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ENUMERATIVE TROPICAL ALGEBRAIC GEOMETRY

GRIGORY MIKHALKIN

Abstract. The paper establishes a formula for enumeration ofcurves of arbitrary genus in toric surfaces. It turns out that suchcurves can be counted by means of certain lattice paths in theNewton polygon. The formula was announced earlier in [17].

The result is established with the help of the so-called tropi-cal algebraic geometry. This geometry allows to replace complextoric varieties with the real space Rn and holomorphic curves withcertain piecewise-linear graphs there.

1. Introduction

Recall the basic enumerative problem in the plane. Let g ≥ 0 andd ≥ 1 be two numbers and let Z = (z1, . . . , z3d−1+g) be a collectionof points zj ∈ CP2 in general position. A holomorphic curve C ⊂

CP2 is parameterized by a Riemann surface C under a holomorphicmap φ : C → CP2 so that C = φ(C). Here we choose the minimalparametrization, i.e. such that no component of C is mapped to apoint by φ. The curve C is irreducible if and only if C is connected.The number N irr

g,d of irreducible curves of degree d and genus g passingthrough Z is finite and does not depend on the choice of zj as long asthis choice is generic.

Similarly we can set up the problem of counting all (not necessarily

irreducible) curves. Define the genus of C ⊂ CP2 to be 12(2 − χ(C).

Note that the genus can take negative values for reducible curves. Thenumber Nmult

g,d of curves of degree d and genus g passing through Z isagain finite and does not depend on the choice of zj as long as thischoice is generic. Figure some (well-known) first few numbers N irr

g,d and

Nmultg,d .

The numbers N irrg,d are known as the Gromov-Witten invariants of

CP2 (see [11]) while the numbers Nmultg,d are sometimes called the mul-

ticomponent Gromov-Witten invariant. One series of numbers deter-mines another by a simple combinatorial relation (see e.g. [3]). Arecursive relation which allows to compute the numbers N irr

g,d (and thus

The author is partially supported by the NSF.1

2 GRIGORY MIKHALKIN

g\d 1 2 3 40 1 1 12 6201 0 0 1 2252 0 0 0 273 0 0 0 1

g\d 1 2 3 4-1 0 3 21 6660 1 1 12 6751 0 0 1 2252 0 0 0 27

Figure 1. N irrg,d and Nmult

g,d .

the numbers Nmultg,d ) was given by Kontsevich. This relation came from

the associativity of the quantum cohomology (see [11]). In the arbi-trary genus case Caporaso and Harris [3] gave an algorithm (bases on adegeneration of CP2) which allows to compute the numbers Nmult

g,d (and

thus the numbers N irrg,d).

The main result of this paper gives a new formula for Nmultg,d . This

number turns out to be the number of certain lattice paths of length3d− 1+ g in the triangle ∆d ⊂ R2 with vertices (0, 0), (d, 0) and (0, d).The paths have to be counted with certain non-negative multiplicities.Furthermore, this formula works not only for CP2 but for other toricsurfaces as well. For other toric surfaces we just have to replace thetriangle ∆d with other convex lattice polygons. The polygon should bechosen so that it determines the corresponding (polarized) toric surface.

The formula comes as an application of the so-called tropical geom-etry whose objects are certain piecewise-linear polyhedral complexesin Rn. These objects are the limits of the amoebas of holomorphiccurves after a certain degeneration of the complex structure. The ideato use these objects for enumeration of holomorphic curves is due toKontsevich.

In [12] Kontsevich and Soibelman proposed a program linking Ho-mological Mirror Symmetry and torus fibrations from the Strominger-Yau-Zaslow conjecture. The relation is provided by passing to theso-called “large complex limit” which deforms a complex structure ona manifold to its worst possible degeneration.

Similar deformations appeared in other areas of mathematics un-der different names. The patchworking in Real Algebraic Geometrywas discovered by Viro [23]. Maslov and his school studied the so-called dequantization of the semiring of positive real numbers (cf. [14]).The limiting semiring is isomorphic to the (max,+)-semiring Rtrop, thesemiring of real numbers equipped with taking the maximum for addi-tion and addition for multiplication.

The semiring Rtrop is known to computer scientists as one of tropicalsemirings, see e.g. [19]. In Mathematics this semiring appears from

ENUMERATIVE TROPICAL ALGEBRAIC GEOMETRY 3

some non-Archimedian fields K under a certain pushing forward to R

of the arithmetic operations in K.In this paper we develop some basic algebraic geometry over Rtrop

with a view towards counting curves. In particular, we rigorously setupsome enumerative problems over Rtrop and prove their equivalence tothe relevant problems of complex and real algebraic geometry. Thereader can refer to Chapter 9 of Sturmfels’ recent book [22] for somefirst steps in tropical algebraic geometry.

We solve the corresponding tropical enumerative problem in R2. Asan application we get a formula counting the number of curves of givendegree and genus in terms of certain lattice paths of a given lengthin the relevant Newton polygon. In particular this gives an interpre-tation of the Gromov-Witten invariants in P2 and P1 × P1 via latticepaths in the triangle and the rectangle respectively. This formula wasannounced in [17]. For the proof we use the patchworking side of thestory which is possible to use since the ambient space is 2-dimensionaland the curves there are hypersurfaces. An alternative approach (ap-plicable to higher dimensions as well) is to use the symplectic fieldtheory of Eliashberg, Givental and Hofer [4]. Generalization of thisformula to higher dimensions is a work in progress. In this paper weonly define the enumerative multiplicity for the 2-dimensional case (inhigher dimensions it is not localized at the vertices).

The main theorems are stated in section 7 and proved in section 8.In section 2 we define tropical curves geometrically (in a way similarto webs of Aharony, Hanany and Kol [1], [2]). In section 3 we ex-hibit them as algebraic objects over the tropical semifield. In section4 we define the tropical enumerative problems in R2, in section 5 re-call those in (C∗)2. Section 7 is auxiliary for section 8 and deals withcertain piecewise-holomorphic piecewise-Lagrangian objects in (C∗)2

called complex tropical curves.The author is grateful to Y. Eliashberg, K. Hori, I. Itenberg, M.

Kapranov, M. Kontsevich, A. Okounkov, B. Sturmfels, R. Vakil, O.Viro and J.-Y. Welschinger for useful discussions.

2. Geometry of tropical curves

In this section we define tropical curves in Rn geometrically andset up the corresponding enumerative problem. The following sectionwill exhibit such curves as one-dimensional varieties over the so-calledtropical semiring.

2.1. Definitions and the first examples. Let Γ be a weighted fi-nite graph. The weights are natural numbers prescribed to the edges.

4 GRIGORY MIKHALKIN

Clearly, Γ is a compact topological space. We make it non compact byremoving the set of all 1-valent vertices V1,

Γ = Γ r V1.

Definition 2.1. A proper map h : Γ → Rn is called a parameterizedtropical curve if it satisfies to the following two conditions.

• For every edge E ⊂ Γ the restriction h|E is an embedding. Theimage h(E) is contained in a line l ⊂ R2 such that the slope ofl is rational.

• For every vertex V ∈ Γ we have the following property. LetE1, . . . , En ⊂ Γ be the edges adjacent to V , let w1, . . . , wn ∈ N

be their weights and let v1, . . . , vn ∈ Zn be the primitive integervectors from V in the direction of the edges. We have

(1)

n∑j=1

wjvj = 0.

We say that two parameterized tropical curves h : Γ → Rn and h′ :Γ′ → Rn are equivalent if there exists a homeomorphism Φ : Γ → Γ′

which respects the weights of the edges and such that h = h′◦Φ. In thispaper we do not distinguish equivalent parameterized tropical curves.

The imageC = h(Γ) ⊂ Rn

is called the (unparameterized) tropical curve. It is a weighted piecewise-linear graph in Rn.

Remark 2.2. In dimension 2 the notion of tropical curve coincides withthe notion of (p, q)-webs introduced by Aharony, Hanany and Kol in[2] (see also [1]).

Example 2.3. Consider the union of three simple rays

Y = {(x, 0) | x ≤ 0} ∪ {(0, y) | y ≤ 0} ∪ {(x, x) | x ≥ 0}.

This graph is a tropical curve since (−1, 0) + (0,−1) + (1, 1) = 0. Aparallel translation of Y in any direction in R2 is clearly also a tropicalcurve. This gives us a 2-dimensional family of curves in R2. Suchcurves are called tropical lines.

Remark 2.4. The term tropical lines is justified in the next sectiondealing with the underlying algebra. So far we would like to note thefollowing properties of this family, see Figure 2.

• For any two points in R2 there is a tropical line passing throughthem.

• Such a line is unique if the choice of these two points is generic.


• Two generic tropical lines intersect in a single point.

Figure 2. Three distinct tropical lines.

Somewhat more complicated tropical curves (corresponding to pro-jective curves of degree 3) are pictured on Figure 3

2.2. The degree of a tropical curve. Let T = {τ1, . . . , τq} ⊂ Zn be

a set of non-zero integer vectors such thatq∑j=1

τj = 0. Suppose that in

this set we do not have positive multiples of each other, i.e. if τj = mτkfor m ∈ N then τj = τk. The degree of a tropical curve C ⊂ Rn takesvalues in such sets according to the following construction.

Note that a tropical curve C ⊂ Rn has a finite number of ends, i.e.unbounded edges (rays). Let τ ∈ Zn be a primitive vector. Its positivemultiple is included in T if and only if there exist the ends of C in thedirection of τ . In such case we include mτ into T , where m is the sumof multiplicities of all such rays.

Definition 2.5. The resulting set T is called the toric degree of C.

Note that the sum of all vectors in T is zero. This follows fromadding the conditions (1) from Definition 2.1 in all vertices of C.

For example the degree of both curves from Figure 5 is {(−1,−1),(2,−1), (−1, 2)} while the degree of both curves from Figure 3 is{(−3, 0), (0,−3), (3, 3)}.

Definition 2.6. If the toric degree of a tropical curve C ⊂ Rn is{(−d, 0), (0,−d), (d, d)} then C is called a tropical projective curve ofdegree d.

The curves from Figure 3 are examples of planar projective cubics.

6 GRIGORY MIKHALKIN

2.3. Genus of a tropical curve and the Riemann-Roch formula.

We say that a tropical curve is reducible if it can be parameterized bya disconnected graph. Otherwise we say that it is irreducible.

Definition 2.7. The genus of a parameterized tropical curve Γ → Rn

is dimH1(Γ) − dimH0(Γ) + 1. In particular, if Γ is connected thenits genus is the first Betti number of Γ. The genus of a tropical curveC ⊂ Rn is the minimum genus among all parameterizations of C.

Note that according to this definition the genus can be negative forreducible curves. E.g. the genus of the union of three lines from Figure2 has genus −2.

If C ⊂ Rn is an embedded 3-valent graph then the parameterizationis unique and is an embedding. However, in general, there might beseveral parameterizations of different genus and taking the minimalvalue is essential.

Example 2.8. The curve on the right-hand side of Figure 3 can be pa-rameterized by a tree once we “resolve” its 4-valent vertex. Therefore,its genus is 0.

Figure 3. A smooth projective tropical cubic and arational (genus 0) projective tropical cubic.

As in the classical complex geometry case the tropical curves aresubject to the constraint of the Riemann-Roch formula. Let x be thenumber of ends of a tropical curve C ⊂ Rn. This number can beconsidered as a tropical counterpart of minus the value of the canonicalclass of the ambient complex variety on the curve.

Let us consider the space of deformation of a parameterized tropicalcurve h : Γ → Rn. There are two things that we can deform, the maph or the graph Γ itself. The second type of deformation can only ariseif Γ has vertices of valence 4 or more.


Recall that (as in Chemistry) the valence of a vertex of a weightedgraph is the sum of multiplicities of the adjacent edges.

Definition 2.9. A parameterized tropical curve h : Γ → Rn is calledsimple if

• Γ is 3-valent,• h is an immersion,• if a, b ∈ Γ are such that h(a) = h(b) then neither a nor b can

be a vertex of Γ.

In this case the image h(Γ) is called a a simple tropical curve.

Note that a 3-valent graph parameterizing a tropical curve in Rn,n ≥ 1, has all its edges of weight 1. A simple parameterization ofC (if it exists) is always of minimal genus. Furthermore, any otherparameterization has a higher genus.

We are ready to formulate the tropical Riemann-Roch theorem forsimple curves.

Theorem 2.10. Suppose that Γ is a 3-valent graph that parameterizesa tropical curve h : Γ → Rn. Then h locally varies in a (real) lineark-dimensional space, where

k ≥ x + (n− 3)(1 − g).

Proof. Let Γ′ ⊂ Γ be a finite tree containing all the vertices of Γ. Notethat the number of finite edges in ΓrΓ′ is g. By an Euler characteristiccomputation we get that the number of finite edges of Γ′ is equal tox− 3 + 2g.

Maps Γ′ → Rn vary in a linear (x−3+2g+n)-dimensional family if wedo not change the slopes of the edges. The (x−3+2g)-dimensional partcomes from varying of the lengths of the edges while the n-dimensionalpart comes from translations in Rn. Such a map is extendable to atropical map Γ → Rn if the pairs of vertices corresponding to the gremaining edges define the lines with the correct slope. This each of theg edges imposes a linear condition of codimension at most n− 1. Thustropical perturbations of Γ → Rn form a linear family of dimension atleast x− 3 + 2g + n− (n− 1)g = x + (n− 3)(1 − g). �

Corollary 2.11. A simple tropical curve C ⊂ Rn locally varies in a(real) linear k-dimensional space, where

k ≥ x + (n− 3)(1 − g).

Consider the general case now and suppose that Γ has vertices ofvalence higher than 3. How much Γ differs from a 3-valent graph ismeasured by the following invariant. Let the overvalence µ(Γ) be the

8 GRIGORY MIKHALKIN

sum of the valences of all vertices of valence higher than 3 minus thenumber of such vertices. Thus µ(Γ) = 0 if and only if no vertex of Γhas valence higher than 3.

We can deform Γ to a 3-valent graph by the following procedurereducing µ. If we have n > 3 edges adjacent to the same vertex then wecan separate them into two groups so that each group contains at least2 edges. Let us insert a new edge E ′ separating these groups as shownin Figure 4. This replaces the initial n-valent vertex with 2 vertices (theendpoints of E ′) of smaller valence. There is a “virtual slope” of E ′

determined by the slopes of the edges in each group. This slope appearsin local perturbation of the tropical map Γ → Rn corresponding to thismodification of Γ. We say that Γj is a partial smoothing of Γ if Γj is

Figure 4. Smoothing a vertex of higher valence.

obtained by a sequence of such procedures starting from Γ.Theorem 2.10 can be generalizes in the following way.

Theorem 2.12. The space of deformations of a parameterized tropicalcurve h : Γ → Rn is locally a cone

C =⋃j

Cj,

where Cj is a convex cone corresponding to tropical curves parameter-ized by partial smoothings Γj of Γ. The union is taken over all possiblepartial smoothings Γj. Unless Cj is empty we have

dim Cj ≥ x+ (n− 3)(1 − g) − µ(Γj).

Proof. The proof of Theorem 2.10 is applicable here as well. But onehas to note that µ(Γj) of the finite length edges of the maximal tree Γ′

j

have zero length and therefore cannot vary. �

Remark 2.13. Curves which vary in a family strictly larger than x +(n − 3)(1 − g) (cf. Theorem 2.10) are called superabundant. In con-trast to the classical case superabundancy can be easily seen geometri-cally. This is the case if some of the cycles of C ⊂ Rn are contained insmaller-dimensional affine-linear subspaces of Rn, e.g. if a spatial curvedevelops a planar cycle. Dimension 2 is the smallest dimension where


one can have a cycle of such type in a genus-minimizing parameteri-zation so superabundancy can only appear in dimension 3 or higher.Thus one may strengthen Theorem 2.10 in the case n = 2 (or g = 0).

Theorem 2.14. A simple parameterized tropical curve h : Γ → Rn

locally varies in a (real) linear k-dimensional space, where

k = x + (n− 3)(1 − g)

if either n = 2 or g = 0.

This theorem also follows from the proof of Theorem 2.10 once wenote that the edges of Γ r Γ′ pose independent conditions in this case.

2.4. Smooth vs. singular curves, multiplicity of a vertex. Sup-pose that a parameterized tropical curve h : Γ → Rn is simple, so thereis no way to smooth Γ by reducing the valence of its vertices. Evenin such case the (unparameterized) curve h(Γ) can be deformed with achange in the combinatorial type of Γ.

Example 2.15. Let Γ be the union of 3 rays in R2 in the direction(−2, 1), (1,−2) and (1, 1) emanating from the origin (pictured on theleft-hand side of Figure 5). This curve is a simple tropical curve ofgenus 0.

It can be obtained as a t = 0 limit of the family of genus 1 curvesgiven by the union of 3 rays in R2 in the direction (−2, 1), (1,−2) and(1, 1) emanating from and (−2t, t), (t,−2t) and (t, t) respectively andthe three intervals [(−2t, t), (t,−2t)], [(−2t, t), (t, t)] and [(t, t), (t,−2t)]as pictured in Figure 5.

Figure 5. Perturbation at a non-smooth 3-valent vertex

Let C = h(Γ) be a simple curve. By a vertex V of C we mean theimage of a (3-valent) vertex of Γ. As in Definition 2.1 let w1, . . . , w3

be their weights of the edges adjacent to V and let v1, . . . , v3 be theprimitive integer vectors in the direction of the edges.

