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.
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 Reference12) 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. Reference3). 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 )Reference12). In the arbitrary genus case Caporaso and Harris Reference3 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 of these numbers (that appear in real algebraic geometry). The number -counterparts 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 Reference13 Kontsevich and Soibelman proposed a program linking homological mirror symmetry and torus fibrations from the Strominger-Yau-Zaslow conjecture Reference26. 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 Reference29. Maslov and his school studied the so-called dequantization of the semiring of positive real numbers (cf. Reference15). 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. Reference20. 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 Reference27 for some first steps in tropical algebraic geometry. See also Reference24, Reference23, Reference25 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 Reference18. 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 Reference4. 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 Reference1, Reference2). 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 Reference8. A somewhat different approach can be found in Reference21.
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.
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 ,
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.
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.
The map can be used to induce a certain structure on from the affine space It is an instance of the so-called . structure. For a graph -affine 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 structures. Then tropical maps (e.g. to -affine 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. )
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. .
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.
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.
The resulting set is called the toric degree of Accordingly, the degree of a parameterized tropical curve . is the degree of its image .
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.
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.
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.
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.
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 .
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.
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.
Let be a 3-valent graph. The space of all tropical curves in the same combinatorial type (up to their equivalence from Definition Equation2.2) is an open convex polyhedral domain in a real affine space, where -dimensional
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 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 -dimensional 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.
The space of all tropical curves in the same combinatorial type (up to their equivalence from Definition Equation2.2) is an open convex polyhedral domain in a real affine space, where -dimensional
where is the number of edges of that are mapped to a point.
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 .
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 vertex with 2 vertices (the endpoints of -valent 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 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 .Equation2.2 let be the weights of the edges adjacent to and let be the primitive integer vectors in the direction of the edges.
The multiplicity of at its 3-valent vertex is Here . is the area of the parallelogram spanned by and Note that .
since by Definition Equation2.2.
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.
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).
A tropical curve is called smooth if is 3-valent, is an embedding and the multiplicity of every vertex of is 1.
A smooth curve does not admit perturbations of different combinatorial types.
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 vertex, -valent 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.
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.
This proposition follows from Proposition 2.14 applied to all possible perturbation of .
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.
Some curves vary in a family strictly larger than “the prescribed dimension” .
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.
Every tropical immersion is regular if is 3-valent. If has vertices of valence higher than 3, then varies in at most the family. -dimensional
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 real affine space that is cut by -dimensional 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 with real values. The rows of this matrix correspond to the edges of -matrix 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 with non-zero elements on its diagonal. -minor
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 . and thus the dimension is at most -minor (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 .
An immersed 3-valent tropical curve locally varies in a (real) linear space, where -dimensional
if either or .
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. ,
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 .
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 idempotent, . This makes . a semiring without the additive zero (the role of such a zero would be played by ).
According to Reference20 the term “tropical” appeared in computer science in honor of Brazil and, more specifically, after Imre Simon (who is a Brazilian computer scientist) by Dominique Perrin. In computer science the term is usually applied to Our semiring -semirings. is by our definition but isomorphic to the the isomorphism is given by -semiring;.
As usual, a (Laurent) polynomial in variables over is defined by
where , , , and is a finite set. Note that is a convex piecewise-linear function. It coincides with the Legendre transform of a function defined on the finite set .
The polyhedron is called the Newton polyhedron of It can be treated as a refined version of the degree of the polynomial . in toric geometry.
For a tropical polynomial in variables we define its variety as the set of points where the piecewise-linear function is not smooth; cf. Reference10, Reference18 and Reference27. In other words, is the corner locus of .
is the set of points in where more than one monomial of reaches its maximal value.
If exactly one monomial of is maximal at then , locally coincides with this monomial and, therefore, is linear and smooth. Otherwise has a corner at .
At first glance this definition might appear to be unrelated to the classical definition of the variety as the zero locus of a polynomial. To see the connection, recall that there is no additive zero in but its rôle is played by ,.
Consider the graph of a tropical polynomial The graph . itself is not a tropical variety in but it can be completed to the tropical variety
see Figure 7.
coincides with the variety of the polynomial in variables (where ,).
