American Mathematical Society

Enumerative tropical algebraic geometry in double-struck upper R squared

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 double-struck upper R Superscript n and holomorphic curves with certain piecewise-linear graphs there.

1. Introduction

Recall the basic enumerative problem in the plane. Let g greater-than-or-equal-to 0 and d greater-than-or-equal-to 1 be two numbers and let script upper Z equals left-parenthesis z 1 comma ellipsis comma z Subscript 3 d minus 1 plus g Baseline right-parenthesis be a collection of points z Subscript j Baseline element-of double-struck upper C double-struck upper P squared in general position. A holomorphic curve upper C subset-of double-struck upper C double-struck upper P squared is parameterized by a Riemann surface upper C overTilde under a holomorphic map phi colon upper C overTilde right-arrow double-struck upper C double-struck upper P squared so that upper C equals phi left-parenthesis upper C overTilde right-parenthesis . Here we choose the minimal parametrization, i.e. such that no component of upper C overTilde is mapped to a point by phi . The curve upper C is irreducible if and only if upper C overTilde is connected. The number upper N Subscript g comma d Superscript i r r of irreducible curves of degree d and genus g passing through script upper Z is finite and does not depend on the choice of z Subscript j 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 upper C subset-of double-struck upper C double-struck upper P squared to be one-half left-parenthesis 2 minus chi left-parenthesis upper C overTilde right-parenthesis right-parenthesis . Note that the genus can take negative values for reducible curves. The number upper N Subscript g comma d Superscript m u l t of curves of degree d and genus g passing through script upper Z is again finite and does not depend on the choice of z Subscript j as long as this choice is generic. Figure 1 lists some (well-known) first few numbers upper N Subscript g comma d Superscript i r r and upper N Subscript g comma d Superscript m u l t .

The numbers upper N Subscript g comma d Superscript i r r are known as the Gromov-Witten invariants of double-struck upper C double-struck upper P squared (see Reference12) while the numbers upper N Subscript g comma d Superscript m u l t 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 upper N Subscript 0 comma d Superscript i r r (and thus the numbers upper N Subscript 0 comma d Superscript m u l t ) 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 double-struck upper C double-struck upper P squared ) which allows one to compute the numbers upper N Subscript g comma d Superscript m u l t (and thus the numbers upper N Subscript g comma d Superscript i r r ).

The main result of this paper gives a new way of computation of these numbers as well as the double-struck upper R -counterparts of these numbers (that appear in real algebraic geometry). The number upper N Subscript g comma d Superscript m u l t turns out to be the number of certain lattice paths of length 3 d minus 1 plus g in the triangle normal upper Delta Subscript d Baseline subset-of double-struck upper R squared with vertices left-parenthesis 0 comma 0 right-parenthesis , left-parenthesis d comma 0 right-parenthesis and left-parenthesis 0 comma d right-parenthesis . The paths have to be counted with certain non-negative multiplicities. Furthermore, this formula works not only for double-struck upper C double-struck upper P squared but for other toric surfaces as well. For other toric surfaces we just have to replace the triangle normal upper Delta Subscript d 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 double-struck upper R Superscript n . 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 left-parenthesis max comma plus right-parenthesis -semiring double-struck upper R Subscript t r o p , the semiring of real numbers equipped with taking the maximum for addition and addition for multiplication.

The semiring double-struck upper R Subscript t r o p is known to computer scientists as one of tropical semirings, see e.g. Reference20. In mathematics this semiring appears from non-Archimedean fields upper K under a certain pushing forward to double-struck upper R of the arithmetic operations in upper K .

In this paper we develop some basic algebraic geometry over double-struck upper R Subscript t r o p with a view towards counting curves. In particular, we rigorously set up some enumerative problems over double-struck upper R Subscript t r o p 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 double-struck upper R squared . 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 double-struck upper P squared and double-struck upper P Superscript 1 Baseline times double-struck upper P Superscript 1 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 double-struck upper R cubed 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 double-struck upper R squared ; in Section 5 recall those in left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis squared . Section 6 is auxiliary to Section 8 and deals with certain piecewise-holomorphic piecewise-Lagrangian objects in left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis squared 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.

2. Tropical curves as graphs in double-struck upper R Superscript n

In this section we geometrically define tropical curves in double-struck upper R Superscript n 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 normal upper Gamma overbar be a weighted finite graph. The weights are natural numbers prescribed to the edges. Clearly, normal upper Gamma overbar is a compact topological space. We make it non-compact by removing the set of all 1-valent vertices script upper V 1 ,

normal upper Gamma equals normal upper Gamma overbar minus script upper V 1 period

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 double-struck upper R Subscript t r o p Baseline equals left-parenthesis double-struck upper R comma max comma plus right-parenthesis as our “ground semifield” for tropical variety; see the next section. The operation max plays the rôle of addition and thus we do not have an additive zero. Non-compactness of normal upper Gamma is caused by this choice. Should we have chosen double-struck upper R Subscript t r o p Baseline union StartSet negative normal infinity EndSet 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 double-struck upper R Superscript n 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 h colon normal upper Gamma right-arrow double-struck upper R Superscript n is called a parameterized tropical curve if it satisfies to the following two conditions.

For every edge upper E subset-of normal upper Gamma the restriction h vertical-bar Subscript upper E Baseline is either an embedding or a constant map. The image h left-parenthesis upper E right-parenthesis is contained in a line l subset-of double-struck upper R squared such that the slope of l is rational.

For every vertex upper V element-of normal upper Gamma we have the following property. Let upper E 1 comma ellipsis comma upper E Subscript m Baseline subset-of normal upper Gamma be the edges adjacent to upper V , let w 1 comma ellipsis comma w Subscript m Baseline element-of double-struck upper N be their weights and let v 1 comma ellipsis comma v Subscript m Baseline element-of double-struck upper Z Superscript n be the primitive integer vectors at the point h left-parenthesis upper V right-parenthesis in the direction of the edges h left-parenthesis upper E Subscript j Baseline right-parenthesis (we take v Subscript j Baseline equals 0 if h left-parenthesis upper E Subscript j Baseline right-parenthesis is a point). We have StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel sigma-summation Underscript j equals 1 Overscript m Endscripts w Subscript j Baseline v Subscript j Baseline equals 0 period EndLayout

We say that two parameterized tropical curves h colon normal upper Gamma right-arrow double-struck upper R Superscript n and h prime colon normal upper Gamma prime right-arrow double-struck upper R Superscript n are equivalent if there exists a homeomorphism normal upper Phi colon normal upper Gamma right-arrow normal upper Gamma prime which respects the weights of the edges and such that h equals h prime ring normal upper Phi . We do not distinguish equivalent parameterized tropical curves. The image

upper C equals h left-parenthesis normal upper Gamma right-parenthesis subset-of double-struck upper R Superscript n

is called the unparameterized tropical curve or just a tropical 1-cycle if no connected component of normal upper Gamma gets contracted to a point. The 1-cycle upper C is a piecewise-linear graph in double-struck upper R Superscript n with natural weights on its edges induced from the weights on normal upper Gamma . If upper E is an edge of upper C , then h Superscript negative 1 Baseline left-parenthesis upper E right-parenthesis is a union of subintervals of the edges of normal upper Gamma . The weight of upper E 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 left-parenthesis p comma q right-parenthesis -webs introduced by Aharony, Hanany and Kol in Reference2 (see also Reference1).

Remark 2.4

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

Let upper E subset-of normal upper Gamma be a compact edge of weight w that is not contracted to a point by h . Such an edge is mapped to a finite straight interval with a rational slope in double-struck upper R Superscript n . Let l be the length of a primitive rational vector in the direction of h left-parenthesis upper E right-parenthesis . We set the length of upper E to be StartFraction StartAbsoluteValue h left-parenthesis upper E right-parenthesis EndAbsoluteValue Over l w EndFraction .