10 GRIGORY MIKHALKIN

Definition 2.16. The multiplicity of C at its 3-valent vertex V isw1w2|v1 × v2|. Here |v1 × v2| is the length of the vector product of v1

and v2 or, alternatively, the area of the parallelogram spanned by v1

and v2. Note that

w1w2|v1 × v2| = w2w3|v2 × v3| = w3w1|v3 × v1|

since v1w1 + v2w2 + v3w3 = 0 by Definition 2.1.

Definition 2.17. A tropical curve C ⊂ Rn is called smooth if it can beparameterized by h : Γ → Rn so that Γ is 3-valent, h is an embeddingand the multiplicity of every vertex of C is 1. A tropical curve is calledsmoothly immersed if it is a simple curve of multiplicity 1.

3. Underlying tropical algebra

In this section we exhibit the tropical curves as algebraic varietieswith respect to a certain algebra and also define higher-dimensionaltropical algebraic varieties in Rn.

3.1. The tropical semifield Rtrop. Consider the semiring Rtrop ofreal numbers equipped with the following arithmetic operations calledtropical in Computer Science:

“x+ y” = max{x, y} “xy” = x + y,

x, y ∈ Rtrop. We use the quotation marks to distinguish the tropicaloperations from the classical ones. Note that addition is idempotent,“x+ x = x”. This makes Rtrop to a semiring without the additive zero(the role of such zero would be played by −∞).

Remark 3.1. According to [19] the term “tropical” appeared in Com-puter Science in honor of Brazil and, more specifically, after Imre Si-mon (who is a Brazilian computer scientist) by Dominique Perrin. InComputer Science the term is usually applied to (min,+) semirings.Our semiring Rtrop is (max,+) by our definition but isomorphic to the(min,+)-semiring, the isomorphism is given by x 7→ −x.

As usual, a (Laurent) polynomial in n variables over Rtrop is definedby

f(x) = “∑j∈A

ajxj” = max

j∈A(< j, x > +aj),

where x = (x1, . . . , xn) ∈ Rn, j = (j1, . . . , jn), xj = xj11 . . . xjnn , < j, x > =

j1x1 + · · · + jnxn and A ⊂ Zn is a finite set. Note that f : Rn → R

is a convex piecewise-linear function. It coincides with the Legendretransform of a function j 7→ −aj defined on the finite set A.


Definition 3.2. The polyhedron ∆ = ConvexHull(A) is called theNewton polyhedron of f . It can be treated as a refined version of thedegree of the polynomial f .

3.2. Tropical hypersurfaces: the variety of a tropical polyno-

mial. For a tropical polynomial f in n variables we define its varietyVf ⊂ Rn as the set of points where the piecewise-linear function f isnot smooth, cf. [9], [17] and [22]. In other words, Vf is the corner locusof f .

Proposition 3.3. Vf is the set of points in Rn where more than onemonomial of f reaches its maximal value.

Proof. If exactly one monomial of f(x) = “∑j∈A

ajxj” = max

j∈A(< j, x >

+aj) is maximal at x ∈ Rn then f locally coincides with this monomialand, therefore, linear and smooth. Otherwise f has a corner at x. �

Remark 3.4. At a first glance this definition might appear to be un-related to the classical definition of the variety as the zero locus of apolynomial. To see a connection recall that there is no additive zero inRtrop, but its role is played bu −∞.

Consider the graph Γf ⊂ Rn×R of a tropical polynomial f : Rn → R.The graph Γf itself is not a tropical variety in Rn+1 but it can becompleted to the tropical variety

Γf = Γf ∪ {(x, y) | x ∈ Vf , y ≤ f(x)}.

Proposition 3.5. Γf coincides with the variety of the polynomial in(n+ 1) variables y + f(x), y ∈ R, x ∈ Rn.

Proof. If (x, y) ∈ Γf then we have y and one of the monomials of fboth reaching the maximal values in “y + f(x)” = max{y, f(x)}. Ifx ∈ Vf and y < f(x) then two monomials of f(x) are reaching themaximal value at the expression “y + f(x)”. �

Note that we have Vf = Γf ∩ {y = t} for t sufficiently close to −∞.This is the sense in which Vf can be thought as a zero locus.

One may argue that Γf itself is a subtropical variety (as in subanalyticvs. analytic sets) while Γf is its tropical closure. Figure 6 sketches thegraph y = “ax2 + bx + c” and its tropical closure.

Definition 3.6. Varieties Vf ⊂ Rn are tropical hypersurfaces.

Remark 3.7. Different tropical polynomials may define the same vari-eties. Recall that a function φ : A → R is called concave if for any


x

y

Figure 6. The graph y = “ax2+bx+c” and its closure,the tropical parabola

(possibly non-distinct) b0, . . . , bn ∈ A ⊂ Rn and any t0, . . . , tn ≥ 0 withn∑k=0

tk = 1 we have

φ(

n∑k=0

tkbk) ≥n∑k=0

tkφ(bk).

We have three types of ambiguities when f 6= g but Vf = Vg.

• g = “xjf”, where zj is a coordinate in Rntrop. Note that in this

case the Newton polyhedron of g is a translate of the Newtonpolyhedron of f .

• g = “cf”, where c ∈ Rtrop is a constant.• The function ∆ ∩ Zn 3 j 7→ aj is not concave, where f =

“∑j∈A

ajxj” and we set aj 7→ −∞ if j /∈ A. Then the variety of f

coincides with the variety of g where g is the smallest concavefunction such that g ≥ f (in other words g is a concave hull off).

Thus to define tropical hypersurfaces it suffices to consider only tropicalpolynomials whose coefficients satisfy to the concavity condition above.

Proposition 3.8 ([17]). The space of all tropical hypersurfaces with agiven Newton polydedron ∆ is a closed convex polyhedral cone M∆ ⊂Rm, m = #(∆ ∩ Zn) − 1. The cone M∆ ⊂ Rm is well-defined up tothe natural action of SLm(Z).

Proof. All convex functions ∆ ∩ Zn → R, j 7→ aj form a closed convex

polyhedral cone M∆ ⊂ Rm+1. But the function j 7→ aj + c definesthe same curve as the function j 7→ aj. To get rid of this ambiguity

we choose j ′ ∈ ∆ ∩ Z2 and define M∆ as the image of M∆ under thelinear projection Rm+1 → Rm, aj 7→ aj − aj′. �


3.3. Compactness of the space of tropical hypersurfaces. Clearly,the cone M∆ is not compact. Nevertheless it gets compactified by thecones M∆′ for all non-empty lattice subpolyhedra ∆′ ⊂ ∆ (includ-ing polygons with the empty interior). Indeed, we have the followingproposition.

Proposition 3.9. Let Ck ⊂ Rn, k ∈ N be a sequence of tropical hy-persurfaces whose Newton polyhedron is ∆. There exists a subsequencewhich converges to a tropical curve C whose Newton polyhedron ∆C iscontained in ∆ (note that C is empty if ∆C is a point). The conver-gence is in the Hausdorff metric when restricted to any compact subsetin Rn. Furthermore, if the Newton polyhedron of C coincides with ∆then the convergence is in the Hausdorff metric in the whole Rn.

Proof. Each Cn is defined by a tropical polynomial fCn(x) = “∑j

aCn

j xj”.

We may assume that the coefficients aCn

j are chosen so that that they

satisfy to the concavity condition and so that maxjaCn

j = 0. This takes

care of the ambiguity in the choice of fCn (since the Newton polyhedronis already fixed).

Passing to a subsequence we may assume that aCn

j converge (to afinite number or −∞) when n → ∞ for all j ∈ ∆ ∩ Zn. By ourassumption one of these limits is 0. Define C to be the variety of“∑a∞j x

j”, where we take only finite coefficients a∞j = limn→∞

anj > −∞.

�

3.4. Lattice subdivision of ∆ associated to a tropical hyper-

surface. A tropical polynomial f defines a lattice subdivision of itsNewton polyhedron ∆ in the following way (cf. [6]). Define the (un-bounded) extended polyhedral domain

∆ = ConvexHull{(j, t) | j ∈ A, t ≤ aj} ⊂ Rn × R.

The projection Rn×R → Rn induces a homeomorphism from the unionof all closed bounded faces of ∆ to ∆.

Definition 3.10. The resulting lattice subdivision Subdivf of ∆ iscalled the subdivision associated to f .

Proposition 3.11. The lattice subdivision Subdivf is dual to the trop-ical hypersurface Vf . Namely, for every k-dimensional polygon ∆′ ∈Subdivf such that ∆′ 6⊂ ∂∆ there is a convex polyhedron V ∆′

f ⊂ Vf .This correspondence has the following properties.

• V ∆′

f is contained in an (n−k)-dimensional affine-linear subspace

L∆′

of Rn orthogonal to ∆′.


• The relative interior U∆′

f of V ∆′

f in L∆′

is not empty.

• Vf =⋃U∆′

f .

• U∆′

f ∩ U∆′′

f = ∅ if ∆′ 6= ∆′′.

Proof. For every ∆′ ∈ Subdivf , ∆′ 6⊂ ∂∆ consider the truncated poly-nomial

f∆′

(x) =∑j∈∆′

ajxj

(recall that f(x) =∑j∈∆

ajxj). Define

(2) V ∆′

f = Vf ∩⋂

∆′′⊂∆′

Vf∆′′ .

Note that for any face ∆′′ ⊂ ∆′ we have the variety Vf∆′′ orthogonal

to ∆′′ and therefore to ∆′. To see that V ∆′

is compact we restate thedefining equation (2) algebraically: V ∆′

is the set of points where allmonomials of f indexed by ∆′ have equal values while the value of anyother monomial of f could only be smaller. �

Example 3.12. Figure 7 shows the subdivisions dual to the curvesfrom Figure 3 and 5.

Figure 7. Lattice subdivisions associated to the curvesfrom Figure 3 and Figure 5.

It was observed in [9], [17] and [22] that Vf is an (n−1)-dimensionalpolyhedral complex dual to the subdivision Subdivf . The complexVf ⊂ Rn is a union of convex (not necessarily bounded) polyhedra orcells of Vf . Each k-cell (even if it is unbounded) of Vf is dual to a

bounded (n − k − 1)-face of ∆, i.e. to an (n − k − 1)-cell of Subdivf .In particular, the slope of each cell of Vf is rational.

In particular, a (n−1) dimensional cell is dual to an interval I ⊂ Rn

whose both ends are lattice points. We define the lattice length of I as#(I ∩ Zn) − 1. (Such length is invariant with respect to SLn(Z).) Wecan treat Vf as a weighted piecewise-linear polyhedral complex in Rn,the weights are natural numbers associated to the (n− 1)-cells. Theyare the lattice lengths of the dual intervals.


Definition 3.13. The combinatorial type of a tropical hypersurfaceVf ⊂ Rn is the equivalence class of all Vg such that Subdivg = Subdivf .

Let S be such a combinatorial type.

Lemma 3.14. All tropical hypersurfaces of the same combinatorialtype S form a convex polyhedral domain MS ⊂ M∆ that is open in itsaffine-linear span.

Proof. The condition Subdivf = S can be written in the following wayin terms of the coefficients of f(x) = “

∑j

ajxj”. For every ∆′ ∈ S the

function j 7→ −aj for j ∈ ∆′ should coincide with some linear functionα : Zn → R such that −aj > α(j) for every j ∈ ∆ r ∆′. �

It turns out that the weighted piecewise-linear complex Vf satisfiesthe balancing property at each (n − 2)-cell, see Definition 3 of [17].Namely, let F1, . . . , Fk be the (n− 1)-cells adjacent to a (n− 2)-cell Gof Vf . Each Fj has a rational slope and is assigned a weight wj. Choosea direction of rotation around G and let cFj

: Zn → Z be linear mapswhose kernels are planes parallel to Fj and such that they are primitive(non-divisible) and agree with the chosen direction of rotation. Thebalancing condition states that

(3)k∑j=1

wjcFj= 0.

As it was shown in [17] this balancing condition at every (n − 2)-cellof a rational piecewise-linear (n − 1)-dimensional polyhedral complexin Rn suffices for such a polyhedral complex to be the variety of sometropical polynomial.

Theorem 3.15 ([17]). A weighted (n−1)-dimensional polyhedral com-plex Π ⊂ Rn is the variety of a tropical polynomial if and only if eachk-cell of Π is a convex polyhedron sitting in a k-dimensional affinesubspace of Rn with a rational slope and Π satisfies to the balancingcondition (3) at each (n− 2)-cell.

This theorem implies that the definitions of tropical curves and trop-ical hypersurfaces agree.

Corollary 3.16. Any tropical curve C ⊂ R2 is a tropical hypersurfacefor some polynomial f . Conversely, any tropical hypersurface in R2

can be parameterized by a tropical curve.

Remark 3.17. Furthermore, the degree of C is determined by the New-ton polygon ∆ of f according to the following recipe. If [a, b] ⊂ Rn


is an interval with a, b ∈ Zn let us denote with |[a, b]| = max{k ∈N | 1

k(b − a) ∈ Zn} its integer length. For each side ∆′ ⊂ ∂∆ we take

the primitive integer outward normal vector |∆′| times to get the degreeof C.

3.5. Tropical varieties and non-Archimendian amoebas. Poly-hedral complexes resulting from tropical varieties appeared in [9] inthe following context. Let K be a complete algebraically closed non-Archimedian field. This means that K is an algebraically closed fieldand there is a valuation val : K∗ → R defined on K∗ = K r {0}such that eval defines a complete metric on K. Recall that a valuationval is a map such that val(xy) = val(x) + val(y) and val(x + y) ≤max{val(x), val(y)}.

Our principal example of such K is a field of Puiseux series withreal powers. To construct K we take the algebraic closure C((t)) ofthe field of Laurent series C((t)). The elements of C((t)) are formalpower series in t a(t) =

∑k∈A

aktk, where ak ∈ C and A ⊂ Q is a subset

bounded from below and contained in an arithmetic progression. Weset val(a(t)) = −minA. We define K to be the completion of C((t))as the metric space with respect to the norm eval.

Let V ⊂ (K∗)n be an algebraic variety over K. The image of Vunder the map Val : (K∗)n → Rn, (z1, . . . , zn) 7→ (val(z1), . . . , val(zn))is called the amoeba of V (cf. [6]). Kapranov [9] has shown that theamoeba of a non-Archimedian hypersurface is the variety of a tropicalpolynomial. Namely, if

∑j∈A

ajzj = 0, 0 6= aj ∈ K is a hypersurface

in (K∗)n then its amoeba is the variety of the tropical polynomial∑j∈A

val(aj)xj.

More generally, if F is a field with a real-valued norm then theamoeba of an algebraic variety V ⊂ (F ∗) is Log(V ) ⊂ Rn, whereLog(z1, . . . , zn) = (log ||z1||, . . . , log ||zn||). Note that Val is such mapwith respect to the non-Archimedian norm eval in K.

Another particularly interesting case is if F = C with the standardnorm ||z|| = zz (see [6],[15],[18], etc.). The non-Archimedian hyper-surface amoebas appear as limits in the Hausdorff metric of Rn fromthe complex hypersurfaces amoebas (see e.g. [16]).

It was noted in [21] that the non-Archimedian approach can be usedfor to define tropical varieties of arbitrary codimension in Rn. Namely,one can define the tropical varieties in Rn to be the images Val(V ) ofarbitrary algebraic varieties V ∈ (K∗)n. This definition allow to avoid


dealing with the intersections of tropical hypersurfaces in non-generalposition. We refer to [21] for relevant discussions.

4. Enumeration of tropical curves in R2

4.1. Simple curves and their lattice subdivisions. Corollary 3.16implies that any tropical curve is a tropical hypersurface, i.e. it isthe variety of a tropical polynomial f : R2 → R. By Remark 3.7 theNewton polygon ∆ of such f is well-defined up to a translation.

Definition 4.1. We call ∆ the degree of a tropical curve in R2.

By Remark 3.17 this degree supplies the same amount of informationas the toric degree from Definition 2.5. We extract two numericalcharacteristics from the polygon ∆ ⊂ R2:

(4) s = #(∂∆ ∩ Z2), l = #(Int ∆ ∩ Z2).

It is convenient also to consider m = #(∆ ∩ Z2) − 1 = s + l − 1. Thenumber m is the dimension of the space of tropical curves of degree ∆(since m+ 1 is the number of coefficients in the corresponding polyno-mials). The number s is the number of unbounded edges of the curveif counted with multiplicities (recall that we denoted the number ofunbounded edges counted simply with x ≤ s). The number l is thegenus of a smooth curve of degree ∆. To see this let us note that everylattice point of ∆ is a vertex of the associated subdivision for a smoothcurve C. Therefore, the homotopy type of C coincides with Int ∆rZ2.Note also that smooth curves are dense in M∆.

Lemma 4.2. A tropical curve C ⊂ R2 is simple (see Definition 2.9) ifand only if it is the variety of a tropical polynomial such that Subdivfis a subdivision into triangles and parallelograms.

Proof. The lemma follows from Proposition 3.11. The 3-valent verticesof C are dual to the triangles of Subdivf while the intersection of edgesare dual to the parallelograms (see e.g. the right-hand side of Figure 3and the corresponding lattice subdivision in Figure 7. �

We have the following formula which expresses the genus of a simpletropical curve Vf in terms of the number r of triangles in Subdivf .

Lemma 4.3. If a curve Vf ⊂ R2 is simple then g(Vf) = r−x2

+ 1.

Proof. Let ∆0 be the number of vertices of Subdivf while ∆1 and ∆2 bethe number of its edges and (2-dimensional) polygons. Out of the ∆2

2-dimensional polygons r are triangles and (∆2−r) are parallelograms.We have

χ(Vf ) = −2∆2 + ∆1.


