# Enumerative tropical algebraic geometry in $\mathbb{R}^2$

By Grigory Mikhalkin

## Abstract

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629–634.

The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the real space and holomorphic curves with certain piecewise-linear graphs there.

## 1. Introduction

Recall the basic enumerative problem in the plane. Let and be two numbers and let be a collection of points in general position. A holomorphic curve is parameterized by a Riemann surface under a holomorphic map so that . Here we choose the minimal parametrization, i.e. such that no component of is mapped to a point by . The curve is irreducible if and only if is connected. The number of irreducible curves of degree and genus passing through is finite and does not depend on the choice of as long as this choice is generic.

Similarly we can set up the problem of counting all (not necessarily irreducible) curves. Define the genus of to be . Note that the genus can take negative values for reducible curves. The number of curves of degree and genus passing through is again finite and does not depend on the choice of as long as this choice is generic. Figure 1 lists some (well-known) first few numbers and .

The numbers are known as the Gromov-Witten invariants of (see Reference 12) while the numbers are sometimes called the multicomponent Gromov-Witten invariant. One series of numbers determines another by a simple combinatorial relation (see e.g. Reference 3). A recursive relation which allows one to compute the numbers (and thus the numbers ) was given by Kontsevich. This relation came from the associativity of the quantum cohomology (see Reference 12). In the arbitrary genus case Caporaso and Harris Reference 3 gave an algorithm (bases on a degeneration of ) which allows one to compute the numbers (and thus the numbers ).

The main result of this paper gives a new way of computation of these numbers as well as the -counterparts of these numbers (that appear in real algebraic geometry). The number turns out to be the number of certain lattice paths of length in the triangle with vertices , and . The paths have to be counted with certain non-negative multiplicities. Furthermore, this formula works not only for but for other toric surfaces as well. For other toric surfaces we just have to replace the triangle by other convex lattice polygons. The polygon should be chosen so that it determines the corresponding (polarized) toric surface.

The formula comes as an application of the so-called tropical geometry whose objects are certain piecewise-linear polyhedral complexes in . These objects are the limits of the amoebas of holomorphic varieties after a certain degeneration of the complex structure. The idea of using these objects for enumeration of holomorphic curves is due to Kontsevich.

In Reference 13 Kontsevich and Soibelman proposed a program linking homological mirror symmetry and torus fibrations from the Strominger-Yau-Zaslow conjecture Reference 26. The relation is provided by passing to the so-called “large complex limit” which deforms a complex structure on a manifold to its worst possible degeneration. Similar deformations appeared in other areas of mathematics under different names. The patchworking in real algebraic geometry was discovered by Viro Reference 29. Maslov and his school studied the so-called dequantization of the semiring of positive real numbers (cf. Reference 15). The limiting semiring is isomorphic to the -semiring , the semiring of real numbers equipped with taking the maximum for addition and addition for multiplication.

The semiring is known to computer scientists as one of tropical semirings, see e.g. Reference 20. In mathematics this semiring appears from non-Archimedean fields under a certain pushing forward to of the arithmetic operations in .

In this paper we develop some basic algebraic geometry over with a view towards counting curves. In particular, we rigorously set up some enumerative problems over and prove their equivalence to the relevant problems of complex and real algebraic geometry. The reader can refer to Chapter 9 of Sturmfels’ recent book Reference 27 for some first steps in tropical algebraic geometry. See also Reference 24, Reference 23, Reference 25 for some of more recent development.

We solve the corresponding tropical enumerative problem in . As an application we get a formula counting the number of curves of given degree and genus in terms of certain lattice paths of a given length in the relevant Newton polygon. In particular this gives an interpretation of the Gromov-Witten invariants in and via lattice paths in a triangle and a rectangle, respectively. This formula was announced in Reference 18. For the proof we use the patchworking side of the story which is possible to use since the ambient space is 2-dimensional and the curves there are hypersurfaces. An alternative approach (applicable to higher dimensions as well) is to use the symplectic field theory of Eliashberg, Givental and Hofer Reference 4. Generalization of this formula to higher dimensions is a work in progress. In this paper we only define the enumerative multiplicity for the 2-dimensional case. There is a similar definition (though no longer localized at the vertices) for multiplicities of isolated curves in higher-dimensional tropical enumerative problems. However, in higher dimensions there might be families of tropical curves (of positive genus) for enumerative problems with finite expected numbers of solutions (this phenomenon already appears for curves in passing through a finite collection of points in general position) which seem to pose a serious problem (that perhaps asks for development of tropical virtual classes).

The main theorems are stated in Section 7 and proved in Section 8. In Section 2 we define tropical curves geometrically (in a way similar to webs of Aharony, Hanany and Kol Reference 1, Reference 2). In Section 3 we exhibit them as algebraic objects over the tropical semifield. In Section 4 we define the tropical enumerative problems in ; in Section 5 recall those in . Section 6 is auxiliary to Section 8 and deals with certain piecewise-holomorphic piecewise-Lagrangian objects in called complex tropical curves. An outline of the approach taken in this paper can also be found in Reference 8. A somewhat different approach can be found in Reference 21.

