Broccoli curves and the tropical invariance of Welschinger numbers
In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points — for arbitrary choices of the directions of the ends of the curves, possibly with higher weights, and also if some of the ends are fixed. In the toric Del Pezzo case we show that these broccoli invariants are equal to the Welschinger invariants (with real and complex conjugate point conditions), thus providing a proof of the independence of Welschinger invariants of the point conditions within tropical geometry. The general case gives rise to a tropical Caporaso-Harris formula for broccoli curves which suffices to compute all Welschinger invariants of the plane.
Key words and phrases:Tropical geometry, enumerative geometry, Welschinger numbers
1.1. Background on tropical Welschinger numbers
Welschinger invariants of real toric unnodal Del Pezzo surfaces count real rational curves, weighted with depending on the nodes of the curve, belonging to an ample linear system and passing through a generic conjugation invariant set of points. It was shown in [Wel03] and [Wel05] that these numbers depend only on the number of real points in , i.e. are invariant under movements of the points in . They can be thought of as real analogues of the numbers of complex rational curves belonging to a fixed linear system and satisfying point conditions, which in the case of are the genus- Gromov-Witten invariants.
By Mikhalkin’s Correspondence Theorem [Mik05], Gromov-Witten invariants of the plane (resp. the complex enumerative numbers of other toric surfaces) can be determined via tropical geometry, by counting tropical curves of a fixed degree and satisfying point conditions. Each tropical curve has to be counted with a “complex multiplicity” which reflects how many complex curves map to it under tropicalization.
Welschinger invariants can be computed via tropical geometry in a similar way: one can define a certain count of tropical curves and prove a Correspondence Theorem stating that this tropical count equals the Welschinger invariant. For the case when consists of only real points, such a Correspondence Theorem is proved in [Mik05], the general case is proved in [Shu06].
If consists of only real points, the tropical curves we have to count to get Welschinger invariants are exactly the same as the ones we need to count to determine complex enumerative numbers — we just have to count them with a different, “real” multiplicity. The lattice path algorithm of [Mik05] enumerates the tropical curves we have to count. If also contains pairs of complex conjugate points, we have to count tropical curves satisfying some more special conditions. The lattice path algorithm is generalized in [Shu06] to an algorithm that computes the corresponding Welschinger invariants.
It follows from the Correspondence Theorem and the fact that Welschinger invariants are independent of the point conditions that the corresponding tropical count is also invariant, i.e. does not depend on the position of the points that we require the tropical curves to pass through.
Still, it is interesting to find an argument within tropical geometry that proves the invariance of the tropical numbers. For the case when consists of only real points, such a statement follows easily since the corresponding tropical count can be shown to be locally invariant, i.e. invariant around a codimension-1 cone of the corresponding moduli space of curves. In addition, such a codimension-1 cone is specified by a -valent vertex of a tropical curve, and it is sufficient to consider the curves locally around this -valent vertex. This tropical invariance statement was proved in [IKS09], and generalized to a relative situation where we count tropical curves with ends of higher weights with their real multiplicity. In [GM07a], tropical curves with ends of higher weights counted with their complex multiplicity are shown to determine relative Gromov-Witten invariants of the plane, i.e. numbers of complex plane curves satisfying point conditions and tangency conditions to a given line . Thus one could imagine that the tropical relative real count corresponds to numbers of real curves satisfying point and tangency conditions. This is true only for real curves near the tropical limit however [Mik05]. The tropical proof of the invariance in this situation thus led to the construction of new tropical invariant numbers whose real counterparts are yet to be better understood.
Also, because of the invariance of the tropical relative real count one can establish a Caporaso-Harris formula for Welschinger invariants for which consists of only real points. Originally, Caporaso and Harris developed their algorithm to determine the numbers of complex curves satisfying point conditions [CH98]. They defined the above mentioned relative Gromov-Witten invariants and specialized one point after the other to lie on the line . Since a curve of degree intersects in points, after some steps the curves become reducible and the line splits off as a component. One then collects the contributions from all the components and thus produces recursive relations among the relative Gromov-Witten invariants that finally suffice to compute the numbers of complex curves satisfying point conditions. A tropical counterpart of this algorithm has been established in [GM07a]. There, one moves one point after the other to the far left part of the plane (but still in general position). The tropical curves then do not become reducible, but in a sense decompose into two parts, leading to recursive relations. The left part, passing through the moved point, is called a floor [BM08]. In [IKS09] the authors use the same idea to specialize points and consider tropical curves decomposing into a floor and another part, only now they have to deal with the real multiplicity for these tropical curves. The formula one thus obtains computes tropical Welschinger numbers which are equal to their classical counterparts by the Correspondence Theorem. Since this formula is recursive it is much more efficient for the computation of Welschinger invariants than the lattice path algorithm mentioned above. There is also work in progress to compute Welschinger invariants without tropical methods [Sol].