Note that 3r + 4(∆2 − r) = 2∆1 − x. Thus, ∆1 = 32r + 2(∆2 − r) + x

2and

g(Vf) = 1 − χ(Vf) = 1 +r − x

2.

�

4.2. Tropical general positions of points in R2.

Definition 4.4. Points p1, . . . , pk ∈ R2 are said to be in general posi-tion tropically if for any tropical curve C ⊂ R2 of genus g and with xends such that k ≤ g + x− 1 and p1, . . . , pk ∈ C we have the followingconditions.

• The curve C is simple (see Definition 2.9).• Points p1, . . . , pk are disjoint from the vertices of C.

Example 4.5. Two distinct points p1, p2 ∈ R2 are in general positiontropically if and only if the slope of the line in R2 passing through p1

and p2 is irrational.

Remark 4.6. Note that we can always find a curve of genus n + 1 −x passing through p1, . . . , pk. For such a curve we can take a non-reducible curve consisting of k lines with rational slope each passingthrough its own point pj. This curve has 2k ends while its genus is1 − k.

Proposition 4.7. Any subset of a set of points in general positiontropically is itself in general position tropically.

Proof. Suppose the points p1, . . . , pj are not in general position. Thenthere is a curve C with x ends of genus j + 2 − x passing throughp1, . . . , pj or of genus j + 1 − x but with a non-generic behavior withrespect to p1, . . . , pj. By Remark 4.6 there is a curve C ′ passing throughpj+1, . . . , pk of genus k − j + 1 − x′. The curve C ∪ C ′ supplies acontradiction. �

Proposition 4.8. For each ∆ ⊂ R2 the set of configurations P ={p1, . . . , pk} ⊂ R2 such that there exists a curve C of degree ∆ suchthat the conditions of Definition 4.4 are violated by C is closed andnowhere dense.

Proof. Note that we have only finitely many combinatorial types oftropical curves of genus g with the Newton polygon ∆. For each suchcombinatorial type we have an (x+g−1)-dimensional family of simplecurves or a smaller-dimensional family otherwise. For a fixed C eachof the (x+ g) points pj can vary in a 1-dimensional family on C or in a0-dimensional family if pj is a vertex of C. Thus the dimension of the

space of “bad” configurations P ∈ Symk(R2) is at most 2k − 1. �


Corollary 4.9. The configurations P = {p1, . . . , pk} in general posi-tion tropically form a dense set which can be obtained as an intersectionof countably many open dense sets in Symk(R2).

4.3. Tropical enumerative problem in R2. To set up an enumer-ative problem we fix the degree, i.e. a polygon ∆ ⊂ R2 with s =#(∂∆ ∩ Z2) and the genus g ∈ Z. Consider a configuration P ={p1, . . . , ps+g−1} ⊂ R2 of s + g − 1 points in tropical general position.Our goal is to count tropical curves C ⊂ R2 of genus g and degree ∆such that C ⊃ P.

Proposition 4.10. There exists only finitely many such C. Further-more, each end of such C is of weight 1, so C has s ends.

Finiteness follows from Lemma 4.20 proved in the next subsection.If C has ends whose weight is greater than 1 then the number of endsis smaller than s and the existence of C contradicts to the generalposition of P. Recall that since P is in general position any such C isalso simple and the vertices of C are disjoint from P.

Example 4.11. Let g = 0 and ∆ be the quadrilateral whose verticesare (0, 0), (1, 0), (0, 1) and (2, 2) (so that the number s of the latticepoints on the perimeter ∂∆ is 4). For a configuration P of 3 points inR2 pictured in Figure 8 we have 3 tropical curves passing. In Figure 9the corresponding number is 2.

Figure 8. Tropical curves through a configuration of3 points, N irr

trop(g,∆) = 5.

Definition 4.12. The multiplicity µ(C) of a tropical curve C ⊂ R2

of degree ∆ and genus g passing via P equals to the product of themultiplicities of all the 3-valent vertices of C (see Definition 2.16).

Definition 4.13. We define the number N irrtrop(g,∆) to be the number

of irreducible tropical curves of genus g and degree ∆ passing via P


2

Figure 9. Tropical curves through another configu-ration of 3 points. Note that the bounded edge in theright-hand curve has weight 2, N irr

trop(g,∆) = 5.

where each such curve is counted with multiplicity µ from Definition4.12. Similarly we define the number Ntrop(g,∆) to be the number ofall tropical curves of genus g and degree ∆ passing via P where eachsuch curve is counted with multiplicity µ from Definition 4.12.

The following theorem is a corollary of Theorem 1. In fact, no proofof it entirely within Tropical Geometry is known.

Theorem 4.14. The numbers Ntrop(g,∆) and N irrtrop(g,∆) are finite

and do not depend on the choice of P.

E.g. the 3-point configurations from Figures 8 and 9 have the samenumber N irr

trop(g,∆).

4.4. Forests in the polygon ∆. Recall that every vertex of C ⊂ R2

corresponds to a polygon of the dual lattice subdivision of the Newtonpolygon ∆ while every edge of C corresponds to an edge of the dualsubdivision SubdivC (see Proposition 3.11). If C ⊂ R2 is a tropicalcurve passing through P then we can mark the k edges of SubdivCdual to p1, . . . , pk. Let Ξ ⊂ ∆ be the union of the marked k edges.

Definition 4.15. The combinatorial type of a tropical curve C passingvia the configuration P is the lattice subdivision SubdivC together withthe graph Ξ ⊂ ∆ formed by the k marked edges of this subdivision.

Proposition 4.16. The graph Ξ ⊂ ∆ is a forest (i.e. a disjoint unionof trees).

Proof. From the contrary, suppose that Ξ contains a cycle Z ⊂ Ξformed by q edges. Without loss of generality we may assume thatp1, . . . , pq are the marked points on the edges of C dual to Z. Weclaim that p1, . . . , pq are not in tropical general position which leads to


a contradiction with Proposition 4.7. To show this we exhibit a curveof non-positive genus with q tails at infinity passing through p1, . . . , pq.

Consider the union D of the (closed) polyhedra from SubdivC thatare enclosed by Z. Let

CD =⋃

∆′⊂DU∆′

,

where U∆′

is the stratum of C dual to ∆′ (see Proposition 3.11). Theset CD can be extended to a tropical curve CD by extending all non-closed bounded edges of CD to infinity. These extensions can inter-sect each other if D is not convex so the Newton polygon of CD isConvexHull(D). In other words, CD is given by a tropical polynomial

fD(x) =∑

j∈ConvexHull(D)

bjxj with some choice of bj ∈ Rtrop (note that

bj = aj if j ∈ D). Consider the polynomial

fZ(x) =∑

j∈∂(ConvexHull(D))

bjxj.

We have VfZ 3 p1, . . . , pq since VfD 3 p1, . . . , pq. On the other hand thegenus of VfZ is non-negative by Lemma 4.3 since no vertex of SubdivfD

is in the interior of D. �

Figure 10. Forests corresponding to the curves pass-ing through the marked points from Figures 8 and 9.

4.5. The tropical curve minus the marked points. The followinglemma strengthens the Proposition 4.16.

Lemma 4.17. Let C be a simple curve of genus g and degree ∆. Sup-pose that C is parameterized by h : Γ → R2.

• Suppose that C passes through a configuration P of s + g − 1points in general position. Then each component K of Γ r

h−1(P) is a tree and the closure of h(K) ⊂ R2 has exactly oneend at infinity.

• Conversely, suppose that P ⊂ C is a finite set such that eachcomponent K of Γrh−1(P) is a tree and the closure of h(K) ⊂R2 has exactly one end at infinity. Then the combinatorial type


of (C,P) is realized by (C,P ′), where C ′ is a curve of genus gand degree ∆ and P ′ is a configuration of x + g − 1 points intropical general position. Here x is the number of ends of C.

Proof. Each component K of C r P in the first part of the statementhas to be a tree. Otherwise we can reduce the genus of C by the sametrick as in Proposition 4.16 keeping the number of ends of C the same.This leads to a contradiction with the assumption that P is in generalposition. Also similarly to Proposition 4.16 we get a contradiction ifh(K) is bounded. If h(K) has more than one end then by Theorem 2.10it can be deformed keeping the marked points from ¯h(K) r h(K) ⊂ Pfixed. This supplies a contradiction with Proposition 4.10.

For the second part let us slightly deform P to bring it to a tropicalgeneral position. We can deform h : Γ → R2 at each component Kindividually to ensure C ′ ⊃ P ′. �

Figure 11. A sketch of a typical component of Γ r h−1(P).

Lemma 4.17 allows to extend the forest Ξ from Proposition 4.16 toa tree X ⊂ ∆ that spans all the vertices of SubdivC in the case whenthe number of points in the configuration P is s + g − 1.

Each parallelogram ∆′ corresponds to an intersection of two edges Eand E ′ of Γ. Lemma 4.17 allows to orient the edges of each componentof Γ r h−1(P) consistently toward the end at infinity. Let U ′ ⊂ R2

be a small disk with a center at E ∩ E ′. Each component of U ′ r

(E ∪ E ′) corresponds to a vertex of ∆′. Two of these components aredistinguished by the orientations of E and E ′. One is adjacent to thesources of the edges while the other is adjacent to the sinks. We connectthe corresponding edges of ∆′ with an edge.

We form the graph X ⊂ ∆ by taking the union of Ξ with such edgesfor all the parallelograms from SubdivC .

Proposition 4.18. The graph X is a tree that contains all the verticesof SubdivC .

Proof. Suppose that SubdivC does not contain parallelograms. ThenX = Ξ. Let K ⊂ X be a component of X . If there exists a vertex


v ∈ SubdivC not contained in K then we can form a 1-parametricfamily of curves of genus g and degree ∆ passing via P. Indeed, letf(x) = “

∑βjx

j be the tropical polynomial that defines C. To get ridof the ambiguity in the choice of f we assume that j runs over onlythe vertices of SubdivC and that βj0 = 0 for a choice of the base indexj0 ∈ K. Let us deform the coefficient βv. If v belongs to componentK ′ of Ξ different from K then we also inductively deform coefficientsat the other vertices of SubdivC that belong to K ′ to make sure thatthe curve C ′ corresponding to the result of deformation still containsP. Clearly the genus of C ′ is still g. Thus we get a contradiction tothe assumption that P is in tropically general position.

We can reduce the general case to this special case by the followingprocedure. For each parallelogram ∆′ ∈ SubdivC consider a point p′

that is obtained by a small shift of the intersection E∩E ′ to the sourcecomponent of U ′ r(E∪E ′). Let P ′ ⊃ P be the resulting configuration.The curve C can be deformed to a curve C ′ ⊃ P ′ with the correspondingforest Ξ′ equal to X . �

Corollary 4.19. If P consists of s + g − 1 points then the forest Ξfrom Proposition 4.16 contains all the vertices of SubdivC .

4.6. Uniqueness in a combinatorial type. Enumeration of tropi-cal curves is easier than that of complex or real curves thanks to thefollowing lemma.

Lemma 4.20. In each combinatorial type of marked tropical curves ofgenus g with x ends there is either one or no curves passing throughP ∈ R2 as long as P is a configuration of x + g − 1 points in generalposition.

Proof. The tropical immersions to R2 of the graph Γ defined by thecombinatorial type are described by linear equations (cf. the proof ofTheorem 2.10). The condition that the image of a particular edge ofΓ contains pj is another linear condition. We have a zero-dimensionalset satisfying to all conditions by Theorem 2.14.

Note that even if the combinatorial type is generic we still may haveno curves passing through P since this linear system of equations de-fined not in the whole Rn but in an open polyhedral domain there byLemma 3.14. �

5. Algebraic curves in (C∗)2

5.1. Enumerative problem in (C∗)2. As in subsection 4.3 we fix anumber g ∈ Z and a convex lattice polygon ∆ ⊂ R2. As before lets = #(∂∆ ∩ Z2). Let P = {p1, . . . , ps+g−1} ∈ (C∗)2 be a configuration


of points in general position. A complex algebraic curve C ⊂ (C∗)2 isdefined by a polynomial f : (C∗)2 → C with complex coefficients. Asin the tropical set-up we refer to the Newton polygon ∆ of f as thedegree of C.

Definition 5.1. We define the number N irr(g,∆) to be the numberof irreducible complex curves of genus g and degree ∆ passing via P.Similarly we define the number N(g,∆) to be the number of all complexcurves of genus g and degree ∆ passing via P.

Note that here we count every relevant complex curve simply, i.e.with multiplicity 1.

Proposition 5.2. For a generic choice of P the numbers N irr(g,∆)and N(g,∆) are finite and do not depend on P.

This proposition is well-known. Nevertheless, later in this sectionwe reduce the invariance of N irr(g,∆) and N(g,∆) to the invariance ofcertain Gromov-Witten numbers.

In modern Mathematics there are two ways to interpret the numbersN irr(g,∆) and N(g,∆). A historically older interpretation is via thedegree of Severi varieties. A more recent interpretation (introduced in[11]) is via the Gromov-Witten invariants. In both interpretations itis convenient to consider the compactification of the problem providedby the toric surface associated to the polygon ∆.

5.2. Toric surfaces and Severi varieties. Recall that a convex poly-gon ∆ defines a compact toric surface CT∆ ⊃ (C∗)2, see e.g. [6]. (Somereaders may be more familiar with the definition of toric surfaces byfans, in our case the fan is formed by the dual cones at the vertices of ∆,see Figure 12.) The sides of the polygon ∆ correspond to the divisorsin CT∆ r (C∗)2. These divisors intersect at the points correspondingto the vertices of ∆. This surface is non-singular if every vertex of ∆is simple, i.e. its neighborhood in ∆ is mapped to a neighborhood ofthe origin in the positive quadrant angle under a composition of anelement of SL2(Z) and a translation in R2. Non-simple vertices of ∆correspond to singularities of CT∆.

Example 5.3. Let ∆d be the convex hull of (d, 0), (0, d) and (0, 0). Wehave CT∆d

= CP2 no matter what is d. If ∆ = [0, r] × [0, s], r, s ∈ N

then CT∆ = CP1 × CP1 no matter what are r and s. All the verticesof such polygons are simple.

If ∆32 = ConvexHull{(0, 0), (2, 1), (1, 2)} then CT∆ = CP2/Z3, where

the generator of Z3 acts on CP2 by [x : y : z] 7→ [x : e2πi3 y : e

4πi3 z].


Figure 12. A polygon and its normal fan.

This action has 3 fixed points which give the singularities of CT∆

32

corresponding to the three (non-simple) vertices of ∆.

In addition to a complex structure (which depends only on the dualfan) the polygon ∆ defines a holomorphic linear bundle H over CT∆.Let L∆ = Γ(H) be the vector space of sections of H. The projectivespace PL∆ is our system of the curves. Note that it can be also con-sidered as the space of all holomorphic curves in CT∆ such that theirhomology class is Poincare dual to c1(H).

Returning to Example 5.3 we note that ∆d gives us the projectivecurves of degree d. The polygon [0, r] × [0, s] gives us the curves of

bidegree (r, s) in the hyperboloid CP1×CP1. The polygon ∆32 gives us

the images in CT∆

32

of the cubic curves in CP2 that are invariant with

respect to the Z3-action.Any curve in L∆ is the closure in CT∆ of the zero sets of polynomials

whose Newton polygon is contained in ∆. Thus dimL∆ = #(∆∩Z2) =s + l (see (4)) and dim PL∆ = s + l − 1. A general curve C ⊂ CT∆

from PL∆ is a smooth curve that is transverse to CT∆ r (C∗)2 (thismeans that it is transverse to all divisors corresponding to the sides of∂∆ and does not pass through their intersection points).

By the genus formula we have g(C) = l = #(Int ∆∩Z2) for a smoothcurve C in PL∆. However singular curves have smaller geometric genus.More precisely let C ⊂ CT∆ be the curve from PL∆ and let C → Cbe its normalization. We define the geometric genus as g(C) = 1

2(2 −

χ(C)). Note that if C is not irreducible then C is disconnected andthen g(C) may take a negative value.

Fix a number g ∈ Z. The curves of genus not greater than g formin the projective space PL∆ an algebraic variety known as the Severivariety of CT∆. This variety may have several components. E.g. ifCT∆ has an exceptional divisor E corresponding to a side ∆E ⊂ ∆then reducible curves E ⊂ C ′ where C ′ corresponds to the polygon


∆′ = ConvexHull((∆ r ∆E) ∩ Z2) form a component (or a union ofcomponents) of the Severi variety. Such components correspond tosmaller polygons ∆′ ⊂ ∆. We are interested only in those componentsthat correspond to the polygon ∆ itself.

Definition 5.4. The irreducible Severi variety Σirrg ⊂ PL∆ correspond-

ing to ∆ of genus g is formed by all irreducible curves whose New-ton polygon is ∆ and genus is not more then g. The Severi varietyΣg ⊂ PL∆ corresponding to ∆ of genus g is formed by all curves whoseNewton polygon is ∆ and genus is not more then g. Clearly, Σg ⊃ Σirr

g .

Note that Σg is empty unless g ≤ l. If g = l we have Σg = PL.If g = l − 1 then Σg is the (generalized) ∆-discriminant variety. Itis sometimes convenient to set δ = l − g. Similarly to [3] it can beshown that Σg is the closure in PL of immersed nodal curves with δordinary nodes. In the same way, Σirr

g is the closure in PL of irreducibleimmersed nodal curves with δ ordinary nodes.

