on the chosen abelian surface. This count includes reducible curves. They managed to compute these numbers for primitive classes C by making an explicit computation for abelian surfaces which are product of two generic elliptic curves;

2. (ii) in [ Reference Halvard Halle and CF Rose15 ], L. Halle and F. Rose use tropical geometry and maps between tropical tori to count the number of genus 2 curves in a fixed class passing through 2 points. They recover the result of [ Reference Bryan and Conan Leung10 ] in the genus 2 case;

3. (iii) finally, in [ Reference Nishinou21 ], Nishinou gives a correspondence theorem that allows one to compute the number of genus g curves passing through g points and that tropicalise to some given trivalent curve. This recovers the result from [ Reference Bryan and Conan Leung10 ] provided one is able to solve the analogous tropical problem, at least for some abelian surface containing curves in the class C.

Let $\mathcal{P}\subset \mathbb{T} A$ be a configuration of g points chosen generically inside $\mathbb{T} A$ . Let $h\;:\;\Gamma\to \mathbb{T} A$ be a tropical curve of genus g in the class C passing through $\mathcal{P}$ . The tropical curve is rigid, i.e. it is not possible to deform it while still passing through the points of $\mathcal{P}$ . We have the following.

The complement $\Gamma\backslash h^{-1}(\mathcal{P})$ has no cycles since it is always possible to deform a cycle using Lemma 3·19. Knowing the complement of the marked points has no cycles, and that its Euler characteristic is $1-g+g=1$ , we deduce that it is connected.

In particular, the set $h^{-1}(\mathcal{P})$ is lifting for the curve $\Gamma$ , and the complement has no cycles. Then, we deduce that each marked point p defines a unique path $\lambda_p$ on $\Gamma\backslash h^{-1}(\mathcal{P})$ : the one that connects the two halfs of the edge that it splits. In other words, it is the unique loop of $(\Gamma\backslash h^{-1}(\mathcal{P}))\cup\{p\}$ . Moreover, this loop realises some homology class inside $\mathbb{T} A$ , which can be either 0 or nonzero, and that we also denote by $\lambda_p$ .

Using Proposition 4·2 and the lifting procedure from the Section 3·2, it is now possible to unfold the complement of the marked points, lifting it to the universal cover $N_\mathbb{R}$ of $\mathbb{T} A$ , and get a planar parametrised tropical curve:

\begin{align*}\tilde{h}\;:\;\widetilde{\Gamma}\longrightarrow N_\mathbb{R}.\end{align*}

If the loop $\lambda_p$ associated to a point p is non-zero, there are two distinct points in $N_\mathbb{R}$ and two corresponding unbounded ends of $\widetilde{\Gamma}$ . Otherwise, the point lifts to a single point and $\lambda_p$ to a loop inside $N_\mathbb{R}$ .

We now focus on enumerative problems involving tropical curves in the torus. One important feature in which it differs from the planar setting, is that as noticed in Remark 3·21, all curves are superabundant: the dimension of their moduli space is greater than the expected dimension. Fix a genus g and a realisable class C. Let $\delta(C)$ be the integer length of the class, i.e. the biggest integer by which C is divisible. Notice that if a curve $\Gamma$ realises the class C, its gcd (equal to the gcd of the weights of its edges) divides $\delta(C)$ . We then consider the two following enumerative problems.

1. (i) How many (irreducible) parametrised tropical curves of genus g in the class C pass through a generic configuration $\mathcal{P}$ of g points ?

2. (ii) For $k|\delta(C)$ , how many irreducible parametrised tropical curves of genus g and gcd k in the class C pass through a generic configuration $\mathcal{P}$ of g points ?

Counting reducible curves with a fixed gcd does not make sense because we only defined the gcd of an irreducible curve.

As usual, the solutions to these enumerative problems need to be counted with some multiplicity in order to get an invariant, i.e. a result that does not depend on the choice of $\mathcal{P}$ provided that it is generic.

The refined multiplicity is a symmetric Laurent polynomial in the variable q, which is the product of the quantum analogs of the vertex multiplicities. Let $h\;:\;\Gamma\to \mathbb{T} A$ be a curve, and $\delta$ be an integer. It is possible to scale the tropical curve by $\delta$ by multiplying all edges weights by $\delta$ . This multiplies the vertex multiplicities by $\delta^2$ . As a simple genus g curve has $2g-2$ vertices, we have

\begin{align*}m_{\delta\Gamma}=\delta^{4g-4}m_\Gamma.\end{align*}

For the refined multiplicity, the relation is a little bit more complicated since the vertex multiplicities are involved in the exponent of the polynomials. The relation is

\begin{align*}m_{\delta\Gamma}^q(q)=[\delta^2]_q^{2g-2}m_{\Gamma}^q(q^{\delta^2}).\end{align*}

Now, let $\mathcal{P}$ be a generic configuration of points on $\mathbb{T} A$ . We set the following:

\begin{align*}N_{g,C,k}^{\textrm{trop}}(\mathbb{T} A,\mathcal{P}) = \sum_{\substack{h(\Gamma)\supset\mathcal{P} \\[5pt] \delta_\Gamma=k}} m_\Gamma, \\[5pt] BG_{g,C,k}(\mathbb{T} A,\mathcal{P}) = \sum_{\substack{h(\Gamma)\supset\mathcal{P} \\[5pt] \delta_\Gamma=k}} m^q_\Gamma. \end{align*}

