# Tropical curves in abelian surfaces I:
enumeration of curves passing through points

Thomas Blomme

###### Abstract

This paper is the first part in a series of three papers devoted to the study
of enumerative invariants of abelian surfaces through the tropical approach.
In this paper, we consider the enumeration of genus $g$ curves of fixed degree
passing through $g$ points. We compute the multiplicity provided by a
correspondence theorem due to T. Nishinou and show that it is possible to
refine this multiplicity in the style of Block-Göttsche to get tropical
refined invariants.

###### :

14N10, 14T90, 11G10, 14K05, 14H99, 14N35

###### keywords:

Enumerative geometry, tropical refined invariants, abelian surfaces
Thomas Blomme, Univerité de Genève
Email<EMAIL_ADDRESS>
Data Statement: I do not have any data to point.

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
    2. 1.2 Results
      1. 1.2.1 Multiplicity of the tropical curves.
      2. 1.2.2 Tropical refined invariants
    3. 1.3 Plan of the paper
  2. 2 Tropical Abelian surfaces
    1. 2.1 Tropical tori
    2. 2.2 Tropical homology and intersection form
    3. 2.3 Degeneration of complex family to a tropical tori
      1. 2.3.1 Abelian surfaces in multiplicative form.
      2. 2.3.2 Line bundles on abelian surfaces
      3. 2.3.3 Deformations.
  3. 3 Tropical curves in abelian surfaces
    1. 3.1 Parametrized tropical curves
      1. 3.1.1 Abstract tropical curves.
      2. 3.1.2 Parametrized tropical curves.
      3. 3.1.3 Degree of a tropical curve.
      4. 3.1.4 Miscellaneous
    2. 3.2 Cutting/lifting procedure
    3. 3.3 Dimension of the moduli space
  4. 4 Enumerative problem and counts
    1. 4.1 Enumerative problem
    2. 4.2 Correspondence theorem
    3. 4.3 Statement of the results
  5. 5 Proof of the results
    1. 5.1 Multiplicity as a product
    2. 5.2 Independence of the choice of the points
    3. 5.3 Independence of the choice of the surface
  6. 6 Examples

##  1 Introduction

###  1.1 Overview

This paper is the first of a series of three papers whose purpose is to study
the enumerative invariants of abelian surfaces. The first paper is dedicated
to presenting the setting of tropical abelian surfaces and tropical curves,
and studying the enumeration of genus $g$ curves passing through $g$ points.
The second paper focuses on the enumeration of genus $g$ curves belonging to a
fixed linear system that pass through $g-2$ points. The third paper provides a
pearl diagram algorithm that enables a concrete computation of the invariants
introduced in the first two papers.

##### Abelian surfaces and tropical tori.

We consider tropical tori. These are quotients of a real vector space
containing some lattice, by some other lattice. Namely, we choose the real
vector space to be $N_{\mathbb{R}}=N\otimes\mathbb{R}$ for some lattice $N$,
and the other lattice is denoted by $\Lambda\subset N_{\mathbb{R}}$. The
inclusion $\Lambda\hookrightarrow N_{\mathbb{R}}$ is denoted by $S$. The case
of surfaces corresponds to the case where both lattices are of rank $2$. This
construction is analog to the construction of complex tori as a quotient of
the algebraic torus $(\mathbb{C}^{*})^{n}$ by a sublattice of maximal rank.
Notice that complex tori are usually presented as the quotient of a complex
vector space by a full-rank additive lattice.

A complex torus (resp. tropical torus) is an abelian variety if it is given
the choice of a polarization, which is a line bundle with positive integer
Chern class. In the complex case, this condition is equivalent to the Riemann
bilinear relation from [12], which gives a criteria for the existence of a
positive integer $(1,1)$-class. In the tropical setting, this condition
translates to the existence of an element $C\in\Lambda\otimes N$ such that the
product $CS^{T}\in(N\otimes N)_{\mathbb{R}}$ is symmetric. Once such a choice
has been made, it is possible to study curves in the class $C$.

##### Enumerative geometry and Gromov-Witten invariants.

Given an integer class $C$ that contains curves, it is natural to try to count
curves in the class that satisfy certain given conditions. Such conditions are
usually fixing the genus, and some geometric conditions such as passing
through points. For instance, in the projective plane $\mathbb{C}P^{2}$, look
at degree $d$ and genus $g$ curves that pass through $3d-1+g$ points. To be
able to consider such enumerative problems, one has to know the dimension of
the moduli space of curves so that we impose the right number of constraints.
In the case of abelian surfaces, the dimension of the moduli space of genus
$g$ curves in a given class $C$ is $g$, meaning we have to impose $g$
constraints to expect a finite number of curves.

For curves in abelian surfaces, there are two main enumerative problem that we
can consider:

  * $\circ$

How many genus $g$ curves in the class $C$ pass through $g$ points ?

  * $\circ$

How many genus $g$ curves in a fixed linear system pass through $g-2$ points ?

In the second case, fixing the linear system imposes a codimension $2$
condition because the group of line bundles on an abelian surface has
dimension $2$. In this paper, we are mainly interested to the first
enumerative problem. The second enumerative problem is adressed in the second
paper of the series.

Sometimes, and it is here the case, the number of solutions to an enumerative
problem does not depend on the choice of the constraints as long as it is
generic. The result is thus called an invariant. In fact, as proved in [8],
these enumerative invariants coincide with Gromov-Witten invariants. The
latter are defined as integrals of cohomology classes in the moduli space of
curves inside the ambiant variety over some virtual fundamental class in the
moduli space of curves. By integrating different cohomology classes, we see
that the enumerative invariants are part of a much bigger family of
invariants, not all of them having an enumerative interpretation. For
instance, one can integrate $\lambda$-class or $\psi$-class. The invariants
with integration of a $\lambda$-class might be related to the refined
invariants introduced in this paper, as it is the case for toric surfaces [7].

Despite the enumerative counts being invariant, their computation often
remains a challenge. The enumerative geometry of curves inside abelian
surfaces has already been studied by J. Bryan and N. Leung [8], and J. Bryan,
B. Oberdieck, R. Pandharipande and Q. Yin [9], although in these cases, the
authors restricted to the case of primitive classes, which are classes $C$
that cannot be expressed as a multiple of a smaller class.

##### Tropical geometry and Correspondence Theorem.

As the numbers we are looking for do not depend on the choice of the
constraints, it is now a standard approach to try to compute them close to the
so-called tropical limit. This was first done in the case of Severi degrees of
toric varieties by G. Mikhalkin in [17]. Close to the tropical limit, the
curves solution to the enumerative problems break into several components
whose structure is encoded in objects called tropical curves. Tropical curves
are finite graphs whose edges have integer slope and satisfy a balancing
condition. Then, a suitable correspondence theorem allows one to recover the
solutions to the enumerative problem close to the tropical limit from the
tropical curves. The computation of the classical invariants has thus been
reduced to a tropical enumerative problem, and the result is obtained by
counting the tropical solutions with suitable multiplicities.

For the case of abelian surfaces, T. Nishinou gives a correspondence theorem
in [19] for genus $g$ curves passing through $g$ points inside an abelian
surface. One of the main results of [19] consists in providing the
multiplicity $m_{\Gamma}^{\mathbb{C}}$ with which to count the tropical
solutions, so that their count gives the number of complex genus $g$ curves in
a fixed class $C$ passing through a generic point configuration inside a
complex abelian variety. The multiplicity provided in [19] lacks an easy
expression such as the one from the toric setting in [17], and the tropical
enumerative problem has yet to be resolved.

In the setting of toric surfaces, the multiplicity provided by the
correspondence theorem from [17] expresses as a product over the vertices of a
tropical curve. In different settings, such as for instance in dimension
bigger than $3$, the generalizations of the correspondence theorem [20] [16]
express the tropical multiplicities as the determinant of some map naturally
associated to each tropical curve. In the case of abelian surfaces, the
correspondence theorem from [19] does not express the multiplicity as a
determinant because unlike all the cases previously considered, all the
tropical curves are superabundant, meaning the dimension of their deformation
space does not match the expected dimension: they vary in dimension $g$ while
the expected dimension is $g-1$. The multiplicity is thus defined a little bit
differently, and though it can still be expressed as a determinant, this
changes its computation. See section 4.2 for more details.

##### Refined invariants.

In the case of toric surfaces, the multiplicity provided by the correspondence
theorem from [17] is expressed as a product over the vertices of the tropical
curve. In [3], F. Block and L. Göttsche proposed to replace the vertex
multiplicities by their quantum analog: an integer $a$ is replaced by
$[a]_{q}=\frac{q^{a/2}-q^{-a/2}}{q^{1/2}-q^{-1/2}}$. This new multiplicity is
a symmetric Laurent polynomial in a new variable $q$. Surprisingly, in many
situations, the count of tropical curves solution to a suitable enumerative
problem using this new multilicity happens not to depend either on the choice
of the connstraints, as proven by I. Itenberg and G. Mikhalkin [15] in the
case of tropical curves in toric surfaces. This may not be expected because
these multiplicities are not provided by a correspondence that guarantees that
the count of tropical curves solutions to an enumerative problem leads to an
invariant. The meaning of these refined invariants in classical geometry is
thus a natural question. Such refined invariants have since been extended to
many other situations. See for instance [10], [22], [1], [21], [5]. The
setting of floor diagrams has also been adapted to enable the computation of
some of these refined invariants in [2].

These refined invariants were first defined purely tropically, and they
possess a curious relationship to various other invariants in classical
geometry. One of their conjectured interpretation is that they should
coincide, in a certain sense, with the refinement of the Euler characteristic
of some relative Hilbert scheme by its Hirzebruch genus [11]. Yet, other
interpretations have since been found. In some situations, they have been
proven to coincide with real refined classical invariants, obtained by
counting real oriented rational curves according to the value of some quantum
index [18], [6]. In other situations, P. Bousseau proved that through the
change of variable $q=e^{iu}$, their generating series is connected to the
generating series of Gromov-Witten invariants with the integration of a
$\lambda$-class [7]. The relation between these different approaches remains
unclear. Yet, one common point in the last approaches is that they both need
to cancel the denominators of the vertex multiplicities, suggesting that the
refined multiplicity should in fact just be $\prod(q^{m_{V}/2}-q^{-m_{V}/2})$.