It follows from the Riemann-Roch formula that Σg and Σirrg have

pure dimension s+ g− 1. The Severi numbers N irr(g,∆) and N(g,∆)can be interpreted as the degrees of Σirr

g and Σg in PL.

Example 5.5. Suppose ∆ = ∆d so that CT∆ = CP2. We have s = 3d

and l = (d−1)(d−2)2

. The number N irr(g,∆) is the number of genus g,degree d (not necessarily irreducible) curves passing through 3d+g−1generic points in CP2.

The formula N(l− 1,∆d) = 3(d− 1)2 is well-known as the degree ofthe discriminant (cf. [6]). (More generally, if CT∆ is smooth then N(l−1,∆) = 6 Area(∆)− 2 Length(∂∆) + # Vert∆, where Length(∂∆) = sis the “lattice length” of ∂∆ and # Vert ∆ is the number of vertices,see [6]).

An elegant recursive formula for N irr(0,∆d) was found by Kontse-vich [11]. Caporaso and Harris [3] found an algorithm for computingN(g,∆d) for arbitrary g. See [23] for computations for some other ra-tional surfaces, in particular, the Hirzebruch surfaces (this correspondsto the case when ∆ is a trapezoid).

5.3. Gromov-Witten invariants. Let ∆ be a polygon with sim-ple vertices so that CT∆ is a smooth 4-manifold. This manifolds isequipped with a symplectic form ω∆ defined by ∆. The linear sys-tem L∆ gives an embedding CT∆ ⊂ PL∆ ≈ CPm. This embeddinginduces ω∆. As we have already seen ∆ also defines a homology classβ∆ ∈ H2(CT∆), it is the homology class of the curves from PL∆.

To define the Gromov-Witten invariants of genus g one takes ageneric almost-complex structure on CT∆ that is compatible with ω∆


and count the number of pseudo-holomorphic curves of genus g viageneric s + g − 1 points in CT∆ in the following sense (see [11] for aprecise definition). 1 Consider the space Mg,s+g−1(CT∆) of all stable(i.e. those with finite automorphism group) parameterized pseudo-holomorphic curves with s + g − 1 marked points. Evaluation at eachmarked point produces a map Mg,s+g−1(CT∆) → CT∆. With the helpof this map we can pull back to Mg,s+g−1(CT∆) any cohomology classin CT∆, in particular the cohomology class of a point. Doing so foreach of the s+ g− 1 point and taking the cup-product of the resultingclasses we get the Gromov-Witten invariant ICT∆

g,s+g−1,βδ< pt⊗

s+g−1>.

The result is invariant with respect to deformations of the almost-complex structure. In many cases it is useful to pass to a genericalmost-complex structure to make sure that for any stable curve C →CT∆ passing through our points we have H1(C,NC/CT∆

) = 0. Butwe have this condition automatically if CT∆ is a smooth Fano surface(or, equivalently, all exceptional divisors have self-intersection −1), cf.e.g. [23] for details. (However, if CT∆ has exceptional divisors of self-intersection −2 and less we need either to perturb the almost-complexstructure or to consider a virtual fundamental class.)

In particular, if CT∆ is CP2 or CP1×CP1 (these are the only smoothtoric surfaces without exceptional divisors, in other words minimalFano) then the Gromov-Witten invariants coincide with the corre-sponding numbers N irr(g,∆). The Gromov-Witten invariant of CP2

of degree d is equal to N irr(g,∆d). The Gromov-Witten invariant ofCP1 × CP1 of bidegree (r, s) is equal to N irr(g, [0, r]× [0, s]).

The following proposition computes the Gromov-Witten invariantsfor non-minimal smooth Fano toric surfaces in terms of the numbersN∆,δ. Let E1, . . . , En be all the sides of ∆ that correspond to theexceptional divisors of CT∆. For a subcollection Ej1, . . . , Ejk of suchexceptional divisors we denote

∆j1,...,jk = ConvexHull((∆ r⋃

Ejk) ∩ Z2).

We have the following proposition.

Proposition 5.6. If CT∆ is a smooth Fano surface then its Gromov-Witten invariant

ICT∆g,s+g−1,βδ

< pt⊗s+g−1

>=n∑k=0

∑j1<···<jk

N irr(g,∆j1,...,jk).

1These are the Gromov-Witten invariants evaluated on the cohomology classesdual to a point, this is the only non-trivial case for surfaces. In this discussion wecompletely ignore the gravitational descendants.


In particular, if CT∆ is a minimal smooth Fano surface (i.e. isomor-phic to CP2 or CP1 × CP1) then

ICT∆g,s+g−1,βδ

< pt⊗s+g−1

>= N irr(g,∆).

Proof. Since CT∆ is Fano no component of Σl−δ has dimension higherthen s+g−1. We have to add the degrees of all components consistingof curves of the type Ej1 ∪ · · · ∪Ejk ∪C, where the Newton polygon ofC is ∆j1,...,jk. �

The Gromov-Witten invariants corresponding to disconnected curves(i.e. to the numbers N(g,∆)) are sometimes called multicomponentGromov-Witten invariants.

6. Statement of the main theorems

6.1. Enumeration of complex curves. Recall that the numbersN irr

trop(g,∆) and Ntrop(g,∆) were introduced in Definition 4.13 and a

priori they depend on a choice of a configuration R ⊂ R2 of s + g − 1points in general position.

Theorem 1. For any generic choice R we have N irrtrop(g,∆) = N irr(g,∆)

and Ntrop(g,∆) = N(g,∆).Furthermore, there exists a configuration P ⊂ (C∗)2 of s+g−1 points

in general position such that for every tropical curve C of genus g anddegree ∆ passing through R we have µ(C) distinct complex curves ofgenus g and degree ∆ passing through P. These curves are distinct fordistinct C and are irreducible if C is irreducible.

The following is an efficient way to compute Ntrop(g,∆) (and there-fore also N(g,∆)). Let us choose a configuration P to be contained inan affine line A ⊂ R2. Furthermore, we make sure that the order ofP coincides with the order on A and that d(pj+1, pj) >> d(pj, pj−1).These conditions ensure that P is in tropically general position as longas the slope of A is irrational. Furthermore, these conditions specifythe combinatorial types of the tropical curves of genus g and degree ∆passing via P (see Definition 4.15). We shall see that for this choice ofP the forests corresponding to such combinatorial types are all pathsconnecting a pair of vertices of ∆. This observation can be used tocompute N(g,∆) once the irrational slope of A is chosen.

6.2. Counting of complex curves by lattice paths.

Definition 6.1. A path γ : [0, n] → R2, n ∈ N, is called a latticepath if γ|[j−1,j], j = 1, . . . , n is an affine-linear map and γ(j) ∈ Z2,j ∈ 0, . . . , n.


Clearly, a lattice path is determined by its values at the integerpoints. Let us choose an auxiliary linear map

λ : R2 → R

that is irrational, i.e. such that λ|Z2 is injective. (If an affine lineA ⊂ R2 is already chosen we take λ to be the orthogonal projection ontoA ⊂ R2.) Let p, q ∈ ∆ be the vertices where α|∆ reaches its minimumand maximum respectively. A lattice path is called λ-increasing if λ◦γis increasing.

Figure 13. All λ-increasing paths for a triangle withvertices (0, 0), (2, 0) and (0, 2) where λ(x, y) = x− εy fora small ε > 0 and δ = 1.

The points p and q divide the boundary ∂∆ into two increasinglattice paths

α+ : [0, n+] → ∂∆ and α− : [0, n−] → ∂∆.

We have α+(0) = α−(0) = p, α+(n+) = α−(n−) = q, n+ + n− =m − l + 3. To fix a convention we assume that α+ goes clockwisearound ∂∆ wile α− goes counterclockwise.

Let γ : [0, n] → ∆ ⊂ R2 be an increasing lattice path such thatγ(0) = p and γ(n) = q. The path γ divides ∆ into two closed regions:∆+ enclosed by γ and α+ and ∆− enclosed by γ and α−. Note thatthe interiors of ∆+ and ∆− do not have to be connected.

We define the positive (resp. negative) multiplicity µ±(γ) of thepath γ inductively. We set µ±(α±) = 1. If γ 6= α± then we take1 ≤ k ≤ n− 1 to be the smallest number such that γ(k) is a vertex of∆± with the angle less than π (so that ∆± is locally convex at γ(k)).

If such k does not exist we set µ±(γ) = 0. If k exist we consider twoother increasing lattice paths connecting p and q γ ′ : [0, n − 1] → ∆and γ′′ : [0, n] → R2. We define γ′ by γ′(j) = γ(j) if j < k andγ′(j) = γ(j + 1) if j ≥ k. We define γ ′′ by γ′′(j) = γ(j) if j 6= k andγ′′(k) = γ(k − 1) + γ(k + 1) − γ(k) ∈ Z2. We set

(5) µ±(γ) = 2 Area(T )µ±(γ′) + µ±(γ′′),

where T is the triangle with the vertices γ(k − 1), γ(k) and γ(k + 1).The multiplicity is always integer since the area of a lattice triangle ishalf-integer.


Note that it may happen that γ ′′(k) /∈ ∆. In such case we use aconvention µ±(γ′′) = 0. We may assume that µ±(γ′) and µ±(γ′′) isalready defined since the area of ∆± is smaller for the new paths. Notethat µ± = 0 if n < n± as the paths γ′ and γ′′ are not longer than γ.

We define the multiplicity of the path γ as the product µ+(γ)µ−(γ).Note that the multiplicity of a path connecting two vertices of ∆ doesnot depend on λ. We only need λ to determine whether a path isincreasing.

Example 6.2. Consider the path γ : [0, 8] → ∆3 depicted on theextreme left of Figure 14. This path is increasing with respect toλ(x, y) = x− εy, where ε > 0 is very small.

Figure 14. A path γ with µ+(γ) = 1 and µ−(γ) = 2.

Let us compute µ+(γ). We have k = 2 as γ(2) = (0, 1) is a lo-cally convex vertex of ∆+. We have γ′′(2) = (1, 3) /∈ ∆3 and thusµ+(γ) = µ+(γ′), since Area(T ) = 1

2. Proceeding further we get µ+(γ) =

µ+(γ′) = · · · = µ+(α+) = 1.Let us compute µ−(γ). We have k = 3 as γ(3) = (1, 2) is a locally

convex vertex of ∆−. We have γ′′(3) = (0, 0) and µ−(γ′′) = 1. Tocompute µ−(γ′) = 1 we note that µ−((γ′)′) = 0 and µ−((γ′)′′) = 1.Thus the full multiplicity of γ is 2.

Recall that we fixed an (irrational) linear function λ : R2 → R andthis choice gives us a pair of extremal vertices p, q ∈ ∆.

Theorem 2. The number Ntrop(g,∆) is equal to the number (countedwith multiplicities) of λ-increasing lattice paths [0, s + g − 1] → ∆connecting p and q.

Furthermore, there exists a configuration R ∈ R2 of s+g−1 points intropical general position such that each λ-increasing lattice path encodesa number of tropical curves of genus g and degree ∆ passing via R oftotal multiplicity µ(γ). These curves are distinct for distinct paths.

Example 6.3. Let us compute N(0,∆) = 5 for the polygon ∆ depictedon Figure 15 in two different ways. Using λ(x, y) = −x + εy for asmall ε > 0 we get the left two paths depicted on Figure 15. Usingλ(x, y) = x + εy we get the three right paths. The corresponding


multiplicities are shown under the path. All other λ-increasing pathshave zero multiplicity.

1 4 1 1 3

Figure 15. Computing N(1,∆) = 5 in two different ways.

In the next example we use λ(x, y) = x − εy as the auxiliary linearfunction.

Example 6.4. Figure 16 shows a computation of the well-known num-ber N(1,∆3). This is the number of rational cubic curves through 8generic points in CP2.

21432

Figure 16. Computing N(0,∆3) = 12.

Example 6.5. Figure 17 shows a computation of a less well-knownnumber N(1,∆4). This is the number of genus 1 quartic curves through12 generic points in CP2.

9 12 3 63 6 4 4 6 6 16

8 12 16 4 8 9 8 4 16 4 8

1 4 16 4 8 2 2 4 3 3 6

Figure 17. Computing N(1,∆4) = 225.


6.3. Enumeration of real curves. Let P = {p1, . . . , ps+g−1} ⊂ (R∗)2

be a configuration of points in general position. We have a total ofN(g,∆) complex curves of genus d and degree ∆ in (C∗)2 passingthrough P. Some of these curves are defined over R while others comein complex conjugated pairs.

Definition 6.6. We define the number N irrR

(g,∆,P) to be the numberof irreducible real curves of genus g and degree ∆ passing via P. Sim-ilarly we define the number NR(g,∆,P) to be the number of all realcurves of genus g and degree ∆ passing via P.

Unlike the complex case the numbers N irrR

(g,∆,P) and NR(g,∆,P)do depend on the choice of P. We have

0 ≤ NR(g,∆,P) ≤ N(g,∆)

whileNR(g,∆,P) ≡ N(g,∆) (mod 2) and, similarly, 0 ≤ N irrR

(g,∆,P) ≤N irr(g,∆) and N irr

R(g,∆,P) ≡ N irr(g,∆) (mod 2).

Tropical geometry allows to compute NR(g,∆,P) and NR(g,∆,P)for some configurations P. Note that one can extract a sign sequence{Sign(pj)}

s−g+1j=1 ⊂ Z2

2 from P = {pj}s−g+1j=1 by taking the coordinatewise

sign. Some signs choices turn out to be equivalent.Accordingly we can enhance the tropical configuration data by the

adding a choice of signs that take values in certain equivalence classesin Z2

2. Equivalence is determined by the slope of the edge containing rj.Suppose that rj belongs to an edge of C whose weight is wj and slopeis (xj, yj) ∈ Z2 (by that we mean that (xj, yj) is a primitive integervector parallel to the edge of rj).

We define the set Sj as the quotient of Z22 by the equivalence relation

(X, Y ) ∼ (X + wjxj, Y + wjyj), X, Y ∈ Z2. Thus if the weight wj iseven then Sj = Z2

2 but if Sj is odd then Sj is a 2-element set.

Definition 6.7. A signed tropical configuration of points

R = {(r1, σ1), . . . , (rs+g−1, σs+g−1)}

is a collection of s − g + 1 points rj in the tropical plane R2 togetherwith a choice of signs σj ∈ Sj.

We denote with |R| = {r1, . . . , rs+g−1} ⊂ R2 the resulting configura-tion after forgetting the signs.

Our next goal is to define the real multiplicity of a tropical curvepassing through a signed configuration. This multiplicity does dependon the choice of signs.

Let h : Γ → R2 be a tropical curve of genus g and degree ∆passing through |R|. Recall that by Lemma 4.17 each component ofΓ r h−1(|R|) is a tree with a single end at infinity (see Figure 11).


We define the real tropical multiplicity µR(T,R) of each componentT of Γ r h−1(|R|) inductively. Let A and B be two 1-valent verticesof T corresponding to marked points ra and rb such that the edgesadjacent to A and B meet at a 3-valent vertex C (see Figure 18).

T’

A B

CC

D D

T

Figure 18. Inductive reduction of the components ofΓ r h−1(|R|) in the definition of real multiplicity

Form a new tree T ′ by removing the edges [A,C] and [B,C] fromT . The number of 1-valent vertices of T ′ is less by one (C becomesa new 1-valent vertex while A and B disappear). By the inductionassumption the real multpilicity of T ′ is already defined for any choiceof signs. All the finite 1-valent vertices of T ′ except for C have theirsigns induced from the signs of T . To completely equip T ′ with thesigns we have to define the sign σd at the edge [C,D].

Suppose that the slopes of [A,C] and [B,C] are (xa, ya) and (xb, yb)and their signs are σa and σb respectively. Suppose that [C,D] is thethird edge adjacent to C. Let Sa, Sb, Sd be the set of equivalence classesof signs corresponding to [A,C], [B,C] and [C,D]. Let wa, wb, wd betheir weights.

Definition 6.8. The sign at C and the real multiplicity of T are definedaccording to the following rules.

• Suppose that wa ≡ wb ≡ 1 (mod 2) and (xa, ya) ≡ (xb, yb)(mod 2). In this case we have Sa = Sb so the signs σa andσb take values in the same set. Note that the weight of [C,D]is even in this case. The sign rd on such edge takes values inSd = Z2

2. If σa = σb then this sign can be presented by twodistinct equivalent elements σ+

d , σ−d ∈ Z2

2. Let T ′+ and T ′

− bethe trees equipped with the corresponding signs. We set

(6) µR(T ) = µR(T ′+) + µR(T ′

−).

If σa 6= σb we set µR(T ) = 0.• Suppose that wa ≡ wb ≡ 1 (mod 2) and (xa, ya) 6≡ (xb, yb)

(mod 2). In this case the weight of [C,D] is odd and the threesets Sa, Sb, Sd are all distinct. The sign σd ∈ Sd is uniquely


determined by the condition that its equivalence class has com-mon elements both with the equivalence class σa and with theequivalence class σb. Let T ′ be the tree equipped with this sign.We set

(7) µR(T ) = µR(T ′).

• Suppose that one of the weights wa and wb is odd and theother is even. E.g. suppose that wa ≡ 1 (mod 2) and wb ≡ 0(mod 2). In this case we have wd ≡ 1 (mod 2) and Sa = Sdwhile Sb = Z2

2. If the equivalence class σa contains σb we setσd = σa and

(8) µR(T ) = 2µR(T ′),

where T ′ is equipped with the sign σd at [C,D]. If the equiva-lence class σa does not contain σb we set µR(T ) = 0.

