Enumeration of rational curves with cross-ratio constraints
aa r X i v : . [ m a t h . AG ] O c t ENUMERATION OF RATIONAL CURVES WITH CROSS-RATIOCONSTRAINTS
ILYA TYOMKIN
Abstract.
In this paper we prove the algebraic-tropical correspondence forstable maps of rational curves with marked points to toric varieties such thatthe marked points are mapped to given orbits in the big torus and in theboundary divisor, the map has prescribed tangency to the boundary divisor,and certain quadruples of marked points have prescribed cross-ratios. In par-ticular, our results generalize the results of Nishinou-Siebert [NS06]. The proofis very short, involves only the standard theory of schemes, and works in ar-bitrary characteristic (including the mixed characteristic case). Introduction
Background.
Enumeration of curves in algebraic varieties is a classical prob-lem that has a long history going back to Ancient Greeks. Many tools have beendeveloped to approach enumerative problems including Schubert calculus, intersec-tion theory, degeneration techniques, quantum cohomology etc.In 1989, Ran [Ran89] proposed a recursive procedure based on degenerationtechniques for enumeration of nodal curves of given degree and genus in the planesatisfying point constraints. Several years later, there was a major break-through inthe problem, when Kontsevich introduced the moduli spaces of stable maps [Kon95,KM94], and used them to get recursive formulae for the number of rational planecurves of degree d passing through 3 d − d marked points such that the first two are mappedto two given lines, the remaining 3 d − Key words and phrases.
Enumeration of rational curves, algebraic and tropical geometry.Partially supported by German-Israeli Foundation under grant agreement 1174-197.6/2011. it was a break-through, and in particular, led to the calculation of Welschingerinvariants in many interesting cases, see e.g., [IKS13, Mik05, Shu06].Tropical geometry is now a rapidly developing field, that includes the study ofthe combinatorial piece-wise linear side of tropical varieties as well as the link be-tween tropical, algebraic, and Berkovich analytic geometry. On one side, manytropical analogs of classical problems have been studied in the recent years. Inparticular, Gathmann and Markwig proved tropical analogs of formulae of Kontse-vich and Caporaso-Harris [GM07, GM08]. On the other side, several new proofs ofknown results such as Brill-Noether Theorem [CDPR12] have been found, as wellas tropical proofs of algebra-geometric statements that used to be beyond the reachof the classical methods, e.g., Zariski’s theorem in positive characteristic [Tyo13].Since 2005, few algebraic proofs of various versions of Mikhalkin’s correspon-dence have been obtained by Nishinou-Siebert [NS06], Shustin [Shu05], the author[Tyo12], Ranganathan [Ran15] and others. However, the proofs are relatively com-plicated and involve techniques such as deformation theory, log-geometry, stacks,rigid analytic spaces etc.; and all but [Tyo12] assume the ground field to be ofcharacteristic zero.1.2.
The goals of the paper, the results, and the approach.
In the currentpaper we study the algebraic-tropical correspondence for stable maps of rationalcurves with marked points to toric varieties such that the marked points are mappedto given orbits of given subtori in the big torus and in the boundary divisor, themap has prescribed tangency to the boundary divisor, and certain quadruples ofmarked points have prescribed cross-ratios. In particular, our results extend theresults of Nishinou-Siebert [NS06].We begin by proving that the canonical tropicalization procedure of [Tyo12] as-sociates to a stable map satisfying the constraints a rational parameterized tropicalcurve satisfying the tropicalization of the constraints.Our first main result is the
Realization theorem (Theorem 4.2). Under certainregularity assumptions we prove that any parameterized tropical curve satisfyingthe tropicalization of the algebraic constraints belongs to the image of the trop-icalization map. We shall emphasize, that we do not assume the tropical curveto be three-valent and our proof works over any complete discretely valued field,including the case of arbitrary small or mixed characteristic.Our second main result is the
Correspondence theorem (Theorem 5.1). Assumingthat the tropicalization of the constraints is tropically general, the characteristicof the residue field is big enough, and the problem is enumerative, we prove thatthe number of algebraic curves satisfying given constraints is equal to the numberof tropical curves satisfying the tropicalization of the constraints and counted withexplicit multiplicities. We also explain (Remark 3) that if the characteristic of theresidue field is arbitrary then the Correspondence theorem is still valid, but thealgebraic curves should be counted with multiplicities too.The proofs are surprisingly short, elementary, and involve no deformation theory,log-geometry, stacks, or rigid analytic spaces. Similarly to [Ran15], we do notuse the degeneration of the target, but unlike the other proofs we use only thestandard scheme theory, and if the characteristic of the residue field is big enoughwe even propose a reformulation in terms of elementary commutative algebra (seeSubsection 6.3).