###  1.2 Results

The present paper provides new results in several directions. Shortly, we give
a concrete formula to compute the complex multiplicity of a parametrized
tropical curve provided by Nishinou’s correspondence theorem [19], we extend
the setting of refined invariants to the case of tropical abelian surfaces.
Concrete computations are enabled by a pearl diagram algorithm presented in
the third paper.

####  1.2.1 Multiplicity of the tropical curves.

We consider a genus $g$ trivalent parametrized tropical curve
$h:\Gamma\to\mathbb{T}A$ (see Section 3 for definitions) passing through a
generic configuration of $g$ points, where
$\mathbb{T}A=N_{\mathbb{R}}/\Lambda$ is a tropical torus. We define the gcd of
the curve $\delta_{\Gamma}$ as the gcd of the weights of the edges. See
section 3 for more details. The following theorem gives a concrete formula to
compute the complex multiplicity provided by Nishinou’s correspondence theorem
[19].

######  Theorem.

4.3 The multiplicity $m_{\Gamma}^{\mathbb{C}}$ provided by Nishinou’s
correspondence theorem splits as follows:

$m_{\Gamma}^{\mathbb{C}}=\delta_{\Gamma}\prod_{V\in V(\Gamma)}m_{V},$

where the vertex multiplicity $m_{V}=|\det(a_{V},b_{V})|$ is the determinant
of two out of the three outgoing slopes, and $\delta_{\Gamma}$ is the gcd of
the tropical curve $\Gamma$.

The appearance of the gcd of the tropical curve may come as a surprise.
However, it can be expected by considering an argument of homogeneity in the
definition given of [19].

The presence of the gcd of the tropical curve means that the complex
multiplicity involves some global information on the curve. This does not
prevent from computing the multiplicity of a unique tropical curve but might
bring complications when trying to count the tropical curves passing through
$g$ points because one has to keep track of this gcd.

The expression as a product over the vertices of the tropical curve suggests
that a refinement as proposed by F. Block and L. Göttsche [3] should also
provide tropical refined invariants in this situation. We thus introduce the
refined multiplicity of a tropical curve to be

$m_{\Gamma}^{q}=\prod_{V\in
V(\Gamma)}[m_{V}]_{q}=\prod_{V}\frac{q^{m_{V}/2}-q^{-m_{V}/2}}{q^{1/2}-q^{-1/2}}\in\mathbb{Z}[q^{\pm
1/2}].$

####  1.2.2 Tropical refined invariants

We give an invariance statement for the count of tropical curves of genus $g$
in a class $C$ passing through a configuration $\mathcal{P}$ of $g$ points
using the previous multiplicities. In fact, one has an even more refined
invariance by only considering the curves having a fixed gcd. This is due to
the fact that through the deformations induced by the moving of the point
configuration $\mathcal{P}$, the gcd of the tropical curves is preserved. In
other words, all the tropical curves passing through the some point
configuration split in different groups according to the value of their gcd,
and we have invariance in each group. Thus, it is possible to count tropical
curves with a fixed gcd and partially get rid of the factor $\delta_{\Gamma}$
in the complex multiplicity. We thus introduce the following counts of
tropical curves:

$\displaystyle N_{g,C,k}(\mathbb{T}A,\mathcal{P})$
$\displaystyle=\sum_{\begin{subarray}{c}h(\Gamma)\supset\mathcal{P}\\\
\delta_{\Gamma}=k\end{subarray}}m_{\Gamma}\in\mathbb{N},$ $\displaystyle
BG_{g,C,k}(\mathbb{T}A,\mathcal{P})$
$\displaystyle=\sum_{\begin{subarray}{c}h(\Gamma)\supset\mathcal{P}\\\
\delta_{\Gamma}=k\end{subarray}}m^{q}_{\Gamma}\in\mathbb{Z}[q^{\pm 1/2}],$
$\displaystyle M_{g,C}(\mathbb{T}A,\mathcal{P})$
$\displaystyle=\sum_{h(\Gamma)\supset\mathcal{P}}m_{\Gamma}=\sum_{k|\delta(C)}N_{g,C,k}(\mathbb{T}A,\mathcal{P})\in\mathbb{N},$
$\displaystyle N_{g,C}(\mathbb{T}A,\mathcal{P})$
$\displaystyle=\sum_{h(\Gamma)\supset\mathcal{P}}\delta_{\Gamma}m_{\Gamma}=\sum_{k|\delta(C)}kN_{g,C,k}(\mathbb{T}A,\mathcal{P})\in\mathbb{N},$
$\displaystyle R_{g,C}(\mathbb{T}A,\mathcal{P})$
$\displaystyle=\sum_{h(\Gamma)\supset\mathcal{P}}\delta_{\Gamma}m^{q}_{\Gamma}=\sum_{k|\delta(C)}kBG_{g,C,k}(\mathbb{T}A,\mathcal{P})\in\mathbb{Z}[q^{\pm
1/2}],$ $\displaystyle BG_{g,C}(\mathbb{T}A,\mathcal{P})$
$\displaystyle=\sum_{h(\Gamma)\supset\mathcal{P}}m^{q}_{\Gamma}=\sum_{k|\delta(C)}BG_{g,C,k}(\mathbb{T}A,\mathcal{P})\in\mathbb{Z}[q^{\pm
1/2}].$

All the counts are over irreducible tropical curves $h:\Gamma\to\mathbb{T}A$
of genus $g$ in the class $C$ that pass through the point configuration
$\mathcal{P}$. The first two counts are over the curves that have a fixed gcd
$k$. The other counts are over all the tropical curves (without gcd
constraints) and can thus be expressed as a linear combination of the first
two ones. The first count is obtained by specializing the second one at $q=1$.
Thus, an invariance for the second count yields an invariance for all the
others. We have the following invariance statements. First, we have an
invariance regarding the point configuration $\mathcal{P}$.

######  Theorem.

4.4 The refined count $BG_{g,C,k}(\mathbb{T}A,\mathcal{P})$ does not depend on
the choice of $\mathcal{P}$ as long as it is generic.

The corresponding invariant is denoted by $BG_{g,C,k}(\mathbb{T}A)$. Then, we
have an invariance concerning the choice of the tropical torus $\mathbb{T}A$.

######  Theorem.

4.6 The refined invariant $BG_{g,C,k}(\mathbb{T}A)$ does not depend on
$\mathbb{T}A$ as long as $\mathbb{T}A$ contains curves in the class $C$ and is
chosen generically among them.

The dependence of $BG_{g,C,k}(\mathbb{T}A)$ in terms of $\mathbb{T}A$ depends
in fact on the lattice generated by realizable classes in the tropical torus.
Choosing bases of $N$ and $\Lambda$, the tropical torus
$N_{\mathbb{R}}/\Lambda$ is defined by the $2\times 2$ real matrix of the
inclusion $S:\Lambda\hookrightarrow N_{\mathbb{R}}$, and the classes that are
realizable are given by integer matrices $C$ such that
$CS^{T}\in\mathcal{S}_{2}(\mathbb{R})$. If $C$ is fixed and $S$ chosen
generically so that this condition is satisfied, there will be no other
matrices satisfying the condition, and the multiples of $C$ will be the only
classes realizable by curves in the tropical torus associated to $S$. If $S$
is not chosen generically, there may be classes $C_{1}$ and $C_{2}$ such that
$C_{1}+C_{2}=C$.

Due to the fact that the dimension of the deformation space of genus $g$
tropical curves in any class is equal to $g$, among the genus $g$ curves
passing through a point configuration $\mathcal{P}$, there are reducible
curves having irreducible components of genera that add up to $g$, splitting
the point constraints among the different components. When deforming the
tropical torus $\mathbb{T}A$, which means changing the matrix $S$, it is
possible to deform the irreducible components whose class keeps being
realizable. This is the case of the irreducible components whose class is
proportional to $C$. However, if for a non-generic $S$ we have a decomposition
of $C$ as a sum of classes that are not proportional to $C$, for instance
$C=C_{1}+C_{2}$, and it is not possible to separately deform the components.
In other words, when deforming $\mathbb{T}A$, a family of irreducible curves
might deform to a reducible curve. The reducible curve gives no contribution
to $BG_{g,C}(\mathbb{T}A)$, but its deformation contributes for nearby
$\mathbb{T}A$. See section 5.3 for more details. Thus, the invariant
$BG_{g,C}(\mathbb{T}A)$ is not the same for any $\mathbb{T}A$.

All the invariants are defined purely tropically. The count $N_{g,C}$ is
interesting because Nishinou’s correspondence theorem [19] ensures that it
matches the number of genus $g$ curves in the class $C$ passing through $g$
points for a complex abelian surface, number that we denote by
$\mathcal{N}_{g,C}$. The meaning of the refined invariants remains open
although one can probably adapt results from [7] to show that they coincide
with Gromov-Witten invariants with insertion of $\lambda$-classes. Moreover,
due to the invariance for each class of curves with fixed gcd, we have many
refined invariants since it is possible to take any linear combination of the
first invariants. Maybe some other choices could have an interpretation in
complex or real geometry.

###  1.3 Plan of the paper

The paper is organized as follows.

  * -

The second section defines tropical tori and a description of the families of
complex tori that tropicalize to a given tropical torus. If one knows the
description of a tropical torus as $\mathbb{T}A=N_{\mathbb{R}}/\Lambda$, that
the degree is the data of a class $C\in\Lambda\otimes N$, and that a class $C$
is realizable if and only if $CS^{T}$ is symmetric, this section can be
skipped in a first lecture.

  * -

In the third section, we give definitions for tropical curves inside tropical
abelian surfaces, a way to relate them to tropical curves in the plane
$N_{\mathbb{R}}$, and we compute the dimension of their deformation space.

  * -

The fourth section is devoted to stating the invariance results of the paper:
first we set the enumerative problems, then we recall Nishinou’s
correspondence theorem, and last we state the invariance results. All these
results are proved in the fifth section.

  * -

Theorem 4.3 relies on the computation of the multiplicity of a planar tropical
curve and is recalled in Appendix.

Acknowledgments. The author is grateful to Hannah Markwig for some helpful
remarks and kindly agreeing to read the paper. Research is supported in part
by the SNSF grant 204125.