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

It is possible to consider abstract tropical curves as graphs equipped with such double-struck upper Z -affine structures. Then tropical maps (e.g. to double-struck upper R Superscript n ) will be maps that respect such structure. Abstract tropical curves have genus (equal to b 1 left-parenthesis normal upper Gamma right-parenthesis ) and the number of punctures (equal to the number of ends of normal upper Gamma ) 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

upper Y equals StartSet left-parenthesis x comma 0 right-parenthesis vertical-bar x less-than-or-equal-to 0 EndSet union StartSet left-parenthesis 0 comma y right-parenthesis vertical-bar y less-than-or-equal-to 0 EndSet union StartSet left-parenthesis x comma x right-parenthesis vertical-bar x greater-than-or-equal-to 0 EndSet subset-of double-struck upper R squared period

This graph (considered as a tautological embedding in double-struck upper R squared ) is a tropical curve since left-parenthesis negative 1 comma 0 right-parenthesis plus left-parenthesis 0 comma negative 1 right-parenthesis plus left-parenthesis 1 comma 1 right-parenthesis equals 0 . A parallel translation of upper Y in any direction in double-struck upper R squared is clearly also a tropical curve. This gives us a 2-dimensional family of curves in double-struck upper R squared . 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 double-struck upper R squared 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 double-struck upper R Superscript n

Let script upper T equals StartSet tau 1 comma ellipsis comma tau Subscript q Baseline EndSet subset-of double-struck upper Z Superscript n be a set of non-zero integer vectors such that sigma-summation Underscript j equals 1 Overscript q Endscripts tau Subscript j Baseline equals 0 . Suppose that in this set we do not have positive multiples of each other, i.e. if tau Subscript j Baseline equals m tau Subscript k for m element-of double-struck upper N , then tau Subscript j Baseline equals tau Subscript k . The degree of a tropical 1-cycle upper C subset-of double-struck upper R Superscript n takes values in such sets according to the following construction.

By our definition a tropical curve h colon normal upper Gamma right-arrow double-struck upper R Superscript n has a finite number of ends, i.e. unbounded edges (rays). Let tau element-of double-struck upper Z Superscript n be a primitive vector. A positive multiple of tau is included in script upper T if and only if there exists an end of normal upper Gamma which is mapped in the direction of tau . In such a case we include m tau into script upper T , where m is the sum of multiplicities of all such rays.

Definition 2.7

The resulting set script upper T is called the toric degree of upper C subset-of double-struck upper R Superscript n . Accordingly, the degree of a parameterized tropical curve h colon normal upper Gamma right-arrow double-struck upper R Superscript n is the degree of its image h left-parenthesis normal upper Gamma right-parenthesis .

Note that the sum of all vectors in script upper T is zero. This follows from adding the conditions Equation1 from Definition Equation2.2 in all vertices of upper C .

For example the degree of both curves from Figure 5 is left-brace left-parenthesis negative 1 comma negative 1 right-parenthesis , left-parenthesis 2 comma negative 1 right-parenthesis , left-parenthesis negative 1 comma 2 right-parenthesis right-brace while the degree of both curves from Figure 3 is StartSet left-parenthesis negative 3 comma 0 right-parenthesis comma left-parenthesis 0 comma negative 3 right-parenthesis comma left-parenthesis 3 comma 3 right-parenthesis EndSet .

Definition 2.8

If the toric degree of a tropical 1-cycle upper C subset-of double-struck upper R Superscript n is StartSet left-parenthesis negative d comma 0 comma ellipsis comma 0 right-parenthesis comma ellipsis comma left-parenthesis 0 comma ellipsis comma 0 comma negative d right-parenthesis comma left-parenthesis d comma ellipsis comma d right-parenthesis EndSet , then upper C is called a tropical projective curve of degree d .

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 h colon normal upper Gamma right-arrow double-struck upper R Superscript n is reducible if normal upper Gamma is disconnected. We say that a tropical 1-cycle upper C subset-of double-struck upper R Superscript n 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 normal upper Gamma right-arrow double-struck upper R Superscript n is dimension upper H 1 left-parenthesis normal upper Gamma right-parenthesis minus dimension upper H 0 left-parenthesis normal upper Gamma right-parenthesis plus 1 . In particular, for irreducible parameterized curves the genus is the first Betti number of normal upper Gamma . The genus of a tropical 1-cycle upper C subset-of double-struck upper R Superscript n is the minimum genus among all parameterizations of upper C .

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

If upper C subset-of double-struck upper R Superscript n 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 h colon normal upper Gamma right-arrow double-struck upper R Superscript n is subject to the constraint coming from the Riemann-Roch formula. Let x be the number of ends of normal upper Gamma .

Remark 2.11

The number negative x is a tropical counterpart of the value of the canonical class of the ambient complex variety on the curve h left-parenthesis normal upper Gamma right-parenthesis . The ambient space double-struck upper R Superscript n corresponds to the torus left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis Superscript n classically. Let upper V subset-of left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis Superscript n be a holomorphic curve with a finite number of ends. The space left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis Superscript n is not compact, but one can always choose a toric compactification double-struck upper C upper T superset-of left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis Superscript n such that every point of the closure upper V overbar superset-of upper V in double-struck upper C upper T intersects not more than one boundary divisor (i.e. a component of double-struck upper C upper T minus left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis Superscript n ). Then every end of upper V can be prescribed a multiplicity equal to the intersection number of the point of upper V overbar and the corresponding boundary divisor. The value of the canonical class of double-struck upper C upper T on upper V overbar equals the sum of these multiplicities.

Definition 2.12

The curves h colon normal upper Gamma right-arrow double-struck upper R Superscript n and h prime colon normal upper Gamma right-arrow double-struck upper R Superscript n (parameterized by the same graph normal upper Gamma ) are said to be of the same combinatorial type if for any edge upper E subset-of normal upper Gamma the segments h left-parenthesis upper E right-parenthesis and h prime left-parenthesis upper E right-parenthesis are parallel.

Note that if two tropical curves normal upper Gamma right-arrow double-struck upper R Superscript n are isotopic in the class of tropical curves (with the same domain normal upper Gamma ), then they are of the same combinatorial type.

The valence of a vertex of normal upper Gamma is the number of adjacent edges regardless of their weights. The graph normal upper Gamma is called 3-valent if every vertex is 3-valent. The parameterized tropical curve h colon normal upper Gamma right-arrow double-struck upper R Superscript n is called 3-valent if normal upper Gamma is 3-valent.

Proposition 2.13

Let normal upper Gamma be a 3-valent graph. The space of all tropical curves normal upper Gamma right-arrow double-struck upper R Superscript n in the same combinatorial type (up to their equivalence from Definition Equation2.2) is an open convex polyhedral domain in a real affine k -dimensional space, where

k greater-than-or-equal-to x plus left-parenthesis n minus 3 right-parenthesis left-parenthesis 1 minus g right-parenthesis period

Proof.

It suffices to prove this for a connected graph normal upper Gamma since different components of normal upper Gamma vary independently, and, furthermore, both sides of the inequality are additive with respect to taking the union of components (note that 1 minus g equals b 0 left-parenthesis normal upper Gamma right-parenthesis minus b 1 left-parenthesis normal upper Gamma right-parenthesis equals chi left-parenthesis normal upper Gamma right-parenthesis ). Let normal upper Gamma prime subset-of normal upper Gamma be a finite tree containing all the vertices of normal upper Gamma . Note that the number of finite edges in normal upper Gamma minus normal upper Gamma prime is g . By an Euler characteristic computation we get that the number of finite edges of normal upper Gamma prime is equal to x minus 3 plus 2 g .

Maps normal upper Gamma prime right-arrow double-struck upper R Superscript n vary in a linear left-parenthesis x minus 3 plus 2 g plus n right-parenthesis -dimensional family if we do not change the slopes of the edges. The left-parenthesis x minus 3 plus 2 g right-parenthesis -dimensional part comes from varying the lengths of the edges while the n -dimensional part comes from translations in double-struck upper R Superscript n . Such a map is extendable to a tropical map normal upper Gamma right-arrow double-struck upper R Superscript n if the pairs of vertices corresponding to the g remaining edges define the lines with the correct slope. Each of the g edges imposes a linear condition of codimension at most n minus 1 . Thus tropical perturbations of normal upper Gamma right-arrow double-struck upper R Superscript n are contained in a linear family of dimension at least x minus 3 plus 2 g plus n minus left-parenthesis n minus 1 right-parenthesis g equals x plus left-parenthesis n minus 3 right-parenthesis left-parenthesis 1 minus g right-parenthesis . 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 normal upper Gamma has vertices of valence higher than 3. How much normal upper Gamma differs from a 3-valent graph is measured by the following characteristic. Let the overvalence o v left-parenthesis normal upper Gamma right-parenthesis be the sum of the valences of all vertices of valence higher than 3 minus the number of such vertices. Thus o v left-parenthesis normal upper Gamma right-parenthesis equals 0 if and only if no vertex of normal upper Gamma has valence higher than 3.