The sums are over the irreducible curves of genus g in the class C, having gcd k for k dividing the integral length $\delta(C)$ of C, that pass through the point configuration $\mathcal{P}$ . Due to the relations between the multiplicities, and as the curves having gcd k are obtained by multiplication of a curve with gcd 1, we always have the following:

\begin{align*} N_{g,C,k}(\mathbb{T} A,\mathcal{P})=(k^2)^{2g-2} N_{g,C/k,1}(\mathbb{T} A,\mathcal{P}) \textrm{ and } BG_{g,C,k}(\mathbb{T} A,\mathcal{P})= [k^2]_q^{2g-2} BG_{g,C/k,1}(\mathbb{T} A,\mathcal{P}).\end{align*}

We can also consider the sums over all the irreducible curves passing through $\mathcal{P}$ , without imposing the value of the gcd of the curves. However, we prove the invariance for the counts with the value of the gcd imposed, which is stronger than the invariance for the counts of all curves. In particular, we can choose multiplicities of the form $f(\delta_\Gamma)m_\Gamma$ and still get an invariant. We set

\begin{align*}M_{g,C}(\mathbb{T} A,\mathcal{P}) & = \sum_{h(\Gamma)\supset\mathcal{P}} m_\Gamma = \sum_{k|\delta(C)} N_{g,C,k}(\mathbb{T} A,\mathcal{P}) \in\mathbb{N}, \\[5pt] N_{g,C}(\mathbb{T} A,\mathcal{P}) & = \sum_{h(\Gamma)\supset\mathcal{P}} \delta_\Gamma m_\Gamma = \sum_{k|\delta(C)}k N_{g,C,k}(\mathbb{T} A,\mathcal{P}) \in\mathbb{N}, \\[5pt] R_{g,C}(\mathbb{T} A,\mathcal{P}) & = \sum_{h(\Gamma)\supset\mathcal{P}} \delta_\Gamma m^q_\Gamma \in \mathbb{Z}[q^{\pm 1/2}], \\[5pt] BG_{g,C}(\mathbb{T} A,\mathcal{P}) & = \sum_{h(\Gamma)\supset\mathcal{P}} m^q_\Gamma = \sum_{k|\delta(C)} BG_{g,C,k}(\mathbb{T} A,\mathcal{P}) \in \mathbb{Z}[q^{\pm 1/2}] . \end{align*}

These are four different sums over the solutions. The first is with multiplicity $m_\Gamma$ and the second with multiplicity $\delta_\Gamma m_\Gamma$ . The third is a refinement of the second since it specializes to it when q is set equal to 1. The multiplicity has been replaced by $\delta_\Gamma m_\Gamma^q$ . The last is also a refined count of solutions, but of the first. The multiplicity does not involve the gcd of the curves, only the vertex multiplicities.

Fig. 7.

The unique genus 2 curve passing through two points.

Example 4·6. We refer to the end of the paper for examples and computations. For now, we just give the contribution of the curve depicted in Figure 7 to the various counts:

\begin{align*}M_{2,2I_2}^{\textrm{trop}} & \to 4\times 4=16,\\[5pt] N_{2,2I_2}^{\textrm{trop}} & \to 2\times 4\times 4=32,\\[5pt] R_{2,2I_2} & \to 2\times (q^{1/2}+q^{-1/2})\times (q^{1/2}+q^{-1/2})=2(q+2+q^{-1}),\\[5pt] BG_{2,2I_2} & \to q+2+q^{-1}. \end{align*}

4·2. Correspondence theorem

In [ Reference Nishinou21 ], Nishinou gives a realisation theorem and a correspondence theorem for genus g irreducible curves passing through g points in general position on the torus that we now recall. More precisely, he considers the following family of complex algebraic tori:

\begin{align*}\mathbb{C} A_t=N_{\mathbb{C}^*}/\Lambda_t,\end{align*}

as in Section 2·3. With the same notations, $\Lambda_t$ is generated by $\alpha_1 t^{s_1},\alpha_2 t^{s_2}\in N_{\mathbb{C}^*}$ . The family tropicalises to the tropical torus

\begin{align*}\mathbb{T} A=N_\mathbb{R}/\Lambda,\end{align*}

with $\Lambda=\textrm{Im} S$ . Let $C\in \Lambda\otimes N$ be a class, that we assume to be primitive, otherwise we divide it by its integer length. Let $(e_1,e_2)$ be a basis of N, and $(e_1^*,e_2^*)$ be the dual basis of M. Assume the slope of h on the edges that intersect $B_1$ is $a_ie_1+b_i e_2$ , and $c_ie_1+d_ie_2$ for $B_2$ . The vectors associated through Poincaré duality are $-b_i e_1^*+a_i e_2^*$ and $-d_i e_1^*+c_ie_2^*$ . Assume that $\Lambda_t\subset N_{\mathbb{C}^*}$ is generated by $(\alpha_{11} t^{s_{11}},\alpha_{12}t^{s_{12}})$ and $(\alpha_{21} t^{s_{21}},\alpha_{22}t^{s_{22}})$ . Then, we consider the following quantity, introduced in [ Reference Nishinou21 ]:

\begin{align*}\sigma(C)=\prod_i \alpha_{11}^{-b_i}\alpha_{12}^{a_i}\prod_j \alpha_{21}^{-d_i}\alpha_{22}^{c_i}.\end{align*}