##  2 Tropical Abelian surfaces

###  2.1 Tropical tori

We follow partially the notations from [13] and use the following definition
of a tropical torus. In all what 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

$\mathbb{T}A=N_{\mathbb{R}}/\Lambda,$

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 $\mathbb{T}A$. The
lattice $N$ lives in fact in the tangent bundle of $\mathbb{T}A$, which is the
trivial bundle with fiber $N_{\mathbb{R}}$. It is possible to fix a basis of
$N$ and define $\mathbb{T}A$ just as a quotient $\mathbb{R}^{2}/\Lambda$, but
this definition does not handle a change of basis, 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

$(\alpha,\gamma)\in
M\times\Lambda\longmapsto\int_{\gamma}\alpha\in\mathbb{R},$

corresponds exactly to the map

$(m,\lambda)\in M\times\Lambda\longmapsto\langle
m,S\lambda\rangle\in\mathbb{R},$

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 [14]. 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:

$\begin{matrix}&&&\Lambda^{2}N\otimes\Lambda^{2}\Lambda&&\\\
&&\Lambda^{2}N\otimes\Lambda&&N\otimes\Lambda^{2}\Lambda&\\\
H_{p,q}(\mathbb{T}A)=&\Lambda^{2}N&&\Lambda\otimes N&&\Lambda^{2}\Lambda\\\
&&N&&\Lambda&\\\ &&&\mathbb{Z}&&\\\ \end{matrix}.$

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 $\Lambda^{2}N$ 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,

$(\lambda\otimes n)\cdot(\lambda^{\prime}\otimes
n^{\prime})=\det(\lambda,\lambda^{\prime})\det(n,n^{\prime}).$

######  Remark 1.

In our case, the tropical homology groups can be seen as the homology group
with value in the homology of a torus
$N_{\mathbb{R}/\mathbb{Z}}=N\otimes\mathbb{R}/\mathbb{Z}$. Thus, they are the
homology groups of some $N_{\mathbb{R}/\mathbb{Z}}$-principal bundle over
$N_{\mathbb{R}}/\Lambda$. Topologically, this space is a $4$-dimensional
torus, which is just the associated complex variety, as depicted in the next
section.

###  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
representation of the form
$\mathbb{C}A=\mathbb{C}^{2}/\Lambda_{\mathrm{add}}$, where
$\Lambda_{\mathrm{add}}$ is the image of a matrix
$\Omega\in\mathcal{M}_{2,4}(\mathbb{C})$. Up to a change of coordinate inside
the lattice $\Lambda_{\mathrm{add}}$ and the vector space $\mathbb{C}^{2}$,
one can always assume that the matrix $\Omega$ is of the form

$\Omega=\begin{pmatrix}I_{2}&Z\\\ \end{pmatrix}.$

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:

$\mathbb{C}A=N_{\mathbb{C}}/\Lambda_{\mathrm{add}}\longrightarrow
N_{\mathbb{C}^{*}}/\Lambda_{\mathrm{mult}},$

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.

The downfall of the multiplicative notation is that we implicitly fix a
preferred sublattice of $\Lambda_{\mathrm{add}}$, 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 [12], a
integer skew-symmetric matrix $Q$ is the Chern class of a positive line bundle
if and only if

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

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:

$Q=\begin{pmatrix}0&-c\\\ c^{T}&0\\\ \end{pmatrix}.$

In our previous notations, this matrix $Q$ can be seen as an application
$N\oplus\Lambda\to M\oplus\Lambda^{*}$, and $c:\Lambda\to M$. In fact, one can
even assume that the matrix $c$ is of the form $\begin{pmatrix}\delta_{1}&0\\\
0&\delta_{2}\\\ \end{pmatrix}$, where $\delta_{1}|\delta_{2}$. As
$Q^{-1}=\begin{pmatrix}0&c^{-1T}\\\ -c^{-1}&0\\\ \end{pmatrix}$, the Riemann
bilinear condition then becomes

$\left\\{\begin{array}[]{l}Zc^{-1}\text{ is symmetric,}\\\
\mathfrak{Im}(-Zc^{-1})\text{ is positive definite.}\\\ \end{array}\right.$

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 realized 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 $C\in
H_{2}(\mathbb{C}A,\mathbb{Z})$ realized 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
$Z_{t}=\frac{1}{2i\pi}(A+S\log t)$, 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 $\frac{1}{2i\pi}(A+S\log t)$ is satisfied close to
the limit. In the exponential notations, our family of tori $\mathbb{C}A_{t}$
is of the form

$\mathbb{C}A_{t}=N_{\mathbb{C}^{*}}/\langle e^{A}t^{S}\rangle,$

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

######  Remark 3.

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

$0\to\ker\pi\to
N_{\mathbb{C}^{*}}/\Lambda_{t}\xrightarrow{\pi=|\bullet|}N_{\mathbb{R}}/\log|\Lambda_{t}|\to
0.$

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:

$H_{1}(\mathbb{C}A_{t},\mathbb{Z})=N\oplus\Lambda_{t}.$

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:

$H^{1}(\mathbb{C}A_{t},\mathbb{Z})=M\oplus\Lambda^{*},$

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

$H^{2}(\mathbb{C}A_{t},\mathbb{Z})=\Lambda^{2}\Lambda^{*}\oplus
M\otimes\Lambda^{*}\oplus\Lambda^{2}M.$

######  Remark 4.

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 tropicalization of the family
$\mathbb{C}A_{t}$ is the tropical torus $\mathbb{T}A=N_{\mathbb{R}}/\Lambda$.
In [19], T. Nishinou constructs 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 periodization 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 Tropical curves in abelian surfaces

###  3.1 Parametrized tropical curves

####  3.1.1 Abstract tropical curves.

An abstract tropical curve is a finite metric graph $\Gamma$ that may possess
several edges of infinite length, called unbounded ends, which all need to be
adjacent to univalent vertices. The number of neighbors of a vertex is called
its valency. Its genus $g$ is equal to its first Betti number, i.e.
$b_{1}(\Gamma)=g$. The vertices have no genus. The set of edges is denoted by
$E(\Gamma)$ and its set of vertices by $V(\Gamma)$. The curve is said to be
trivalent if every vertex has three neighbors. The length of an edge $e$ is
denoted by $l_{e}$. An isomorphism between two abstract tropical curves is an
isometry. A marked abstract tropical curve is an abstract tropical curve with
the choice of $n$ points on it, labeled by $[\\![1;n]\\!]$. By refining the
graph structure, we can assume that the marked points are vertices of the
graph, which are generically bivalent.

####  3.1.2 Parametrized tropical curves.

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

######  Definition 3.1.

A parametrized 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

  * -

$h$ is affine with slope in $N$ on each edge of $\Gamma$. For an oriented edge
$e$, its slope $u_{e}$ is of the form $w_{e}u^{\prime}_{e}$ where
$u^{\prime}_{e}\in N$ is a primitive vector, and $w_{e}$ a positive integer
called the weight of the edge.

  * -

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

$\sum_{e\ni V}w_{e}u_{e}=0,$

when each $e$ is oriented with $V$ as its source.

######  Remark 5.

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

|  | $2$$2$
---|---|---
$(a)$ | $(b)$ | $(c)$
Figure 1: Three examples of tropical curves in tropical tori. The first and
third are of genus $2$, and the second of genus $5$.

######  Example 3.1.

On Figure 1, we can see three different examples of tropical curves in
different tropical tori $\mathbb{T}A$.  $\lozenge$

####  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 realizes a $1$-chain

$\sum_{e\in E(\Gamma)}u_{e}e\in C_{1}(\mathbb{T}A,N).$

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

######  Definition 3.2.

The class $C$ in $\Lambda\otimes N$ realized by a parametrized 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 6.

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 3.2.

Let $C^{\prime}$ be the class defined in the proposition. We check that
$C^{\prime}$ 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 realized by tropical curves are of a very specific kind: they
satisfy the following property.

######  Proposition 7.

A class $C\in\Lambda\otimes N$ can be realized by a tropical curve if and only
if $CS^{T}$ is symmetric.

######  Proof 3.3.

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 [18], 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 next section for more details.

######  Remark 8.

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 realizable 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 realizable classes is some
subset of the intersection between an hyperplane and the lattice
$\Lambda\otimes N$. Generically, there will be none. Alternatively, the
realizability of a given class $C$ imposes a codimension $1$ constraint on
$S$.

######  Example 3.4.

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
on 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
$S=\begin{pmatrix}a&b\\\ b&a\\\ \end{pmatrix}$, and the curve represents the
class $C=\begin{pmatrix}1&0\\\ 0&1\\\ \end{pmatrix}$. We easily check that
$CS^{T}$ is symmetric.

  2. $(b)$

We have $S=\begin{pmatrix}12&2\\\ 3&8\\\ \end{pmatrix}$, and the curve
represents the class $C=\begin{pmatrix}2&0\\\ 0&3\\\ \end{pmatrix}$, and we
have again $CS^{T}$ symmetric.

  3. $(c)$

We have $S=\begin{pmatrix}4&-1\\\ -2&3\\\ \end{pmatrix}$, and the curve
represents the class $C=\begin{pmatrix}2&1\\\ 0&1\\\ \end{pmatrix}$, 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
$N_{\mathbb{R}}$.  $\lozenge$

######  Example 3.5.

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
$S=\begin{pmatrix}a(\alpha_{1}+\alpha_{2})+b\alpha_{1}&a(\alpha_{1}+\alpha_{2})+c\alpha_{2}\\\
a(\beta_{1}+\beta-2)+b\beta_{1}&a(\beta_{1}+\beta_{2})+c\beta_{2}\\\
\end{pmatrix}$, and $C=\begin{pmatrix}\alpha_{1}&\alpha_{2}\\\
\beta_{1}&\beta_{2}\\\ \end{pmatrix}$. We can check that $CS^{T}$ is always
symmetric. We indeed have a genus $2$ tropical curve in the class $C$, as
depicted on 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.  $\lozenge$

| $3$$2$
---|---
$(a)$ | $(b)$
Figure 2:  Examples of genus $2$ tropical curve.

####  3.1.4 Miscellaneous

We end with some definition that intervenes at different moments of the paper
regarding the enumerative problems and multiplicities.

######  Definition 3.3.

  * -

For an irreducible tropical curve $h:\Gamma\to\mathbb{T}A$, its gcd
$\delta_{\Gamma}$ is the gcd of the weights of its edges.

  * -

An irreducible curve is said to be primitive if its gcd is $1$.