Now let us discuss the situation when does not only contain real points, but also pairs of complex conjugate points. As already mentioned, also here a Correspondence Theorem exists to relate these Welschinger invariants to a certain count of tropical curves, and one can count the tropical curves with a generalized lattice path algorithm [Shu06]. In addition, it follows of course again from the Correspondence Theorem together with the Welschinger Theorem that the tropical count is invariant. However, the tropical count is no longer locally invariant in the moduli space, and thus there was no known tropical proof for the (global) invariance of the tropical count. Even worse, if we try to generalize the tropical count to relative numbers, i.e. to curves with ends of higher weight, then these numbers are no longer invariant. However, one can still pick a special configuration of points, namely the result after applying the Caporaso-Harris algorithm as many times as possible. Then each point is followed by a point which is far more left, and the curves totally decompose into floors. They can then be counted by means of floor diagrams. Although the tropical relative count is not invariant, the floor diagram count leads to a Caporaso-Harris type formula which is sufficient to compute all Welschinger invariants of the plane [ABLdM11].
1.2. The content of this paper
The aim of this paper is to give a tropical proof of the invariance of tropical Welschinger numbers for real and complex conjugate points. As an additional result this will allow us to construct corresponding tropical invariants in the relative setting (or more generally for any choice of directions for the ends of the curve). Using this result, we can then establish a Caporaso-Harris formula for rational curves in a much simpler way than in [ABLdM11].
The key idea to achieve this is to modify (and in fact also simplify) the class of tropical curves that we count in order to obtain the invariants. This modification is small enough so that the (weighted) number of these curves through given points remains the same in the toric Del Pezzo case, but big enough so that their count becomes locally invariant in the moduli space.
Let us explain this modification in more detail. For this it is important to distinguish between odd and even edges of a tropical curve, i.e. edges whose weight is odd resp. even. In our pictures we will always draw odd edges as thin lines and even edges as thick lines. Moreover, we will draw real points as thin dots and complex points (i.e. those corresponding to a pair of complex conjugate points in the algebraic case) as thick dots. All our curves will be of genus zero.
The tropical curves that are usually counted to obtain the Welschinger invariants — we will call them Welschinger curves — then have the property that each connected component of even edges is connected to the rest of the curve at exactly one point (we can think of such a component as an end tree). Moreover, real points cannot lie on end trees, and each complex point is either on an end tree or at a 4-valent vertex [Shu06]. Below on the left we have drawn a typical (schematic) picture of such a Welschinger curve, with the end trees marked blue. Note that the marking lying on a point is itself an edge, so that the 4-valent complex markings away from the end trees look like 3-valent vertices in the picture.
We now change this condition slightly to obtain a different class of curves that we call broccoli curves: each connected component of even edges can now be connected to the rest of the curve at several points, of which exactly one is a 3-valent vertex without marking as before (the “broccoli stem”), and the remaining ones are complex points (the “broccoli florets”). The even part of the curve (the “broccoli part”) may not contain any marked points in its interior, whereas away from this part we can have real points at 3-valent and complex points at 4-valent vertices as before. The picture above on the right shows a typical schematic example of a broccoli curve, with the broccoli part drawn in green. Note that, in contrast to Welschinger curves, complex points are always at 4-valent vertices in broccoli curves.
Broccoli curves have the advantage that their count (with suitably defined multiplicities) is locally invariant in the moduli space, similarly to the situation mentioned above when we count complex curves or Welschinger curves through only real points. Hence counting these curves we obtain well-defined broccoli invariants — even for curves with directions of the ends for which the corresponding Welschinger count would not be invariant of the position of the points.
In addition, we show that in the toric Del Pezzo case broccoli invariants equal Welschinger numbers, thereby giving a new and entirely tropical proof of the invariance of Welschinger numbers. We prove this by constructing bridges between broccoli curves and Welschinger curves which show that their numbers must be equal. To illustrate this concept of bridges in an easy example we have drawn in the picture below a Welschinger curve (which is not a broccoli curve) and a broccoli curve (which is not a Welschinger curve) of degree through the same two real and three complex points. They can be connected by the bridge drawn below those curves: starting from the Welschinger curve we first split the vertical end of weight into two edges of weight until the rightmost complex point becomes 4-valent (in the picture at the bottom), and then split the other end of weight in a similar way until we arrive at the broccoli curve.