For a version free of coordinates choices, for $t=1$ , we have $j\;:\;\Lambda_1\to N_{\mathbb{C}^*}$ . Thus, we also have the following map, also denoted by $\sigma$ :

\begin{align*}\begin{array}{rccl}\sigma: & \Lambda\otimes N\simeq\Lambda_1\otimes N & \longrightarrow & \mathbb{C}^* \\[5pt] & \lambda_1\otimes n & \longmapsto & \textrm{PD}(n)\cdot j(\lambda_1) \end{array},\end{align*}

where $\textrm{PD}(n)$ is the vector Poincaré dual to n, $j(\lambda_1)\in N_{\mathbb{C}^*}$ , and the action of M on $N_{\mathbb{C}^*}$ is the evaluation of the monomial: $m\cdot z^n=z^{\langle m,n\rangle}\in\mathbb{C}^*$ . The complex number $\sigma(C)$ is then the value of $\sigma$ on the class C. We now have the following realisation theorem.

Theorem. (Nishinou [ Reference Nishinou21 ]). Let $h\;:\;\Gamma\to \mathbb{T} A$ be a simple tropical curve realising the class $\delta C$ , and let $\delta_\Gamma$ be the gcd of $\Gamma$ . The curve is realisable if and only if

\begin{align*}\sigma\left(\frac{\delta C}{\delta_\Gamma}\right)=(\!-\!1)^{\sum m_V/\delta_\Gamma}.\end{align*}

The relation from Nishinou’s realisation’s theorem is obtained by making the product of Menelaus relations at each vertex of the tropical curve, and is equivalent to the curve being “phase-realisable”. Here, Menelaus relation means relation (46) in proposition 39 from [ Reference Mikhalkin20 ]. It is the relation between the phases of the edges adjacent to a vertex for a tropical curve that is realisable. Recall that phases are complex numbers associated to edges of a tropical curve when given a family of complex curves that approximates the tropical curve.

The necessary condition from Nishinou’s theorem corresponds in fact to the necessary condition that the class C of the curve satisfies Riemann bilinear relations for members of the family $\mathbb{C} A_t$ . The left-hand side only depends on the abelian surface and the choice of the curve class. The right-hand side seems to depend on the tropical curve and suggests that not all tropical curves can be approximated, since there is a condition to satisfy. This did not appear in Section 2·3, as we do not have any condition on the tropical curve. This is due to the following lemma, that asserts that the right-hand side of the equality in Nishinou’s theorem is always 1.

Lemma 4·7. Let $h\;:\;\Gamma\to\mathbb{T} A$ be a simple parametrised tropical curve, and let $\delta_\Gamma$ be the gcd of the curve. Then

\begin{align*}\sum_{V\in V(\Gamma)}\frac{m_V}{\delta_\Gamma}\equiv 0 \mod 2.\end{align*}

Proof. First, assume that $\delta_\Gamma$ is even. Then, as for any vertex the multiplicity is divisible by $\delta_\Gamma^2$ , each term in the sum is still even, and $\sum_{V\in V(\Gamma)}({m_V}/{\delta_\Gamma})$ is also even.

Assume that $\delta_\Gamma$ is odd. Then, the sum has the same parity as $\sum_{V\in V(\Gamma)}m_V$ . This sum is equal mod 2 to the intersection number $C\cdot C$ . To see this, consider the curve $\Gamma$ and a small translate of it. The intersection points come from the vertices and double points:

1. (i) each double point contributes two intersection points with the same multiplicity;

2. (ii) each trivalent vertex contributes one intersection points of multiplicity $m_V$ .

As the intersection form on $H_{1,1}(\mathbb{T} A)$ is even, since it is given in the basis $(\lambda_i\otimes e_j)$ by the matrix

\begin{align*}\begin{pmatrix}0\;\;\;\; & 1\;\;\;\; & 0\;\;\;\; & 0 \\[5pt] 1\;\;\;\; & 0\;\;\;\; & 0\;\;\;\; & 0 \\[5pt] 0\;\;\;\; & 0\;\;\;\; & 0\;\;\;\; & 1 \\[5pt] 0\;\;\;\; & 0\;\;\;\; & 1\;\;\;\; & 0 \end{pmatrix},\end{align*}

the result follows.

In other words, when we choose a Mumford family of complex abelian surfaces, it has to satisfy Riemann conditions. Riemann conditions are already satisfied in modulus since we have a tropical abelian surface. The condition given in Nishinou’s theorem is that arguments should also satisfy Riemann condition.