NUMERATION OF RATIONAL CURVES WITH CROSS-RATIO CONSTRAINTS 3
Roughly speaking the proof of the Realization theorem consists of the followingsteps: First, we introduce convenient coordinates and describe the moduli spaceof the constrained stable maps that tropicalize to a given parameterized tropicalcurve as a fiber of an explicitly constructed map Θ. This way the set of algebraiccurves we are interested in becomes the set of integral points in a given fiber ofΘ. On the tropical side, we construct a linear map θ that controls the deformationtheory of the parameterized tropical curve. Finally, we show that the map Θ is adeformation of the map of algebraic tori associated to θ , which allows us to controlthe fibers of Θ, and to deduce the result. The Correspondence theorem then followsfrom the Realization theorem, using an easy combinatorial lemma (Lemma 5.4).We assume that the reader is familiar with the theory of schemes, basic toricgeometry, commutative algebra, and knows the definition of parameterized tropicalcurves. However, we remind the necessary notions from tropical geometry, in par-ticular the canonical tropicalization procedure of [Tyo12], in Section 3. We madea lot of effort to introduce intuitive and self-explaining notation, and summarizedmost of it in Subsection 2.1. We shall especially emphasize § Acknowledgements.
This research was initiated while the author was visiting theCentre Interfacultaire Bernoulli in 2014 in the framework of the special program on
Tropical geometry in its complex and symplectic aspects . I am very grateful to theorganizers Grigory Mikhalkin and Ilia Itenberg for inviting me, and to CIB for itshospitality and fantastic research atmosphere. I would also like to thank EugeniiShustin, Hannah Markwig, and Michael Temkin for helpful discussions.
Contents
1. Introduction 11.1. Background 11.2. The goals of the paper, the results, and the approach 2Acknowledgements 32. Preliminaries 42.1. Conventions and Notation 42.2. The cross-ratio 62.3. The objectives of the paper 72.4. A toy example 73. Tropicalization 84. Realization 104.1. The space of rational curves with given tropicalization 114.2. The space of morphisms with given tropicalization 13
ILYA TYOMKIN
Preliminaries
Conventions and Notation.
Multi-index.
Finite collections of objects with given index set are denotedwith bold letters, e.g., O = ( O , . . . , O s ), = (1 , . . . , α = ( α γ ), γ ∈ E .2.1.2. Valuations.
We fix a complete discretely valued field F having algebraicallyclosed residue field k and group of values Z , and its algebraic closure F . Thevaluation on F extending the valuation on F is denoted by ν : F → Q ∪ {∞} . Suchvaluation exists and unique by [Bou72, Corollary 2, p.425]. For a finite intermediateextension F ⊆ K ⊆ F , we denote by K o and K oo its ring of integers and maximalideal respectively, and by π a uniformizer of K o . The ramification index ν ( π ) − isdenoted by e K . To simplify the notation, the ring of integers of F is denoted by R .If X is a Noetherian integral scheme then any reduced codimension-one sub-scheme Y ⊂ X whose generic point is regular in X defines a discrete valuation onthe field of rational functions K ( X ) - the order of vanishing along Y . We denotethe latter valuation by ord Y .2.1.3. Toric geometry.