Proposition 2.14

The space of all tropical curves normal upper Gamma right-arrow double-struck upper R Superscript n in the same combinatorial type (up to their equivalence from Definition Equation2.2) is an open convex polyhedral domain in a real affine k -dimensional space, where

k greater-than-or-equal-to x plus left-parenthesis n minus 3 right-parenthesis left-parenthesis 1 minus g right-parenthesis minus o v left-parenthesis normal upper Gamma right-parenthesis minus c comma

where c is the number of edges of normal upper Gamma 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 double-struck upper R Superscript n , 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 o v greater-than 0 .

Note that o v plus c can be interpreted as the overvalence of the image h left-parenthesis normal upper Gamma right-parenthesis .

2.5. Changing the combinatorial type of normal upper Gamma

Sometimes we can deform normal upper Gamma and h colon normal upper Gamma right-arrow double-struck upper R Superscript n by the following procedure reducing o v . If we have n greater-than 3 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 upper E prime separating these groups as shown in Figure 4. This replaces the initial n -valent vertex with 2 vertices (the endpoints of upper E prime ) of smaller valence. There is a “virtual slope” of upper E prime determined by the slopes of the edges in each group. This is the slope to appear in local perturbation of the tropical map h colon normal upper Gamma right-arrow double-struck upper R Superscript n (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 normal upper Gamma which works even for some 3-valent vertices.

Example 2.15

Let normal upper Gamma be the union of three rays in double-struck upper R squared in the direction left-parenthesis negative 2 comma 1 right-parenthesis , left-parenthesis 1 comma negative 2 right-parenthesis and left-parenthesis 1 comma 1 right-parenthesis 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 t right-arrow 0 limit of the family of genus 1 curves given by the union of three rays in double-struck upper R squared in the direction left-parenthesis negative 2 comma 1 right-parenthesis , left-parenthesis 1 comma negative 2 right-parenthesis and left-parenthesis 1 comma 1 right-parenthesis emanating from left-parenthesis minus 2 t comma t right-parenthesis , left-parenthesis t comma minus 2 t right-parenthesis and left-parenthesis t comma t right-parenthesis , respectively, and the three intervals left-bracket left-parenthesis minus 2 t comma t right-parenthesis comma left-parenthesis t comma minus 2 t right-parenthesis right-bracket , left-bracket left-parenthesis minus 2 t comma t right-parenthesis comma left-parenthesis t comma t right-parenthesis right-bracket and left-bracket left-parenthesis t comma t right-parenthesis comma left-parenthesis t comma minus 2 t right-parenthesis right-bracket as pictured in Figure 5.

Let upper V be a 3-valent vertex of normal upper Gamma . As in Definition Equation2.2 let w 1 comma w 2 comma w 3 be the weights of the edges adjacent to upper V and let v 1 comma v 2 comma v 3 be the primitive integer vectors in the direction of the edges.

Definition 2.16

The multiplicity of upper C at its 3-valent vertex upper V is w 1 w 2 StartAbsoluteValue v 1 logical-and v 2 EndAbsoluteValue . Here StartAbsoluteValue v 1 logical-and v 2 EndAbsoluteValue is the area of the parallelogram spanned by v 1 and v 2 . Note that

w 1 w 2 StartAbsoluteValue v 1 logical-and v 2 EndAbsoluteValue equals w 2 w 3 StartAbsoluteValue v 2 logical-and v 3 EndAbsoluteValue equals w 3 w 1 StartAbsoluteValue v 3 logical-and v 1 EndAbsoluteValue

since v 1 w 1 plus v 2 w 2 plus v 3 w 3 equals 0 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.

Definition 2.17

We say that h prime colon normal upper Gamma prime right-arrow double-struck upper R Superscript n is a perturbation of h colon normal upper Gamma right-arrow double-struck upper R Superscript n if there exists a family h prime Subscript t Baseline colon normal upper Gamma prime right-arrow double-struck upper R Superscript n , t greater-than 0 , in the same combinatorial type as h prime and the pointwise limit h prime 0 equals limit Underscript t right-arrow 0 Endscripts h prime Subscript t such that h prime 0 left-parenthesis normal upper Gamma prime right-parenthesis coincides with h left-parenthesis normal upper Gamma right-parenthesis (as tropical 1-cycles).

Definition 2.18

A tropical curve h colon normal upper Gamma right-arrow double-struck upper R Superscript n is called smooth if normal upper Gamma is 3-valent, h is an embedding and the multiplicity of every vertex of upper C is 1.

Proposition 2.19

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

Proof.

Suppose that h prime Subscript t Baseline colon normal upper Gamma prime right-arrow double-struck upper R Superscript n is a perturbation of a smooth curve h colon normal upper Gamma right-arrow double-struck upper R Superscript n . Since h is an embedding and h left-parenthesis normal upper Gamma right-parenthesis equals h prime 0 left-parenthesis normal upper Gamma prime right-parenthesis , we have a map

psi colon normal upper Gamma prime right-arrow normal upper Gamma period

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

Thus psi must be a homeomorphism near the inner points of the edges of normal upper Gamma prime . Let a element-of normal upper Gamma be a vertex and let upper U contains-as-member a be its small neighborhood in normal upper Gamma . Note that psi Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis is connected since a is 3-valent (otherwise we can divide the adjacent edges to a into two groups with zero sums of the primitive integer vectors).

Suppose that psi Superscript negative 1 Baseline left-parenthesis a right-parenthesis is not a point. Then psi Superscript negative 1 Baseline left-parenthesis a right-parenthesis is a graph which has three distinguished vertices that are adjacent to the edges of psi Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis minus psi Superscript negative 1 Baseline left-parenthesis a right-parenthesis . The graph upper A equals psi prime Subscript t Baseline left-parenthesis psi Superscript negative 1 Baseline left-parenthesis upper U right-parenthesis right-parenthesis must be contained in the affine 2-plane in double-struck upper R Superscript n containing the ends upper A . This follows from the balancing condition for upper A .

The 3-valent vertices of upper A have multiplicities from Definition 2.16. Since upper A is planar, we can extend the definition of the multiplicity to higher-valent vertices as follows. Let upper V element-of upper A be a k -valent vertex, v 1 comma ellipsis comma v Subscript k Baseline be the primitive integer vectors in the directions of the adjacent edges to upper A numbered consistently with the cyclic order in the ambient 2-plane and let w 1 comma ellipsis comma w Subscript k Baseline be the corresponding weights. We set the multiplicity of upper V to be

product Underscript l equals 2 Overscript k minus 1 Endscripts StartAbsoluteValue v Subscript l Baseline logical-and sigma-summation Underscript j equals 1 Overscript l Endscripts v Subscript j Baseline EndAbsoluteValue period

It is easy to see that the multiplicity of a in normal upper Gamma is equal to the sum of multiplicities of all the vertices of upper A . The multiplicities of all vertices are positive integers. Therefore, the multiplicity of a is greater than 1 unless psi Superscript negative 1 Baseline left-parenthesis a right-parenthesis 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 h colon normal upper Gamma right-arrow double-struck upper R Superscript n is locally a cone

script upper C equals union Underscript j Endscripts script upper C Subscript j Baseline comma

where script upper C Subscript j is a polyhedral convex cone corresponding to tropical curves parameterized by perturbations normal upper Gamma Subscript j Baseline right-arrow double-struck upper R Superscript n of h colon normal upper Gamma right-arrow double-struck upper R Superscript n of a given combinatorial type. The union is taken over all possible combinatorial types of perturbations. We have

dimension script upper C Subscript j Baseline greater-than-or-equal-to x plus left-parenthesis n minus 3 right-parenthesis left-parenthesis 1 minus g right-parenthesis minus o v left-parenthesis normal upper Gamma Subscript j Baseline right-parenthesis minus c comma

where c is the number of the edges of normal upper Gamma Subscript j that are mapped to a point.

Proof.

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

Remark 2.21

Not all conceivable perturbations of h colon normal upper Gamma right-arrow double-struck upper R Superscript n are realized as the following example shows. Let upper C 1 subset-of double-struck upper R squared times StartSet 0 EndSet subset-of double-struck upper R cubed be a tropical 1-cycle of genus 1. Let upper C 2 equals StartSet y EndSet times double-struck upper R subset-of double-struck upper R cubed be a vertical line such that y is a point on upper C 1 such that upper C 1 minus StartSet y EndSet is contractible. The curve

upper C equals upper C 1 union upper C 2

has a 4-valent vertex that cannot be perturbed (since any such perturbation would force upper C 1 out of the plane double-struck upper R squared times StartSet 0 EndSet ). Thus any 3-valent perturbation of the tautological embedding upper C 1 union upper C 2 subset-of double-struck upper R cubed 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” x plus left-parenthesis n minus 3 right-parenthesis left-parenthesis 1 minus g right-parenthesis minus o v minus c .