It should be noted that this example is a particularly simple bridge as it connects a Welschinger curve to a unique corresponding broccoli curve. In general, traversing bridges will involve creating and resolving higher-valent vertices of curves along 1-dimensional families — and as there are usually several possibilities for such resolutions this means that bridges may ramify on their way from the Welschinger to the broccoli side. Bridge curves will be assigned a multiplicity (in a similar way as for Welschinger and broccoli curves), and at each point of the bridge it is just the weighted number of incoming Welschinger and outgoing broccoli curves that is the same — not necessarily the absolute number of them. In particular, bridges do in general not provide a bijection between Welschinger and broccoli curves, in fact not even a well-defined map in either direction.
Another technical thing to note is that we have twice split an even end of weight 2 into two odd ends of weight on the bridge above. This might look like a discontinuous change in the underlying graph of the tropical curve. In order to avoid this inconvenience we will usually parametrize even ends of Welschinger curves as two ends of half the weight (which we call double ends). This way no further end splitting takes place on bridges.
It would certainly be very interesting to see if one could prove a Correspondence Theorem for broccoli curves that relates these tropical curves directly to certain real algebraic ones. So far there is no such statement known; in particular there is no algebraic counterpart to broccoli invariants for directions of the ends of the curves when the corresponding Welschinger number is not an invariant.
This paper is organized as follows. In section 2 we review basic notions of tropical curves and their moduli spaces. In particular, we introduce the notion of oriented curves (i.e. tropical curves with the edges oriented in a certain way), a tool which simplifies proofs in the rest of the paper. The next three sections are dedicated to the different kinds of tropical curves mentioned above: section 3 deals with broccoli curves; the main result here is theorem 3.6 which states that the counts of broccoli curves do not depend on the position of the points. In a very analogous way, section 4 considers Welschinger curves and shows that their counts yield the Welschinger invariants. We then introduce bridge curves in section 5 and use them in corollary 5.16 to prove that Welschinger and broccoli invariants agree in the toric Del Pezzo case, and thus that the Welschinger invariants then do not depend on the choice of point conditions (corollary 5.17). Finally, the existence of well-defined broccoli invariants also in the relative case enables us to prove a Caporaso-Harris formula for Welschinger invariants of the plane in section 6.
We would like to thank Eugenii Shustin and Inge Sandstad Skrondal for helpful discussions. Part of this work was accomplished at the Mathematical Sciences Research Institute (MSRI) in Berkeley, CA, USA, during the one-semester program on tropical geometry in fall 2009, and part at the Mittag-Leffler Institute in Stockholm, during the semester program in spring 2011 on “Algebraic Geometry with a View towards Applications”. The authors would like to thank both institutes for hospitality and support. In particular, Andreas Gathmann was supported by the Simons Professorship of the MSRI.
2. Oriented marked curves
Let us start by introducing the tropical curves that we will deal with in this paper. As all our curves will be tropical we usually drop this attribute in the notation. All curves will be in (parametrized and labeled in the sense of [GKM09] section 4), connected, and of genus . Let us quickly recall the definition of these tropical curves, already making the distinction between real and complex markings resp. odd and even edges that we will later need to consider real enumerative invariants.
Definition 2.1 (Marked curves).
Let . An -marked (plane tropical) curve is a tuple for some such that:
is a connected rational metric graph, with unbounded edges allowed, and such that each vertex has valence at least 3. The unbounded edges of will be called the ends of .
is a continuous map that is integer affine linear on each edge of , i.e. on each edge it is of the form for some and . If we parametrize starting at the vertex the vector in this equation will be denoted and called the direction (vector) of starting at . For an end we will also write instead of , where is the unique vertex of . We say that an edge is contracted if its direction is .
At each vertex of the balancing condition
is a labeling of the contracted ends, a labeling of the non-contracted ends of . We call the markings or marked ends; more specifically the ends are called the real markings, the ends the complex markings of . The other ends are called the unmarked ends; the collection of their directions will be called the degree of . We denote the number of vectors in by .
The set of all -marked curves of degree will be denoted .
Definition 2.2 (Even and odd edges, weights).
Let be a marked curve.
A vector in will be called even if both its coordinates are even, and odd otherwise. We say that an edge of is even resp. odd if its direction vector is even resp. odd.
If we write the direction vector of an edge of as a non-negative multiple of a primitive integral vector we call this number the weight of . Note that is even resp. odd if and only if its weight is even resp. odd.