## 2. Tropical curves as graphs in $\mathbb{R}^n$

In this section we geometrically define tropical curves in and set up the corresponding enumerative problem. We postpone the algebraic treatment of the tropical curves (which explains the term “tropical” among other things) until the next section.

### 2.1. Definitions and the first examples

Let be a weighted finite graph. The weights are natural numbers prescribed to the edges. Clearly, is a compact topological space. We make it non-compact by removing the set of all 1-valent vertices ,

Somewhat more complicated tropical curves (corresponding to projective curves of degree 3) are pictured on Figure 3.

### 2.2. The degree of a tropical curve in ${\mathbb{R}}^n$

Let be a set of non-zero integer vectors such that . Suppose that in this set we do not have positive multiples of each other, i.e. if for , then . The degree of a tropical 1-cycle takes values in such sets according to the following construction.

By our definition a tropical curve has a finite number of ends, i.e. unbounded edges (rays). Let be a primitive vector. A positive multiple of is included in if and only if there exists an end of which is mapped in the direction of . In such a case we include into , where is the sum of multiplicities of all such rays.

Note that the sum of all vectors in is zero. This follows from adding the conditions Equation 1 from Definition 2.2 in all vertices of .

For example the degree of both curves from Figure 5 is , , while the degree of both curves from Figure 3 is .

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

### 2.3. Genus of tropical curves and tropical $1$-cycles

We say that a tropical curve is reducible if is disconnected. We say that a tropical 1-cycle is reducible if it can be presented as a union of two distinct tropical 1-cycles. Clearly, every reducible 1-cycle can be presented as an image of a reducible parameterized curve.

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

If is an embedded 3-valent graph, then the parameterization is unique. However, in general, there might be several parameterizations of different genus and taking the minimal value is essential.

### 2.4. Deformations of tropical curves within their combinatorial type

As in the classical complex geometry case the deformation space of a tropical curve is subject to the constraint coming from the Riemann-Roch formula. Let be the number of ends of .

Note that if two tropical curves are isotopic in the class of tropical curves (with the same domain ), then they are of the same combinatorial type.

The valence of a vertex of is the number of adjacent edges regardless of their weights. The graph is called 3-valent if every vertex is 3-valent. The parameterized tropical curve is called 3-valent if is 3-valent.

Consider the general case now and suppose that has vertices of valence higher than 3. How much differs from a 3-valent graph is measured by the following characteristic. Let the overvalence be the sum of the valences of all vertices of valence higher than 3 minus the number of such vertices. Thus if and only if no vertex of has valence higher than 3.

Note that can be interpreted as the overvalence of the image .

### 2.5. Changing the combinatorial type of $\Gamma$

Sometimes we can deform and by the following procedure reducing . If we have edges adjacent to the same vertex, then we can separate them into two groups so that each group contains at least 2 edges. Let us insert a new edge separating these groups as shown in Figure 4. This replaces the initial -valent vertex with 2 vertices (the endpoints of ) of smaller valence. There is a “virtual slope” of determined by the slopes of the edges in each group. This is the slope to appear in local perturbation of the tropical map (if such a perturbation exists). Note that the weight of the new edge does not have to be equal to 1.

There is another modification of a tropical curve near its vertex by changing the combinatorial type of which works even for some 3-valent vertices.

Let be a 3-valent vertex of . As in Definition 2.2 let be the weights of the edges adjacent to and let be the primitive integer vectors in the direction of the edges.

Note that the multiplicity of a vertex is always divisible by the product of the weights of any two out of the three adjacent edges.

Proposition 2.13 can be generalized in the following way to incorporate possible perturbations.

### 2.6. Superabundancy and regularity

Some curves vary in a family strictly larger than “the prescribed dimension” .

In contrast to the classical case tropical superabundancy can be easily seen geometrically. By the proof of Proposition 2.13 the superabundancy appears if the cycles of the graph do not provide transversal conditions for the length of the bounded edges of the subtree . This is the case if some of the cycles of are contained in smaller-dimensional affine-linear subspaces of , e.g. if a non-trivial cycle of gets contracted or if a spatial curve develops a planar cycle. More generally, this is the case if several non-degenerate “spatial” cycles combine to a degenerate “flat” cycle.

Clearly, no irreducible tropical curve of genus 0 can be superabundant since it has no cycles. Furthermore, tropical immersions of 3-valent graphs to the plane are never superabundant as the following proposition shows.

## 3. Underlying tropical algebra

In this section we exhibit the tropical curves as algebraic varieties with respect to a certain algebra and also define some higher-dimensional tropical algebraic varieties in .

### 3.1. The tropical semifield ${{\mathbb{R}}_{\operatorname {trop}}}$

Consider the semiring of real numbers equipped with the following arithmetic operations called tropical in computer science:

. We use the quotation marks to distinguish the tropical operations from the classical ones. Note that addition is idem