Definition 2.22

A parameterized tropical curve h colon normal upper Gamma right-arrow double-struck upper R Superscript n 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 x plus left-parenthesis n minus 3 right-parenthesis left-parenthesis 1 minus g right-parenthesis minus o v minus c . 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 normal upper Gamma do not provide transversal conditions for the length of the bounded edges of the subtree normal upper Gamma prime . This is the case if some of the cycles of upper C subset-of double-struck upper R Superscript n are contained in smaller-dimensional affine-linear subspaces of double-struck upper R Superscript n , e.g. if a non-trivial cycle of normal upper Gamma 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 double-struck upper R squared are never superabundant as the following proposition shows.

Proposition 2.23

Every tropical immersion h colon normal upper Gamma right-arrow double-struck upper R squared is regular if normal upper Gamma is 3-valent. If  normal upper Gamma has vertices of valence higher than 3, then h colon normal upper Gamma right-arrow double-struck upper R squared varies in at most the left-parenthesis x plus g minus 2 right-parenthesis -dimensional family.

Proof.

Recall the proof of Propositions 2.13 and 2.14. Once again we may assume that normal upper Gamma is connected. Let upper V element-of normal upper Gamma be any vertex.

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

The space of deformation of h within the same combinatorial type is open in a k -dimensional real affine space that is cut by g hyperplanes in double-struck upper R Superscript l plus n where l is the number of bounded edges of normal upper Gamma prime . Each of these g hyperplanes is non-trivial if h is an immersion and normal upper Gamma 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 g times left-parenthesis l plus 2 right-parenthesis -matrix with real values. The rows of this matrix correspond to the edges of normal upper Gamma minus normal upper Gamma prime while the first l columns correspond to the edges of normal upper Gamma prime (the remaining two columns correspond to translations in double-struck upper R squared ). To show that the rank of this matrix is g in the 3-valent case, we exhibit an upper-triangular g times g -minor with non-zero elements on its diagonal.

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

If normal upper Gamma is not 3-valent, then the number of bounded edges of normal upper Gamma prime is x minus 3 plus 2 g minus o v . This number is the same as the number of vertices of normal upper Gamma prime other than upper V . We can do the construction of the minor as above but only using one edge of normal upper Gamma minus normal upper Gamma prime at every vertex of normal upper Gamma other than upper V . In such a way we can get a non-degenerate left-parenthesis g minus o v right-parenthesis times left-parenthesis g minus o v right-parenthesis -minor and thus the dimension is at most x minus 3 plus 2 g minus o v minus left-parenthesis g minus o v right-parenthesis plus 2 equals x minus 1 plus g (2 comes from translations in double-struck upper R squared ). If there exists a vertex of valence higher than 3, then we may choose such a vertex for the root upper V of the tree normal upper Gamma prime and this gives a non-degenerate minor of size strictly larger than left-parenthesis g minus o v right-parenthesis .

Corollary 2.24

An immersed 3-valent tropical curve h colon normal upper Gamma right-arrow double-struck upper R Superscript n locally varies in a (real) linear k -dimensional space, where

k equals x plus left-parenthesis n minus 3 right-parenthesis left-parenthesis 1 minus g right-parenthesis

if either n equals 2 or g equals 0 .

Remark 2.25

There exist superabundant tropical immersions normal upper Gamma right-arrow double-struck upper R squared if normal upper Gamma 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 normal upper Gamma to be their union in double-struck upper R squared so that our tropical curve normal upper Gamma right-arrow double-struck upper R squared is a tautological embedding. We have x equals 18 , g equals 22 , o v equals 39 , c equals 0 . Therefore x plus g minus 1 minus o v minus c equals 0 , yet our configuration varies at least in a 3-dimensional family (since we can apply any translation and homothety in double-struck upper R squared without changing the slopes of our lines).

Clearly, there also exist superabundant immersed 3-valent tropical curves in double-struck upper R Superscript n , n greater-than 2 . E.g. if h colon normal upper Gamma right-arrow double-struck upper R squared is a (regular) tropical immersion of a 3-valent graph normal upper Gamma , then its composition with the embedding double-struck upper R squared subset-of double-struck upper R Superscript n , n greater-than 2 , 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 double-struck upper R Superscript n .

3.1. The tropical semifield double-struck upper R Subscript t r o p

Consider the semiring double-struck upper R Subscript t r o p of real numbers equipped with the following arithmetic operations called tropical in computer science:

single-turned-comma-quotation-mark single-turned-comma-quotation-mark x plus y quotation-mark equals max left-brace x comma y right-brace comma single-turned-comma-quotation-mark single-turned-comma-quotation-mark x y quotation-mark equals x plus y comma

x comma y element-of double-struck upper R Subscript t r o p Baseline . We use the quotation marks to distinguish the tropical operations from the classical ones. Note that addition is idempotent, single-turned-comma-quotation-mark single-turned-comma-quotation-mark x plus x equals x quotation-mark . This makes double-struck upper R Subscript t r o p a semiring without the additive zero (the role of such a zero would be played by negative normal infinity ).

Remark 3.1

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 left-parenthesis min comma plus right-parenthesis -semirings. Our semiring double-struck upper R Subscript t r o p is left-parenthesis max comma plus right-parenthesis by our definition but isomorphic to the left-parenthesis min comma plus right-parenthesis -semiring; the isomorphism is given by x right-arrow from bar negative x .

As usual, a (Laurent) polynomial in n variables over double-struck upper R Subscript t r o p is defined by

f left-parenthesis x right-parenthesis equals single-turned-comma-quotation-mark single-turned-comma-quotation-mark sigma-summation Underscript j element-of upper A Endscripts a Subscript j Baseline x Superscript j Baseline quotation-mark equals max Underscript j element-of upper A Endscripts left-parenthesis mathematical left-angle j comma x mathematical right-angle plus a Subscript j Baseline right-parenthesis comma

where x equals left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis element-of double-struck upper R Superscript n , j equals left-parenthesis j 1 comma ellipsis comma j Subscript n Baseline right-parenthesis , x Superscript j Baseline equals x 1 Superscript j 1 Baseline period period period x Subscript n Superscript j Super Subscript n , mathematical left-angle j comma x mathematical right-angle equals j 1 x 1 plus dot dot dot plus j Subscript n Baseline x Subscript n and upper A subset-of double-struck upper Z Superscript n is a finite set. Note that f colon double-struck upper R Superscript n Baseline right-arrow double-struck upper R is a convex piecewise-linear function. It coincides with the Legendre transform of a function j right-arrow from bar minus a Subscript j defined on the finite set upper A .

Definition 3.2

The polyhedron normal upper Delta equals upper C o n v e x upper H u l l left-parenthesis upper A right-parenthesis is called the Newton polyhedron of f . It can be treated as a refined version of the degree of the polynomial f in toric geometry.

3.2. Tropical hypersurfaces: The variety of a tropical polynomial

For a tropical polynomial f in n variables we define its variety upper V Subscript f Baseline subset-of double-struck upper R Superscript n as the set of points where the piecewise-linear function f is not smooth; cf. Reference10, Reference18 and Reference27. In other words, upper V Subscript f is the corner locus of f .

Proposition 3.3

upper V Subscript f is the set of points in double-struck upper R Superscript n where more than one monomial of f reaches its maximal value.

Proof.

If exactly one monomial of f left-parenthesis x right-parenthesis equals single-turned-comma-quotation-mark single-turned-comma-quotation-mark sigma-summation Underscript j element-of upper A Endscripts a Subscript j Baseline x Superscript j Baseline quotation-mark equals max Underscript j element-of upper A Endscripts left-parenthesis mathematical left-angle j comma x mathematical right-angle plus a Subscript j Baseline right-parenthesis is maximal at x element-of double-struck upper R Superscript n , then f locally coincides with this monomial and, therefore, is linear and smooth. Otherwise f has a corner at x .

Remark 3.4

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 double-struck upper R Subscript t r o p , but its rôle is played by negative normal infinity .

Consider the graph normal upper Gamma Subscript f Baseline subset-of double-struck upper R Superscript n times double-struck upper R of a tropical polynomial f colon double-struck upper R Superscript n Baseline right-arrow double-struck upper R . The graph normal upper Gamma Subscript f itself is not a tropical variety in double-struck upper R Superscript n plus 1 but it can be completed to the tropical variety

normal upper Gamma overbar Subscript f Baseline equals normal upper Gamma Subscript f Baseline union StartSet left-parenthesis x comma y right-parenthesis vertical-bar x element-of upper V Subscript f Baseline comma y less-than-or-equal-to f left-parenthesis x right-parenthesis EndSet semicolon

see Figure 7.