Then, choosing g families of points $\mathcal{P}_t$ inside $\mathbb{C} A_t$ that tropicalise to a set $\mathcal{P}$ of g points inside $\mathbb{T} A$ which are in generic position, Nishinou gives a formula to count the number of irreducible curves inside $\mathbb{C} A_t$ passing through $\mathcal{P}_t$ that tropicalise to a given simple tropical curve passing through $\mathcal{P}$ . The multiplicity $m_\Gamma^\mathbb{C}$ of a tropical curve given by the correspondence theorem from [ Reference Nishinou21 ]. Let $h\;:\;\Gamma\to \mathbb{T} A$ be a trivalent tropical curve passing through $\mathcal{P}$ , and let $\Gamma'$ be the graph $\Gamma$ subdivided by the points of $h^{-1}(\mathcal{P})$ . First, choose an orientation of every edge $e\in E(\Gamma')$ . Then, we have a map

\begin{align*}\begin{array}{rccl}\Theta \;: & \bigoplus_{v\in V(\Gamma^{\prime})}N & \longrightarrow & \bigoplus_{e\in E(\Gamma^{\prime})}N/N_e \oplus\bigoplus_{1}^g N\\[5pt] & (\phi_v) & \longmapsto & \left( \phi_e,\phi_{v_i}\right) \\[5pt] \end{array},\end{align*}

where $N/N_e$ is the 1-dimensional lattice obtained by quotienting N by the primitive slope of the edge e, $\phi_e=\phi_{\partial^+ e}-\phi_{\partial^- e}\in N/N_e$ is the difference between the extremities if the oriented edge e projected to the quotient, and $v_i$ is the vertex of $\Gamma'$ associated to a marked point $p_i\in\mathcal{P}$ . The dimension of the domain is $2|V(\Gamma')|=2|V(\Gamma)|+2g$ while the dimension of the codomain is $|E(\Gamma')|+2g=|E(\Gamma)|+3g$ . As $\Gamma$ is of genus g and is trivalent, we know that

\begin{align*}\left\{ \begin{array}{l}3|V(\Gamma)|=2|E(\Gamma)|, \\[5pt] |V(\Gamma)|-|E(\Gamma)|=1-g. \end{array}\right.\end{align*}

Hence,

\begin{align*}(|E(\Gamma)|+3g)-(2|V(\Gamma)|+2g)=|V(\Gamma)|-|E(\Gamma)|+g=1,\end{align*}

and contrarily to the usual planar setting, domain and codomain do not have the same dimension. In fact, this map is not surjective precisely because of the superabundancy of the tropical curves. The multiplicity is defined as follows.

Definition 4·8. The multiplicity of a trivalent tropical curve $h\;:\;\Gamma\to A$ passing through $\mathcal{P}$ is

\begin{align*}m^\mathbb{C}_\Gamma=|\ker\Theta_{\mathbb{C}^*}|\prod_{e\in E(\Gamma')} w_e,\end{align*}

where $\Theta_{\mathbb{C}^*}$ is the map $\Theta\otimes\mathbb{C}^*$ .

Remark 4·9. In the usual correspondence theorems, such as in [ Reference Nishinou and Siebert22 ] and [ Reference Tyomkin25 ], we have a similar map between lattices. This map is used to compute the number of possible phases that one can put on the tropical curve. A phase is a complex number such that at each vertex, the Menelaus condition is satisfied. See [ Reference Mikhalkin19, Reference Nishinou and Siebert22 ] for more details. The difference is that in those last cases, domain and codomain have the same dimension, and the multiplicity is then equal to the determinant of the map $\Theta$ between lattices. Here, we cannot take the determinant as the two spaces do not have the same dimension. Nevertheless, $|\ker\Theta_{\mathbb{C}^*}|$ is also equal to the cardinal of the torsion part of $\textrm{coker}\Theta$ , as we see by applying the tensor product to the following small exact sequence:

\begin{align*}0\to \mathbb{Z}^{2 |V(\Gamma)|+2g} \xrightarrow{\Theta} \mathbb{Z}^{|E(\Gamma)|+3g} \to \mathbb{Z} \oplus G \to 0,\end{align*}

with G the torsion part of the cokernel. Applying the functor $-\otimes\mathbb{C}^*$ , we get

\begin{align*}0\to \textrm{Tor}(G,\mathbb{C}^*) \to (\mathbb{C}^*)^{2 |V(\Gamma)|+2g} \xrightarrow{\Theta_{\mathbb{C}^*}} (\mathbb{C}^*)^{|E(\Gamma)|+3g} \to \mathbb{C}^* \to 0,\end{align*}

and as $\mathbb{C}^*$ is divisible, we have $\textrm{Tor}(G,\mathbb{C}^*)\simeq G$ . The cokernel can then be computed using exterior powers of the map between the lattices.

We now have the following correspondence theorem.

Theorem. (Nishinou [ Reference Nishinou21 ]). Let $h\;:\;\Gamma\to \mathbb{T} A$ be a simple tropical curve passing through $\mathcal{P}$ . The number of irreducible parametrised curves passing through $\mathcal{P}_t$ that tropicalise to $(h,\Gamma)$ is

\begin{align*}m_\Gamma^\mathbb{C}=|\ker \Theta_{\mathbb{C}^*}|\prod_{e\in E(\Gamma')} w_e.\end{align*}

In particular, if $\mathcal{N}_{g,C}$ denotes the number of complex genus g curves in the class C passing through a generic configuration of g points inside an abelian surface, then

\begin{align*}N_{g,C}=\mathcal{N}_{g,C}.\end{align*}

4·3. Statement of the results

Using Proposition 4·2 to relate the tropical curve inside $\mathbb{T} A$ to a planar tropical curve inside $N_\mathbb{R}$ , we can compute the multiplicity and get a formula analog to the multiplicity of a planar tropical curve provided by the correspondence theorem of Mikhalkin [ Reference Mikhalkin19 ].