When drawing a marked curve we will usually only show the image , together with the image points of the markings. These image points will be drawn as small dots for real markings and as big dots for complex markings. The other edges will always be displayed as thin lines for odd edges and as thick lines for even edges. Unmarked contracted edges would not be visible in these pictures, but (although allowed) they will not play a special role in this paper.
Using convention 2.3, the picture on the right shows a -marked plane curve of degree . It has two 3-valent vertices and one 4-valent vertex. The thick edge has direction starting at the complex marking. For clarity we have labeled all the ends in the picture, but in the future we will usually omit this as the actual labeling will not be relevant for most of our arguments.
Note that our set is precisely the moduli space of -marked plane labeled tropical curves of [GKM09] definition 4.1. As such it is a polyhedral complex, and in fact even a tropical variety (see [GKM09] proposition 4.7). In this paper we will not need its structure as a tropical variety however, but only consider as an abstract polyhedral complex with polyhedral structure induced by the combinatorial types of the curves. Let us quickly establish this notation.
Definition 2.6 (Combinatorial types).
Let be a marked curve. The combinatorial type of is the data of the non-metric graph , together with the labeling of the ends and the directions of all edges. For such a combinatorial type we denote by the subspace of of all marked curves of type .
Remark 2.7 ( as a polyhedral complex).
In the same way as in [GM08] example 2.13 the moduli spaces are abstract polyhedral complexes in the sense of [GM08] definition 2.12, i.e. they can be obtained by glueing finitely many real polyhedra along their faces. The open cells of these complexes are exactly the subspaces , where runs over all combinatorial types of curves in . The curves in such a cell (i.e. for a fixed combinatorial type) are parametrized by the position in of a chosen root vertex and the lengths of all bounded edges (which need to be positive). Hence can be thought of as an open polyhedron whose dimension is equal to 2 plus the number of bounded edges in the combinatorial type . We will call this dimension the dimension of the type .
Let us now consider enumerative questions for our curves. In addition to the usual incidence conditions we want to be able to require that some of the unmarked ends are fixed, i.e. map to a given line in . To count such curves we will now introduce the corresponding evaluation maps. Moreover, to be able to compensate for the overcounting due to the labeling of the non-fixed unmarked ends we will define the group of permutations of these ends that keep the degree fixed.
Definition 2.8 (Evaluation maps and ).
Let , let be a collection of vectors in , and let .
The evaluation map (with set of fixed ends ) on is defined to be
In our pictures we will indicate ends that we would like to be considered fixed with a small orthogonal bar at the infinite side.
We denote by the subgroup of the symmetric group of all permutations such that for all and for all .
For the case of no fixed ends we denote simply by and by .
As in [GM08] example 3.3 these evaluation maps are morphisms of polyhedral complexes in the sense that they are continuous maps that are linear on each cell of . Note that acts on by permuting the unmarked ends, and that is invariant under this operation. By definition, if
then the inverse image consists of all -marked curves of degree that pass through at the marked point for all and map the -th unmarked end to the line for all . We call a collection of conditions for .
Of course, when counting curves we must assume that the conditions we impose are in general position so that the dimension of the space of curves satisfying them is as expected. Let us define this notion rigorously.
Definition 2.10 (General and special position of points).
Let , and let be a morphism of polyhedral complexes (as e.g. the evaluation map of definition 2.8 (a)). Then the union , taken over all cells of such that the polyhedron has dimension at most , is called the locus of points in special position for . Its complement is denoted the locus of points in general position for .
Note that the locus of points in general position for a morphism is by definition the complement of finitely many polyhedra of positive codimension in . In particular, it is a dense open subset of .
Let be a polyhedral subcomplex, and let . Then a collection of conditions as in remark 2.9 is in general position for if and only if for each curve in satisfying the conditions and every small perturbation of these conditions we can still find a curve of the same combinatorial type satisfying them.
Collections of conditions in general position for the evaluation map have a special property that will be crucial for the rest of the paper: in [GM08] remark 3.7 it was shown that every 3-valent curve through a collection of points in general position for the evaluation map without fixed ends has the property that each connected component of contains exactly one unmarked end. For the purposes of this paper we need the following generalization of this statement to curves that are not necessarily 3-valent and evaluation maps that may have fixed ends.
Let be a polyhedral subcomplex, and let be a collection of conditions in general position for the evaluation map . Assume that there is a curve satisfying these conditions. Then:
Each connected component of has at least one unmarked end with .