• Suppose that wa ≡ wb ≡ 0 (mod 2). Then wd is even andSa = Sb = Sd = Z2

2. If σa = σb we set σd = σa and

(9) µR(T ) = 4µR(T ′),

where T ′ is equipped with the sign σd at [C,D]. If σa 6= σb weset µR(T ) = 0.

Let R be a signed configuration of s + g − 1 points and h : Γ → R2

be a tropical curve C = h(Γ) of genus g and degree ∆ passing via |R|.The real multiplicity of a tropical curve passing is the product

µR(C,R) =∏T

µR(T ),

where T runs over all the components of Γ r h−1(|R|).

Proposition 6.9. The real multiplicity µR(C,R) is never greater thanand has the same parity as the multiplicity of C from Definition 4.12.

Proof. The proposition follows directly from Definition 6.8 by induc-tion. The multiplicity of a three-valent vertex is odd if and only if allthree adjacent edges have odd weights. This multiplicity is at least 2 ifone of the adjacent edges has even weight. This multiplicity is at least4 if all three adjacent edges have even weights. �

Similarly we have a real tropical enumerative number once the genusg, the degree ∆ and a signed configuration R of s + g − 1 points isfixed.

Definition 6.10. We define the number N irrtrop,R(g,∆,R) to be the

number of irreducible tropical curves of genus g and degree ∆ passingvia |R| counted with real multiplicities µR(C,R). Similarly we define


the number Ntrop,R(g,∆,R) to be the number of all tropical curves ofgenus g and degree ∆ passing via |R| counted with real multiplicitiesµR(C,R).

Theorem 3. Suppose that R is a signed configuration of s+g−1 pointsin tropically general position. Then there exists a configuration P ⊂(R∗)2 of s+g−1 real points in general position such that N irr

R(g,∆,P) =

N irrtrop,R(g,∆,R) and NR(g,∆,P) = Ntrop,R(g,∆,R).Furthermore, for every tropical curve C of genus g and degree ∆

passing through |R| we have µR(C,R) distinct real curves of genus gand degree ∆ passing through P. These curves are distinct for distinctC and are irreducible if C is irreducible.

Example 6.11. Let us choose the signs of R so that every σj contains(+,+) ∈ Z2

2 in its equivalence class. Let g = 0 and ∆ be the quadri-lateral whose vertices are (0, 0), (1, 0), (0, 1) and (2, 2) as in Example4.11. Then we have Ntrop,R(g,∆,R) = 3 for the configuration of 3points from Figure 8 and Ntrop,R(g,∆,R) = 5 for the configuration of3 points from Figure 9.

For other choices of signs of R we can get Ntrop,R(g,∆,R) = 1 forFigure 9 while Ntrop,R(g,∆,R) = 3 for Figure 8 for any sign choices.

6.4. Counting of real curves by lattice paths. Theorem 2 can bemodified to give the relevant count of real curves. In order to do thiswe need to define the real multiplicity of a lattice path γ : [0, n] → ∆connecting the vertices p and q once γ is equipped with signs.

Suppose γ(j)− γ(j− 1) = (yj, xj) ∈ Z2, j = 1, . . . , n. Let wj ∈ N bethe GCD of yj and xj. Similarly to the previous subsection we defineSj to be the set obtained from Z2

2 by taking the quotient under theequivalence relation (X, Y ) ∼ (X + xj, Y + yj), X, Y ∈ Z2. Let

σ = {σj}nj=1, σj ∈ Sj

be any choice of signs.We set

(10) µR

±(γ, σ) = a(T )µR

±(γ′, σ′) + µR

±(γ′′, σ′′).

The definition of the new paths γ ′, γ′′ and the triangle T is the sameas in subsection 13. The sign sequence for γ ′′ is σ′′

j = σj, j 6= k, k + 1,σ′′k = σk+1, σ

′′k+1 = σk. The sign sequence for γ ′ is σ′

j = σj, j < k,σ′j = σj+1, j > k. We define the sign σ′

k and the function a(T ) (in away similar to Definition 6.8) as follows.

• If all sides of T are odd we set a(T ) = 1 and define the sign σ ′k

(up to the equivalence) by the condition that the three equiva-lence classes of σk, σk+1 and σ′

k do not share a common element.


• If all sides of T are even we set a(T ) = 0 if σk−1 6= σk. Inthis case we can ignore γ ′ (and its sequence of signs). We seta(T ) = 4 if σk = σk+1. In this case we define σ′

k = σk = σk+1.• Otherwise we set a(T ) = 0 if the equivalence classes of σk andσk+1 do not have a common element. We set a(T ) = 2 if theydo. In the latter case we define the equivalence class of σ ′

k by thecondition that σk, σk+1 and σ′

k have a common element. Thereis one exception to this rule. If the even side is γ(k+1)−γ(k−1)then there are two choices for σ′

k satisfying the above condition.In this case we replace a(T )µR

±(γ′) in (10) by the sum of thetwo multiplicities of γ′ equipped with the two allowable choicesfor σ′

k (note that this agrees with a(T ) = 2 in this case).

Similar to subsection 13 we define µR

±(α±) = 1 and µR(γ, σ) = µR

+(γ, σ)µR

−(γ, σ).As before λ : R2 → R is a linear map injective on Z2 and p and q arethe extrema of λ|∆.

Theorem 4. For any choice of σj ∈ Z22, j = 1, . . . , s + g − 1 there

exists a configuration of s + g − 1 of generic points in the respectivequadrants such that the number of real curves among the N∆,δ relevantcomplex curves is equal to the number of λ-increasing lattice paths γ :[0, s+ g − 1] → ∆ connecting p and q counted with multiplicities µR.

Furthermore, each λ-increasing lattice path encodes a number of trop-ical curves of genus g and degree ∆ passing via R of total real multi-plicity µR(γ, σ). These curves are distinct for distinct paths.

Example 6.12. Here we use the choice σj = (+,+) so all the pointszj are in the positive quadrant (R>0)

2 ⊂ (R∗)2. The first count of N∆,1

from Example 6.3 gives a configuration of 3 real points with 5 realcurves. The second count gives a configuration with 3 real curves asthe real multiplicity of the last path is 1. Note also that the second pathon Figure 15 changes its real multiplicity if we reverse its direction.

Example 6.4 gives a configuration of 9 generic points in RP2 with all12 nodal cubics through them real. Example 6.5 gives a configurationof 12 generic points in RP2 with 217 out of the 225 quartics of genus1 real. The path in the middle of Figure 17 has multiplicity 9 butreal multiplicity 1. A similar computation shows that there exists aconfiguration of 11 generic points in RP2 such that 564 out of the 620irreducible quartic through them are real.

6.5. Different types of real nodes and the Welschinger invari-

ant. Let V ⊂ (C∗)2 be a curve defined over R. In other words it is acurve invariant with respect to the involution of complex conjugation.


Suppose that V is nodal, i.e. all singularities of V are ordinary doublepoints (nodes).

There are three types of nodes of V :

• Hyperbolic. These are the real nodes that locally are inter-sections of a pair of real branches. Such nodes are given byequation z2 − w2 = 0 for a choice of local real coordinates(z, w).

• Elliptic. These are the real nodes that locally are intersectionsof a pair of imaginary branches. Such nodes are given by equa-tion z2 + w2 = 0 for a choice of local real coordinates (z, w).

• Imaginary. These are nodes at non-real points of V . Such nodescome in complex conjugate pairs.

This distinction was used in [25] in order to get a real curve countinginvariant with respect to the initial configuration P ⊂ (R∗)2. Indeed,let us modify the real enumerative problem from Definition 6.6 in thefollowing way. Let V ⊂ (C∗)2 be a real nodal curve. Let e(V ) be thenumber of real elliptic nodes of V . We prescribe to V the sign equalto (−1)e(V ). As usual we fix a genus g, a degree ∆ and a configurationP ∈ (R∗)2 of s+ g − 1 points in general position.

Definition 6.13 (see [25]). We define the number N irrR,W (g,∆,P) to

be the number of irreducible real curves of genus g and degree ∆passing via P counted with signs. Similarly we define the numberNR,W (g,∆,P) to be the number of all real curves of genus g and de-gree ∆ passing via P counted with signs.

Theorem 5 (Welschinger [25]). If g = 0 and CT∆ is smooth then thenumber N irr

R,W (g,∆,P) does not depend on the choice of P.

We call this number the Welschinger invariant. Theorem 3 can bemodified to compute N irr

R,W (g,∆,P) and NR,W (g,∆,P).

Let C ⊂ R2 be a simple tropical curve. Recall that Definition 2.16assigns a multiplicity multV (C) to every 3-valent vertex V ∈ C.

Definition 6.14. We define

multR,WV (C) = (−1)

multV (C)−1

2

if multV (C) is odd and multR,WV (C) = 0 if multV (C) is even.

The tropical Welschinger sign of C is the product of multR,WV (C)

over all 3-valent vertices of C.

As usual let us fix a genus g, a degree ∆ and a configuration R ⊂ R2

of s+g−1 points in tropically general position. Define N irrtrop,R(g,∆,R)

to be the number of irreducible tropical curves of genus g and degree


∆ passing via R counted with the Welschinger sign. In a similar waydefine Ntrop,R(g,∆,R) to be the number of all tropical curves of genusg and degree ∆ passing via R counted with the Welschinger sign.

Theorem 6. Suppose that R ⊂ R2 is a configuration of s + g − 1points in tropically general position. Then there exists a configurationP ⊂ (R∗)2 of s+ g − 1 real points in general position such that

NR,W (g,∆,P) = N irrtrop,R(g,∆,R)

andNR,W (g,∆,P) = Ntrop,R(g,∆,R).

Furthermore, for every tropical curve C of genus g and degree ∆passing through R we have a number of distinct real curves of genusg and degree ∆ passing through P with the total sum of signs equal toνR(C). These curves are distinct for distinct C and are irreducible ifC is irreducible.

Example 6.15. Let g = 0 and ∆ be the quadrilateral with vertices(0, 0), (1, 0), (0, 1) and (2, 2). In Figure 8 we have 3 real curves, two ofthem have the sign +1 and one has the sign −1. In Figure 9 we have2 real curves, one of them has the sign +1 and one has the sign 0. Inboth cases we have Ntrop,R(g,∆,R) = 1.

Theorem 2 can be adjusted to computeNtrop,R(g,∆,R) = 1 by latticepaths. Let γ : [0, n] → ∆ be a lattice path connecting the vertices pand q of ∆. Let us introduce the multiplicity νR inductively, in amanner similar to our definition of the multiplicity µ(γ). Namely weset νR(γ) = νR

+(γ)νR+(γ). To define νR

±(γ) we repeat the definition ofµ±(γ) but replace (5) with

νR

±(γ) = b(T )νR

±(γ′) + νR

±(γ′′).

Here we define b(T ) = 0 if at least one side of T is even and b(T ) =

(−1)#(Int T∩Z2) otherwise. The paths γ ′, γ′′ and the triangle T are thesame as in the inductive definition of µ±.

Theorem 7. For any choice of an irrational linear map λ : R2 → R

there exists a configuration P of s + g − 1 of generic points in (R∗)2

such that the number of real curves of genus g and degree ∆ passingthrough P counted with the tropical Welschinger sign is equal to thenumber of λ-increasing lattice paths γ : [0, s+ g− 1] → ∆ connecting pand q counted with multiplicities νR.

Furthermore, there exists a configuration R ∈ R2 of s+g−1 points intropical general position such that each λ-increasing lattice path encodesa number of tropical curves of genus g and degree ∆ passing via R with


the sum of signs equal to νR(γ). These curves are distinct for distinctpaths.

Example 6.16. In example 6.3 we have NR,W (∆,P) = 1 for somechoice of P. In example 6.4 we have N irr

R,W (0,∆3,P) = 8 for some

choice of P. In example 6.5 we have N irrR,W (1,∆4,P) = 93 for some

choice of P. Note that by Theorem 5 the number N irrR,W (0,∆3,P) = 8

does not depend on the choice of P.

The following observation is due to Itenberg, Kharlamov and Shustin[8]. If λ(x, y) = x− εy and ∆ = ∆d of ∆ = [0, d1]× [0, d2] then νR(γ) ≥0 for any λ-increasing path γ and, furthermore, any tropical curveencoded by γ by Theorem 7 has a non-negative tropical Welschingersign. It is easy to show that there exist λ-increasing paths that encodeirreducible tropical curves of non-zero tropical Welschinger signs. Weget the following corollary for any d, d1, d2 ∈ N.

Corollary 8. For any generic configuration P ⊂ RP2 of 3d− 1 pointsthere exists an irreducible rational curve V ⊂ RP2 of degree d passingthrough P.

For any generic configuration P ⊂ RP1×RP1 of 2d1 +2d2−1 pointsthere exists an irreducible rational curve V ⊂ RP1 × RP1 of bidegree(d1, d2) passing through P.

With the help of Theorem 7 Itenberg, Kharlamov and Shustin in[8] have obtained a non-trivial lower bound for the number of suchrational curves. In particular, they have shown that for any genericconfiguration P of 3d− 1 points in RP2 there exists at least d!

2rational

curves of degree d passing via P.

7. Connection between classical and tropical geometries

7.1. Degeneration of complex structure on (C∗)2. Let t > 1 be areal number. We have the following self-diffeomorphism (C∗)2 → (C∗)2

(11) Ht : (z, w) 7→ (|z|1

log tz

|z|, |w|

1log t

w

|w|).

For each t this map induces a new complex structure on (C∗)2.Here is a description of the complex structure induced by Ht in

logarithmic polar coordinates (C∗)2 ≈ R2 × iT 2. (This identificationis induced by the holomorphic logarithm Log from the identificationC2 ≈ R2 × iR2.) If v is a vector tangent to iT 2 we set Jtv = 1

log(t)iv.

Note that Jtv is tangent to R2.Clearly, a curve Vt is holomorphic with respect to Jt if and only

if Vt = Ht(V ), where V is a holomorphic curve with respect to the


standard complex structure, i.e. Je-holomorphic. Let Log : (C∗)2 → R2

be the map defined by Log(z, w) = (log |z|, log |w|). We have

Log ◦Ht = Logt .

Note that Ht corresponds to a log(t)-contraction (x, y) 7→ ( xlog(t)

, ylog(t)

)

under Log.

7.2. Complex tropical curves in (C∗)2. There is no limit (at leastin the usual sense) for the complex structures Jt, t→ ∞. Nevertheless,as in section 6.3 of [16] we can define the J∞-holomorphic curves whichhappen to be the limits of Jt-holomorphic curves, t→ ∞.

There are several way to define them. An algebraic definition is theshortest and involves varieties over a non-Archimedian field. Let K bethe field of the (real-power) Puiseux series a =

∑j∈Ia

ajtj, where Ia ⊂ R

is a bounded from below set contained in a finite union of arithmeticprogression. The field K is algebraically closed and of characteristic0. The field K has a non-Archimedian valuation val(a) = −min Ia,val(a+ b) ≤ val(a) + val(b).

As usual, we set K∗ = K r {0}. The multiplicative homomorphismval : K∗ → R can be “complexified” to w : K∗ → C∗ ≈ R × S1 bysetting w(a) = eval(a)+i arg(aval(a)). Applying this map coordinatewise weget the map W : (K∗)2 → R2. The image of an algebraic curve VKunder W turns out to be a J∞-holomorphic curve (cf. Theorem 7.1).

Note the following special case. Let

VK = {z ∈ (C∗)2 |∑

j∈Vert∆

ajzj},

where z ∈ (K∗)2, j runs over all vertices of ∆ and aj is such thatval(aj) = 0. By Kapranov’s theorem [9] Val(VK) = Log(W (VK)) isthe tropical curve Cf defined by f(x) = “

∑j∈Vert∆

xj”, x ∈ R2. Thus

Cf is a union of rays starting from the origin and orthogonal to thesides of ∆, in other words it is the 1-skeleton of the normal fan to ∆.However, W (VK) ⊂ (C∗)2 depends on the argument of the leading termof aj ∈ K∗. Thus different choices of W (VK) give different phases forthe lifts of Cf . We may translate Cf in R2 so that it has a vertex ina point x ∈ R2 instead of the origin. Corresponding translations ofW (VK) give a set of possible lifts.

This allows to give a more geometric description of J∞-curves. Theyare certain 2-dimensional objects in (C∗)2 which project to tropicalcurves under Log. Namely, let C ⊂ R2 be a tropical curve, x ∈ C be apoint and U 3 x ∈ R2 be a convex neighborhood such that U ∩ C is a


cone over x (i.e. for every y ∈ C ∩ U we have [x, y] ⊂ C). Note that ifx is a point on an open edge then it is dual (in the sense of 3.4) to asegment in ∆. If x is a vertex of C then it is dual to a 2-dimensionalpolygon in ∆. In both cases we denote the dual polygon with ∆′. Wesay that a 2-dimensional polyhedron V∞ ⊂ (C∗)2 is (C ∩U)-compatibleif Log−1(U)∩V∞ = Log−1(U)∩W , where W is a translation of W (VK)while VK = {z ∈ (C∗)2 |

∑j

ajzj}, j runs over some lattice points of ∆′

and val(aj) = 0.

Theorem 7.1. Let V∞ ⊂ (C∗)2 be a polyhedral subcomplex. The fol-lowing conditions are equivalent.