Theorem 4·10. The multiplicity $m_\Gamma^\mathbb{C}$ provided by Nishinou’s correspondence theorem splits as follows:

\begin{align*}m_\Gamma^\mathbb{C}=\delta_\Gamma\prod_{V\in V(\Gamma)}m_V = \delta_\Gamma m_\Gamma,\end{align*}

where $m_\Gamma$ is the first multiplicity from Definition 4·3, and $\delta_\Gamma$ is the gcd of the tropical curve $\Gamma$ .

The gcd $\delta$ of the weights of the edges is not necessarily constant on the moduli space $\mathcal{M}_g(\mathbb{T} A,C)$ , nor for the solutions of the enumerative problems. However, it is constant on the connected components of the complement of some codimension 1 substrata. And this suffices to provide invariance for the counts of solutions with a fixed gcd. The proof Theorem 4·10 is postponed to Section 5·1.

Fig. 8.

Two tropical curves with different marking.

Just as in the usual planar case, it is natural to try to replace the vertex multiplicity $m_V$ by its quantum analog $[m_V]_q=({q^{m_V/2}-q^{-m_V/2}})/({q^{1/2}-q^{-1/2}})$ , in an attempt to obtain tropical refined invariants similar to Block-Göttsche refined invariants for toric surfaces. This justifies the definition of the refined tropical multiplicities from Definition 4·3.

Recall that we defined several refined tropical counts: $BG_{g,C,k}(\mathbb{T} A,\mathcal{P})$ , $R_{g,C}(\mathbb{T} A,\mathcal{P})$ and $BG_{g,C}(\mathbb{T} A,\mathcal{P})$ . We then have the following invariance statements:

The proof is in Section 5·2. The corollary follows from the expression of the respective enumerative counts in terms of $BG_{g,C,k}(\mathbb{T} A,\mathcal{P})$ , which is invariant by Theorem 4·13, evaluating at $q=1$ for the integer valued counts. We remove the $\mathcal{P}$ from the notation to denote the corresponding invariants. We now partially get rid of the dependence in $\mathbb{T} A$ .

The proof is in Section 5·3. Notice that if $\mathbb{T} A$ is not chosen generically, we still have invariants for $\mathbb{T} A$ since the proof of Theorem 4·13 does not require a generic choice of $\mathbb{T} A$ .

Due to the preceding remark, the refined invariant depends on the choice of the matrix S, and more precisely on the cone of realisable classes inside the the group $\{C\in \Lambda\otimes N \textrm{ such that }C S^T\in\mathcal{S}_2(\mathbb{R})\}$ .

5·1. Multiplicity as a product

Proof of Theorem 4·10. For each edge $e\in E(\Gamma')$ directed by some primitive vector $u_e$ , we use $\det\!(u_e,-)$ as a coordinate on $N/N_e$ . The map $\Theta$ starts from a lattice of rank $2|V(\Gamma')|=6g-4$ to a lattice of rank $|E(\Gamma')|+g=6g-3$ . It is not surjective. In fact, its image always lies in the kernel of the following linear form defined on the codomain:

\begin{align*}\varphi(\phi_e,p_i)=\sum_e \frac{w_e}{\delta}\phi_e,\end{align*}

where $\delta$ is the gcd of the curve. Indeed, for each $v\in V(\Gamma')$ , we have

\begin{align*}\varphi\circ\Theta(p_v)=\frac{1}{\delta}\left( w_1 \det\!(u_1,\varepsilon_1 p_v)+ w_2 \det\!(u_2,\varepsilon_2 p_v) + w_3 \det\!(u_3,\varepsilon_3 p_v)\right),\end{align*}

where $\varepsilon_i=\pm 1$ whether $p_v$ is the head or the tail of $e_i$ . This is 0 by the balancing condition. This does not happen in the planar case due to the presence of unbounded ends. As $\Theta$ is injective, taking a basis of the domain and codomain, we have a short exact sequence,

\begin{align*}0\to \mathbb{Z}^{6g-4}\xrightarrow{\Theta} \mathbb{Z}^{6g-3}\to \mathbb{Z}\oplus G\to 0,\end{align*}

where G is a torsion group. We want to compute the cardinal of this torsion group. By taking the long exact sequence of Tor, we obtain

\begin{align*}0\to \textrm{Tor}(G,\mathbb{C}^*)\to (\mathbb{C}^*)^{6g-4}\xrightarrow{\Theta_{\mathbb{C}^*}} (\mathbb{C}^*)^{6g-3}\to \mathbb{C}^*\to 0.\end{align*}