If the combinatorial type of has dimension and every vertex of that is not adjacent to a marking is 3-valent then every connected component of as in (a) has exactly one unmarked end with .
Consider a connected component of and denote by its closure in . We can consider as a graph, having a certain number of unbounded fixed ends, unbounded non-fixed ends, and bounded ends (i.e. 1-valent vertices) at markings of . The statement of part (a) of the lemma is that , with equality holding in case (b). For an example, in the picture below on the right consists of the solidly drawn lines; the curve continues in some way behind the dashed lines. Recall that fixed ends are indicated by small bars at the infinite sides. Hence in our example we have , , and .
By the same argument as in remark 2.7, the graph as well as the map is fixed by the position of a root vertex in and the lengths of all bounded edges of . But an easy combinatorial argument shows that the number of bounded edges of is equal to , with the sum taken over all vertices that are not adjacent to a marking. Hence and its image can vary with real parameters in .
On the other hand, together with fixes coordinates in the image of the evaluation map, namely the positions of the fixed ends and the markings in .
Hence is impossible: then these coordinates of the evaluation map would vary with fewer than coordinates of , meaning that the image of on the cell of cannot be full-dimensional and thus cannot have been in general position. This proves (a). But in case (b) is impossible as well: then by assumption we have for all as above, and thus one could fix a position for the fixed ends and markings at in and still obtain a -dimensional family for and . As a movement in this family does not change anything away from this means that is not injective on the cell of corresponding to . But is surjective on this cell as is in general position. This is a contradiction since by assumption the source and the target of the restriction of to the cell corresponding to have the same dimension. ∎
The important consequence of lemma 2.13 (b) is that — whenever it is applicable — it means that there is a unique way to orient every unmarked edge of so that it points towards the unique unmarked non-fixed end of the component of containing the edge. The picture on the right shows this for the curve of example 2.4. Note that the arrow will always point inwards on fixed ends, and outwards on non-fixed ends.
To be able to talk about this concept in the future we will now introduce the notion of oriented curves.
Definition 2.15 (Oriented marked curves).
An oriented -marked curve is an -marked curve as in definition 2.1 in which each unmarked edge of is equipped with an orientation (which we will draw as arrows in our pictures). In accordance with our above idea, the subset of all such that the unmarked end is oriented inwards is called the set of fixed ends of . The space of all oriented -marked curves with a given degree and set of fixed ends will be denoted ; for the case of no fixed ends we write also as . We denote by the obvious forgetful map that disregards the information of the orientations.
Obviously, our constructions and results for non-oriented curves carry over immediately to the oriented case: is a polyhedral complex with cells corresponding to the combinatorial types of the oriented curves (which now include the data of the orientations of all edges). The forgetful map is a morphism of polyhedral complexes that is injective on each cell. There are evaluation maps on as in definition 2.8 (a) that are morphisms of polyhedral complexes; by abuse of notation we will write them as in the unoriented case as .
So far we have allowed any choice of orientations on the edges of our curves in . To ensure that the orientations are actually as explained in remark 2.14 we will now allow only certain types of vertices. In the rest of the paper we will study various kinds of oriented marked curves — broccoli curves in section 3, Welschinger curves in section 4, and bridge curves in section 5 — that differ mainly in their allowed vertex types. The following definition gives a complete list of all vertex types that will occur anywhere in this paper.
Definition 2.17 (Vertex types and multiplicities).
We say that a vertex of an oriented -marked curve is of a certain type if the number, parity (even or odd), and orientation of its adjacent edges is as in the following table. In addition, two arrows pointing in the same direction (as in the types (6b) and (8)) require these odd edges to be two unmarked ends with the same direction, and an arc (as in the types (6a) and (9)) means that these two odd edges must not be two unmarked ends with the same direction. Hence the type (6) splits up into the two subtypes (6a) and (6b). All other types in the list are mutually exclusive.
In addition, each vertex of one of the above types is assigned a multiplicity that can also be read off from the table. Here, the number denotes the “complex vertex multiplicity” in the sense of Mikhalkin [Mik05], i.e. the absolute value of the determinant of two of the adjacent directions. For the type (8) it is the absolute value of the determinant of the two even adjacent directions.
If consists only of vertices of the above types, we denote by the number of vertices in of a given type . In addition, we then define the multiplicity of to be
where the second product is taken over all vertices of . Although some of the vertex multiplicities are complex numbers, the following lemma shows that the curve multiplicity is always real. In fact, the complex vertex multiplicities are just a computational trick that makes the “sign factor”, i.e. the power of , the same for all the vertex types (2) to (6) (which will be the most important ones), leading to easier proofs in the rest of the paper.