  * -

For an irreducible curve $h:\Gamma\to\mathbb{T}A$ and an integer $k$, we have
a tropical curve $kh:k\Gamma\to\mathbb{T}A$ where the curve $k\Gamma$ is
$\Gamma$ where the lengths of the edges have been divided by $k$, and the
slope of $kh$ on the edges is the slope of $h$ multiplied by $k$. This curve
is denoted by $k\Gamma$.

In particular, $\delta_{k\Gamma}=k\delta_{\Gamma}$, and every irreducible
parametrized tropical curve $h:\Gamma\to\mathbb{T}A$ can be written as
$\delta_{\Gamma}\Gamma^{\prime}$ for a primitive parametrized tropical curve
$\Gamma^{\prime}$. For a reducible curve, we can speak about the gcd of each
of its irreducible components.

###  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 $g$ parametrized 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:

  * -

If $e$ is disconnecting, we have $\Gamma-e=\Gamma_{1}\sqcup\Gamma_{2}$ and we
reparametrize $h:\Gamma\to\mathbb{T}A$ by $\Gamma_{1}\sqcup\Gamma_{2}$, which
is a reducible curve, with components of respective genera $g_{1}$ and $g_{2}$
with $g_{1}+g_{2}=g$.

  * -

If $e$ is not disconnecting, $\Gamma-e$ is a connected curve of genus $g-1$.

######  Definition 3.4.

A subset $\mathcal{Q}\subset\Gamma$ of points located on the edges of $\Gamma$
is said to be lifting if the image of the morphism
$h_{*}:\pi_{1}(\Gamma\backslash\mathcal{Q})\to\pi_{1}(\mathbb{T}A)=\Lambda\simeq\mathbb{Z}^{2}$
is trivial.

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

$\tilde{h}:\Gamma\backslash\mathcal{Q}\longrightarrow N_{\mathbb{R}}.$

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

$\tilde{h}:\widetilde{\Gamma}_{\mathcal{Q}}\longrightarrow N_{\mathbb{R}},$

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

######  Example 3.6.

On Figure 3 we see a tropical curve of genus $5$ in the class
$\begin{pmatrix}2&0\\\ 0&2\\\ \end{pmatrix}$ in some tropical torus. We choose
$5$ points on it so that the complement has no cycle. This choice is depicted
on $(a)$. Then we unfold it to the universal cover and consider a connected
component of the complement of the marked point. We get $(b)$. Finally, we
prolong to infinity the edges cut by a point of the lifting set to get a
tropical curve in $N_{\mathbb{R}}$ on $(c)$. Notice that the unbounded end of
the lifted curve are paired together.  $\lozenge$

$\times$$\times$$\times$$\times$$\times$ | $\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$ |
---|---|---
$(a)$ | $(b)$ | $(c)$
Figure 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 9.

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

$\det(n_{i},e_{i})+\det(-n_{i},f_{i})=\det(n_{i},S\gamma).$

######  Proof 3.7.

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 parametrize the loop $\gamma$ by the path $\tilde{\gamma}$:

$\det(n_{i},S\gamma)=\sum_{k}\det(n_{i},l_{k}u_{e_{k}}),$

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.

######  Remark 10.

For a tropical curve $\Gamma\to N_{\mathbb{R}}$ and $e$ an unbounded end of
$\Gamma$ of slope $n_{e}$, the evaluation of $\det(n_{e},-)$ at any point of
the end is called the moment of the end. Using the balancing condition, we get
the tropical Menelaus theorem, proved in [18]. It asserts that the sum of the
moments of the unbounded ends is $0$.

Using the preceding remark, another way to reformulate the result is then that
the moments of the unbounded edges of $\widetilde{\Gamma}$ is constrained by
the tropical abelian surface $\mathbb{T}A$.

######  Proposition 11.

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

$\det(n_{i},e_{i})+\det(-n_{i},f_{i})=\det(n_{i},S\gamma).$

######  Proof 3.8.

The first part is obvious, since a lifting set keeps being lifting for a small
deformation. Moreover, the preceding lemma ensures that, as $\mathbb{T}A$ is
fixed, the relation between the moments of unbounded ends keeps being
satisfied through the deformation. Conversely, the relation between the
moments ensures that projecting the curve from $N_{\mathbb{R}}$ to
$\mathbb{T}A$, it is possible to glue them together and get a curve inside
$\mathbb{T}A$.

######  Example 3.9.

On Figure 4 we depict a tropical curve in a tropical torus and a small
deformation of the curve. The deformation clearly lifts to a small deformation
of the lifted curve inside $N_{\mathbb{R}}$, taking for instance as a lifting
set the points of intersection with the boundary of the parallelogram.
$\lozenge$

Figure 4: A small deformation of a tropical curve inside a tropical torus.

######  Lemma 12.

Let $h:\Gamma\to\mathbb{T}A$ be a parametrized tropical curve. A disconnecting
edge of $\Gamma$ has slope $0$ under $h$.

######  Proof 3.10.

Let $e$ be a disconnecting edge for $\Gamma$ and let $\mathcal{Q}$ be a
lifting set for $\Gamma$ such that $\Gamma\backslash\mathcal{Q}$ is connected.
As $e$ is disconnecting, it cannot contain a point of $\mathcal{Q}$. Moreover,
as $e$ is disconnecting for $\Gamma$, it also is for
$\widetilde{\Gamma}_{\mathcal{Q}}$. Finally, for any pair of ends
$\\{e_{i},f_{i}\\}$ associated to a point $p_{i}\in\mathcal{P}$ belong to the
same component of $\widetilde{\Gamma}-e$, otherwise $e$ would not be
disconnecting for $\Gamma$. Then, the balancing condition on either component
of $\widetilde{\Gamma}-e$ ensures that $e$ has slope $0$.

###  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
realizable in $\mathbb{T}A$: $CS^{T}\in\mathcal{S}_{2}(\mathbb{R})$. 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 $\mathcal{M}_{g}(\mathbb{T}A,C)$.

######  Definition 3.5.

An irreducible parametrized tropical curve $h:\Gamma\to\mathbb{T}A$ is said to
be simple if:

  * -

$\Gamma$ is trivalent,

  * -

$h$ is an immersion.

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

######  Lemma 13.

If $h:\Gamma\to\mathbb{T}A$ is a simple parametrized tropical curve that has
no contracted edges, and $\gamma$ is a cycle in $\Gamma$, it is always
possible to deform the map along $\gamma$.

######  Proof 3.11.

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

$u_{i+1}=u_{i}+v_{i}.$

Then, move the vertex in the direction $v_{i}$:

$V_{i}(t)=V_{i}+\frac{t}{\det(u_{i},u_{i+1})}v_{i}.$

As

$\displaystyle\det(u_{i+1},V_{i+1}(t)-V_{i}(t))$
$\displaystyle=\det(u_{i+1},V_{i+1}-V_{i})+t\det\left(u_{i+1},\frac{v_{i+1}}{\det(u_{i},u_{i+1})}-\frac{v_{i}}{\det(u_{i},u_{i+1})}\right)$
$\displaystyle=0+t\frac{\det(u_{i+1},v_{i+1})}{\det(u_{i+1},u_{i+2})}-t\frac{\det(u_{i+1},v_{i})}{\det(u_{i},u_{i+1})}$
$\displaystyle=t-t$ $\displaystyle=0,$

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

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

|
---|---
$(a)$ | $(b)$
Figure 5: Deformation of a contractible cycle and a non-contractible one

######  Proposition 14.

We have the following facts:

  1. (i)

If $\mathcal{M}_{g,C}^{\Gamma}$ is a combinatorial type with $\Gamma$ simple,
the space of its deformations is of dimension $g$.

  2. (ii)

Let $h:\Gamma\to\mathbb{T}A$ be a parametrized tropical curve with $h$ an
immersion but $\Gamma$ not necessarily trivalent. Then $\Gamma$ can be
deformed into a nearby tropical curve that is trivalent.

  3. (iii)

With assumptions as in $(ii)$, the space of its deformation is of dimension at
most $g-1$.

######  Proof 3.12.

  1. (i)

Let $h:\Gamma\to\mathbb{T}A$ be a simple tropical. Let $\mathcal{Q}$ be a
lifting set for $\Gamma$, and assume that the curve $\widetilde{\Gamma}$ is
rational. The set $\mathcal{Q}$ is of size $g$, the curve $\widetilde{\Gamma}$
has genus $0$ and $2g$ unbounded ends. As it is trivalent, it has a space of
deformation of dimension $2g-1$.

By assumption, there are no contracted edge. Then, the only relation between
the moment of the unbounded edges is provided by the tropical Menelaus theorem
[18]:

$\sum_{i=1}^{2g}\mu_{i}=0,$

where the scalar $\mu_{i}$ denote the moments of the unbounded ends of
$\widetilde{\Gamma}$, indexed from $1$ to $2g$. Indeed, for any pair of
unbounded ends, it is possible to move the chord between them, modifying only
their moments, proving that the functions $\mu_{i}$ span a space of dimension
exactly $2g-1$, since there are $2g$ unbounded ends.

Thus, the $g$ conditions in Proposition 11 impose in fact $g-1$ conditions,
since the sum of them is $0$ by tropical Menelaus theorem. Then, the space of
deformations of $\Gamma$ is of dimension $2g-1-(g-1)=g$.

  2. (ii)

To prove the second statement, assume that a curve $h:\Gamma\to\mathbb{T}A$ is
not trivalent, but $h$ is still an immersion. Let $V$ be a vertex which is not
trivalent, and choose a loop in $\Gamma$ that contains the vertex $V$, and
such that at any vertex of the loop, the two directions of the edges of the
loop are not colinear. This is possible because $h$ is an immersion. Then,
move a little bit the loop in one direction, keeping fixed all edges incident
to the loop. This modifies the combinatorial type at each non-trivalent
vertex. By induction, we can assume that there are no non-trivalent vertices
left. As in [4], the only case where this is not possible is when $V$ is a
quadrivalent vertex with a loop on it: an edge whose ends coincide but has
non-zero slope. Such an edge would realize a circle, and this cannot happen
since $\mathbb{T}A$ does not contain any elliptic curve.

  3. (iii)

Assume that there exists a family of deformations of $h:\Gamma\to A$ of
dimension at least $g$. Combining this family with a deformation of $\Gamma$
into a trivalent curve, we would obtain a trivalent curve that varies in a
space of dimension greater than $g+1$. This is impossible by the first point.
Hence, $\Gamma$ cannot vary in a space of dimension greater than $g$.