We have seen that the image of $\Theta$ lies in $\ker\varphi\subset\mathbb{Z}^{6g-3}$ . The latter is a primitive lattice, that possesses a Plücker vector, obtained by making the wedge product of a basis, yielding an element of $\Lambda^{6g-4}\mathbb{Z}^{6g-3}$ . The latter is defined up to sign. Considering the exterior power $\Lambda^{6g-4}\Theta\;:\;\Lambda^{6g-4}\mathbb{Z}^{6g-4}\simeq\mathbb{Z}\to \Lambda^{6g-4}\mathbb{Z}^{6g-3}\simeq\mathbb{Z}^{6g-3}$ . It maps a generator of $\Lambda^{6g-4}\mathbb{Z}^{6g-4}$ to some multiple of the Plücker vector of the $\ker\varphi$ , in which the image of the map lies. We are in fact interested in the absolute value of the proportionality constant, that yields the index of $\textrm{Im}\Theta$ in $\ker\varphi$ . Let us rephrase it. Choosing any bases of the lattices, $\Theta$ is given by an integer matrix. Choosing suitable bases, it is possible to assume the matrix is in Smith normal form:

\begin{align*}\begin{pmatrix}d_1\;\;\;\; & 0\;\;\;\; & \cdots\;\;\;\; & 0 \\[5pt] 0\;\;\;\; & d_2\;\;\;\; & \;\;\;\; & \vdots \\[5pt] \vdots\;\;\;\; & \;\;\;\; & \ddots\;\;\;\; & \vdots \\[5pt] 0\;\;\;\; & \cdots\;\;\;\; & \cdots\;\;\;\; & d_{6g-4} \\[5pt] 0\;\;\;\; & \cdots\;\;\;\; & \cdots\;\;\;\; & 0 \end{pmatrix}.\end{align*}

The cokernel of $\Theta$ is $\mathbb{Z}\oplus G$ where $G=\bigoplus_1^{6g-4}\mathbb{Z}/d_i\mathbb{Z}$ , and $|G|=\prod d_i$ . The map $\varphi$ sends a point in $\mathbb{Z}^{6g-3}$ to its last coordinate. The Plücker vector of $\ker\varphi$ is $e_1\wedge\dots\wedge e_{6g-4}\in\Lambda^{6g-4}\mathbb{Z}^{6g-3}$ , and the image of the generator of $\Lambda^{6g-4}\mathbb{Z}^{6g-4}$ is $\bigwedge (d_i e_i)=\left(\prod d_i\right)\bigwedge e_i$ . We are interested in the constant $d_1\cdots d_{6g-4}$ , which is both the cardinal of the torsion part G of the cokernel, and the integral length of the image of a generator of $\Lambda^{6g-4}\mathbb{Z}^{6g-4}$ by $\Lambda^{6g-4}\Theta$ . If the matrix is not given in bases that make it a diagonal matrix, the integer $\prod d_i$ is equal to the gcd of the maximal minors of the matrix.

As there is a relation between the coordinates corresponding to $\bigoplus_{E(\Gamma')}N/N_e$ , any maximal minor obtained by forgetting a row from the coordinates of $\bigoplus_1^g N$ is 0. The only non-zero maximal minors come from forgetting one of the edges of $\Gamma'$ . Let $e_0$ be such an edge. The matrix of $\Theta$ contains blocks of the following form:

\begin{align*}\begin{array}{|c|}\hline\det\!(u_e,-) \\[5pt] \hline I_2 \\[5pt] \hline\det\!(u_e,-) \\[5pt] \hline\end{array}\;,\end{align*}

where columns are the coordinates of a marked point p, and rows are its position, and the coordinates corresponding to the two adjacent edges. The block $I_2$ is the only non-zero block on the row of the matrix of $\Theta$ . In particular, we can use it to develop each maximal minors with respect to these rows. Then, as the complement of $\mathcal{P}$ inside the curve $\Gamma$ is a tree, we can assume that the edges have been oriented toward the edge whose row we are forgetting. We have already develop the minor with respect to the rows of the marked points. Now, let V be a vertex adjacent to two leafs of the tree, with primitive directing vectors $u_1$ and $u_2$ , and outgoing slope $u_e$ . The balancing condition is $w_e u_e=w_1u_1+w_2u_2$ . The corresponding block is

\begin{align*}\begin{array}{|c|}\hline\det\!(u_e,-) \\[5pt] \hline\det\!(u_1,-) \\[5pt] \hline\det\!(u_2,-) \\[5pt] \hline\end{array}\;,\end{align*}

where columns are coordinates for the copy of N corresponding to V, and the rows to the adjacent edges. As V is adjacent to two leafs, the last two rows of the block are the only non-zero elements in the row of the minor. Thus, we can develop with respect to these two rows and get the smaller determinant where the vertex has been deleted, and the determinant multiplied by $\det\!(u_1,u_2)={m_V}/{w_1w_2}$ . In the end, we get $({1}/{\prod_{e\neq e_0} w_e})\prod_V m_V=({w_{e_0}}/{\prod_e w_e})\prod_V m_V$ .

Recall $\delta$ is the gcd of the weights of the edges. Assume that $\delta=1$ . By Bezout’s theorem, there exists some integer numbers $\alpha_e$ such that $\sum \alpha_e w_e=1$ . Thus, we have

\begin{align*}\sum \alpha_{e_0}\frac{w_{e_0}}{\prod_{e\in E(\Gamma')}w_e}\prod_V m_V=\frac{1}{\prod_{e\in E(\Gamma')}w_e}\prod_V m_V,\end{align*}

which is also an integer. The gcd of the $({w_{e_0}}/{\prod_{e\in E(\Gamma')}w_e})\prod_V m_V$ is then equal to $({1}/{\prod_{e\in E(\Gamma')}w_e})\prod_V m_V$ . Multiplying by $\prod_{e\in E(\Gamma')}w_e$ , we get the result. If $\delta>1$ , we consider the tropical curve where all the weights have been divided by $\delta$ . It has the same matrix $\Theta$ and the same minors. Let $w_e=\delta \widehat{w_e}$ , where $\widehat{w_e}$ are the weights of the curve divided by $\delta$ . For a vertex V, its multiplicity in the curve divided by $\delta$ is $\widehat{m_V}=({m_V}/{\delta^2})$ . The value of the minors are

\begin{align*}\frac{\widehat{w_{e_0}}}{\prod_{e\in E(\Gamma')}\widehat{w_e}}\prod_V \widehat{m_V}.\end{align*}