Every oriented marked curve that has only vertices of the types in definition 2.17 has a real multiplicity.
Let be a vertex of , and denote by the adjacent unmarked edges (so depending on the type of the vertex). Pick’s theorem implies that the complex vertex multiplicity as in definition 2.17 satisfies . By checking all vertex types we thus see that in each case
Now every unmarked edge is adjacent to exactly two vertices if it is bounded, and adjacent to exactly one vertex if it is unbounded. Hence
where the sum is taken over all unmarked edges. ∎
The picture of example 2.4 and remark 2.14 shows an oriented marked curve with . Its vertices , , , labeled from left to right, are of the types (1), (3), and (6), respectively, so that e.g. . The vertex is also of type (6a). The multiplicities of the vertices are , , and . As all unmarked ends of have weight the multiplicity of is thus .
Lemma 2.20 (Uniqueness of the orientation of curves).
Let the notations and assumptions be as in lemma 2.13 (b). If there is a way to make into an oriented curve with vertices of the types (1) to (7) and so that the orientations of the unmarked ends are as given by , this must be the orientation that lets each unmarked edge point towards the unique unmarked and non-fixed end in the component of containing it.
By lemma 2.13 (b) there is a unique orientation on pointing on each unmarked edge towards the unmarked and non-fixed end in the component of containing the edge. Now assume that we have any orientation on with vertices of types (1) to (7). Denote by the subgraph of where these two orientations differ; we have to show that .
Note that is a bounded subgraph since the orientation on the ends is fixed by . Moreover, cannot contain an edge adjacent to a marking since all possible vertex types (1), (5), (6), and (7) with markings require the orientation on the adjacent edges precisely as in remark 2.14. So if is non-empty it must have a 1-valent vertex somewhere that is not adjacent to a marking. This can only be a vertex of the types (2), (3), or (4), and the condition of being 1-valent means that the two orientations differ at exactly one adjacent edge. But this is impossible since both orientations have the property that they have one adjacent edge pointing outwards and two pointing inwards at this vertex. ∎
We will end this section by computing the dimensions of the cells of .
Let be an oriented marked curve all of whose vertices are of the types listed in definition 2.17. Let be the combinatorial type of . Then the cell of corresponding to has dimension
By remark 2.7 it suffices to show that the number of bounded edges of is equal to
This is easily proven by induction on the number of vertices in : if has only one vertex (and thus no bounded edge) it has to be one of the types in definition 2.17, and the statement is easily checked in all of these cases. If the curve has more than one vertex we cut it at any bounded edge into two parts and , making the cut edge unbounded in both parts. Note that the cut edge points inward for one part, and thus becomes a fixed end for this part. If for , then , , , , and for . The number of bounded edges of is now just the number of bounded edges in and plus , i.e. by induction equal to
as well as
3. Broccoli curves
In this section we will introduce the most important type of curves considered in this paper: the broccoli curves. We define corresponding numbers, and show that they are independent of the chosen point conditions.
Broccoli curves can be defined with or without orientation. Both definitions have their advantages: the oriented one is easier to state and local at the vertices, whereas the unoriented one is easier to visualize (as one does not need to worry about orientations at all). So let us give both definitions and show that they agree for enumerative purposes.
Definition 3.1 (Broccoli curves).
Let , let be a collection of vectors in , and let .
An oriented curve all of whose vertices are of the types (1) to (6) of definition 2.17 is called an oriented broccoli curve.
Let . Consider the subgraph of of all even edges (including the markings). The 1-valent vertices of as well as the with are called the stems of . We say that is an unoriented broccoli curve (with set of fixed ends ) if
all complex markings are adjacent to 4-valent vertices;
every connected component of has exactly one stem.
The picture below shows an oriented broccoli curve in which every allowed vertex type appears. We have labeled the vertices with their types. Note that by forgetting the orientations of the edges (and thus also disregarding the vertex types) one obtains an unoriented broccoli curve. Its subgraph of even edges consists of all markings and thick edges. It has four connected components , and each component has exactly one stem: the non-fixed unmarked end in , the vertex of type (3) in , and the unique vertices in and .
Of course, to count these curves we have to fix the right number of conditions to get a finite answer. This dimension condition follows e.g. for oriented broccoli curves from lemma 2.21: we must have since .
Proposition 3.3 (Equivalence of oriented and unoriented broccoli curves).
Let , let be a collection of vectors in , and let such that . Moreover, let be a collection of conditions in general position for (see example 2.12).
