Enumerative tropical algebraic geometry in

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

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 ,

Remark 2.1.

Removal of the 1-valent vertices is due to a choice we made in the algebraic side of the treatment. In this paper we chose the semifield as our “ground semifield” for tropical variety; see the next section. The operation plays the rôle of addition and thus we do not have an additive zero. Non-compactness of is caused by this choice. Should we have chosen instead for our ground semifield, we would not need to remove the 1-valent vertices but then we would have to consider tropical toric compactification of the ambient space as well. The approach of this paper is chosen for the sake of simplicity. The other approach has its own advantages and will be realized in a forthcoming paper.

Definition 2.2.

A proper map is called a parameterized tropical curve if it satisfies to the following two conditions.

For every edge the restriction is either an embedding or a constant map. The image is contained in a line such that the slope of is rational.

For every vertex we have the following property. Let be the edges adjacent to , let be their weights and let be the primitive integer vectors at the point in the direction of the edges (we take if is a point). We have

We say that two parameterized tropical curves and are equivalent if there exists a homeomorphism which respects the weights of the edges and such that . We do not distinguish equivalent parameterized tropical curves. The image

is called the unparameterized tropical curve or just a tropical 1-cycle if no connected component of gets contracted to a point. The 1-cycle is a piecewise-linear graph in with natural weights on its edges induced from the weights on . If is an edge of , then is a union of subintervals of the edges of . The weight of is the sum of the weights of these edges.

Remark 2.3.

In dimension 2 the notion of tropical curve coincides with the notion of -webs introduced by Aharony, Hanany and Kol in Reference 2 (see also Reference 1).

Remark 2.4.

The map can be used to induce a certain structure on from the affine space . It is an instance of the so-called -affine structure. For a graph such a structure is equivalent to a metric for every edge of . Here is a way to obtain such a metric for the edges that are not contracted to a point.

Let be a compact edge of weight that is not contracted to a point by . Such an edge is mapped to a finite straight interval with a rational slope in . Let be the length of a primitive rational vector in the direction of . We set the length of to be .

Note that also has non-compact edges (they result from removing 1-valent vertices from ). Such edges are mapped to unbounded straight intervals by .

It is possible to consider abstract tropical curves as graphs equipped with such -affine structures. Then tropical maps (e.g. to ) will be maps that respect such structure. Abstract tropical curves have genus (equal to ) and the number of punctures (equal to the number of ends of ) and form the moduli space in a manner similar to that of the classical Riemann surfaces. This point of view will be developed in a forthcoming paper.

Example 2.5.

Consider the union of three simple rays

This graph (considered as a tautological embedding in ) is a tropical curve since . A parallel translation of in any direction in is clearly also a tropical curve. This gives us a 2-dimensional family of curves in . Such curves are called tropical lines.

Remark 2.6.

The term tropical line is justified in the next section dealing with the underlying algebra. So far we would like to note the following properties of this family; see Figure 2.

For any two points in there is a tropical line passing through them.

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

Two generic tropical lines intersect in a single point.

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

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.

Definition 2.7.

The resulting set is called the toric degree of . Accordingly, the degree of a parameterized tropical curve is the degree of its image .

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 .

Definition 2.8.

If the toric degree of a tropical 1-cycle is , then is called a tropical projective curve of degree .

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

2.3. Genus of tropical curves and tropical -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.

Definition 2.9.

The genus of a parameterized tropical curve is . In particular, for irreducible parameterized curves the genus is the first Betti number of . The genus of a tropical 1-cycle is the minimum genus among all parameterizations of .

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.

Example 2.10.

The tropical 1-cycle on the right-hand side of Figure 3 can be parameterized by a tree once we “resolve” its 4-valent vertex to make the parameterization domain into a tree. Therefore, its genus is 0.

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 .

Remark 2.11.