As in the case $\delta=1$ , their gcd is

\begin{align*}\frac{1}{\prod_{e\in E(\Gamma')}\widehat{w_e}}\prod_V \widehat{m_V}=\delta\frac{1}{\prod_{e\in E(\Gamma')}w_e}\prod_V m_V,\end{align*}

as in Remark 4·11. Multiplying by the weight of the bounded edges, the result follows.

5·2. Independence of the choice of the points

Before getting to the proof of invariance, we notice the following:

1. (i) the invariance for the enumerative problem where you do not fix the value of the gcd of the curves can be deduced from the invariance of the enumerative problem where the gcd is fixed. Indeed, you can sort out the curves solving the problem according to the value of their gcd. If you have invariance for each possible value of the gcd, you globally have invariance;

2. (ii) conversely, assume you have invariance for the problem where you do not fix the value of the gcd. A curve of gcd $\delta$ in the class C is obtained by scaling by $\delta$ a primitive curve in the class $C/\delta$ . Therefore, initialising with primitive classes C (for which all the curves realising the class are also primitive), and proceeding by induction, one can also recover the invariance for the enumerative problem counting curves with a fixed gcd.

Proof of Theorem 4·13. Let $(\mathcal{P}_s)_{s\in [0;1]}$ be a generic path between two generic point configurations $\mathcal{P}_0$ and $\mathcal{P}_1$ inside the tropical torus $\mathbb{T} A=N_\mathbb{R}/\Lambda$ . By composing different paths, we can always assume that only one of the points in the configuration moves. We intend to prove the invariance for the refined count $BG_{g,C,k}(\mathbb{T} A,\mathcal{P}_s)$ when moving one point.

If $\mathcal{P}_s$ is a generic configuration, and $\Gamma$ is a simple tropical curve with gcd k passing through $\mathcal{P}_s$ , considering as lifting set $\{p\in\mathcal{P}_s\;:\;\lambda_p\neq 0\}$ , it is easy to see that a small deformation of the point configuration leads to a small deformation of the curve $\Gamma$ which stays in the same combinatorial type. In fact, only the cycle $\lambda_{p_s}$ moves, where $p_s$ is the moving point, as depicted in Figure 9. As the multiplicity is constant on the combinatorial type, we get that the refined count of solutions is locally invariant. We now have to check what happens when the point configuration crosses a wall, where the configuration $\mathcal{P}_s$ is not generic anymore.

Fig. 9.

Displacement of a point and the corresponding deformation of the cycle that it contains.

Fig. 10.

Displacement of a point and the corresponding deformation across the different walls. In (a), a marked point that coincides with a vertex, in (b), the flattening of a cycle, and in (c), the appearance of a quadrivalent vertex.

When the point configuration becomes non-generic, we have at least a quadrivalent vertex that appears. We notice that when a point moves, the only part of the curve that moves is the cycle $\lambda_p$ containing it, and the edges adjacent to it vary their lengths. Let p be the moving marked point. We distinguish two cases according to whether $\lambda_p=0$ or $\lambda_p\neq 0$ .

1. (i) If $\lambda_p=0$ , then the deformation only affects the cycle of the lifted curve $\tilde{h}\;:\;\widetilde{\Gamma}\to N_\mathbb{R}$ . The walls are dealt with using the invariance statement from [ Reference Itenberg and Mikhalkin17 ]. However, to be able to glue back the unbounded ends and get a curve inside $\mathbb{T} A$ , we have to assume tat the marked points sitting on the unbounded ends stay on the unbounded ends. In this case, the invariance exactly amounts to the local invariance statement from [ Reference Itenberg and Mikhalkin17 ]. One has the wall corresponding to a quadrivalent vertex, a marked point (not belonging to an unbounded end) meeting a vertex, or a cycle degenerating to a pair of parallel edges linking two quadrivalent vertices. These walls are depicted in Figure 10.

   Let us deal with the remaining case. If a marked point q sitting on an unbounded end meets a vertex, there are three adjacent combinatorial types according to which edge adjacent to the vertex contains the marked point. This corresponds to a wall of type (a) in Figure 10. On one of of them, the complement $\Gamma\backslash h^{-1}(\mathcal{P}_s)$ is not simply connected anymore, and therefore cannot happen. In other words, the marked point q has to stay on the cycle $\lambda_q$ , and then moves from to the bounded edge belonging to $\lambda_p\cap\lambda_q$ . The multiplicity is the same on both sides, and the two adjacent combinatorial types contribute solutions for $\mathcal{P}_{s\pm\varepsilon}$ . To represent the deformation, we choose in the cutting set not q but some point on the same edge as q, on the other side of the edges regarding the moving vertex.