######  Remark 15.

Proposition 14 proves that all parametrized 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 realizable, 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

$3g-3+2-2g=g-1.$

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 reparametrized by a parametrized tropical curve
$h^{\prime}:\Gamma^{\prime}\to\mathbb{T}A$, where $h^{\prime}$ is an
immersion, and $\Gamma^{\prime}$ is not trivalent, or has smaller genus.

######  Remark 16.

If $g=\deg C+1$, the dimension of the deformation space is easy to compute
since curves are generic sections of a line bundle on $\mathbb{T}A$. Thus,
deformations of a curve are corner locus of another section of the same line
bundle, hence a dimension $\deg C-1=g-2$, or a translate of it, hence a
“$+2$”, giving the expected $g$.

######  Remark 17.

In a sense, computing the dimension of the moduli space amounts to computing
the dimension of the moduli space of planar tropical curves which are subject
to some boundary conditions assuring that the gluing pass to the quotient. The
superabundancy comes from the fact that the position of the boundary points is
constrained by the tropical Menelaus theorem, as used in the proof of
Proposition 14.

##  4 Enumerative problem and counts

###  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 14 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.

  * -

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 $g_{1}+\cdots+g_{r}=g+r$.

  * -

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.

######  Example 4.1.

On Figure 6 $(b)$ we see a curve in the class $C=\begin{pmatrix}2&0\\\ 0&2\\\
\end{pmatrix}$ which is the union of two reducible components of genus $2$,
and on $(a)$ an irreducible curve of genus $4$. Both curves vary in spaces of
dimension $4$. This is especially easy for the reducible curves since the only
possibility of deformation is to translate each of the reducible components.
$\lozenge$

|
---|---
$(a)$ | $(b)$
Figure 6: On $(a)$ an irreducible curve of genus $4$ and on $(b)$ 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 realizable 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:

  * -

In [8], without tropical geometry, J. Bryan and N. 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 realizable 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.

  * -

In [13], 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 [8] in the genus $2$ case.

  * -

Finally, in [19], T. Nishinou gives a correspondence theorem that allows one
to compute the number of genus $g$ curves passing through $g$ points and that
tropicalize to some given trivalent curve. This recovers the result from [8]
provided one is able to solve the analog 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.

######  Proposition 18.

Let $h:\Gamma\to\mathbb{T}A$ be an irreducible tropical curve of genus $g$ in
the class $C$ passing through $\mathcal{P}$. Then:

  * -

the tropical curve is simple and rigid,

  * -

the complement $\Gamma\backslash h^{-1}(\mathcal{P})$ is a tree: it is
connected and has no cycles.

######  Proof 4.2.

If a curve $h:\Gamma\to\mathbb{T}A$ is not simple, then either $h$ is not an
immersion, or $\Gamma$ is not trivalent. If $h$ is not an immersion, merging
the edges that have a common vertex and the same image, it is possible to
reparametrize the curve by an immersion with a new graph of smaller genus or
vertices of higher valence. In either case, Proposition 14 tells us that the
curve varies in a space of dimension strictly smaller than $g$. Thus, a curve
passing through a generic configuration $\mathcal{P}$ has to be simple. If the
curve was not rigid, it would be possible to deform the curve $\Gamma$ in a
$1$-parameter family, and get a curve that is not simple, contradicting the
genericity of $\mathcal{P}$. Hence, the curve is rigid.

The complement $\Gamma\backslash h^{-1}(\mathcal{P})$ has no cycles since it
is always possible to deform a cycle using lemma 13. 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 realizes some homology
class inside $\mathbb{T}A$, which can be either $0$ or nonzero, and that we
also denote by $\lambda_{p}$.

Using Proposition 18 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
parametrized tropical curve:

$\tilde{h}:\widetilde{\Gamma}\longrightarrow N_{\mathbb{R}}.$

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 15, all curves are superabundant: the dimension of their moduli
space is greater than the expected dimension. Fix a genus $g$ and a realizable
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 $\Gamma$
realizes 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.

  * -

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

  * -

For $k|\delta(C)$, how many irreducible parametrized 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.

######  Definition 4.1.

Let $h:\Gamma\to\mathbb{T}A$ be a simple tropical curve. For a vertex $V$ of
$\Gamma$ with outgoing slopes $a_{V}$, $b_{V}$ and $-a_{V}-b_{V}$, let
$m_{V}=|\det(a_{V},b_{V})|$ be its multiplicity. Then, we set:

  * $\bullet$

the usual multiplicity $m_{\Gamma}=\prod_{V\in V(\Gamma)}m_{V}\in\mathbb{N}$,

  * $\bullet$

the refined multiplicity $m^{q}_{\Gamma}=\prod_{V\in
V(\Gamma)}[m_{V}]_{q}=\prod_{V\in
V(\Gamma)}\frac{q^{m_{V}/2}-q^{-m_{V}/2}}{q^{1/2}-q^{-1/2}}\in\mathbb{Z}[q^{\pm
1/2}]$.

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. The
multiplication by $\delta$ multiplies the vertex multiplicities by
$\delta^{2}$. As a simple genus $g$ curve has $2g-2$ vertices, we have

$m_{\delta\Gamma}=\delta^{4g-4}m_{\Gamma}.$

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

$m_{\delta\Gamma}^{q}(q)=[\delta^{2}]_{q}^{2g-2}m_{\Gamma}^{q}(q^{\delta^{2}}).$

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

$\displaystyle
N_{g,C,k}^{\mathrm{trop}}(\mathbb{T}A,\mathcal{P})=\sum_{\begin{subarray}{c}h(\Gamma)\supset\mathcal{P}\\\
\delta_{\Gamma}=k\end{subarray}}m_{\Gamma},$ $\displaystyle
BG_{g,C,k}(\mathbb{T}A,\mathcal{P})=\sum_{\begin{subarray}{c}h(\Gamma)\supset\mathcal{P}\\\
\delta_{\Gamma}=k\end{subarray}}m^{q}_{\Gamma}.$

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:

$N_{g,C,k}(\mathbb{T}A,\mathcal{P})=(k^{2})^{2g-2}N_{g,C/k,1}(\mathbb{T}A,\mathcal{P})\text{
and
}BG_{g,C,k}(\mathbb{T}A,\mathcal{P})=[k^{2}]_{q}^{2g-2}BG_{g,C/k,1}(\mathbb{T}A,\mathcal{P}).$

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

$\displaystyle M_{g,C}(\mathbb{T}A,\mathcal{P})$
$\displaystyle=\sum_{h(\Gamma)\supset\mathcal{P}}m_{\Gamma}=\sum_{k|\delta(C)}N_{g,C,k}(\mathbb{T}A,\mathcal{P})\in\mathbb{N},$
$\displaystyle N_{g,C}(\mathbb{T}A,\mathcal{P})$
$\displaystyle=\sum_{h(\Gamma)\supset\mathcal{P}}\delta_{\Gamma}m_{\Gamma}=\sum_{k|\delta(C)}kN_{g,C,k}(\mathbb{T}A,\mathcal{P})\in\mathbb{N},$
$\displaystyle R_{g,C}(\mathbb{T}A,\mathcal{P})$
$\displaystyle=\sum_{h(\Gamma)\supset\mathcal{P}}\delta_{\Gamma}m^{q}_{\Gamma}\in\mathbb{Z}[q^{\pm
1/2}],$ $\displaystyle BG_{g,C}(\mathbb{T}A,\mathcal{P})$
$\displaystyle=\sum_{h(\Gamma)\supset\mathcal{P}}m^{q}_{\Gamma}=\sum_{k|\delta(C)}BG_{g,C,k}(\mathbb{T}A,\mathcal{P})\in\mathbb{Z}[q^{\pm
1/2}].$

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.

######  Remark 19.

Perhaps some other choices of multiplicities would yield invariant counts that
are related to invariant counts on complex or real abelian surfaces, similarly
to the relation between refined invariants in toric varieties and refined
counts of real curves, as investigated in [18] and [6].

######  Example 4.3.

We refer to the end of the paper for examples and computations, once the pearl
diagram algorithm has been described. For now, we just give the contribution
of the curve depicted on Figure 7 to the various counts:

$\displaystyle M_{2,2I_{2}}^{\mathrm{trop}}$ $\displaystyle\to 4\times 4=16,$
$\displaystyle N_{2,2I_{2}}^{\mathrm{trop}}$ $\displaystyle\to 2\times 4\times
4=32,$ $\displaystyle R_{2,2I_{2}}$ $\displaystyle\to
2\times(q^{1/2}+q^{-1/2})\times(q^{1/2}+q^{-1/2})=2(q+2+q^{-1}),$
$\displaystyle BG_{2,2I_{2}}$ $\displaystyle\to q+2+q^{-1}.$

$\lozenge$

$\times$$+$$2$$2$$2$$2$ Figure 7: The unique genus $2$ curve passing through
two points.

###  4.2 Correspondence theorem

In [19], T. Nishinou gives a realization 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:

$\mathbb{C}A_{t}=N_{\mathbb{C}^{*}}/\Lambda_{t},$

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
tropicalizes to the tropical torus

$\mathbb{T}A=N_{\mathbb{R}}/\Lambda,$

with $\Lambda=\mathrm{Im}S$. 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
$a_{i}e_{1}+b_{i}e_{2}$, and $c_{i}e_{1}+d_{i}e_{2}$ for $B_{2}$. The vectors
associated through Poincaré duality are $-b_{i}e_{1}^{*}+a_{i}e_{2}^{*}$ and
$-d_{i}e_{1}^{*}+c_{i}e_{2}^{*}$. 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 [19]:

$\sigma(C)=\prod_{i}\alpha_{11}^{-b_{i}}\alpha_{12}^{a_{i}}\prod_{j}\alpha_{21}^{-d_{i}}\alpha_{22}^{c_{i}}.$

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 $\sigma$:

$\begin{array}[]{rccl}\sigma:&\Lambda\otimes N\simeq\Lambda_{1}\otimes
N&\longrightarrow&\mathbb{C}^{*}\\\ &\lambda_{1}\otimes
n&\longmapsto&\mathrm{PD}(n)\cdot j(\lambda_{1})\\\ \end{array},$

where $\mathrm{PD}(n)$ 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 realization
theorem.