• V∞ = W (VK), where VK ⊂ (K∗)2 is an algebraic curve.• C = Log(V∞) is a tropical curve such that for every x ∈ C there

exists a small open convex neighborhood x ∈ U ⊂ R2 such thatLog−1(U) ∩ V∞ is (C ∩ U)-compatible.

• V∞ is the limit when k → ∞ in the Hausdorff metric of asequence of Jtk-holomorphic curves Vtk with lim

k→∞tk = ∞.

Definition 7.2. Curves satisfying to any of the equivalent conditionsof Theorem 7.1 are called complex tropical curves.

Theorem 7.1 allows to think of complex tropical curves both astropical curves equipped with a phase, i.e. a lifting to (C∗)2, and asJ∞-holomorphic curves, i.e. as limits of Jt-holomorphic curves whent→ ∞.

Proof of Theorem 7.1. 1 =⇒ 2. Let f : (K∗)2 → K be the polynomialdefining VK . The image C = Log(V∞) is a tropical curve by Kapranov’sTheorem [9]. Let x ∈ C ⊂ (K∗)2. The lowest t-powers of f(x) comeonly from the polygon ∆′ ⊂ ∆ dual to the stratum containing x. Thecompatibility curve is given by the sum of the ∆′-monomials of f .

2 =⇒ 1. Consider the subdivision of ∆ dual to the tropical curveC. The compatibility condition gives us a choice of monomials for eachpolygon ∆′ ⊂ ∆ in the subdivision dual to C. However, the choiceis not unique, due to the higher t-power contributions. On the otherhand, a monomial corresponds to a lattice point of ∆ which may belongto several subpolygons in the subdivision.

We have to choose the coefficients for the monomials so that theywould work for all subpolygons of the subdivision. Let j ∈ ∆∩Z2. Thecoefficient aj is a Puiseux series in t. The number val(aj) is definedby the equation for the tropical curve C. We set aj = αjt

val(aj), whereαj ∈ C∗, i.e. our coefficient Puiseux series are actually monomials.


Namely, let ∆′ be a polygon in the subdivision of ∆ dual to C. Apoint x on the corresponding stratum of C is compatible withW ({f∆′

=0}) for a polynomial f∆′

over K with the Newton polygon ∆′. Thecurve W ({f∆′

= 0}) coincides with the curve W ({f∆′

min = 0}), wherethe polynomial f∆′

min is obtained by replacing each coefficient seriesaj = aj(t) of f∆′

with its lower t-power monomial αjtval(aj). The poly-

nomial f∆′

min is well-defined up to a multiplication by a constant. Fora proper choice of constants for different ∆′ the coefficients given bydifferent f∆′

min for the same j ∈ ∆ ∩ Z2 agree.1 =⇒ 3. This implication is a version of the so-called Viro patch-

working [24] in real algebraic geometry. If f : (K∗)2 → K is the poly-nomial defining VK then we can construct the sequence Vtk ⊂ (C∗)2

in the following way, see [17]. Let f tk : (C∗)2 → C be the complexpolynomial obtained from f by plugging t = tk to the Puiseux seriescoefficients of f . We set Vk to be the image of the zero set of f tk bythe self-diffeomorphism Htk defined in (11).

3 =⇒ 1. Propositions 3.9 and 8.2 imply that Log(V∞) is a tropicalcurve. Let αj ∈ Rtrop be the coefficients of the tropical polynomialdefining Log(V∞). To find a presentaion V∞ = W (VK) we take thepolynomial with coefficients βjt

αj ∈ K, βj ∈ S1.To find βj we note that if Vk is a Jtk -holomorphic curve then H−1

tk(Vk)

is (honestly) holomorphic and is given by a complex polynomial fk =∑j

aVk

j zj. To get rid of the ambiguity resulting from multiplication by

a constant we may assume that aVk

j′ = 1 for a given j ′ ∈ ∆∩Z2 and for

all sufficiently large k. We take βj = limk→∞

Arg(aVk

j ). �

Let Arg : (C∗)2 → S1×S1 be defined by Arg(z1, z2) = (arg(z1), arg(z2)).

Proposition 7.3. Suppose that V∞ ⊂ (C∗)2 is a complex tropicalcurve, C = Log(V∞) ⊂ R2 is the corresponding “absolute value” tropi-cal curve and x ∈ C is either a vertex dual to a polygon ∆′ ⊂ ∆ or apoint on an open edge dual to an edge ∆′ ⊂ ∆. We have Arg(Log−1(x)∩V∞) = Arg(V ′) for some holomorphic curve V ′ ⊂ (C∗)2 with the New-ton polygon ∆′.

This proposition follows from the second characterization of J∞-holomorphic curves in Theorem 7.1.

7.3. Parameterized complex tropical curves and their genus.

Definition 7.4. A map ψ : V∞ → V∞ is called a parameterized complextropical curve if


• V∞ is homeomorphic to a surface;• for any x ∈ C = Log(V∞) the map ψ|(Log ◦ψ)−1(x) :

(Log ◦ψ)−1(x) → V∞ ∩ Log−1(x) ⊂ Log−1(x) ≈ S1 × S1

is conjugate to the map Arg : V ′ → S1 × S1 from Proposition7.3;

• for any x ∈ C = Log(V∞) there is a small convex neighborhoodx ∈ U ⊂ R2 with the following property. For every componentV ′ of (Log ◦ψ)−1(U) there exists a complex tropical curve V ′′ ⊂(C∗)2 such that ψ(V ′) = V ′′ ∩ Log−1(U).

Definition 7.5. The genus g(V∞) of a complex tropical curve V∞ is

the minimal genus of V∞ among all parameterizations V∞ → V∞. Asusual, for the case when V∞ is disconnected we use a convention

g(V∞) = 1 +∑V ′

(g(V ′) − 1),

where V ′ runs over all connected components of V∞.

Proposition 7.6. If V∞ ⊂ (C∗)2 is a complex tropical curve and C =Log(V∞) ⊂ R2 is the corresponding tropical curve then g(C) ≤ g(V∞).

Proof. Any parameterized complex tropical curve V∞ → V∞ defines aparameterized tropical curve h : Γ → C with g(V∞) ≥ g(C). �

Remark 7.7. It can happen that g(C) < g(V∞). For example take V a∞

to be the limiting curve for the family 1+x+y+at−1xy, where a ∈ C,|a| = 1, and t > 0, t → ∞. The image C = Log(V a

∞) does not dependon the choice of a. We have g(C) = −1. If a = 1 then g(V a

∞) = −1,otherwise g(V a

∞) = 0.

Proposition 7.8. For any open edge E ⊂ C the inverse image (Log ◦ψ)−1(E)is a collection E of disjoint annuli. The image ψ(E) is a collection ofdisjoint open holomorphic annuli in (C∗)2. The restriction of ψE toeach component of the image is a covering map. The total degree ofthese coverings equals to the weight of E.

Proof. Denote with ∆′ the (1-dimensional) polygon dual to E in thesubdivision of ∆ associated to C. By definition V∞ is ∆′-compatible.Any holomorphic curve of degree ∆′ is a collection of disjoint annuli in(C∗)2. �

Definition 7.9. Let ψ : V∞ → V∞ be a parameterized complex tropicalcurve. Let E be an open edge of the tropical curve Log(V∞). Theelementary cylinder A of V∞ is a component of (Log ◦ψ)−1(E). Itsweight is the degree of the covering ψA : A→ ψ(A).


Definition 7.10. A parameterized complex tropical curve ψ : V∞ →V ◦∞ ⊂ (C∗)2 parameterized by is called simple if C = Log(V∞) is a

simple tropical curve, g(V∞) = g(C) and for any open edge E ⊂ C theinverse image (Log ◦ψ)−1(E) is connected.

Remark 7.11. Recall that we denote with ∆1 a triangle with vertices(0, 0), (1, 0) and (1, 0). A complex tropical curve of degree ∆1 is called acomplex tropical projective line (cf. Definition 2.6). Note that any suchcurve is simple. Any two such curves can be identified by a (multiplica-tive) translation in (C∗)2. Furthermore, if P ⊂ (C∗)2 is a configurationof 2 points such that Log(P) ⊂ R2 is tropically generic then there existsa unique complex tropical projective line passing through P.

Let C ⊃ Log(P) be a tropical curve in R2 of genus g and degree ∆.Let µ be the multiplicity of C (see Definition 4.12) and let x ≤ s bethe number of ends of C. Let P ⊂ (C∗)2 be a configuration of x+g−1points such that Log(P) ⊂ R2 is in tropically general position.

Let V∞ be a simple complex tropical curve passing via P. Sinceqqq = Log(P) is in tropically general position each pj ∈ P sits onan elementary cylinder. We prescribe the multiplicity to V∞ equal tothe product of the weights of all its elementary cylinders that containpoints from P. Then we have the following proposition.

Proposition 7.12. There are µ simple complex tropical curves in (C∗)2

of genus g and degree ∆ such that they project to C and pass via P.

Let ∆ be a triangle and C ⊂ R2 be a tropical curve of degree ∆ withno bounded edges. Such curve has genus 0 and 3 unbounded edges.(Note that if #(∂∆ ∩ Z2) > 3 then some of the unbounded edges haveweight greater than 1.) Let V∞ ⊂ (C∗)2 be a simple complex tropicalcurve V∞ projecting to C.

Lemma 7.13. The curve V∞ ⊂ (C∗)2 lifts to a complex tropical pro-jective line under some linear map M∆ : (C∗)2 → (C∗)2 that is self-covering of degree 2 Area(∆).

Clearly, we have deg(M∆) = 2 Area(∆) of such lifts.

Proof of Lemma 7.13. Consider an affine-linear surjection ∆1 → ∆.Let

L∆ : R2 → R2

be the linear part of this map. Note that det(L∆) = 2 Area(∆). Thematrix of L∆ written multiplicatively defines a map

(12) M∆ : (C∗)2 → (C∗)2.


Alternatively we can define M∆ as the map covered by L∆ ⊗ C :C2 → C2 under exp : C2 → (C∗)2. We have deg(M∆) = det(L∆) =2 Area(∆).

Note that M∆ extends to a holomorphic map

(13) M∆ : CP2 → CT∆.

If ∆′ ⊂ ∂∆ is a side of ∆ then (M∆)−1(∆′) consists of 2 Area(∆)/l(∆′)components, where l(∆′) = #(∆′ ∩ Z2) − 1 is the integer length of ∆′.

Since V∞ is simple we have V∞ homeomorphic to sphere puncturedin 3 points. Each puncture corresponds to a side of ∆. A loop aroundthe puncture in V∞ wraps around CT∆′ l(∆′) times. Therefore, V∞ liftsunder M∆. �

Proof of Theorem 7.12. Note that C is simple since Log(P) is in trop-ically general position.

We start with a special case. Suppose (as in Lemma 7.13) that g = 0,∆ is a triangle and C has 3 ends. Then P consists of two points.

By Lemma 7.13 any simple complex tropical curve over C is an imageof a complex tropical projective line under M∆. The set (M∆)−1(P)consists of 2 deg(M∆) = 4 Area(∆) points. Each pair projecting todifferent points under M∆ can be connected with a complex tropicalprojective line. We have a total of (deg(M∆))2 of such lines and eachof them projects to a simple complex tropical curve of degree ∆ underM∆. Conversely, each such simple complex tropical curve lifts to ddistinct lines. Therefore, we have deg(M∆) = 2 Area(∆) distinct simplecomplex tropical curves projecting to C and passing via P.

Now we are ready to treat general case. By Lemma 4.17 each com-ponent K of Γ r h−1(P) is a tree which contains one end at infinity.Here h : Γ → C is the parametrization of C. Let A,B ∈ P be twopoints that connect to the same 3-valent vertex in K as in Figure 18.Let ∆′ ⊂ ∆ be the subpolygon dual to this 3-valent vertex of C. Byour special case we have 2 Area(∆′)-compatible special case and canproceed inductively. �

7.4. Polynomials to define complex tropical curves. Let VK ⊂(K∗)2 be a curve given by a non-Archimedian polynomial

fK(z) =∑j

αjzj ,


z ∈ K2, j ∈ ∆, αj ∈ K∗. Let V∞ = W (VK) ⊂ (C∗)2 be the corre-sponding complex tropical curve. Denote with f the tropical polyno-mial defining the projection Log(V ) ⊂ R2. Note that by Kapranov’stheorem [9] f(x) = “c

∑j

val(αj)xj”.

Proposition 7.14. If j and j ′ are vertices of Subdivf then the ratiow(αj)

w(αj′ )depends only on V∞ and does not depend on the choice of VK

such that W (VK) = V∞.

Proof. Since ∆ is connected we need to prove the proposition only inthe case when j and j ′ are connected with an edge ∆′ ∈ Subdivf . Thecomplex tropical curve V∞ is ∆′-compatible, therefore the propositionfollows from the special case when ∆ = ∆′ is 1-dimensional.

After an automorphism in (C∗)2 we may assume that ∆ = ∆′ = [0, l],l ∈ N. Then V∞ is a collection of the elementary cylinders each defined

by equation z = zk and of weight wk. We havew(αj)

w(αj′ )=

∏zwk

k . �

If j 7→ − val(αj) is strictly convex then to recover a complex tropicalcurve w(VK) it sufficed to know only aj = w(αj).

Proposition 7.15. Let f(z) =∑j∈∆

ajzj, be a formal sum of monomials,

aj ∈ C∗, where j runs over some lattice points of ∆. Suppose thatj 7→ − log |aj| is strictly convex. Then f defines a complex tropicalcurve V∞ ⊂ (C∗)2 of degree ∆. This correspondence satisfies to thefollowing properties.

• Two different such polynomials define different complex tropicalcurves.

• Complex tropical curves defined in this way form an open anddense set in the space of all complex tropical curves of degree∆.

Proof. Form a polynomial

fK(x, y) =∑

(j,k)∈∆∩Z2

ajktlog |ajk|xjyk

defined over K. Define V∞ = w({(x, y) ∈ (K∗)2 | fK(x, y) = 0}. Tosee that different polynomials define different curves it suffices to notethat for the corresponding tropical polynomial

ftrop(x) = “∑j∈∆

log |aj|xj”

the subdivision Subdivftrop contains all lattice points of ∆ as its vertices.In particular, no lattice point of ∆ is contained in the interior of an


edge of Subdivftrop . For different polynomials we have different ∆′-compatible elements for some edge ∆′ ∈ Subdivftrop .

On the other hand, any complex tropical curve can be obtainedfrom a non-Archimedian curve VK ⊂ (K∗)2 given by a polynomialfK(z) =

∑j

αjzj such that the function j 7→ − val(αj) is convex. The

polynomials with strictly convex functions j 7→ − val(αj) are open anddense among all such polynomials. �

Remark 7.16. If VK ⊂ (K∗)2 is given by a polynomial fK(z) =∑j

αjzj

such that the function j 7→ − val(αj) is not strictly convex then the col-lection of complex numbers aj = w(αj) does not necessarily determinethe complex tropical curve w(VK).

E.g. the polynomials fK(z, w) = z2 +3z+2 and f ′K(z, w) = z2+z+1

(treated as polynomials over K) produce the same collection of threecomplex numbers, namely a(0,0) = a(1,0) = a(2,0) = 1. However we havew(VK) = {(z, w) ∈ (C∗)2 | z = −1} and w(V ′

K) = {(z, w) ∈ (C∗)2 | z =

−12±

√3

2}.

8. Counting holomorphic curves by tropical curves

8.1. Complex amoebas in R2 and the key lemma. Let P ={p1, . . . , pn} ⊂ (C∗)2 be generic points such that the points

r1 = Log(p1), . . . , rn = Log(pn)

are in general position tropically. Denote R = {r1, . . . , rn} ⊂ R2.Recall that we fix a Newton polygon ∆ ⊂ R2 and a genus g ∈ N.There are N(g,∆) holomorphic curves passing through P as long asn = s + g − 1. By Proposition 4.10 there are finitely many tropicalcurves

(14) C1, . . . , Cm ⊂ R2

of genus g with the Newton polygon ∆ and passing through R. Notethat m depends on the choice of the points xj (unlike the invariantnumber Ntrop(g,∆) ≥ m).

Proposition 8.1. For generic t we have N(g,∆) Jt-holomorphic curvespassing through P.

Proof. Indeed, this number equals to the number of holomorphic (i.e.Je-holomorphic) curves through the points Ht(p1), . . . , Ht(pn). (Thesepoints are in general position for generic t since they are for t = 1.) �


Let Nε(Cj) be the ε-neighborhood of Cj for ε > 0. Recall (see [6])that the amoeba of a curve V ⊂ (C∗)2 is its image

A = Log(V ) ⊂ R2.

Note that if V is a Jt-holomorphic curve then Log(V ) can be obtainedfrom the amoeba of some holomorphic curve by the log(t)-contraction.This allows us to speak of the Newton polygons of Jt-holomorphiccurves.

Proposition 8.2. If V is a Jt-holomorphic curve whose Newton poly-gon is ∆ then its amoeba A = Log(V ) contains a tropical curve C withthe same Newton polygon ∆.

Proof. If t = 1, i.e. C is holomorphic with respect to the standardholomorphic structure, then the statement follows from the theorem ofPassare and Rullgard [18]. Recall (see [18]) that that if V is a complexcurve defined by a polynomial f(z, w) =

∑j,k

aj,kzjwk then the spine

of its amoeba A is a tropical curve defined by a tropical polynomialN∞f (x, y)“

∑j,k

bj,kxjyk”, where

bj,k =1

(2πi)2

∫

Log−1(r)

log |f(z, w)|dz

z

dw

w

and r ∈ R2 r A is any point such that its index is (j, k). If there areno points in R2 r A of index (j, k) then the monomial xjyk is omittedfrom N∞

f .It is shown in [18] that the tropical hypersurface defined by N∞

f iscontained in A. Clearly, the Newton polygon of N∞

f is ∆. To finish theproof we note that the image of a tropical curve under a homothety istropical with the same Newton polygon. �

By Proposition 8.1 Theorem 1 follows from the following lemma.