If then we have , and one of the monomials of both reaching the maximal values in If . and then two monomials of , reach the maximal value at the expression .
Note that we have for sufficiently close to This is the sense in which . can be thought of as a zero locus.
One may argue that itself is a subtropical variety (as in subanalytic vs. analytic sets) while is its tropical closure. Figure 7 sketches the graph and its tropical closure.
Varieties are called tropical hypersurfaces associated to .
Different tropical polynomials may define the same varieties. To see this, let us first extend the notion of concavity to those functions which are only defined on a finite set -valued We call a function . concave if for any (possibly non-distinct) and any with and we have
We have three types of ambiguities when but .
where , is a coordinate in Note that in this case the Newton polyhedron of . is a translate of the Newton polyhedron of .
where , is a constant.
The function is not concave, where and we set if Then the variety of . coincides with the variety of where is the smallest concave function such that (in other words is a concave hull of ).
Thus to define tropical hypersurfaces, it suffices to consider only tropical polynomials whose coefficients satisfy the concavity condition above.
Proposition 3.8 (Reference17).
The space of all tropical hypersurfaces with a given Newton polyhedron is a closed convex polyhedral cone , The cone . is well-defined up to the natural action of .
All concave functions form a closed convex polyhedral cone But the function . defines the same curve as the function To get rid of this ambiguity, we choose . and define as the image of under the linear projection .
Clearly, the cone is not compact. Nevertheless it gets compactified by the cones for all non-empty lattice subpolyhedra (including polygons with the empty interior). Indeed, we have the following proposition.
Let , be a sequence of tropical hypersurfaces whose Newton polyhedron is , There exists a subsequence which converges to a tropical hypersurface . whose Newton polyhedron is contained in (note that is empty if is a point). The convergence is in the Hausdorff metric when restricted to any compact subset in Furthermore, if the Newton polyhedron of . coincides with then the convergence is in the Hausdorff metric in the whole of ,.
Each is defined by a tropical polynomial We may assume that the coefficients . are chosen so that they satisfy the concavity condition and so that This takes care of the ambiguity in the choice of . (since the Newton polyhedron is already fixed).
Passing to a subsequence, we may assume that converge (to a finite number or when ) for all By our assumption one of these limits is 0. Define . to be the variety of where we take only finite coefficients .
A tropical polynomial defines a lattice subdivision of its Newton polyhedron in the following way (cf. Reference6). Define the (unbounded) extended polyhedral domain
The projection induces a homeomorphism from the union of all closed bounded faces of to .
The resulting lattice subdivision of is called the subdivision associated to .
The lattice subdivision is dual to the tropical hypersurface Namely, for every . polyhedron -dimensional there is a convex closed (perhaps unbounded) polyhedron This correspondence has the following properties. .
is contained in an affine-linear subspace -dimensional of orthogonal to .
The relative interior of in is not empty.
is compact if and only if .
For every consider the truncated polynomial ,
(recall that Define ).
Note that for any face we have the variety orthogonal to (since moving in the direction orthogonal to does not change the value of and therefore to -monomials) To verify the last item of the proposition, we restate the defining equation .Equation2 algebraically: is the set of points where all monomials of indexed by have equal values while the value of any other monomial of could only be smaller.
It was observed in Reference10, Reference18 and Reference27 that is an polyhedral complex dual to the subdivision -dimensional The complex . is a union of convex (not necessarily bounded) polyhedra or cells of Each . (even if it is unbounded) of -cell is dual to a bounded of -face i.e. to an , of -cell In particular, the slope of each cell of . is rational.
In particular, an cell is dual to an interval -dimensional both of whose ends are lattice points. We define the lattice length of as (Such a length is invariant with respect to . We can treat .) as a weighted piecewise-linear polyhedral complex in the weights are natural numbers associated to the ; They are the lattice lengths of the dual intervals. -cells.
The combinatorial type of a tropical hypersurface is the equivalence class of all such that .
Let be such a combinatorial type.
All tropical hypersurfaces of the same combinatorial type form a convex polyhedral domain that is open in its affine-linear span.
The condition can be written in the following way in terms of the coefficients of For every . the function for should coincide with some linear function such that for every .