2. (ii) Now, assume that $\lambda_p\neq 0$ . It means that now we move a chord (path between two unbounded ends) in the lifted curve. We proceed similarly. As long as the chord does not meet any other vertex sitting on an unbounded end, the invariance follows from the statements of [ Reference Itenberg and Mikhalkin17 ]. If the chord meets some marked point q on an unbounded end, we do not take q in the lifting set anymore but some other point on the edge, not on the side of the chord. The chords $\lambda_p$ and $\lambda_q$ overlap on a segment, and the marked point q stays on the chord $\lambda_q$ . The multiplicities also stay the same on both combinatorial type, and they provide solutions for $\mathcal{P}_{s\pm\varepsilon}$ respectively.

Finally, for each kind of wall, the gcd of the curves in the adjacent combinatorial types are the same. The result follows.

5·3. Independence of the choice of the surface

We now consider the dependence of the refined invariants when changing the lattice inclusion $j\;:\;\Lambda\hookrightarrow N_\mathbb{R}$ , i.e. changing the matrix S so that the relation $C S^T\in\mathcal{S}_2(\mathbb{R})$ for a fixed C keeps being satisfied.

The subtlety is that for a non-generic matrix $S_{t_*}$ , there are several decompositions $C=C_1+C_2$ , and the reducible curves on $\mathbb{T} A_{t_*}$ can be deformed to irreducible curves in the class C for nearby matrices $S_t$ , $t\in]t_*-\varepsilon;\; t_*+\varepsilon[$ . Let $h\;:\;\Gamma\to\mathbb{T} A_{t_*}$ be a genus g parametrised tropical curve in the class C passing through the point configuration $\mathcal{P}_{t_*}$ . As $\mathcal{P}_{t_*}$ is generic, and we do not assume irreducibility but only the genus, the irreducible components of $\Gamma$ are simple, and their genera add up to g.

1. (i) If $\Gamma$ is irreducible. Use the cutting process to lift it to a curve inside $N_\mathbb{R}$ satisfying point constraints and moments constraints so that the gluing is possible. Then, as the curve is regular, a small deformation of the constraints yields a small deformation of the curve, and the multiplicity stays the same. In particular, a deformation of the gluing condition yields a small deformation of the curve and it provides solutions in the same combinatorial type for all $t\in]t_*-\varepsilon;\; t_*+\varepsilon[$ .

2. (ii) If $\Gamma$ is reducible but with irreducible components that realise some class proportional to C, we reduce to the previous case by deforming each of the corresponding components. Thus, it is not possible to deform them to an irreducible simple tropical curves: assume that the curve has two components. A deformation into an irreducible curve would result from the smoothing of a unique intersection point between the irreducible components: at least one to get a connected curve, and at most one to keep the right genus. Then, either the edge resulting from the smoothing is disconnecting and thus has slope 0, or it is not disconnecting but has length 0 due to the moment condition. See next point for more details.

3. (iii) Assume that $\Gamma=\Gamma_1\sqcup\cdots\sqcup\Gamma_r$ is reducible with r connected components, not all of them realising a class proportional to C. In particular, the matrix $S_{t_*}$ is not generic. We assume that none of the classes is proportional to C, since the previous points allow one to deform separately these components. Then, choose a minimal subset $\mathcal{Q}$ among the intersection points between the components $\Gamma_i$ , and such that the graph obtained from $\Gamma$ adding edges corresponding to the intersection points has genus g, and the new components realise classes proportional to C. We obtain new parametrised tropical curves that have some contracted edges. The edges corresponding to the intersection points in $\mathcal{Q}$ are contracted since they are disconnecting. When deforming the torus and the curve along with it, each node gets resolved in one of the two possible ways, with an edge that is not contracted anymore.

Example 5·5. Consider the class $C=\left(\begin{smallmatrix}2 & 0 \\[5pt] 0 & 3 \end{smallmatrix}\right)$ inside the torus for $S=\left(\begin{smallmatrix}8 & 0 \\[5pt] 0 & 6 \end{smallmatrix}\right)$ . The corresponding tropical torus contains curves in the classes $\left(\begin{smallmatrix}1 & 0 \\[5pt] 0 & 0 \end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}0 & 0 \\[5pt] 0 & 1 \end{smallmatrix}\right)$ , which are not realisable for a small deformation of the torus. Consider the curve depicted in Figure 11 (b), which is comprised of 5 irreducible components, all of genus 1. The set $\mathcal{Q}$ is depicted with a $\times$ . Then, deforming the matrix S, it is possible to get a curve of genus 5 by smoothing the points of $\mathcal{Q}$ , and get the curve in Figure 11 (c).

Consider the graph $\mathcal{T}$ whose vertices are the components $\Gamma_j$ , and vertices are linked by an edge if the corresponding components share a point $q\in\mathcal{Q}$ . The genus of the curve obtained by merging the components is $\sum g(\Gamma_i)+g(\mathcal{T})$ . As g is already equal to to the sum of the genera of the $\Gamma_j$ , the genus of $\mathcal{T}$ is 0. In other words, smoothing too many intersection points yields a curve of the wrong genus.

Fig. 11.

In (b) a reducible curve with components in classes that do not survive the torus deformation, and in (c) the corresponding deformation for a specific set of intersection points between the irreducible components, giving the graph from (a).

Fig. 12.

In (b) a reducible curve with components in classes that do not survive the torus deformation, and in (c) the corresponding deformation for a specific set of intersection points between the irreducible components, giving the graph from (a).