Lemma 8.3. There exist a sufficiently small ε > 0 and a sufficientlylarge t > 0 with the following properties.

a: If V is a Jt-holomorphic curve of genus g and degree ∆ passingthrough P then its amoeba Log(V ) is contained in Nε(Cj) forsome j = 1, . . . , n.

b: The multiplicity of each Cj from (14) (see Definition 4.12) isequal to the number of the Jt-holomorphic curves V of genusg and degree ∆ passing through P and such that Log(V ) iscontained in Nε(Cj).


8.2. Proof of Lemma 8.3.a. A holomorphic curve V ⊂ (C∗)2 is givenby a polynomial

f(z, w) =∑

(j,k)∈∆

aj,kzjwk.

To a curve V ⊂ (C∗)2 we may associate its tropicalization V trop ⊂ R2

given by the tropical polynomial

f trop(x, y) = max(j,k)∈∆

jx+ ky + log |aj,k|.

Proposition 8.6 and the following lemma imply Lemma 8.3.a.

Lemma 8.4. The amoeba Log(V ) is contained in the ε-neighborhoodof V trop, where ε = log(#(∆ ∩ Z2)) is the number of monomials of f .

Proof. Suppose that (x0, y0) is not contained in the ε-neighborhood ofV trop. Then there exists (j, k) such that

(15) jx0 + ky0 + log |aj,k| > j ′x0 + k′y0 + log |aj′,k′| + ε

for any (j, k) 6= (jmax, kmax). Indeed, the distance from (x, y) to the linejx+ky+log |aj,k| = j ′x+k′y+log |aj′,k′| is greater than ε and the norm ofthe gradient of the function (j−j ′)x+(k−k′)y is at least 1 (as j, j ′, k, k′

are all integers). Let (z, w) be a point such that Log(z, w) = (x0, y0).Exponentiating the inequality (15) we get |aj′,k′zj

′

wk′

| < 1m|aj,kzjwk|.

By the triangle inequality f(z, w) 6= 0. �

Corollary 8.5. The amoeba Log(Vt) of a Jt-holomorphic curve Vt =Ht(V ) is contained in the ε-neighborhood of V trop, where ε = logt(#(∆∩Z2)).

The corollary is obtained by applying the log t-contraction to Lemma8.4.

Let Vk ⊂ (C∗)2, k ∈ N, be a sequence of curves passing through Pand such that Vk is a Jtk-holomorphic curve for some tk > 0, wheretk → ∞, k → ∞. As in the previous subsection we assume that theholomorphic curve H−1

tk(Vk) is of genus g and has the Newton polygon

∆ for each k. Denote with Ak = Log(Vk) the amoeba of Vk.Lemma 8.3.a follows from Corollary 8.5 and the following proposi-

tion.

Proposition 8.6. There is a subsequence Vkα, α ∈ N, such that the

sets Akα⊂ R2 converge in the Hausdorff metric in R2 to some tropical

curve Cj from (14).

Proof. Consider the tropicalizations V tropk . By Proposition 3.9 we can

extract a subsequence which converges to a tropical curve C. To prove


the proposition it suffices to show that the Newton polyhedron of C isa tropical curve passing through R of genus g whose Newton polygonis ∆. Proposition 3.9 and Corollary 8.5 ensure convergence in theHausdorff metric in R2.

We have C ⊃ R since Vk ⊃ P and thus Ak ⊃ R. The degree of Cis a subpolygon ∆′ ⊂ ∆ since C is the limit of curves of degree ∆. Wewant to prove that ∆′ = ∆.

Choose a disk DR ⊂ R2 of radius R so large that DR contains allvertices of C. Furthermore, making R larger if needed we may assumethat the extension of the exterior edges of C ∩DR beyond DR do notintersect.

The Newton polygon of V tropk is ∆. Therefore, it has s ends. By

Proposition 3.9 V tropk ∩ DR approximates C ∩ DR. Therefore, for a

large k V tropk ⊃ R′ where R′ is a configuration of s + g − 1 points

in tropically general position obtained by a small deformation of R.We have ∆′ = ∆ if and only if V trop

k r DR is a disjoint union of rays(each going to ∞). If not, C r DR has a bounded edge connecting apoint of V trop

k ∩ ∂DR with a vertex of V tropk outside of DR. A change

of the length of this edge produces a deformation of V tropk such that

all curves in the family pass via R′. This contradicts to the tropicalgeneral position of R′.

Note that the genus of C cannot be smaller than g, otherwise theconfiguration R is not in general position. The genus of curves V trop

k

may be larger than g even though the genus of Vk is g. However,the genus of their limit C cannot be larger than g by Proposition 7.6.Therefore, the genus of C is g. Thus C has to be one of Ck from(14). �

8.3. Proof of Lemma 8.3.b. Let C ⊂ R2 be one of the tropicalcurves Cj from (14) and µ be its multiplicity from Definition 4.12.Denote with SubdivC the lattice subdivision of ∆ dual to C. Let

(16) ftrop =∑

j∈∆∩Z2

βjxj

be the tropical polynomial that defines C.By Theorem 7.12 there are µ complex tropical curves projecting to

C and passing via P. Let V be one of them. Let us fix a “referencevertex” j0 ∈ ∆ so that the complex coefficient at the correspondingmonomial will always be 1. We may assume that βj0 = 0.

By Propositions 7.14 we have a well-defined coefficients aj ∈ C forthe vertices j of SubdivC . Note that since C ⊃ R and R is in tropically


general position the number of ends of C is s and therefore

∂∆ ∩ Z2 ⊂ Vert(SubdivC).

The coefficients aj correspond to the complex tropical curve V . Hereaj0 = 1 and − log(aj) = βj, where j runs over all vertices of SubdivCis the tropical polynomial that defines C.

Proposition 8.7. The coefficients aj do not depend on the choice ofV . Thus these coefficients depend only on C and P.

Proof. We repeat the induction steps from the proof of Proposition7.12. By Lemma 4.17 each component K of Γrh−1(P) is a tree whichcontains one end at infinity.

Let A,B ∈ P be two points that connect to the same 3-valent vertexin K as in Figure 18. Then the points A,B are contained in the edges ofC dual to the edges [j, j ′], [j ′, j ′′] such that the triangle with the verticesj, j ′, j ′′ belongs to SubdivC . By Propositions 7.14 we may recover

aj

aj′

andaj′

aj′. Therefore, we know the ratio

aj

aj′′and we can proceed further

by induction. �

Consider the polynomial

ft(z) =∑

j∈SubdivC

arg(aj)t− log |aj |.

The sum is taken over all vertices j of SubdivC . Define Vt ⊂ (C∗)2 by

Vt = Ht({z ∈ (C∗)2 | ft(z) = 0}.

For each t > 1 the curve Vt is Jt-holomorphic. For large values of t wehave Log(Vt) ⊂ Nε(C).

For large values of t we may consider the curve Vt as an approximatesolution to the problem of finding the Jt holomorphic curves of genusg and degree ∆ via P. Indeed, most likely Vt is smooth (and thereforeof genus #(∆ ∩ Z2) ≥ g) and does not contain P. However it is closeto a (singular) curve of genus g and is very close to the configurationP. We need to find a genuine solution near this approximate one forlarge values t >> 1.

Recall that the amoebas of the (Jt-holomorphic) curves we searchhave to be contained in a small neighborhood of C while the curvesthemselves have to contain P. For large t >> 1 this implies that suchcurve can be presented in the form Vt = Ht({f

ζt (z) = 0), where

f ζt =∑

j∈∆∩Z2

arg(ζj)t− log |ζj |zj


and ζ ∈ Cn, n = #(∆ ∩ Z2) − 1, are such that |ζj − aj| < ε′j forj ∈ Vert(SubdivC) while − log |ζj| − βj < εj for j /∈ Vert(SubdivC).Here ε′j > 0 is some collection of small numbers. All ζ that comply tothese conditions form a polydisk D ⊂ Cn.

Using the subdivision SubdivC we can essentially localize the prob-lem. We need to find the number of such choices for ζ that V ζ

t ⊃ Pand Vt has genus g. Let ∆′ ⊂ SubdivV be a 2-dimensional subpolygonwhose dual vertex is v′ ∈ C. Then we denote with U ′(∆′) ⊂ R2 a smallopen neighborhood of v′. Let ∆′ ⊂ SubdivV be an edge whose dualedge is E ′ ∈ C. In this case we denote with U ′(∆′) ⊂ R2 a small openneighborhood of E ′. We assume that ε > 0 in Lemma 8.3 is so smallthat

Nε ⊂⋃

∆′∈SubdivC

U ′(∆′).

Proposition 8.8. Suppose that V ζt ⊃ P is a curve of genus g and t is

large.

• If ∆′ is a triangle then V ζt ∩ Log−1(U ′(∆′)) is a rational curve

with 3 ends.• If ∆′ is a parallelogram then V ζ

t ∩ Log−1(U ′(∆′)) splits to twoirreducible components. Each of the components is an embeddedannulus.

• If ∆′ is an edge then V ζt ∩Log−1(U ′(∆′)) is an irreducible com-

ponent that is an immersed annulus.

Conversely, suppose that t is large, ζ ∈ D and V ζt satisfies to the

conditions above. Then the genus of V tζ is g. In addition we have

Aζt = Log(V ζ

t ) ⊂ Nε(C).

Proof. Recall that each point of ∂∆ ∩ Z2 is a vertex of SubdivC . IfV ζt satisfies to the conditions of Proposition 8.8 then no component ofV ζt ∩Log−1(U ′(∆′)) has positive genus. The condition on the number of

ends of V ζt ∩ Log−1(U ′(∆′)) guarantees that the genus of V ζ

t coincideswith the genus of C. Any other choice gives a larger genus. Thecondition Aζ

t = Log(V ζt ) ⊂ Nε(C) automatically holds for large values

of t because of our constraints on ζ. �

The localization is possible thanks to the following patchworkingprinciple, cf. [24]. A deformation of coefficients of monomials of index

not contained in ∆′ has little effect on V ζt ∩ Log−1(U ′(∆′)).

Proposition 8.9. For any ε′ > 0 there exists T > 1 such that forevery t ≥ T we have the following property. If ζ ′ ∈ D is such that


ζ ′j = ζj for every j ∈ ∆′ then V ζ′

t ∩ Log−1(U ′(∆′)) is contained in the

ε′-neighborhood of V ζt ∩ Log−1(U ′(∆′)).

Proof. After an automorphism of (C∗)2 we may assume that βj = 0for j ∈ ∆′. In this case we can take a disk around the origin for U ′.In addition in this case the power of t at zj in the expression f ζt =∑j∈∆∩Z2

arg(ζj)t− log |ζj |zj is negative if j 6∈ ∆′. Thus for a sufficiently

large t the difference of coefficients of f ζt and f ζ′

t is arbitrary small.

Furthermore, if Log(z) ∈ U ′ then f ζt (z)− f ζ′

t (z) can be made arbitrarysmall by increasing t. �

Lemma 8.10. Let ∆′ ⊂ R2 be a triangle with vertices k0, k1, k2 ∈ Z2.For any choice bk0 , bk1 , bk2 ∈ C∗ there exist 2 Area(∆′) distinct choicesof coefficients {bj}, j ∈ (∆ ∩ Z2) r Vert(∆), such that the curve

V b = {z ∈ (C∗)2 |∑

j∈∆∩Z2

bjzj = 0}

is a rational (i.e. genus 0) curve of degree ∆′ with 3 ends at infinity.Suppose that the asymptotic directions of V b corresponding to the

sides [k0, k1] and [k0, k2] are chosen. Out of the 2 Area(∆′) distinct

choices of coefficients above 2 Area(∆′)l1l2

agree with the choice of the direc-

tions, where l1 = #([k0, k1] ∩ Z2) − 1 and l2 = #([k0, k2] ∩ Z2) − 1.Suppose that the asymptotic directions of the curve V b correspond-

ing to all three sides of ∆′ are chosen then we have one or nonesuch coefficient choices. Out of the total of l0l1l2 choices of the as-ymptotic directions, 2 Area(∆′) have one such choice of coefficients,l0 = #([k1, k2] ∩ Z2) − 1.

Proof. Consider the endomorphism M∆′ : CP2 → CT∆′ of degree ∆′

defined in (13). As in Lemma 7.13 any rational curve V of degree ∆′

with 3 ends is an image of a line in (C∗)2.Consider the closure V of V in CT∆′. Since bk0 , bk1, bk2 are fixed we

have l1 possibilities for the (unique) intersection point p1 = V ∩CT[k0,k1]

and l2 possibilities for the point p2 = V ∩ CT[k0,k2]. The points p1

and p2 have 2 Area(∆′)l1

and 2 Area(∆′)l2

inverse images under the map M∆′.Connecting different liftings for different choices of p1 and p2 we get(2 Area(∆′))2 different lines in CP2 that project to 2 Area(∆′) differentrational curves in CT∆′. If t is large enough then the amoeba of anysuch curve is contained in a small neighborhood of C �

Lemma 8.11. Let ∆′ ⊂ R2 be a parallelogram with vertices k0, k1, k2, k3 ∈Z2, k1 − k0 = k3 − k2. For any choice bk0 , bk1 , bk2 ∈ C∗ and ε′ > 0 there


exist l1l2 = (#([k0, k1]∩Z2)− 1)(#([k0, k2]∩Z2)− 1) choices of coeffi-cients {bj}, j ∈ (∆′ ∩ Z2) r {k0, k1, k2}, such that the curve

{z ∈ (C∗)2 |∑

j∈∆′∩Z2

bjzj = 0}

is a reducible curve of genus −1 and degree ∆′ that splits to two irre-ducible components of degrees [k0, k1] and [k0, k2] respectively.

Suppose that the asymptotic directions of the curve corresponding tothe sides [k0, k1] and [k0, k2] are chosen. There is a unique choice ofcoefficients {bj} above that agrees with the choice of these directions.

Proof. Any such curve is a union of two holomorphic cylinders takenwith multiplicities l1 and l2 respectively. There are l1 choices for thefirst cylinder and l2 choices for the second. �

Lemma 8.12. Let ∆′ = [k′, k′′] ∈ SubdivC be an edge, ∆′ 6⊂ ∂∆. If tis sufficiently large then there exist (l′)2 different choices of coefficientsζ ′ ∈ D such that ζ ′j = ζj if j = k′, k′′ or j ∈ ∆ r ∆′ and the curve

{z ∈ Log−1(U ′) |∑

j∈(∆∩Z2)

bjzj = 0}

is an immersed cylinder. Here l′ = #(∆′ ∩Z2)− 1 is the integer lengthof ∆′.

Proof. It suffices to check the lemma for a particular model. We mayassume that ∆′ = [(0, 0), (l′, 0)]. Let ∆ be the parallelogram withvertices (0,−1), (0, 0), (l′, 0) and (l′, 1). Let g = 0 and j0 = (0, 0).We have s + g − 1 = 3. We may choose P = {p1, p2, p3} so thatthe only tropical curve C passing via R = Log(P) has SubdivC 3[(0, 0), (l′, 0)] or so that SubdivC 3 [(0,−1), (l′, 1)]. Both choices canbe made so that the forest Ξ from Proposition 4.16 consists of theedges [(0,−1), (0, 0)], [(0,−1), (l′, 0)] and [(l′, 0), (l′, 1)]. The points Pdetermine the coefficients ζ(0,0), ζ(0,−1), ζ(l′,0), ζ(l′,1) since ∂∆∩Z2 consistsonly of the vertices of ∆. We need to determine the coefficients at thepoints j ∈ Int(∆). Note also that there are no lattice points inside[(0,−1), (l′, 1)].

To compute N irr(0,∆) we may use both configurations. For the firstchoice of P we have N irr(g,∆) = N , where N is the number of choicesof ζ ′ we need to find. Indeed, in each triangle T ′ ∈ SubdivC we havea unique choice of coefficients with a rational curve in Log−1(U ′(T ′))compatible with a chosen asymptotic direction corresponding to theside [(0, 0), (l′, 0)] by Lemma 8.10.

For the second choice of P we may use Lemma 8.10. We haveN irr(0,∆) = (l′)2 and, therefore, N = l′. �


Lemma 8.13. Let ∆′ = [k′, k′′] ∈ Ξ be an edge. If t is sufficientlylarge then there exist l′ = #(∆′∩Z2)−1 different choices of coefficientsζ ′ ∈ D such that ζ ′j = ζj if j = k′, k′′ or j ∈ ∆ r ∆′ and the curve

{z ∈ Log−1(U ′) |∑

j∈(∆∩Z2)

bjzj = 0}

is an immersed cylinder that contains a point from P.

Proof. The proof is a small modification of the proof of the previ-ous lemma. We just choose one of the configuration so that the edge[(0, 0), (l′, 0)] is contained in Ξ. �

We are ready to start the proof of Lemma 8.3.b. We choose thecoefficients of ζ so that V ζ

t passes via P and satisfies to the conditionsof Proposition 8.8.