It turns out that the weighted piecewise-linear complex satisfies the balancing property at each see Definition 3 of -cell;Reference18. Namely, let be the adjacent to a -cells -cell of Each . has a rational slope and is assigned a weight Choose a direction of rotation around . and let be linear maps whose kernels are planes parallel to and such that they are primitive (non-divisible) and agree with the chosen direction of rotation. The balancing condition states that
As was shown in Reference18, this balancing condition at every of a rational piecewise-linear -cell polyhedral complex in -dimensional suffices for such a polyhedral complex to be the variety of some tropical polynomial.
Theorem 3.15 (Reference17).
A weighted polyhedral complex -dimensional is the variety of a tropical polynomial if and only if each of -cell is a convex polyhedron sitting in a affine subspace of -dimensional with a rational slope and satisfies the balancing condition Equation3 at each -cell.
This theorem implies that the definitions of tropical curves and tropical hypersurfaces agree if .
Any tropical curve is a tropical hypersurface for some polynomial Conversely, any tropical hypersurface in . can be parameterized by a tropical curve.
Furthermore, the degree of is determined by the Newton polygon of according to the following recipe. For each side we take the primitive integer outward normal vector and multiply it by the lattice length of to get the degree of .
Polyhedral complexes resulting from tropical varieties appeared in Reference10 in the following context. Let be a complete algebraically closed non-Archimedean field. This means that is an algebraically closed field and there is a valuation defined on such that defines a complete metric on Recall that a valuation . is a map such that and .
Our principal example of such a is a field of Puiseux series with real powers. To construct we take the algebraic closure , of the field of Laurent series The elements of . are formal power series in where , and is a subset bounded from below and contained in an arithmetic progression. We set We define . to be the completion of as the metric space with respect to the norm .
Let be an algebraic variety over The image of . under the map , is called the amoeba of (cf. Reference6). Kapranov Reference10 has shown that the amoeba of a non-Archimedean hypersurface is the variety of a tropical polynomial. Namely, if , is a hypersurface in then its amoeba is the variety of the tropical polynomial ,.
More generally, if is a field with a real-valued norm, then the amoeba of an algebraic variety is where , Note that . is such a map with respect to the non-Archimedean norm in .
Another particularly interesting case is if with the standard norm (see Reference6, Reference16, Reference19, etc.). The non-Archimedean hypersurface amoebas appear as limits in the Hausdorff metric of from the complex hypersurfaces amoebas (see e.g. Reference17).
It was noted in Reference23 that the non-Archimedean approach can be used to define tropical varieties of arbitrary codimension in Namely, one can define the tropical varieties in . to be the images of arbitrary algebraic varieties This definition allows one to avoid dealing with the intersections of tropical hypersurfaces in non-general position. We refer to .Reference23 for relevant discussions.
Corollary 3.16 states that any tropical 1-cycle in is a tropical hypersurface, i.e. it is the variety of a tropical polynomial By Remark .3.7 the Newton polygon of such is well defined up to a translation.
We call the degree of a tropical curve in .
The number is the number of unbounded edges of the curve if counted with multiplicities (recall that we denoted the number of unbounded edges “counted simply” with The number ). is the genus of a smooth tropical curve of degree To see this, let us note that every lattice point of . is a vertex of the associated subdivision for a smooth curve Therefore, the homotopy type of . coincides with Note also that smooth curves are dense in ..
There is a larger class of tropical curves in whose behavior is as simple as that of smooth curves.
A parameterized tropical curve is called simple if it satisfies all of the following conditions.
The graph is 3-valent.
The map is an immersion.
For any the inverse image consists of at most two points.
If , are such that , then neither , nor can be a vertex of .
A tropical 1-cycle is called simple if it admits a simple parameterization.
A simple tropical 1-cycle admits a unique simple tropical parameterization. The genus of a simple 1-cycle coincides with the genus of its simple parameterization. Furthermore, any of its non-simple parameterizations has a strictly larger genus.
By Definition 4.2, has only 3- and 4-valent vertices, where 4-valent vertices are the double points of a simple immersion. Any other parametrization would have to have a 4-valent vertex in the parameterizing graph.