Proposition 3.5

normal upper Gamma overbar Subscript f coincides with the variety of the polynomial in left-parenthesis n plus 1 right-parenthesis variables single-turned-comma-quotation-mark single-turned-comma-quotation-mark y plus f left-parenthesis x right-parenthesis quotation-mark (where y element-of double-struck upper R , x element-of double-struck upper R Superscript n ).

Proof.

If left-parenthesis x comma y right-parenthesis element-of normal upper Gamma Subscript f , then we have y and one of the monomials of f both reaching the maximal values in single-turned-comma-quotation-mark single-turned-comma-quotation-mark y plus f left-parenthesis x right-parenthesis quotation-mark equals max left-brace y comma f left-parenthesis x right-parenthesis right-brace . If x element-of upper V Subscript f and y less-than f left-parenthesis x right-parenthesis , then two monomials of f left-parenthesis x right-parenthesis reach the maximal value at the expression single-turned-comma-quotation-mark single-turned-comma-quotation-mark y plus f left-parenthesis x right-parenthesis quotation-mark .

Note that we have upper V Subscript f Baseline equals normal upper Gamma overbar Subscript f Baseline intersection StartSet y equals t EndSet for t sufficiently close to negative normal infinity . This is the sense in which upper V Subscript f can be thought of as a zero locus.

One may argue that normal upper Gamma Subscript f itself is a subtropical variety (as in subanalytic vs. analytic sets) while normal upper Gamma overbar Subscript f is its tropical closure. Figure 7 sketches the graph y equals single-turned-comma-quotation-mark single-turned-comma-quotation-mark a x squared plus b x plus c quotation-mark and its tropical closure.

Definition 3.6

Varieties upper V Subscript f Baseline subset-of double-struck upper R Superscript n are called tropical hypersurfaces associated to f .

Remark 3.7

Different tropical polynomials may define the same varieties. To see this, let us first extend the notion of concavity to those double-struck upper R -valued functions which are only defined on a finite set upper A subset-of double-struck upper R Superscript n . We call a function phi colon upper A right-arrow double-struck upper R concave if for any (possibly non-distinct) b 0 comma ellipsis comma b Subscript n Baseline element-of upper A subset-of double-struck upper R Superscript n and any t 0 comma ellipsis comma t Subscript n Baseline greater-than-or-equal-to 0 with sigma-summation Underscript k equals 0 Overscript n Endscripts t Subscript k Baseline equals 1 and sigma-summation Underscript k equals 0 Overscript n Endscripts t Subscript k Baseline b Subscript k element-of upper A we have

phi left-parenthesis sigma-summation Underscript k equals 0 Overscript n Endscripts t Subscript k Baseline b Subscript k Baseline right-parenthesis greater-than-or-equal-to sigma-summation Underscript k equals 0 Overscript n Endscripts t Subscript k Baseline phi left-parenthesis b Subscript k Baseline right-parenthesis period

We have three types of ambiguities when f not-equals g but upper V Subscript f Baseline equals upper V Subscript g .

g equals single-turned-comma-quotation-mark single-turned-comma-quotation-mark x Subscript j Baseline f quotation-mark , where x Subscript j is a coordinate in double-struck upper R Subscript t r o p Superscript n . Note that in this case the Newton polyhedron of g is a translate of the Newton polyhedron of f .

g equals single-turned-comma-quotation-mark single-turned-comma-quotation-mark c f quotation-mark , where c element-of double-struck upper R Subscript t r o p is a constant.

The function normal upper Delta intersection double-struck upper Z Superscript n Baseline contains-as-member j right-arrow from bar a Subscript j is not concave, where f equals single-turned-comma-quotation-mark single-turned-comma-quotation-mark sigma-summation Underscript j element-of upper A Endscripts a Subscript j Baseline x Superscript j Baseline quotation-mark and we set a Subscript j Baseline right-arrow from bar negative normal infinity if j not-an-element-of upper A . Then the variety of f coincides with the variety of g where g is the smallest concave function such that g greater-than-or-equal-to f (in other words g is a concave hull of f ).

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 normal upper Delta is a closed convex polyhedral cone script upper M Subscript normal upper Delta Baseline subset-of double-struck upper R Superscript m , m equals number-sign left-parenthesis normal upper Delta intersection double-struck upper Z Superscript n Baseline right-parenthesis minus 1 . The cone script upper M Subscript normal upper Delta Baseline subset-of double-struck upper R Superscript m is well-defined up to the natural action of upper S upper L Subscript m Baseline left-parenthesis double-struck upper Z right-parenthesis .

Proof.

All concave functions normal upper Delta intersection double-struck upper Z Superscript n Baseline right-arrow double-struck upper R comma j right-arrow from bar a Subscript j Baseline form a closed convex polyhedral cone script upper M overTilde Subscript normal upper Delta Baseline subset-of double-struck upper R Superscript m plus 1 . But the function j right-arrow from bar a Subscript j Baseline plus c defines the same curve as the function j right-arrow from bar a Subscript j . To get rid of this ambiguity, we choose j prime element-of normal upper Delta intersection double-struck upper Z Superscript n and define script upper M Subscript normal upper Delta as the image of script upper M overTilde Subscript normal upper Delta under the linear projection double-struck upper R Superscript m plus 1 Baseline right-arrow double-struck upper R Superscript m Baseline comma a Subscript j Baseline right-arrow from bar a Subscript j Baseline minus a Subscript j prime Baseline .

3.3. Compactness of the space of tropical hypersurfaces

Clearly, the cone script upper M Subscript normal upper Delta is not compact. Nevertheless it gets compactified by the cones script upper M Subscript normal upper Delta prime for all non-empty lattice subpolyhedra normal upper Delta prime subset-of normal upper Delta (including polygons with the empty interior). Indeed, we have the following proposition.

Proposition 3.9

Let upper C Subscript k Baseline subset-of double-struck upper R Superscript n , k element-of double-struck upper N , be a sequence of tropical hypersurfaces whose Newton polyhedron is normal upper Delta . There exists a subsequence which converges to a tropical hypersurface upper C whose Newton polyhedron normal upper Delta Subscript upper C is contained in normal upper Delta (note that upper C is empty if normal upper Delta Subscript upper C is a point). The convergence is in the Hausdorff metric when restricted to any compact subset in double-struck upper R Superscript n . Furthermore, if the Newton polyhedron of upper C coincides with normal upper Delta , then the convergence is in the Hausdorff metric in the whole of double-struck upper R Superscript n .

Proof.

Each upper C Subscript k is defined by a tropical polynomial f Superscript upper C Super Subscript k Superscript Baseline left-parenthesis x right-parenthesis equals single-turned-comma-quotation-mark single-turned-comma-quotation-mark sigma-summation Underscript j Endscripts a Subscript j Superscript upper C Super Subscript k Superscript Baseline x Superscript j Baseline quotation-mark . We may assume that the coefficients a Subscript j Superscript upper C Super Subscript k are chosen so that they satisfy the concavity condition and so that max Underscript j Endscripts a Subscript j Superscript upper C Super Subscript k Baseline equals 0 . This takes care of the ambiguity in the choice of f Superscript upper C Super Subscript k (since the Newton polyhedron is already fixed).

Passing to a subsequence, we may assume that a Subscript j Superscript upper C Super Subscript k converge (to a finite number or negative normal infinity ) when k right-arrow normal infinity for all j element-of normal upper Delta intersection double-struck upper Z Superscript n . By our assumption one of these limits is 0. Define upper C to be the variety of single-turned-comma-quotation-mark single-turned-comma-quotation-mark sigma-summation a Subscript j Superscript normal infinity Baseline x Superscript j Baseline quotation-mark comma where we take only finite coefficients a Subscript j Superscript normal infinity Baseline equals limit Underscript k right-arrow normal infinity Endscripts a Subscript j Superscript upper C Super Subscript k Baseline greater-than negative normal infinity .

3.4. Lattice subdivision of normal upper Delta associated to a tropical hypersurface

A tropical polynomial f defines a lattice subdivision of its Newton polyhedron normal upper Delta in the following way (cf. Reference6). Define the (unbounded) extended polyhedral domain

normal upper Delta overTilde equals upper C o n v e x upper H u l l left-brace left-parenthesis j comma t right-parenthesis vertical-bar j element-of upper A comma t less-than-or-equal-to a Subscript j Baseline right-brace subset-of double-struck upper R Superscript n Baseline times double-struck upper R period

The projection double-struck upper R Superscript n Baseline times double-struck upper R right-arrow double-struck upper R Superscript n induces a homeomorphism from the union of all closed bounded faces of normal upper Delta overTilde to normal upper Delta .