The number is a tropical counterpart of the value of the canonical class of the ambient complex variety on the curve . The ambient space corresponds to the torus classically. Let be a holomorphic curve with a finite number of ends. The space is not compact, but one can always choose a toric compactification such that every point of the closure in intersects not more than one boundary divisor (i.e. a component of ). Then every end of can be prescribed a multiplicity equal to the intersection number of the point of and the corresponding boundary divisor. The value of the canonical class of on equals the sum of these multiplicities.

Definition 2.12.

The curves and (parameterized by the same graph ) are said to be of the same combinatorial type if for any edge the segments and are parallel.

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.

Proposition 2.13.

Let be a 3-valent graph. The space of all tropical curves in the same combinatorial type (up to their equivalence from Definition 2.2) is an open convex polyhedral domain in a real affine -dimensional space, where

Proof.

It suffices to prove this for a connected graph since different components of vary independently, and, furthermore, both sides of the inequality are additive with respect to taking the union of components (note that ). Let be a finite tree containing all the vertices of . Note that the number of finite edges in is . By an Euler characteristic computation we get that the number of finite edges of is equal to .

Maps vary in a linear -dimensional family if we do not change the slopes of the edges. The -dimensional part comes from varying the lengths of the edges while the -dimensional part comes from translations in . Such a map is extendable to a tropical map if the pairs of vertices corresponding to the remaining edges define the lines with the correct slope. Each of the edges imposes a linear condition of codimension at most . Thus tropical perturbations of are contained in a linear family of dimension at least . They form an open convex polyhedral domain there defined by the condition that the lengths of all the edges are positive.

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.

Proposition 2.14.

The space of all tropical curves in the same combinatorial type (up to their equivalence from Definition 2.2) is an open convex polyhedral domain in a real affine -dimensional space, where

where is the number of edges of that are mapped to a point.

Proof.

The proof is similar to that of Proposition 2.13. If the image of an edge is a point in , then we cannot vary its length. Similarly we are lacking some degrees of freedom (with respect to the set-up of Proposition 2.13) if .

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

2.5. Changing the combinatorial type of

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.

Example 2.15.

Let be the union of three rays in in the direction , and emanating from the origin (pictured on the left-hand side of Figure 5). This curve is a simple tropical curve of genus 0.

It can be obtained as a limit of the family of genus 1 curves given by the union of three rays in in the direction , and emanating from , and , respectively, and the three intervals , and as pictured in Figure 5.

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.

Definition 2.16.

The multiplicity of at its 3-valent vertex is . Here is the area of the parallelogram spanned by and . Note that

since by Definition 2.2.

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.

Definition 2.17.

We say that is a perturbation of if there exists a family , , in the same combinatorial type as and the pointwise limit such that coincides with (as tropical 1-cycles).

Definition 2.18.

A tropical curve is called smooth if is 3-valent, is an embedding and the multiplicity of every vertex of is 1.

Proposition 2.19.

A smooth curve does not admit perturbations of different combinatorial types.

Proof.

Suppose that is a perturbation of a smooth curve . Since is an embedding and , we have a map

Note that the weight of every edge from is 1 since otherwise the endpoints of multiple edges would have multiplicity greater than 1. Thus the inverse image of every open edge of under is a single edge of .

Thus must be a homeomorphism near the inner points of the edges of . Let be a vertex and let be its small neighborhood in . Note that is connected since is 3-valent (otherwise we can divide the adjacent edges to into two groups with zero sums of the primitive integer vectors).

Suppose that is not a point. Then is a graph which has three distinguished vertices that are adjacent to the edges of . The graph must be contained in the affine 2-plane in containing the ends . This follows from the balancing condition for .

The 3-valent vertices of have multiplicities from Definition 2.16. Since is planar, we can extend the definition of the multiplicity to higher-valent vertices as follows. Let be a -valent vertex, be the primitive integer vectors in the directions of the adjacent edges to numbered consistently with the cyclic order in the ambient 2-plane and let be the corresponding weights. We set the multiplicity of to be

It is easy to see that the multiplicity of in is equal to the sum of multiplicities of all the vertices of . The multiplicities of all vertices are positive integers. Therefore, the multiplicity of is greater than unless is a point.