Proposition 4.3 allows us to switch back and forth between parameterized tropical curves and tropical 1-cycles in the case of simple curves in Thus we refer to them just as simple tropical curves. In a sense they are a tropical counterpart of nodal planar curves in classical complex geometry. .
More generally, every tropical 1-cycle admits a parameterization by an immersion of genus not greater than Start from an arbitrary parameterization . To eliminate an edge . such that is contracted to a point, we take the quotient of by for a new domain of parameterization. This procedure does not change the genus of .
Therefore, we may assume that does not have contracting edges. This is an immersion away from such vertices of for which there exist two distinct adjacent edges with Changing the graph . by identifying the points on and with the same image can only decreases the genus of (if and were distinct edges connecting the same pair of vertices). Inductively we get an immersion.
A tropical curve is simple (see Definition 4.2) if and only if it is the variety of a tropical polynomial such that is a subdivision into triangles and parallelograms.
The lemma follows from Proposition 3.11. The 3-valent vertices of are dual to the triangles of while the intersection of edges is dual to the parallelograms (see e.g. the right-hand side of Figure 3 and the corresponding lattice subdivision in Figure 8).
We have the following formula which expresses the genus of a simple tropical curve in terms of the number of triangles in .
Lemma 4.6 (Cf. Reference8).
If a curve is simple, then .
Let be the number of vertices of while and are the numbers of its edges and (2-dimensional) polygons. Out of the 2-dimensional polygons are triangles and are parallelograms.
Note that Thus, . and
Points are said to be in general position tropically if for any tropical curve of genus and with ends such that and we have the following conditions.
The curve is simple (see Definition 4.2).
Inverse images are disjoint from the vertices of .
Two distinct points are in general position tropically if and only if the slope of the line in passing through and is irrational.
Note that we can always find a curve with passing through For such a curve we can take a reducible curve consisting of . affine (i.e. classical) lines in with rational slope each passing through its own point This curve has . ends while its genus is .
Any subset of a set of points in tropically general position is itself in tropically general position.
Suppose the points are not in general position. Then there is a curve with ends of genus passing through or of genus but with a non-generic behavior with respect to By Remark .4.9 there is a curve passing through of genus The curve . supplies a contradiction.
For each the set of configurations such that there exists a curve of degree such that the conditions of Definition 4.7 are violated by is closed and nowhere dense.
By Remark 4.4 it suffices to consider only immersed tropical curves We have only finitely many combinatorial types of tropical curves of genus . with the Newton polygon since there are only finitely many lattice subdivisions of By Proposition .2.23 for each such combinatorial type we have an family of simple curves or a smaller-dimensional family of non-simple curves. For a fixed -dimensional each of the points can vary in a 1-dimensional family on or in a 0-dimensional family if is a vertex of Thus the dimension of the space of “bad” configurations . is at most .
The configurations in general position tropically form a dense set which can be obtained as an intersection of countably many open dense sets in .
To set up an enumerative problem, we fix the degree, i.e. a polygon with and the genus, i.e. an integer number , Consider a configuration . of points in tropical general position. Our goal is to count tropical curves of genus such that and has degree .
There exist only finitely many such curves Furthermore, each end of . is of weight 1 in this case, so has ends.
Finiteness follows from Lemma 4.22 proved in the next subsection. If has ends whose weight is greater than 1, then the number of ends is smaller than and the existence of contradicts the general position of Recall that since . is in general position, any such is also simple and the vertices of are disjoint from .
Let and let be the quadrilateral whose vertices are , , and (so that the number of the lattice points on the perimeter is 4). For a configuration of three points in pictured in Figure 9 we have three tropical curves passing. In Figure 10 the corresponding number is two.
The multiplicity of a tropical curve of degree and genus passing via equals the product of the multiplicities of all the 3-valent vertices of (see Definition 2.16).
We define the number to be the number of irreducible tropical curves of genus and degree passing via where each such curve is counted with the multiplicity from Definition 4.15. Similarly we define the number to be the number of all tropical curves of genus and degree passing via Again each curve is counted with the multiplicity . from Definition 4.15.
The numbers and are finite and do not depend on the choice of .
Recall that every vertex of a tropical curve corresponds to a polygon in the dual lattice subdivision of the Newton polygon while every edge of corresponds to an edge of the dual subdivision