Definition 3.10

The resulting lattice subdivision upper S u b d i v Subscript f of normal upper Delta is called the subdivision associated to f .

Proposition 3.11

The lattice subdivision upper S u b d i v Subscript f is dual to the tropical hypersurface upper V Subscript f . Namely, for every k -dimensional polyhedron normal upper Delta prime element-of upper S u b d i v Subscript f there is a convex closed (perhaps unbounded) polyhedron upper V Subscript f Superscript normal upper Delta prime Baseline subset-of upper V Subscript f Baseline subset-of double-struck upper R Superscript n . This correspondence has the following properties.

upper V Subscript f Superscript normal upper Delta prime is contained in an left-parenthesis n minus k right-parenthesis -dimensional affine-linear subspace upper L Superscript normal upper Delta prime of double-struck upper R Superscript n orthogonal to normal upper Delta prime .

The relative interior upper U Subscript f Superscript normal upper Delta prime of upper V Subscript f Superscript normal upper Delta prime in upper L Superscript normal upper Delta prime is not empty.

upper V Subscript f Baseline equals union upper U Subscript f Superscript normal upper Delta prime .

upper U Subscript f Superscript normal upper Delta prime Baseline intersection upper U Subscript f Superscript normal upper Delta double-prime Baseline equals normal empty-set if normal upper Delta prime not-equals normal upper Delta double-prime .

upper V Subscript f Superscript normal upper Delta prime is compact if and only if normal upper Delta prime subset-of normal upper Delta .

Proof.

For every normal upper Delta prime element-of upper S u b d i v Subscript f , consider the truncated polynomial

f Superscript normal upper Delta prime Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript j element-of normal upper Delta Superscript prime Baseline Endscripts a Subscript j Baseline x Superscript j

(recall that f left-parenthesis x right-parenthesis equals sigma-summation Underscript j element-of normal upper Delta Endscripts a Subscript j Baseline x Superscript j ). Define

StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel upper V Subscript f Superscript normal upper Delta Super Superscript prime Superscript Baseline equals upper V Subscript f Baseline intersection intersection Underscript normal upper Delta double-prime subset-of normal upper Delta Superscript prime Baseline Endscripts upper V Subscript f Sub Superscript normal upper Delta Sub Super Superscript double-prime Sub Superscript Subscript Baseline period EndLayout

Note that for any face normal upper Delta double-prime subset-of normal upper Delta prime we have the variety upper V Subscript f Sub Superscript normal upper Delta double-prime orthogonal to normal upper Delta double-prime (since moving in the direction orthogonal to normal upper Delta double-prime does not change the value of normal upper Delta double-prime -monomials) and therefore to normal upper Delta prime . To verify the last item of the proposition, we restate the defining equation Equation2 algebraically: upper V Subscript f Superscript normal upper Delta prime is the set of points where all monomials of f indexed by normal upper Delta prime have equal values while the value of any other monomial of f could only be smaller.

Example 3.12

Figure 8 shows the subdivisions dual to the curves from Figures 3 and 5.

It was observed in Reference10, Reference18 and Reference27 that upper V Subscript f is an left-parenthesis n minus 1 right-parenthesis -dimensional polyhedral complex dual to the subdivision upper S u b d i v Subscript f . The complex upper V Subscript f Baseline subset-of double-struck upper R Superscript n is a union of convex (not necessarily bounded) polyhedra or cells of upper V Subscript f . Each k -cell (even if it is unbounded) of upper V Subscript f is dual to a bounded left-parenthesis n minus k minus 1 right-parenthesis -face of normal upper Delta overTilde , i.e. to an left-parenthesis n minus k minus 1 right-parenthesis -cell of upper S u b d i v Subscript f . In particular, the slope of each cell of upper V Subscript f is rational.

In particular, an left-parenthesis n minus 1 right-parenthesis -dimensional cell is dual to an interval upper I subset-of double-struck upper R Superscript n both of whose ends are lattice points. We define the lattice length of upper I as number-sign left-parenthesis upper I intersection double-struck upper Z Superscript n Baseline right-parenthesis minus 1 . (Such a length is invariant with respect to upper S upper L Subscript n Baseline left-parenthesis double-struck upper Z right-parenthesis .) We can treat upper V Subscript f as a weighted piecewise-linear polyhedral complex in double-struck upper R Superscript n ; the weights are natural numbers associated to the left-parenthesis n minus 1 right-parenthesis -cells. They are the lattice lengths of the dual intervals.

Definition 3.13

The combinatorial type of a tropical hypersurface upper V Subscript f Baseline subset-of double-struck upper R Superscript n is the equivalence class of all upper V Subscript g such that upper S u b d i v Subscript g Baseline equals upper S u b d i v Subscript f .

Let script upper S be such a combinatorial type.

Lemma 3.14

All tropical hypersurfaces of the same combinatorial type script upper S form a convex polyhedral domain script upper M Subscript script upper S Baseline subset-of script upper M Subscript normal upper Delta that is open in its affine-linear span.

Proof.

The condition upper S u b d i v Subscript f Baseline equals script upper S can be written in the following way in terms of the coefficients of f left-parenthesis x right-parenthesis equals single-turned-comma-quotation-mark single-turned-comma-quotation-mark sigma-summation Underscript j Endscripts a Subscript j Baseline x Superscript j Baseline quotation-mark . For every normal upper Delta prime element-of script upper S the function j right-arrow from bar minus a Subscript j for j element-of normal upper Delta prime should coincide with some linear function alpha colon double-struck upper Z Superscript n Baseline right-arrow double-struck upper R such that minus a Subscript j Baseline greater-than alpha left-parenthesis j right-parenthesis for every j element-of normal upper Delta minus normal upper Delta prime .

It turns out that the weighted piecewise-linear complex upper V Subscript f satisfies the balancing property at each left-parenthesis n minus 2 right-parenthesis -cell; see Definition 3 of Reference18. Namely, let upper F 1 comma ellipsis comma upper F Subscript k Baseline be the left-parenthesis n minus 1 right-parenthesis -cells adjacent to a left-parenthesis n minus 2 right-parenthesis -cell upper G of upper V Subscript f . Each upper F Subscript j has a rational slope and is assigned a weight w Subscript j . Choose a direction of rotation around upper G and let c Subscript upper F Sub Subscript j Subscript Baseline colon double-struck upper Z Superscript n Baseline right-arrow double-struck upper Z be linear maps whose kernels are planes parallel to upper F Subscript j and such that they are primitive (non-divisible) and agree with the chosen direction of rotation. The balancing condition states that

StartLayout 1st Row with Label left-parenthesis 3 right-parenthesis EndLabel sigma-summation Underscript j equals 1 Overscript k Endscripts w Subscript j Baseline c Subscript upper F Sub Subscript j Baseline equals 0 period EndLayout

As was shown in Reference18, this balancing condition at every left-parenthesis n minus 2 right-parenthesis -cell of a rational piecewise-linear left-parenthesis n minus 1 right-parenthesis -dimensional polyhedral complex in double-struck upper R Superscript n suffices for such a polyhedral complex to be the variety of some tropical polynomial.

Theorem 3.15 (Reference17).

A weighted left-parenthesis n minus 1 right-parenthesis -dimensional polyhedral complex normal upper Pi subset-of double-struck upper R Superscript n is the variety of a tropical polynomial if and only if each k -cell of normal upper Pi is a convex polyhedron sitting in a k -dimensional affine subspace of double-struck upper R Superscript n with a rational slope and normal upper Pi satisfies the balancing condition Equation3 at each left-parenthesis n minus 2 right-parenthesis -cell.

This theorem implies that the definitions of tropical curves and tropical hypersurfaces agree if n equals 2 .

Corollary 3.16

Any tropical curve upper C subset-of double-struck upper R squared is a tropical hypersurface for some polynomial f . Conversely, any tropical hypersurface in double-struck upper R squared can be parameterized by a tropical curve.

Remark 3.17

Furthermore, the degree of upper C is determined by the Newton polygon normal upper Delta of f according to the following recipe. For each side normal upper Delta prime subset-of partial-differential normal upper Delta we take the primitive integer outward normal vector and multiply it by the lattice length of normal upper Delta prime to get the degree of upper C .