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

Proposition 2.20.

The space of deformations of a parameterized tropical curve is locally a cone

where is a polyhedral convex cone corresponding to tropical curves parameterized by perturbations of of a given combinatorial type. The union is taken over all possible combinatorial types of perturbations. We have

where is the number of the edges of that are mapped to a point.

Proof.

This proposition follows from Proposition 2.14 applied to all possible perturbation of .

Remark 2.21.

Not all conceivable perturbations of are realized as the following example shows. Let be a tropical 1-cycle of genus 1. Let be a vertical line such that is a point on such that is contractible. The curve

has a 4-valent vertex that cannot be perturbed (since any such perturbation would force out of the plane ). Thus any 3-valent perturbation of the tautological embedding has to have an edge mapping to a point.

This phenomenon is related to the so-called superabundancy phenomenon.

2.6. Superabundancy and regularity

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

Definition 2.22.

A parameterized tropical curve is called regular if the space of the curves of this combinatorial type (which is a polyhedral domain in an affine space by Proposition 2.14) has dimensions . Otherwise it is called superabundant.

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.

Proposition 2.23.

Every tropical immersion is regular if is 3-valent. If  has vertices of valence higher than 3, then varies in at most the -dimensional family.

Proof.

Recall the proof of Propositions 2.13 and 2.14. Once again we may assume that is connected. Let be any vertex.

We may choose an order on the vertices of so that it is consistent with the distance from , i.e. so that the order of a vertex is greater than the order of a vertex whenever is strictly further from than . The balancing condition for implies the following maximum principle for : any vertex of is either adjacent to an unbounded edge of or is connected with a bounded edge to a higher order vertex. Inductively one may choose a maximal tree so that this maximum principle also holds for . Note that the set of vertices of coincides with the set of vertices of . Note also that our choice of order on this set gives the orientation on the edges of : every edge is directed from a smaller to a larger vertex.

The space of deformation of within the same combinatorial type is open in a -dimensional real affine space that is cut by hyperplanes in where is the number of bounded edges of . Each of these hyperplanes is non-trivial if is an immersion and is 3-valent, since then there can be no parallel edges adjacent to the same vertex.

We have regularity if these hyperplanes intersect transversely. The hyperplanes are given by a -matrix with real values. The rows of this matrix correspond to the edges of while the first columns correspond to the edges of (the remaining two columns correspond to translations in ). To show that the rank of this matrix is in the 3-valent case, we exhibit an upper-triangular -minor with non-zero elements on its diagonal.

For each edge of we include the column corresponding to the (bounded) edge of directed toward the highest endpoint of . If is 3-valent, different edges of correspond under this construction to different edges of . This produces the required -minor.

If is not 3-valent, then the number of bounded edges of is . This number is the same as the number of vertices of other than . We can do the construction of the minor as above but only using one edge of at every vertex of other than . In such a way we can get a non-degenerate -minor and thus the dimension is at most (2 comes from translations in ). If there exists a vertex of valence higher than 3, then we may choose such a vertex for the root of the tree and this gives a non-degenerate minor of size strictly larger than .

Corollary 2.24.

An immersed 3-valent tropical curve locally varies in a (real) linear -dimensional space, where

if either or .

Remark 2.25.

There exist superabundant tropical immersions if is not 3-valent. A nice example is given by the Pappus theorem configuration that is a union of 9 lines; see Figure 6.

Assume that the nine Pappus lines have rational slopes and take to be their union in so that our tropical curve is a tautological embedding. We have , , , . Therefore , yet our configuration varies at least in a 3-dimensional family (since we can apply any translation and homothety in without changing the slopes of our lines).

Clearly, there also exist superabundant immersed 3-valent tropical curves in , . E.g. if is a (regular) tropical immersion of a 3-valent graph , then its composition with the embedding , , is superabundant.

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

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