######  Theorem.

(Nishinou [19]) Let $h:\Gamma\to\mathbb{T}A$ be a simple tropical curve
realizing the class $\delta C$, and let $\delta_{\Gamma}$ be the gcd of
$\Gamma$. The curve is realizable if and only if

$\sigma\left(\frac{\delta C}{\delta_{\Gamma}}\right)=(-1)^{\sum
m_{V}/\delta_{\Gamma}}.$

The relation from Nishinou’s realization’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-realizable”. Here, Menelaus relation
means the relation (46) in Proposition 39 from [18]. It is the relation
between the phases of the edges adjacent to a vertex for a tropical curve that
is realizable.

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 $\mathbb{C}A_{t}$. In the condition given
in section 2.3, we do not have any condition on the tropical curve, i.e. the
right-hand side does not appear. 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 20.

Let $h:\Gamma\to\mathbb{T}A$ be a simple parametrized tropical curve, and let
$\delta_{\Gamma}$ be the gcd of the curve. Then

$\sum_{V\in V(\Gamma)}\frac{m_{V}}{\delta_{\Gamma}}\equiv 0\mod 2.$

######  Proof 4.4.

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 $\sum_{V\in V(\Gamma)}\frac{m_{V}}{\delta_{\Gamma}}$ 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:

  * -

each double point contributes two intersection points with the same
multiplicity,

  * -

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

$\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\ \end{pmatrix},$

the result follows.

Then, choosing $g$ families of points $\mathcal{P}_{t}$ inside
$\mathbb{C}A_{t}$ that tropicalize 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 tropicalize 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 [19]. Let
$h:\Gamma\to\mathbb{T}A$ be a trivalent tropical curve passing through
$\mathcal{P}$, and let $\Gamma^{\prime}$ be the graph $\Gamma$ subdivided by
the points of $h^{-1}(\mathcal{P})$. First, choose an orientation of every
edge $e\in E(\Gamma^{\prime})$. Then, we have a map

$\begin{array}[]{rccl}\Theta:&\bigoplus_{v\in
V(\Gamma^{\prime})}N&\longrightarrow&\bigoplus_{e\in
E(\Gamma^{\prime})}N/N_{e}\oplus\bigoplus_{1}^{g}N\\\
&(\phi_{v})&\longmapsto&\left(\phi_{e},\phi_{v_{i}}\right)\\\ \end{array},$

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 $\Gamma^{\prime}$ associated to a
marked point $p_{i}\in\mathcal{P}$. The dimension of the domain is
$2|V(\Gamma^{\prime})|=2|V(\Gamma)|+2g$ while the dimension of the codomain is
$|E(\Gamma^{\prime})|+2g=|E(\Gamma)|+3g$. As $\Gamma$ is of genus $g$ and is
trivalent, we know that

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

Hence,

$(|E(\Gamma)|+3g)-(2|V(\Gamma)|+2g)=|V(\Gamma)|-|E(\Gamma)|+g=1,$

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 4.2.

The multiplicity of a trivalent tropical curve $h:\Gamma\to A$ passing through
$\mathcal{P}$ is

$m^{\mathbb{C}}_{\Gamma}=|\ker\Theta_{\mathbb{C}^{*}}|\prod_{e\in
E(\Gamma^{\prime})}w_{e},$

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

######  Remark 21.

In the usual correspondence theorems, such as in [20] and [23], 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
[17], [20] 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 $\mathrm{coker}\Theta$, as we see by applying the tensor
product to the following small exact sequence:

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

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

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

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

We now have the following correspondence theorem.

######  Theorem.

(Nishinou [19]) Let $h:\Gamma\to\mathbb{T}A$ be a simple tropical curve
passing through $\mathcal{P}$. The number of irreducible parametrized curves
passing through $\mathcal{P}_{t}$ that tropicalize to $(h,\Gamma)$ is

$m_{\Gamma}^{\mathbb{C}}=|\ker\Theta_{\mathbb{C}^{*}}|\prod_{e\in
E(\Gamma^{\prime})}w_{e}.$

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

$N_{g,C}=\mathcal{N}_{g,C}.$

###  4.3 Statement of the results

Using Proposition 18 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 [17].

######  Theorem 4.3.

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

$m_{\Gamma}^{\mathbb{C}}=\delta_{\Gamma}\prod_{V\in
V(\Gamma)}m_{V}=\delta_{\Gamma}m_{\Gamma},$

where $m_{\Gamma}$ is the first multiplicity from Definition 4.1, and
$\delta_{\Gamma}$ is the gcd of the tropical curve $\Gamma$.

######  Remark 22.

The presence of the factor $\delta$ is natural: assume the formula has been
proven when $\delta$ is $1$. To get a curve with gcd $\delta>1$, we multiply
all the weights of the edges by $\Delta$. As there are $3g-3+g=4g-3$ bounded
edges on $\Gamma^{\prime}$, the formula from Definition 4.2 is multiplied by
$\delta^{4g-3}$ since $\Theta$ is unchanged, only the weights of the bounded
edges changes. Meanwhile, as $m_{V}$ becomes $\delta^{2}m_{V}$ when the
weights are multiplied by $\delta$, the product $\prod_{V}m_{V}$ is multiplied
by $\delta^{2V(\Gamma)}=\delta^{4g-4}$. Thus, we need an additional $\delta$
to have the same exponent of homogenization.

######  Example 4.5.

  * -

Let us consider the genus $2$ curve passing through $2$ points on Figure 7.
Its gcd is $2$ and both vertices have multiplicity $4$. The Theorem asserts
that its complex multiplicity is $2\times 4\times 4=32$. Alternatively, the
subdivided curve has $5$ edges of weight $2$. We can see that there is a
unique way of putting phases of the curves, which amounts to compute
$|\ker\Theta_{\mathbb{C}^{*}}|=1$. The multiplicity is still $32$.

  * -

Consider the two curves on Figure 8. Both have multiplicity $4$, since the two
vertices have multiplicity $2$, and its gcd is $1$. However, the formula
$|\ker\Theta_{\mathbb{C}^{*}}|$ does not yield the result in the same way. For
the curve in $(a)$, the edge of weight $2$ is marked and counts double in the
edges weights product. We can check that $|\ker\Theta_{\mathbb{C}^{*}}|$ is
$1$ since there is a unique way to put phases on the edges. For the curve on
$(b)$, there are two ways to put a phase on the edge of weight $2$, meaning
$|\ker\Theta_{\mathbb{C}^{*}}|=2$, and it contributes only once in the product
of weights, also leading to $4$. This is natural because the count of
solutions cannot depend on the choice of the points, since it is equal to some
invariant $N_{g,C}$ using the correspondence Theorem.

$\lozenge$

$\times$$+$$2$$2$ | $+$$+$$2$$2$
---|---
$(a)$ | $(b)$
Figure 8: Two tropical curves with different marking.

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 4.3 is postponed to section 5.1.

Just as in the usual planar case, it is natural to try to replace the vertex
multiplicity $m_{V}$ by its quantum analog
$[m_{V}]_{q}=\frac{q^{m_{V}/2}-q^{-m_{V}/2}}{q^{1/2}-q^{-1/2}}$, 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 4.1.

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:

######  Theorem 4.4.

The refined count $BG_{g,C,k}(\mathbb{T}A,\mathcal{P})$ does not depend on the
choice of $\mathcal{P}$ as long as it is generic.

######  Corollary 4.5.

The counts $N_{g,C,k}^{\mathrm{trop}}(\mathbb{T}A,\mathcal{P})$,
$R_{g,C}(\mathbb{T}A,\mathcal{P})$, $BG_{g,C}(\mathbb{T}A,\mathcal{P})$,
$M_{g,C}(\mathbb{T}A,\mathcal{P})$ and $N_{g,C}(\mathbb{T}A,\mathcal{P})$ do
not depend on the choice of $\mathcal{P}$ as long as it is generic.

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 4.4,
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 $\mathbb{T}A$.

######  Theorem 4.6.

The refined invariant $BG_{g,C,k}(\mathbb{T}A)$ does not depend on
$\mathbb{T}A$ as long as $\mathbb{T}A$ contains curves in the class $C$ and is
chosen generically among them.

######  Corollary 4.7.

The invariants $N_{g,C,k}^{\mathrm{trop}}(\mathbb{T}A)$,
$R_{g,C}(\mathbb{T}A)$, $BG_{g,C}(\mathbb{T}A)$, $M_{g,C}(\mathbb{T}A)$ and
$N_{g,C}(\mathbb{T}A)$ do not depend on the choice of $\mathbb{T}A$ as long as
$\mathbb{T}A$ contains curves in the class $C$ and is chosen generically among
them.

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 4.4 does not require a generic choice of $\mathbb{T}A$.

######  Remark 23.

If $\mathbb{T}A$ is not chosen generically, the class $C$ might decompose as a
sum $C_{1}+C_{2}$. Then, there are solutions which are reducible curves having
two components in the classes $C_{1}$ and $C_{2}$. For instance, in the
computation of the invariants of [8], the authors choose the abelian surface
to be a product of two elliptic curves $S_{1}\times S_{2}$ and count curve in
the class $[S_{1}]+n[S_{2}]$. All the curves in this class are reducible with
all their components being genus $1$ curves. Tropically, this would mean that
all the tropical curve would have refined multiplicity $1$ since there are no
vertices. Curves in the classes $C_{1}$ and $C_{2}$ are not deformable along a
deformation of $\mathbb{T}A$, since the intersection points between the
components would become a pair of trivalent vertices. The only classes that
continue to exist are the one colinear with $C$.

Due to the preceding remark, the refined invariant depends on the choice of
the matrix $S$, and more precisely on cone of realizable classes inside the
the group $\\{C\in\Lambda\otimes N\text{ such that
}CS^{T}\in\mathcal{S}_{2}(\mathbb{R})\\}$.