3.5. Tropical varieties and non-Archimedean amoebas

Polyhedral complexes resulting from tropical varieties appeared in Reference10 in the following context. Let upper K be a complete algebraically closed non-Archimedean field. This means that upper K is an algebraically closed field and there is a valuation v a l colon upper K Superscript asterisk Baseline right-arrow double-struck upper R defined on upper K Superscript asterisk Baseline equals upper K minus StartSet 0 EndSet such that e Superscript v a l defines a complete metric on upper K . Recall that a valuation v a l is a map such that v a l left-parenthesis x y right-parenthesis equals v a l left-parenthesis x right-parenthesis plus v a l left-parenthesis y right-parenthesis and v a l left-parenthesis x plus y right-parenthesis less-than-or-equal-to max left-brace v a l left-parenthesis x right-parenthesis comma v a l left-parenthesis y right-parenthesis right-brace .

Our principal example of such a upper K is a field of Puiseux series with real powers. To construct upper K , we take the algebraic closure ModifyingAbove double-struck upper C left-parenthesis left-parenthesis t right-parenthesis right-parenthesis With bar of the field of Laurent series double-struck upper C left-parenthesis left-parenthesis t right-parenthesis right-parenthesis . The elements of ModifyingAbove double-struck upper C left-parenthesis left-parenthesis t right-parenthesis right-parenthesis With bar are formal power series in t a left-parenthesis t right-parenthesis equals sigma-summation Underscript k element-of upper A Endscripts a Subscript k Baseline t Superscript k , where a Subscript k Baseline element-of double-struck upper C and upper A subset-of double-struck upper Q is a subset bounded from below and contained in an arithmetic progression. We set v a l left-parenthesis a left-parenthesis t right-parenthesis right-parenthesis equals minus min upper A . We define upper K to be the completion of ModifyingAbove double-struck upper C left-parenthesis left-parenthesis t right-parenthesis right-parenthesis With bar as the metric space with respect to the norm e Superscript v a l .

Let upper V subset-of left-parenthesis upper K Superscript asterisk Baseline right-parenthesis Superscript n be an algebraic variety over upper K . The image of upper V under the map upper V a l colon left-parenthesis upper K Superscript asterisk Baseline right-parenthesis Superscript n Baseline right-arrow double-struck upper R Superscript n , left-parenthesis z 1 comma ellipsis comma z Subscript n Baseline right-parenthesis right-arrow from bar left-parenthesis v a l left-parenthesis z 1 right-parenthesis comma ellipsis comma v a l left-parenthesis z Subscript n Baseline right-parenthesis right-parenthesis is called the amoeba of upper V (cf. Reference6). Kapranov Reference10 has shown that the amoeba of a non-Archimedean hypersurface is the variety of a tropical polynomial. Namely, if sigma-summation Underscript j element-of upper A Endscripts a Subscript j Baseline z Superscript j Baseline equals 0 , 0 not-equals a Subscript j Baseline element-of upper K is a hypersurface in left-parenthesis upper K Superscript asterisk Baseline right-parenthesis Superscript n , then its amoeba is the variety of the tropical polynomial sigma-summation Underscript j element-of upper A Endscripts v a l left-parenthesis a Subscript j Baseline right-parenthesis x Superscript j .

More generally, if upper F is a field with a real-valued norm, then the amoeba of an algebraic variety upper V subset-of left-parenthesis upper F Superscript asterisk Baseline right-parenthesis is upper L o g left-parenthesis upper V right-parenthesis subset-of double-struck upper R Superscript n , where upper L o g left-parenthesis z 1 comma ellipsis comma z Subscript n Baseline right-parenthesis equals left-parenthesis log StartAbsoluteValue EndAbsoluteValue z 1 StartAbsoluteValue EndAbsoluteValue comma ellipsis comma log StartAbsoluteValue EndAbsoluteValue z Subscript n Baseline StartAbsoluteValue EndAbsoluteValue right-parenthesis . Note that upper V a l is such a map with respect to the non-Archimedean norm e Superscript v a l in upper K .

Another particularly interesting case is if upper F equals double-struck upper C with the standard norm StartAbsoluteValue EndAbsoluteValue z StartAbsoluteValue EndAbsoluteValue equals StartRoot z z overbar EndRoot (see Reference6, Reference16, Reference19, etc.). The non-Archimedean hypersurface amoebas appear as limits in the Hausdorff metric of double-struck upper R Superscript n 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 double-struck upper R Superscript n . Namely, one can define the tropical varieties in double-struck upper R Superscript n to be the images upper V a l left-parenthesis upper V right-parenthesis of arbitrary algebraic varieties upper V element-of left-parenthesis upper K Superscript asterisk Baseline right-parenthesis Superscript n . This definition allows one to avoid dealing with the intersections of tropical hypersurfaces in non-general position. We refer to Reference23 for relevant discussions.

4. Enumeration of tropical curves in double-struck upper R squared

4.1. Simple curves and their lattice subdivisions

Corollary 3.16 states that any tropical 1-cycle in double-struck upper R squared is a tropical hypersurface, i.e. it is the variety of a tropical polynomial f colon double-struck upper R squared right-arrow double-struck upper R . By Remark 3.7 the Newton polygon normal upper Delta of such f is well defined up to a translation.

Definition 4.1

We call normal upper Delta the degree of a tropical curve in double-struck upper R squared .

By Remark 3.17 this degree supplies the same amount of information as the toric degree from Definition 2.7. We extract two numerical characteristics from the polygon normal upper Delta subset-of double-struck upper R squared :

StartLayout 1st Row with Label left-parenthesis 4 right-parenthesis EndLabel s equals number-sign left-parenthesis partial-differential normal upper Delta intersection double-struck upper Z squared right-parenthesis comma l equals number-sign left-parenthesis upper I n t normal upper Delta intersection double-struck upper Z squared right-parenthesis period EndLayout

The number s 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 x less-than-or-equal-to s ). The number l is the genus of a smooth tropical curve of degree normal upper Delta . To see this, let us note that every lattice point of normal upper Delta is a vertex of the associated subdivision for a smooth curve upper C . Therefore, the homotopy type of upper C coincides with upper I n t normal upper Delta minus double-struck upper Z squared . Note also that smooth curves are dense in script upper M Subscript normal upper Delta .

There is a larger class of tropical curves in double-struck upper R squared whose behavior is as simple as that of smooth curves.

Definition 4.2

A parameterized tropical curve h colon normal upper Gamma right-arrow double-struck upper R squared is called simple if it satisfies all of the following conditions.

The graph normal upper Gamma is 3-valent.

The map h is an immersion.

For any y element-of double-struck upper R Superscript n the inverse image h Superscript negative 1 Baseline left-parenthesis y right-parenthesis consists of at most two points.

If a comma b element-of normal upper Gamma , a not-equals b , are such that h left-parenthesis a right-parenthesis equals h left-parenthesis b right-parenthesis , then neither a nor b can be a vertex of normal upper Gamma .

A tropical 1-cycle upper C subset-of double-struck upper R squared is called simple if it admits a simple parameterization.

Proposition 4.3

A simple tropical 1-cycle upper C subset-of double-struck upper R squared 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.

Proof.

By Definition 4.2, upper C 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 double-struck upper R squared . 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.

Remark 4.4

More generally, every tropical 1-cycle upper C subset-of double-struck upper R squared admits a parameterization by an immersion of genus not greater than g left-parenthesis upper C right-parenthesis . Start from an arbitrary parameterization h colon normal upper Gamma right-arrow double-struck upper R squared . To eliminate an edge upper E subset-of normal upper Gamma such that h left-parenthesis upper E right-parenthesis is contracted to a point, we take the quotient of normal upper Gamma by upper E for a new domain of parameterization. This procedure does not change the genus of normal upper Gamma .

Therefore, we may assume that h colon normal upper Gamma right-arrow double-struck upper R squared does not have contracting edges. This is an immersion away from such vertices of normal upper Gamma for which there exist two distinct adjacent edges upper E 1 comma upper E 2 with h left-parenthesis upper E 1 right-parenthesis intersection h left-parenthesis upper E 2 right-parenthesis not-equals normal empty-set . Changing the graph normal upper Gamma by identifying the points on upper E 1 and upper E 2 with the same image can only decreases the genus of normal upper Gamma (if upper E 1 and upper E 2 were distinct edges connecting the same pair of vertices). Inductively we get an immersion.

Lemma 4.5

A tropical curve upper C subset-of double-struck upper R squared is simple (see Definition 4.2) if and only if it is the variety of a tropical polynomial such that upper S u b d i v Subscript f is a subdivision into triangles and parallelograms.

Proof.

The lemma follows from Proposition 3.11. The 3-valent vertices of upper C are dual to the triangles of upper S u b d i v Subscript f 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 upper V Subscript f in terms of the number r of triangles in upper S u b d i v Subscript f .