Then the forgetful map of definition 2.15 gives a bijection between oriented and unoriented -marked broccoli curves through with degree and set of fixed ends .
We have to prove three statements.
maps oriented to unoriented broccoli curves through : Let be an oriented broccoli curve. The list of allowed vertex types for implies immediately that then satisfies condition (i) of definition 3.1.
To show (ii) let be a connected component of . If contains no vertex of type (4) it can only be a single marking (types (1) or (5)) or a single unmarked edge with possibly attached markings (vertex types (3) together with (6), (3) with a fixed unmarked end, or (6) with a non-fixed unmarked end), and in each of these cases condition (ii) is satisfied. If there are vertices of type (4) they must form a tree in , and obviously every such tree made up from type (4) vertices has exactly one outgoing end. This unique outgoing end must be a non-fixed end of or connected to a type (3) vertex, hence in any case it leads to a stem. On the other hand, the incoming ends of the tree must be fixed ends of or connected to a type (6) vertex, i.e. they never lead to a stem. Consequently, satisfies condition (ii).
is surjective on the set of curves through : Let be an unoriented broccoli curve through with set of fixed ends . Then by (i) the curve has 4-valent vertices at the complex markings, so by [GM08] proposition 2.11 the combinatorial type of has dimension , with the sum taken over all vertices that are not adjacent to a complex marking. But as is in general position this dimension cannot be less than . So we see that all vertices without adjacent complex marking are 3-valent, and that the combinatorial type of has dimension equal to . Hence we can apply lemma 2.13 (b) again to conclude that there is an orientation on that points on each edge towards the unique non-fixed unmarked end in .
It remains to be shown that with this orientation the only vertex types occurring in are (1) to (6). For this, note that for a vertex
as we have said above, is 4-valent if there is a complex marking at , and 3-valent otherwise;
by the construction of the orientation, all edges at are oriented outwards if there is a marking at , and exactly one edge is oriented outwards otherwise;
by the balancing condition, it is impossible that exactly one edge at is odd.
With these restrictions, the only possible vertex types besides (1) to (6) would be the ones in the picture below.
To exclude these three cases, note that in all of them would be contained in a connected component of that contains at least one unmarked edge. So let us consider such a component, and let be a vertex where meets the complement of . Then there must be an odd as well as an unmarked even edge in at , so by the balancing condition as above there are exactly two odd edges and one even unmarked edge at . Hence is a stem if and only if there is no marking at . So a connection in from a point in the interior of to a non-fixed unmarked end can only be via a stem — which is unique by (ii). This means that every point in the interior of must be connected in to the stem. In particular, the interior of can have no further markings, which rules out the first two vertex types in the picture above. The third vertex type is impossible since this would have to be the stem and thus the connection from to the non-fixed unmarked end, which does not match with the orientation of the even edge. ∎
Let us now make the obvious definition of the enumerative invariants corresponding to broccoli curves. Proposition 3.3 tells us that it does not matter whether we count oriented or unoriented broccoli curves. We choose the oriented ones here as their definition is easier. So we make the convention that from now on a broccoli curve will always mean an oriented broccoli curve.
We denote by the closure of the space of all broccoli curves in ; this is obviously a polyhedral subcomplex. By lemma 2.21 it is non-empty only if the dimension condition is satisfied. Moreover, in this case it is of pure dimension , and its maximal open cells correspond exactly to the broccoli curves in .
Definition 3.5 (Broccoli invariants).
As above, let , let be a collection of vectors in , and let such that . Moreover, let be a collection of conditions in general position for broccoli curves, i.e. for the evaluation map . Then we define the broccoli invariant
where the sum is taken over all broccoli curves in with degree , set of fixed ends , and . The group as in definition 2.8 (b) takes care of the overcounting of curves due to relabeling the non-fixed unmarked ends. The sum is finite by the dimension statement of notation 3.4, and the multiplicity is as in definition 2.17.
The main result of this section — and in fact the most important point that distinguishes our new invariants from the otherwise quite similar Welschinger invariants that we will study in section 4 — is that broccoli invariants are always independent of the choice of conditions .
The broccoli invariants are independent of the collection of conditions . We will thus usually write them simply as (or for ).
The proof follows from a local study of the moduli space . Compared to the one for ordinary tropical curves in [GM07b] theorem 4.8 it is very similar in style and conceptually not more complicated; there are just (many) more cases to consider because we have to distinguish orientations as well as even and odd edges.