######  Example 4.6.

Consider the class $C=\begin{pmatrix}1&0\\\ 0&1\\\ \end{pmatrix}$, which are
curves as depicted on Figure 1 $(a)$. If the torus $\mathbb{T}A$ is not chosen
generically, the genus $2$ curve might decompose as a union of two tropical
elliptic curves.There are no trivalent vertices anymore. Thus, we do not have
any irreducible curve to count anymore, and the value of the invariant is now
$0$, while it is $1$ for a generic choice of $\mathbb{T}A$.  $\lozenge$

##  5 Proof of the results

###  5.1 Multiplicity as a product

######  Proof 5.1 (Proof of Theorem 4.3).

For each edge $e\in E(\Gamma^{\prime})$ 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 $2|V(\Gamma^{\prime})|=6g-4$ to a lattice of
rank $|E(\Gamma^{\prime})|+g=6g-3$. It is not surjective. In fact, its image
always lie in the kernel of the following linear form defined on the codomain:

$\varphi(\phi_{e},p_{i})=\sum_{e}\frac{w_{e}}{\delta}\phi_{e},$

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

$\varphi\circ\Theta(p_{v})=\frac{1}{\delta}\left(w_{1}\det(u_{1},\varepsilon_{1}p_{v})+w_{2}\det(u_{2},\varepsilon_{2}p_{v})+w_{3}\det(u_{3},\varepsilon_{3}p_{v})\right),$

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,

$0\to\mathbb{Z}^{6g-4}\xrightarrow{\Theta}\mathbb{Z}^{6g-3}\to\mathbb{Z}\oplus
G\to 0,$

where $G$ is a torsion group. We want to compute the cardinal of this torsion
group, because

$0\to\mathrm{Tor}(G,\mathbb{C}^{*})\to(\mathbb{C}^{*})^{6g-4}\xrightarrow{\Theta_{\mathbb{C}^{*}}}(\mathbb{C}^{*})^{6g-3}\to\mathbb{C}^{*}\to
0.$

In fact, consider 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. Let us rephrase it. In suitable basis, the matrix of $\Theta$ has
the following form:

$\begin{pmatrix}d_{1}&0&\cdots&0\\\ 0&d_{2}&&\vdots\\\
\vdots&&\ddots&\vdots\\\ 0&\cdots&\cdots&d_{6g-4}\\\ 0&\cdots&\cdots&0\\\
\end{pmatrix}.$

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
$\bigoplus_{E(\Gamma^{\prime})}N/N_{e}$, 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
$\Gamma^{\prime}$. Let $e_{0}$ be such an edge. The matrix of $\Theta$
contains blocks of the following form:

$\begin{array}[]{|c|}\hline\cr\det(u_{e},-)\\\ \hline\cr I_{2}\\\
\hline\cr\det(u_{e},-)\\\ \hline\cr\end{array},$

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
$w_{e}u_{e}=w_{1}u_{1}+w_{2}u_{2}$. The corresponding block is

$\begin{array}[]{|c|}\hline\cr\det(u_{e},-)\\\ \hline\cr\det(u_{1},-)\\\
\hline\cr\det(u_{2},-)\\\ \hline\cr\end{array},$

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 $\det(u_{1},u_{2})=\frac{m_{V}}{w_{1}w_{2}}$. In the end, we get
$\frac{1}{\prod_{e\neq
e_{0}}w_{e}}\prod_{V}m_{V}=\frac{w_{e_{0}}}{\prod_{e}w_{e}}\prod_{V}m_{V}$.

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

$\sum\alpha_{e_{0}}\frac{w_{e_{0}}}{\prod_{e\in
E(\Gamma^{\prime})}w_{e}}\prod_{V}m_{V}=\frac{1}{\prod_{e\in
E(\Gamma^{\prime})}w_{e}}\prod_{V}m_{V},$

which is also an integer. The gcd of the $\frac{w_{e_{0}}}{\prod_{e\in
E(\Gamma^{\prime})}w_{e}}\prod_{V}m_{V}$ is then equal to
$\frac{1}{\prod_{e\in E(\Gamma^{\prime})}w_{e}}\prod_{V}m_{V}$. Multiplying by
$\prod_{e\in E(\Gamma^{\prime})}w_{e}$, 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 $\widehat{m_{V}}=\frac{m_{V}}{\delta^{2}}$. The value
of the minors are

$\frac{\widehat{w_{e_{0}}}}{\prod_{e\in
E(\Gamma^{\prime})}\widehat{w_{e}}}\prod_{V}\widehat{m_{V}}.$

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

$\frac{1}{\prod_{e\in
E(\Gamma^{\prime})}\widehat{w_{e}}}\prod_{V}\widehat{m_{V}}=\delta\frac{1}{\prod_{e\in
E(\Gamma^{\prime})}w_{e}}\prod_{V}m_{V},$

as in Remark 22. Multiplying by the weight of the bounded edges, the result
follows.

######  Remark 24.

Following [19], the cardinal of $\mathrm{Coker}\Theta$ is equal to the number
a possible phases of the tropical curve. The computation of the multiplicity
done in the proof is more in the spirit of [20]. It should also be possible to
compute to be $\delta\frac{\prod m_{V}}{\prod w_{e}}$ in a way similar to the
computation of the multiplicity in the planar setting from [17], by directly
computing the possible number of possible phases on the edges.

###  5.2 Independence of the choice of the points

Before getting to the proof of invariance, we notice the following. The
invariance for the enumerative problem without imposing the gcd of the curves
can be deduced from the invariance of the enumerative problems where the gcd
is involved. Conversely, a curve of gcd $\delta$ in the class $C$ is obtained
by multiplying by $\delta$ a primitive curve in the class $C/\delta$.
Therefore, initializing with primitive classes $C$ for which all the curves
realizing the class are also primitive, one can also recover the invariance of
the counts with a fixed gcd from the invariance of the global enumerative
problem.

######  Proof 5.2 (Proof of Theorem 4.4).

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 on 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.

$\times$$\times$ | $\times$$\times$
---|---
$(a)$ | $(b)$
Figure 9: Displacement of a point and the corresponding deformation of the
cycle that it contains.

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
$\lambda_{p}\neq 0$.

  * $\bullet$

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 [15]. 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 [15]. 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 on Figure 10.

$+$$+$$+$ | $\times$$\times$$\times$ |
---|---|---
$(a)$ | $(b)$ | $(c)$
Figure 10: Displacement of a point and the corresponding deformation across
the different walls. On $(a)$, a marked point that coincides with a vertex, on
$(b)$, the flattening of a cycle, and on $(c)$, the appearance of a
quadrivalent vertex.

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 $(a)$ on 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.

  * $\bullet$

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 [15]. 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.

######  Remark 25.

Notice that we do not need the degenerations to occur one at a time. As the
picture is purely local, they can happen at the same time. The only
restriction is that a chord or a loop cannot approach a marked point from both
sides at the same time: the edges containing it would contract to a point, and
as the path $\mathcal{P}_{s}$ is chosen generically, it stays outside some
fixed codimension $2$ sets, including the ones where the marked point sits at
the place where the edge is contracted.

######  Remark 26.

Notice that going through the degenerations, we get that the gcd of the
weights of the unbounded ends is preserved. This comes from the fact that in
the moduli space of genus $g$ curves, the only way to change the gcd of the
curves, is to pass through the codimension $1$ strata where a cycle is
contracted to a point. And this is impossible by the genericity assumption on
$\mathcal{P}_{s}$.

However, curves with different gcd can happen. For instance, consider the wall
$(b)$ on Figure 10, and all the edges of the curve be of weight divisible by
$2$. Then, to produce a curve with a different gcd, deform the pair of
parallel edges so that the two resulting edges have an odd weight. This is not
considered in the proof because the slope of the edges stays the same along
the deformation.

###  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 $CS^{T}\in\mathcal{S}_{2}(\mathbb{R})$ for a
fixed $C$ keeps being satisfied.

######  Proof 5.3 (Proof of Theorem 4.6).

Let $C\in\Lambda\otimes N$ be a class. Up to a change of basis of both $N$ and
$\Lambda$, we can assume that $C$ is of the form $\begin{pmatrix}d&0\\\
0&dn\\\ \end{pmatrix}=dC_{0}$. Thus the condition on
$S=\begin{pmatrix}s_{11}&s_{12}\\\ s_{21}&s_{22}\\\ \end{pmatrix}$ is that
$ns_{12}=s_{21}$. We choose a generic path $S_{t}=(s_{ij}(t))_{ij}$ between
two generic choices of matrices $S_{0}$ and $S_{1}$ such that this relation is
satisfied along the path. Here, a matrice $S$ is said to be generic if
$\\{C^{\prime}\in\Lambda\otimes N\text{ such that
}C^{\prime}S^{T}\in\mathcal{S}_{2}(\mathbb{R})\\}=\mathbb{Z}C_{0}$. In other
words, $C$ and its multiples are the only classes realizable. The path $S_{t}$
might have to cross the set of non-generic matrices to go from $S_{0}$ to
$S_{1}$. Let $\Lambda_{t}=\mathrm{Im}S_{t}$ and
$\mathbb{T}A_{t}=N_{\mathbb{R}}/\Lambda_{t}$.

In addition to the path of matrices, for each $t$ we can choose a
configuration of points $\mathcal{P}_{t}$ such that $\mathcal{P}_{t}$ varies
continuously. Moreover, we can assume that for each $t$, the configuration
$\mathcal{P}_{t}$ is generic as a point configuration inside $\mathbb{T}A_{t}$
so that we know that the curves passing through $\mathcal{P}_{t}$ are simple.
Notice that the choice $\mathcal{P}_{t}$ is generic inside $\mathbb{T}A_{t}$
even when the matrix $S_{t}$ is not generic. Such a path of configurations
exists because for every matrix $S$, the set of generic configurations is a
dense open subset.

Using the cutting process, curves inside $\mathbb{T}A_{t}$ are obtained from
curves inside $N_{\mathbb{R}}$ satisfying some gluing condition. If we impose
furthermore to pass through $\mathcal{P}_{t}$, the curves in $\mathbb{T}A_{t}$
passing through $\mathcal{P}_{t}$ are in bijection with the curves inside
$N_{\mathbb{R}}$ passing through some fixed points (the lift of
$\mathcal{P}_{t}$) and whose ends satisfy some moment conditions (for the
gluing). When moving $t$, the solutions move. To prove the invariance, we only
need to show that for each value $t_{*}$, on a neighborhood of $t_{*}$, the
solutions to the enumerative problem deform and give refined counts that are
equal.

######  Remark 27.

The difference with usual proofs of tropical invariance, such as the proof of
Theorem 4.4, is that here we avoid almost any kind of wall due to the
genericity of the point configuration $\mathcal{P}_{t}$ inside
$\mathbb{T}A_{t}$. In fact, we are proving that the refined invariants
$BG_{g,C,k}(\mathbb{T}A)$ are locally constant on the moduli space of tropical
torus with curves in the class $C$. As it is connected, it is therefore
constant.