Recall that C ⊃ R ∈ R2. Let X ⊂ ∆ be the tree given by Proposition4.18. Without loss of generality we may assume that j0 ⊂ Ξ. Thenumber ζj0 = 1 is already determined. Let us orient X so that j0 isits only source. For a vertex j of X we denote with j ′ the vertex of Xsuch that [j ′, j] is a positively oriented edge of X .

We partite the points of (∆ ⊂ Z2) r {j0} into groups G∆′ ⊂ ∆ ∩ Z2

corresponding to the 1- and 2-dimensional subpolygons ∆′ ∈ SubdivCusing the following rules.

• Suppose that ∆′ = [j ′, j] is an edge of the forest Ξ from Propo-sition 4.16. We let

G∆′ = ([j ′, j] ∩ Z2) r {j ′}.

• Suppose that [j ′, j] is a diagonal of a parallelogram ∆′ ∈ SubdivCwith the other vertices k1, k2. We let

G∆′ = (∆′ ∩ Z2) r ([j ′, k1] ∪ [j ′, k2]).

• Suppose that [j1, j2] ∈ SubdivC is an edge such that [j1, j2] r

{j1, j2} is disjoint fromG∆′ for all parallelograms ∆′ ⊂ SubdivC .We let

G∆′ = ([j1, j2] ∩ Z2) r {j1, j2}.

• Suppose that ∆′ ∈ SubdivC is a triangle with vertices k0, k1, k2.We let

G∆′ = (Int(Delta′) ∩ Z2).

Clearly, the groups G∆′ are disjoint. Furthermore, since X contains allvertices of SubdivC we have⋃

∆′∈SubdivC

G∆′ = (∆ ⊂ Z2) r {j0}.


We say that ζ ∈ D is ∆′-compatible in the following cases.

• If ∆′ = [j ′, j] is an edge of the forest Ξ we require that V ζt ∩

Log−1(U ′(∆′)) is an immersed cylinder that contains a pointfrom P.

• If ∆′ is a parallelogram we require that V ζt ∩ Log−1(U ′(∆′)) is

a union of two immerse cylinders.• If ∆′ = [j1, j2] is an edge such that [j1, j2] r {j1, j2} is disjoint

from G∆′ for all parallelograms ∆′ ⊂ SubdivC we require thatV ζt ∩ Log−1(U ′(∆′)) is an immersed cylinder.

• If ∆′ is a triangle we require that V ζt ∩Log−1(U ′(∆′)) is a rational

curve with 3 ends.• If G∆′ = ∅ then any ζ ∈ D is by default ∆′-compatible.

Lemma 8.14. There exists an order on the polygons ∆′ ∈ SubdivCsuch that if ∆′ is greater than ∆′′ then G∆′ is disjoint from ∆′′.

Proof. We assign the lowest order to the edges [j1, j2] ∈ SubdivC suchthat [j1, j2] r {j1, j2} is disjoint from G∆′ for all parallelograms ∆′ ⊂SubdivC . Their mutual order is chosen arbtrarily.

As the next step we order the edges of the tree X so that it agreeswith the chosen orientation (recall that this is the orientation such thatthe only sink is j0). Note that each such edge is either an edge ∆′ of Ξor a diagonal of a parallelogram ∆′ ∈ SubdivC .

Finally, we assign the highest order to the triangles ∆′ ∈ SubdivC .Again their mutual order can be chosen arbitrarily. �

Let ∆′1, . . . ,∆

′N be the positive-dimensional polygons from SubdivC

enumerated according to an order given by Lemma 8.14.

Definition 8.15. The inner multiplicity µ′k of a polygon ∆′

k is thefollowing number.

• If ∆′k is an edge then µ′

k = (lk)2, where the integer length lk is

defines as #(∆′k ∩ Z2).

• If ∆′k is a triangle then µ′

k =2 Area(∆′

k)

l′k

, where l′k is the product

of the integer lengths of the 3 sides of ∆′k. Note that in this

case µ′k is often non-integer.

• If ∆′k is a paralellogram then µ′

k = 1.

For any edge E of SubdivC contained entirely in ∂∆ we have GE =∅ since P is in tropically general position (and thus x = s for C).

ThereforeN∏u=1

µ′u equals to the multiplicity µ of the tropical curve C ⊂

R2. Lemma 8.3.b follows inductively from the following proposition.


Proposition 8.16. For any δ > 0 there exists T > 1 such that for anyt ≥ T we have the following property.

Suppose that ζ ∈ D is chosen compatible with ∆′1, . . . ,∆

′k−1. Unless

∆′ is a triangle there exists µ′k distinct choices of ζ ′ ∈ D such that if

t ≥ T we have the following properties.

• The parameter ζ ′ is compatible with ∆′1, . . . ,∆

′k.

• We have ζ ′j = ζj if j ∈ G∆′

u, u > k.

• For u < k We have V ζ′

t ∩ Log−1(U ′(∆′u)) is contained in a δ-

neighborhood of V ζt ∩ Log−1(U ′(∆′

u)).

If ∆′k is a triangle then we have either one or none such choices de-

pending on the choice of asymptotic directions of V ζ′

t ∩ Log−1(U ′(∆′k))

corresponding to the sides ∆′u of ∆′

k, u < k. We have a total of l′k ofpossibilities for this choice of asymptotic direction. For 2 Area(∆′) ofthese l′k choices we have a unique choice of coefficients.

Proof. We have the coefficient ζ ′j = ζj already chosen for j ∈ G∆′

u,

u > k.Suppose that j ∈ G∆′

k. Let us vary the corresponding coefficients ζ ′j

within D, i.e. within the disk |ζ ′j−aj | < ε′j for j ∈ Vert(SubdivC)∩G∆′

k

while − log |ζ ′j| − βj < εj for j /∈ Vert(SubdivC) ∩ G∆′

k. Denote the

corresponding #(G∆′

k)-dimensional disk with Dk.

Inductively, Proposition 8.16 givesk−1∏u=1

µ′u choices of coefficients ζ ′j,

j ∈ G∆′

u, u < k. By Proposition 8.9 and Lemma 8.14 we can choose

T > 1 so large that for any choice of

η = {ζ ′j}j∈G∆′

k

∈ Dk

the curve V ζ′

t ∩Log−1(U ′(∆′u)) is contained in a small neighborhood of

V ζt ∩ Log−1(U ′(∆′

u)).This yields a map

Φk : Dk → Ml,s,

where l = #(Int ∆ ∩ Z2) and Ml,s ⊃ Ml,s is the Deligne-Mumfordcompactification (see e.g. [7])

Φk : η 7→ V ζ′

t .

The map Φk is a priori only partially defined (for η such that V ζ′

t hassingularities other than nodes).

The map Φk is arbitrary close to the map Ψk : η 7→ V ζt . Here ζj for

j ∈ G∆′

kis defined by η and ζj for j /∈ G∆′

kis fixed for some generic

values with ζ ∈ D. The map Ψk : Dk → Ml,s is globally defined and


intersects the stratum of ∆′k-compatible curves in µ′

k points by Lemma8.10, 8.11, 8.12 and 8.13. Thus we have the same for the map Φk. �

This finishes the proof of Lemma 8.3.b and Theorem 1.

8.4. Real curves: proof of Theorem 3 and 6. A signed configu-ration R determines its lift P ∈ (R∗)2. Indeed, for every r ∈ R theinverse image Log−1(r) ∩ (R∗)2 consists of 4 points, each in its ownquadrant parameterized by the corresponding sign σ.

We detect the real curves among the µ holomorphic curves fromLemma 8.3 inductively as in the proof of Proposition 8.16. For that wesuppose that ζ ∈ D is chosen so that ζj ∈ R, j ∈ ∆ ∩ Z2 and we needto count the number µ′

k,R of ∆′k-compatible choices of ζ ′j ∈ R, j ∈ G∆′

k.

Clearly, µ′k,R ≡ µ′

k (mod 2) if ∆′k is not a triangle. Thus if ∆′

k is aparallelogram then µ′

k,R = µ′k = 1.

If ∆′k 6⊂ Ξ is an edge then µ′

k,R is equal to 1 or 4 (according tothe parity of µ′

k). Indeed, if µ′k is odd then out of the l′ asymptotic

directions of V ζ′

t ∩ Log−1(U ′(∆′k)) there is one that corresponds to a

real quadrant of (R∗)2.If µ′

k is even then there are two such choices. for each of them thereare two real branches in the corresponding quadrant. Two additionalchoices come from the choice of merging of the asymptotic directions(corresponding to the 2-dimensional polygons from SubdivC adjacentto ∆′)

If ∆′k ⊂ Ξ is an edge then µ′

k,R is equal to 1 or 4 or 0. Indeed, if µ′k is

odd then the point p of P corresponding to ∆′k sits of the real branch

of V ζ′

t .If µ′

k is even then µ′k,R = 0 if the two real quadrants corresponding

to the possible real asymptotic directions are disjoint from p ∈ (R∗)2.If p belongs to one of these quadrants then 2 choices come from thechoice of merging of the asymptotic directions and 2 from the choiceof the real branch that contains p.

Suppose that ∆′k is a triangle. If all sides of ∆′

k have odd integerlength then µ′

k = 1 since only one choice of asymptotic directions of

V ζ′

t ∩ Log−1(U ′(∆′k)) is real. Otherwise we need only to consider the

case when real asymptotic directions exist, i.e. µ′u 6= 0 for all u < k.

If only one side of ∆′k has even integer length then µ′

k = 1. Inthis case there are 2 distinct real choices of asymptotic directions of

V ζ′

t ∩ Log−1(U ′(∆′k)). Each of the two choices corresponds to 1 real

choice of η ∈ Dk.If all sides of ∆′

k are even then we have 8 distinct real choices of

asymptotic directions of V ζ′

t ∩Log−1(U ′(∆′k)). However, only 4 of them


correspond to real choice of η ∈ Dk. Thus in the last case we haveµ′k,R = 1

2.

It remains to note that the real inner multiplicities agree with Defi-nition 6.8. This finishes the proof of Theorem 3.

To prove Theorem 6 we note that if SubdivC contains an edge ofan even integer length then C contributes zero to the Welschinger in-variant. Indeed, for each such edge E we have two real branches of

V ζ′

t ∩ Log−1(U ′(E)). For one choice of the merging of the asymptoticdirections of the real branches we get a hyperbolic node. For the otherchoice we have an elliptic node. This implies that the Welschingerinvariant is independent of the signs of R.

If all edges of SubdivC have odd integer length then ellpitic nodes

can appear only from V ζ′

t ∩Log−1(U ′(∆′)) where ∆′ is a triangle. This

part of V ζ′

t has a total of #(∆′ ∩ Z2) nodes. None of these nodes canbe real hyperbolic since the restriction M∆′ |(R∗)2 : (R∗)2 → (R∗)2 (see12) is injective if all sides of ∆′ have odd integer length. Therefore the

multiplicity multR,WV (C) from Definition 6.14 gives the right count for

Theorem 6.

8.5. Counting by lattice paths: proof of Theorems 2, 4 and

7. Recall that we have a linear map λ : R2 → R injective on Z2. LetL ⊂ R2 be an affine line (in the classical sense) orthogonal to the kernelof λ.

We choose a configuration R = {r1, . . . , rs+g−1 ⊂ L so that the orderof ru coincides with the linear order on L. Furthermore, we choose eachrk so that the distance from rk to rk−1 is much larger than the distancefrom rk−1 to rk−2. Such configuration can be chosen in a tropicallygeneral position since the slope of L is irrational.

Let C ⊂ R2 be a tropical line of genus g and degree ∆ passing viaR. Let Ξ be the forest Ξ from Proposition 4.16.

Lemma 8.17. We have C ∩ L = R.

Proof. Let K be a component of Γ r h−1(R). Suppose that h(K)intersects L at a point not from R. One of the components of K r

h−1(L) would yield a bounded graph with edges at L contained in ahalf-plane. Clearly such graph can not be ballanced. �

Corollary 8.18. The forest Ξ ⊂ ∆ is a λ-increasing path that connectsvertices p and q as in Theorem 2.

Proof. By Lemma 8.17 the vertices of Ξ corrspond to the componentsof Lr R. �


This corollary allows to enumerate all tropical curves of genus g anddegree ∆ passing via P by the corresponding paths. Suppose that sucha path γ is chosen.

The path γ determines the slope of the edges that contain pointsfrom R. Suppose that Eu 3 ru. We need to compute the number ofways to extend these edges to a tropical curve of genus g and degree∆. Such extensions are independent in the half-planes H+ and H−bounded by L.

We compute these numbers inductively. Let us extend E1 and E2

in H+. Since the distance between r1 and r2 is the smallest if theseextensions intersect in H+ they do so before intersecting the extensionof any other edge Eu. Each such intersection can give a 3-valent vertexof C. Otherwise these extension pass through the intersection pointwithout an interaction. The first case is dual to a triangle in the dualsubdivision. The second case is dual to a parallelogram. Then weconsider the intersection of the result with the extension of E3 and soon. After incorporating the extension of Es+g−1 the resulting curve hasto have the ends orthogonal to the positive part of ∂∆, i.e. given bythe path α+.

Note that our inductive procedure agrees with the definition of thepositive multiplicity of γ. Similarly, the negative multiplicity of γagrees with a choice of extensions to H−. Note that our procedurenecessarily gives a tropical curve C of degree ∆ parameterized byh : Γ → C so that each component of Γ r h−1(R) is a tree withone end at infinity. Therefore, the genus of C is g.

Theorems 2, 4 and 7 follow from Theorems 1, 3 and 6 respectively.

References

[1] O. Aharony, A. Hanany, Branes, superpotentials and superconformal fixed

points, http://arxiv.org hep-th/9704170.[2] O. Aharony, A. Hanany, B. Kol, Webs of (p, q) 5-branes, five dimensional field

theories and grid diagrams, http://arxiv.org hep-th/9710116.[3] L. Caporaso, J. Harris, Counting plane curves of any genus, Invent. Math. 131

(1998), 345-392.[4] Y. Eliashberg, A. Givental, H. Hofer, Introduction to symplectic field theory,

GAFA 2000, Special Volume, Part II, 560–673.[5] M. Forsberg, M. Passare, A. Tsikh, Laurent determinants and arangements of

hyperplane amoebas, Advances in Math. 151 (2000), 45–70.[6] I. M. Gelfand, M. M. Kapranov, A. V. Zelevinski, Discriminants, resultants

and multidimensional determinants, Birkhauser Boston 1994.[7] J. Harris, D. Morrison, Moduli of curves, Graduate Texts in Mathematics, 187.

Springer-Verlag, New York, 1998.[8] I. Itenberg, V. Kharlamov, E. Shustin, Welschinger invariant and enumeration

of real plane rational curves, http://arxiv.org/abs/math.AG/0303378.


[9] M. M. Kapranov, Amoebas over non-Archimedian fields, Preprint 2000.[10] A. G. Khovanskii, Newton polyhedra and toric varieties (in Russian),

Funkcional. Anal. i Prilozen. 11 (1977), no. 4, 56-64.[11] M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology and

enumerative geometry, Comm. Math. Phys. 164 (1994), 525-562.[12] M. Kontsevich, Y. Soibelman, Homological mirror symmetry and torus fi-

brations, Symplectic geometry and mirror symmetry (Seoul, 2000), 203–263,World Sci. Publishing, River Edge, NJ, 2001.

[13] A. G. Kouchnirenko,Polyedres de Newton et nombres de Milnor, Invent. Math.,32 (1976), 1 - 31.

[14] G.L. Litvinov and V.P. Maslov, The correspondence principle for idempotent

calculus and some computer applications, in Idempotency, J. Gunawardena(Editor), Cambridge Univ. Press, Cambridge, 1998, 420-443.

[15] G. Mikhalkin, Real algebraic curves, moment map and amoebas, Ann. of Math.151 (2000), 309 - 326.

[16] G. Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hy-

persurfaces, http://arxiv.org math.GT/0205011 Preprint 2002, to appear inTopology.

[17] G. Mikhalkin, Counting curves via lattice paths in polygons, C. R. Math. Acad.Sci. Paris 336 (2003), no. 8, 629–634.

[18] M. Passare, H. Rullgard, Amoebas, Monge-Ampere measures, and triangula-

tions of the Newton polytope. Preprint, Stockholm University, 2000.[19] J.-E. Pin, Tropical semirings, Idempotency (Bristol, 1994), 50–69, Publ. New-

ton Inst., 11, Cambridge Univ. Press, Cambridge, 1998.[20] Z. Ran, Enumerative geometry of singular plane curves, Invent. Math. 97

(1989), 447-465.[21] J. Richter-Gebert, B. Sturmfels, T. Theobald, First steps in tropical geometry,

http://arXiv.org math.AG/0306366.[22] B. Sturmfels, Solving systems of polynomial equations, CBMS Regional Con-

ference Series in Mathematics, AMS Providence, RI 2002.[23] R. Vakil, Counting curves on rational surfaces, Manuscripta Math. 102 (2000),

53-84.[24] O.Ya. Viro, Gluing of plane real algebraic curves and constructions of curves

of degrees 6 and 7, Lecture Notes in Math., 1060 (1984) 187-200.[25] J.-Y. Welschinger, Invariants of real rational symplectic 4-manifolds and lower

bounds in real enumerative geometry, C. R. Math. Acad. Sci. Paris 336 (2003),no. 4, 341–344.

Department of Mathematics, University of Utah, Salt Lake City,

UT 84112, USA

Department of Mathematics, University of Toronto, 100 St. George

St., Toronto, Ontario, M5S 3G3 Canada

St. Petersburg Branch of Steklov Mathematical Institute, Fontanka

27, St. Petersburg, 191011 Russia

E-mail address : [email protected]

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