By definition, the multiplicity of a curve depends only on its combinatorial type. So it is obvious that the function is locally constant on the open subset of of conditions in general position for broccoli curves, and may jump only at the image under of the boundary of top-dimensional cells of . This image is a union of polyhedra in of positive codimension. It suffices to show that the function is locally constant around a cell in this image of codimension 1 in since any two top-dimensional cells of can be connected to each other through codimension-1 cells.
So let be a combinatorial type in of dimension such that is injective on and thus maps this cell to a unique hyperplane in . As in the picture on the right let be the open subset consisting of together with all adjacent top-dimensional cells of . To prove the theorem we will show that for a point in a neighborhood of the sum of the multiplicities of the curves in does not depend on , i.e. is the same on both sides of . In our picture this would just mean that , where denote the multiplicities of , respectively.
Actually, we will show this in a slightly different form: to each codimension-0 type in we will associate a so-called -sign that is or depending on the side of on which lies (it will be if ). So in the picture above on the right we could take and . We then obviously have to show that , where the sum is taken over all top-dimensional cells adjacent to .
To prove this, we will start by listing all codimension-1 combinatorial types in . They are obtained by shrinking the length of a bounded edge in a broccoli curve to zero, thereby merging two vertices into one. Depending on the merging vertex types we distinguish the following cases:
a vertex (1) merging with a vertex (2)/(3), leading to a 4-valent vertex with one real marking, two outgoing edges, and one incoming edge.
a vertex (2)/(3)/(4) merging with a vertex (2)/(3)/(4), leading to a 4-valent vertex with no marking, one outgoing edge, and three incoming edges.
a vertex (5)/(6) merging with a vertex (2)/(3)/(4), leading to a 5-valent vertex with one complex marking, three outgoing edges, and one incoming edge.
More precisely, noting that by the balancing condition it is impossible to have exactly one odd edge at a vertex, the cases (A), (B), and (C) split up into the following possibilities depending on the orientation and parity of the adjacent edges.
Next, we will list the adjacent codimension-0 types in (called resolutions) that make up in the cases (A), (B), and (C). In this picture, the dashed lines can be even or odd depending on which of the subcases (A), (B), (C) we are in. The vectors will be used in the computations below; they are always meant to be oriented outwards (i.e. not necessarily in the direction of the orientation of the edge), so that in case (A) and in the cases (B) and (C).
Note that the allowed vertex types for broccoli curves fix the orientation of the newly inserted bounded edge in all these resolutions; it is already indicated in the picture above. Moreover, the requirement that there cannot be exactly one odd edge at a vertex fixes the parity of the new bounded edge in all cases except (B1) and (C1). In the (B1) and (C1) cases, there are two possibilities: the four vectors can either be all the same in (in which case the new bounded edge joining and is even in all three types I, II, III; we call this case (B) and (C), respectively), or they make up two non-zero equivalence classes in (in which case the new bounded edge is even in exactly one of the types I, II, III; we call this case (B) and (C), respectively). In the (B) and (C) cases, we can assume by symmetry that the even bounded edge occurs in type I. So in total we now have 18 codimension-1 cases (A1), …, (A4), (B), (B), (B2),…(B6), (C), (C), (C2),…(C6) to consider, and in each of these cases we know the resolutions together with all parities and orientations of all edges of the curves — in particular, with the vertex types of and (as in the picture above). For example, in case (B6) the new bounded edge must be even in all three resolutions. Hence in all three resolutions all edges are even, and thus both vertices and are of type (4).
The following table lists the vertex types for and for all resolutions I, II, III of all codimension-1 cases. The symbol “—” means that the required vertex type is not allowed in broccoli curves and thus that a corresponding codimension-0 cell does not exist. The columns labeled and will be explained below.
|codim-1||resolution I||resolution II|
|codim-1||resolution I||resolution II||resolution III|
|codim-1||resolution I||resolution II||resolution III|
Let us now determine the -sign of the resolutions above, i.e. figure out which of them occur on which side of . To do this we set up the system of linear equations determining the lengths of the bounded edges of the curve in terms of the positions of the markings in . For such a given position of the markings (on the one or on the other side of ), a given resolution type is then possible if and only if the required length of the new bounded edge is positive.
More concretely, let be the length of the newly created bounded edge, and denote by in the cases (A) and (C) the required image point for the marking. In the cases (A) and (C) the end is fixed, so to determine the existing resolutions we may assume that there is another marking on the end at a distance of on the graph that is required to map to a point . In the case (B) the ends , , and are fixed, so we do the same then with lengths and points . As an example, these notions are illustrated for the resolution I in the following picture.