Lemma 4.6 (Cf. Reference8).

If a curve upper V Subscript f Baseline subset-of double-struck upper R squared is simple, then g left-parenthesis upper V Subscript f Baseline right-parenthesis equals StartFraction r minus x Over 2 EndFraction plus 1 .

Proof.

Let normal upper Delta 0 be the number of vertices of upper S u b d i v Subscript f while normal upper Delta 1 and normal upper Delta 2 are the numbers of its edges and (2-dimensional) polygons. Out of the normal upper Delta 2 2-dimensional polygons r are triangles and left-parenthesis normal upper Delta 2 minus r right-parenthesis are parallelograms.

We have

chi left-parenthesis upper V Subscript f Baseline right-parenthesis equals minus 2 normal upper Delta 2 plus normal upper Delta 1 period

Note that 3 r plus 4 left-parenthesis normal upper Delta 2 minus r right-parenthesis equals 2 normal upper Delta 1 minus x . Thus, normal upper Delta 1 equals three-halves r plus 2 left-parenthesis normal upper Delta 2 minus r right-parenthesis plus StartFraction x Over 2 EndFraction and

g left-parenthesis upper V Subscript f Baseline right-parenthesis equals 1 minus chi left-parenthesis upper V Subscript f Baseline right-parenthesis equals 1 plus StartFraction r minus x Over 2 EndFraction period

4.2. Tropical general positions of points in double-struck upper R squared

Definition 4.7

Points p 1 comma ellipsis comma p Subscript k Baseline element-of double-struck upper R squared are said to be in general position tropically if for any tropical curve h colon normal upper Gamma right-arrow double-struck upper R squared of genus g and with x ends such that k greater-than-or-equal-to g plus x minus 1 and p 1 comma ellipsis comma p Subscript k Baseline element-of h left-parenthesis normal upper Gamma right-parenthesis we have the following conditions.

The curve h colon normal upper Gamma right-arrow double-struck upper R squared is simple (see Definition 4.2).

Inverse images h Superscript negative 1 Baseline left-parenthesis p 1 right-parenthesis comma ellipsis comma h Superscript negative 1 Baseline left-parenthesis p Subscript k Baseline right-parenthesis are disjoint from the vertices of upper C .

k equals g plus x minus 1 .

Example 4.8

Two distinct points p 1 comma p 2 element-of double-struck upper R squared are in general position tropically if and only if the slope of the line in double-struck upper R squared passing through p 1 and p 2 is irrational.

Remark 4.9

Note that we can always find a curve with g plus x minus 1 equals k passing through p 1 comma ellipsis comma p Subscript k Baseline . For such a curve we can take a reducible curve consisting of k affine (i.e. classical) lines in double-struck upper R squared with rational slope each passing through its own point p Subscript j . This curve has 2 k ends while its genus is 1 minus k .

Proposition 4.10

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

Proof.

Suppose the points p 1 comma ellipsis comma p Subscript j Baseline are not in general position. Then there is a curve upper C with x ends of genus j plus 2 minus x passing through p 1 comma ellipsis comma p Subscript j Baseline or of genus j plus 1 minus x but with a non-generic behavior with respect to p 1 comma ellipsis comma p Subscript j Baseline . By Remark 4.9 there is a curve upper C prime passing through p Subscript j plus 1 Baseline comma ellipsis comma p Subscript k Baseline of genus k minus j plus 1 minus x prime . The curve upper C union upper C prime supplies a contradiction.

Proposition 4.11

For each normal upper Delta subset-of double-struck upper R squared the set of configurations script upper P equals StartSet p 1 comma ellipsis comma p Subscript k Baseline EndSet subset-of double-struck upper R squared such that there exists a curve upper C of degree normal upper Delta such that the conditions of Definition 4.7 are violated by upper C is closed and nowhere dense.

Proof.

By Remark 4.4 it suffices to consider only immersed tropical curves h colon normal upper Gamma right-arrow double-struck upper R squared . We have only finitely many combinatorial types of tropical curves of genus g with the Newton polygon normal upper Delta since there are only finitely many lattice subdivisions of normal upper Delta . By Proposition 2.23 for each such combinatorial type we have an left-parenthesis x plus g minus 1 right-parenthesis -dimensional family of simple curves or a smaller-dimensional family of non-simple curves. For a fixed upper C each of the k points p Subscript j can vary in a 1-dimensional family on upper C or in a 0-dimensional family if p Subscript j is a vertex of upper C . Thus the dimension of the space of “bad” configurations script upper P element-of upper S y m Superscript k Baseline left-parenthesis double-struck upper R squared right-parenthesis is at most 2 k minus 1 .

Corollary 4.12

The configurations script upper P equals StartSet p 1 comma ellipsis comma p Subscript k Baseline EndSet in general position tropically form a dense set which can be obtained as an intersection of countably many open dense sets in upper S y m Superscript k Baseline left-parenthesis double-struck upper R squared right-parenthesis .

4.3. Tropical enumerative problem in double-struck upper R squared

To set up an enumerative problem, we fix the degree, i.e. a polygon normal upper Delta subset-of double-struck upper R squared with s equals number-sign left-parenthesis partial-differential normal upper Delta intersection double-struck upper Z squared right-parenthesis , and the genus, i.e. an integer number g . Consider a configuration script upper P equals StartSet p 1 comma ellipsis comma p Subscript s plus g minus 1 Baseline EndSet subset-of double-struck upper R squared of s plus g minus 1 points in tropical general position. Our goal is to count tropical curves h colon normal upper Gamma right-arrow double-struck upper R squared of genus g such that h left-parenthesis normal upper Gamma right-parenthesis superset-of script upper P and has degree normal upper Delta .

Proposition 4.13

There exist only finitely many such curves h colon normal upper Gamma right-arrow double-struck upper R squared . Furthermore, each end of upper C equals h left-parenthesis normal upper Gamma right-parenthesis is of weight 1 in this case, so normal upper Gamma has s ends.

Finiteness follows from Lemma 4.22 proved in the next subsection. If upper C has ends whose weight is greater than 1, then the number of ends is smaller than s and the existence of upper C contradicts the general position of script upper P . Recall that since script upper P is in general position, any such upper C is also simple and the vertices of upper C are disjoint from script upper P .

Example 4.14

Let g equals 0 and let normal upper Delta be the quadrilateral whose vertices are left-parenthesis 0 comma 0 right-parenthesis , left-parenthesis 1 comma 0 right-parenthesis , left-parenthesis 0 comma 1 right-parenthesis and left-parenthesis 2 comma 2 right-parenthesis (so that the number s of the lattice points on the perimeter partial-differential normal upper Delta is 4). For a configuration script upper P of three points in double-struck upper R squared pictured in Figure 9 we have three tropical curves passing. In Figure 10 the corresponding number is two.

Definition 4.15

The multiplicity m u l t left-parenthesis upper C right-parenthesis of a tropical curve upper C subset-of double-struck upper R squared of degree normal upper Delta and genus g passing via script upper P equals the product of the multiplicities of all the 3-valent vertices of upper C (see Definition 2.16).

Definition 4.16

We define the number upper N Subscript t r o p Superscript i r r Baseline left-parenthesis g comma normal upper Delta right-parenthesis to be the number of irreducible tropical curves of genus g and degree normal upper Delta passing via script upper P where each such curve is counted with the multiplicity m u l t from Definition 4.15. Similarly we define the number upper N Subscript t r o p Baseline left-parenthesis g comma normal upper Delta right-parenthesis to be the number of all tropical curves of genus g and degree normal upper Delta passing via script upper P . Again each curve is counted with the multiplicity m u l t from Definition 4.15.

The following proposition is a corollary of Theorem 1 formulated below in Section 7.

Proposition 4.17

The numbers upper N Subscript t r o p Baseline left-parenthesis g comma normal upper Delta right-parenthesis and upper N Subscript t r o p Superscript i r r Baseline left-parenthesis g comma normal upper Delta right-parenthesis are finite and do not depend on the choice of script upper P .

E.g. the 3-point configurations from Figures 9 and 10 have the same number upper N Subscript t r o p Superscript i r r Baseline left-parenthesis g comma normal upper Delta right-parenthesis .

4.4. Forests in the polygon normal upper Delta

Recall that every vertex of a tropical curve upper C subset-of double-struck upper R squared corresponds to a polygon in the dual lattice subdivision of the Newton polygon normal upper Delta while every edge of upper C corresponds to an edge of the dual subdivision upper S u b d i v Subscript upper C