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 $g$ parametrized 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 $g$.

  * $\bullet$

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
$t\in]t_{*}-\varepsilon;t_{*}+\varepsilon[$.

  * $\bullet$

If $\Gamma$ is reducible but with irreducible components that realize some
class proportional to $C$, we reduce to the previous case by deforming each
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.

  * $\bullet$

Assume that $\Gamma=\Gamma_{1}\sqcup\cdots\sqcup\Gamma_{r}$ is reducible with
$r$ connected components, not all of them realizing 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
realize classes proportional to $C$. We obtain new parametrized 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 5.4.

Consider the class $C=\begin{pmatrix}2&0\\\ 0&3\\\ \end{pmatrix}$ inside the
torus for $S=\begin{pmatrix}8&0\\\ 0&6\\\ \end{pmatrix}$. The corresponding
tropical torus contains curves in the classes $\begin{pmatrix}1&0\\\ 0&0\\\
\end{pmatrix}$ and $\begin{pmatrix}0&0\\\ 0&1\\\ \end{pmatrix}$, which are not
realizable for a small deformation of the torus. Consider the curve depicted
on Figure 11 $(b)$, which is comprised of $6$ 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 $6$ by smoothing the
points of $\mathcal{Q}$, and get the curve on Figure 11 $(c)$.  $\lozenge$

$\Gamma_{5}$$\Gamma_{1}$$\Gamma_{3}$$\Gamma_{2}$$\Gamma_{4}$ | $\times$$\times$$\times$$\times$$\Gamma_{1}$$\Gamma_{2}$$\Gamma_{3}$$\Gamma_{4}$$\Gamma_{5}$ |
---|---|---
$(a)$ | $(b)$ | $(c)$
Figure 11:  On $(b)$ a reducible curve with components in classes that do not
survive the torus deformation, and on $(c)$ the corresponding deformation for
a specific set of intersection points between the irreducible components,
giving the graph from $(a)$.

Consider the graph $\mathcal{T}$ consisting of the components $\Gamma_{j}$,
where vertices are linked by an edge if the corresponding components share a
point $q\in\mathcal{Q}$. This graph is a tree, otherwise the new graph would
not have the right genus.

######  Example 5.5.

The graph $\mathcal{T}$ corresponding to the example 5.4 is drawn on Figure
11. On Figure 12 is drawn a second example with the same base curve but a
different set of node $\mathcal{Q}$.  $\lozenge$

$\Gamma_{5}$$\Gamma_{1}$$\Gamma_{3}$$\Gamma_{4}$$\Gamma_{2}$ | $\times$$\times$$\times$$\times$$\Gamma_{1}$$\Gamma_{2}$$\Gamma_{3}$$\Gamma_{4}$$\Gamma_{5}$ |
---|---|---
$(a)$ | $(b)$ | $(c)$
Figure 12:  On $(b)$ a reducible curve with components in classes that do not
survive the torus deformation, and on $(c)$ the corresponding deformation for
a specific set of intersection points between the irreducible components,
giving the graph from $(a)$.

Let $\Gamma_{i}$ be a leaf of $\mathcal{T}$ and apply the cutting process to
it, for a cutting set that does not contain any point on the edge containing
the intersection point $q$. The lifted curve inside $N_{\mathbb{R}}$ is
subject to some point constraints, and the gluing conditions from
$\mathbb{T}A_{t_{*}}$. Those are no longer satisfied when moving $t$ because
the condition $C_{i}S_{t}^{T}$ is no longer satisfied. If $C_{i}S_{t}^{T}$ was
symmetric, the gluing would be satisfied and the node would be deformable.
Disconnect the edge containing $q$ and replace it by two unbounded ends. The
new curve has genus one less, and now has the right number of constraints if
we do not impose the two new unbounded ends to meet. The deformation forces
the two ends to separate, and thus the appearance of a new bounded edge, and
the combinatorial type is prescribed by the deformation of $\mathbb{T}A_{t}$.

Proceeding inductively, we lift the curves corresponding to the vertices of
$\mathcal{T}$ and prune the tree. When removing the vertex corresponding to a
component, we cut open the curve at the intersection point and replace the
bounded edge containing the intersection point by two unbounded ends, whose
difference of moments is fixed by the leaf.

######  Example 5.6.

In the example of Figure 12, it means that when deformaing the torus, we first
deform the components $\Gamma_{2}$, $\Gamma_{4}$ and $\Gamma_{5}$. Knowing the
deformation of $\Gamma_{2}$, we deduce the deformation of $\Gamma_{3}$, and
finally the deformation of $\Gamma_{1}$.  $\lozenge$

For each choice of set $\mathcal{Q}$, among the $2^{|\mathcal{Q}|}$ adjacent
combinatorial types resulting from the smoothing of the nodes of
$\mathcal{Q}$, we only have two adjacent combinatorial types that are obtained
by deformation of the torus. Clearly, the two deformations yield solutions for
$\mathbb{T}A_{t_{\ast}-\varepsilon}$ and $\mathbb{T}A_{t_{\ast}-\varepsilon}$
respectively, and they have equal multiplicities since each intersection point
becomes a pair of trivalent vertices with equal multiplicities on either side
of the wall.

As the reducibility can be read on the graph, the result follows: if the tree
$\mathcal{T}$ is not connected, it only yields reducible curves in
$\mathbb{T}A_{t}$ for nearby $t$. Otherwise, we get irreducible solutions.

######  Remark 28.

If the tropical torus $\mathbb{T}A$ is non-generic, there is still a relation
between the invariants $BG_{g,C}(\mathbb{T}A)$ and $BG_{g,C}$, which is the
invariant for a generic choice of $\mathbb{T}A$. The difference between both
invariants comes from the appearance of reducible curves with classes not
proportional to $C$: the value of $BG_{g,C}(\mathbb{T}A)$ depends on the
lattice $\\{C^{\prime}:C^{\prime}S^{T}\in\mathcal{S}_{2}(\mathbb{R})\\}$.

######  Remark 29.

The refined multiplicity of the tropical curves resulting from the smoothing
of a reducible curve depends on the intersection index between the curves.
With the refined multiplicity, having two intersection points of index $1$ is
not the same as having a point of intersection index $2$, since the first
yield a contribution $1$ to the refined multiplicity, while the second yield a
$q+2+q^{-1}$. Surprisingly, the invariance statement tells us that the total
count over all smoothings of all the reducible curve is invariant.

##  6 Examples

Computation of the invariants requires to solve the tropical enumerative
problem. Thus, we postpone the sophisticated examples to the third paper which
provides an algorithm for the computation of the invariants. For sake of
completeness, we include two examples to illustrate.

$2$ Figure 13:  A curve of genus $4$ in a long hexagon. It has one horizontal
loop, and three vertical loops. Two of them make only one vertical round while
the middle one makes three.

######  Example 6.1.

Assume that the class is $\begin{pmatrix}1&0\\\ 0&n\\\ \end{pmatrix}$ and that
we are looking for curves of genus $g$ passing through $g$ points. Assuming
that the abelian surface is represented by a fundamental domain that is a long
hexagon and that the points constraints are horizontally spread, all the
curves are of the following form: a loop that goes around the horizontal
direction, and $g-1$ loops that go around the vertical direction. Each of the
vertical loops is marked by a point, and one of the horizontal edges linking
the vertical loops has a marked point. See Figure 13 for an example of such a
curve.

To choose such a curve, we need to choose the marked point that belongs to an
horizontal edge and dispatch the degree $n$ among the $g-1$ vertical loops.
For a choice of dispatching $a_{1}+\cdots+a_{g-1}=n$, we have several
possibilities. Choosing $k_{i}|a_{i}$ such that the loop makes $k_{i}$ rounds
around the vertical direction, the $i$-th marked point has $k_{i}$ possible
positions, and the loop has two vertices of multiplicity
$\frac{a_{i}}{k_{i}}$. Thus, we get

$\displaystyle N_{g,(1,n)}=$ $\displaystyle
g\sum_{a_{1}+\cdots+a_{g-1}=n}\sum_{k_{i}|a_{i}}k_{i}\left(\frac{a_{i}}{k_{i}}\right)^{2}$
$\displaystyle=$ $\displaystyle
g\sum_{a_{1}+\cdots+a_{g-1}=n}\prod_{1}^{g-1}a_{i}\sigma_{1}(a_{i}).$

We recover the result from [8] giving the generating series of these numbers:

$\sum_{n=1}^{\infty}N_{g,(1,n)}y^{n}=gDG_{2}(y)^{g-1},$

where $G_{2}(y)$ is the Eisenstein series, and
$D=y\frac{\mathrm{d}}{\mathrm{d}y}$. For the refined invariants, we get

$\displaystyle BG_{g,(1,n)}=$ $\displaystyle
g\sum_{a_{1}+\cdots+a_{g-1}=n}\sum_{k_{i}|a_{i}}k_{i}\left[\frac{a_{i}}{k_{i}}\right]^{2}.$

$\lozenge$

| $2$$2$$2$$2$$2$ | $2$$2$ | $2$$2$
---|---|---|---
$\delta_{\Gamma}$ | $2$ | $1$ | $1$
$m_{\Gamma}$ | $16$ | $4$ | $4$
$m^{q}_{\Gamma}$ | $\begin{subarray}{c}q^{3}+2q^{2}+3q+4\\\ +3q^{-1}+2q^{-2}+q^{-3}\end{subarray}$ | $q+2+q^{-1}$ | $q+2+q^{-1}$
markings | $2$ | $4$ | $4$

|  | $2$
---|---|---
$\delta_{\Gamma}$ | $1$ | $1$
$m_{\Gamma}$ | $1$ | $4$
$m^{q}_{\Gamma}$ | $1$ | $q+2+q^{-1}$
markings | $8$ | $4$

Figure 14: Genus $2$ curves and their multiplicities.

######  Example 6.2.

We compute the invariants for genus $2$ curves in the class
$\begin{pmatrix}2&0\\\ 0&2\\\ \end{pmatrix}$. The combinatorial types of the
curves are represented on Figure 14. For each of them, we compute the number
of them that passes through $2$ fixed marked points. Thus, we recover

$N_{2,(2,2)}=64+16+16+8+16=120,$

and

$BG_{2,(2,2)}=2q^{3}+4q^{2}+18q+40+18q^{-1}+4q^{-2}+2q^{-3}.$

$\lozenge$

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