In this paper M and N denote a pair of dual lattices of finiterank, and T N := Spec( Z [ M ]) the corresponding torus. For a ring A we identify the A -points χ ∈ T N ( A ) and the homomorphisms χ : M → A × . We denote the basechange of T N → Spec( Z ) to Spec( A ) by T N,A := Spec( A [ M ]). For a sublattice L ⊆ N we denote the annihilator of L in M by L . For an abelian group G , wedenote N G := N ⊗ Z G and M G := M ⊗ Z G , e.g., ( Z n ) R = R n .2.1.4. Graphs.
In this paper all graphs are finite. The set of vertices of a graph Γ isdenoted by V (Γ), and the set of edges by E (Γ). The valency (or degree ) of a vertex w is denoted by deg( w ). In metric trees, the geodesic path connecting vertices w and w ′ is denoted by [ w, w ′ ], and the length of an edge γ by | γ | .2.1.5. Tropical curves.
We follow the conventions of [Tyo12]. In particular, theinfinite vertices of tropical curves are totally ordered. We use notation V f (Γ) (resp. V ∞ (Γ)) for the set of finite (resp. infinite) vertices, and E b (Γ) (resp. E ∞ (Γ)) forthe set of bounded (resp. unbounded) edges of Γ.In this paper, the infinite vertices are always denoted by u , . . . , u r , the corre-sponding unbounded edges (or ends) by e , . . . , e r , and the finite vertices attachedto the unbounded edges by v , . . . , v r . Notice that while u , . . . , u r are distinct, NUMERATION OF RATIONAL CURVES WITH CROSS-RATIO CONSTRAINTS 5 u u u u u u v = v = v v v = v γ γ e e e e e e Figure 1.
A rational tropical curve and its rooted tree structure. v , . . . , v r need not (and usually will not) be distinct, see Figure 1. To avoid con-fusion with the set of ends denoted by e , we use letter γ when referring to edges ingeneral. Similarly, we use letter w when referring to vertices in general.We work exclusively with N Q -parameterized Q -tropical curves , i.e., parameter-ized tropical curves whose bounded edges have rational lengths, and whose finitevertices are mapped to rational points of N R . Since N R is the open tropical torus,we remove the infinite vertices from the tropical curves mapped to it without warn-ing. We say that a N Q -parameterized Q -tropical curve h : (Γ; e ) → N R is definedover a field K if | γ | ∈ ν ( K × ) for all γ ∈ E b (Γ) and e K h ( w ) ∈ N for all w ∈ V f (Γ).2.1.6. The rooted tree structure and the induced combinatorics.
Let Γ be a rationaltropical curve, and u , . . . , u r its infinite vertices. We declare u r to be the root, andorient the edges away from the root. The corresponding partial order on the treeΓ for which the root is minimal, is denoted by (cid:22) . The tail and the head functionsare denoted by t : E (Γ) → V (Γ) and h : E (Γ) → V (Γ) respectively.For w ∈ V f (Γ), we set E + w := { γ | t ( γ ) = w } and I ∞ w := { i | u i ≻ w } . Then { E + w } w ∈ V f (Γ) is a partition of E (Γ) \ { e r } , and | E + w | = deg( w ) −
1. We also definea function ι : E (Γ) \ { e r } → { , . . . , r − } by setting ι ( γ ) to be the minimal index i for which u i (cid:23) h ( γ ). Then ι : E + w → { , . . . , r − } is injective for any w ∈ V f (Γ),hence induces a total order on each E + w .We call an edge γ ∈ E (Γ) essential if it belongs to some E + w and is neithermaximal nor minimal in it, and inessential otherwise; e.g., if deg( w ) = 3 for all w ∈ V f (Γ) then all edges are inessential. The set of essential edges is denotedby E es (Γ). Finally, we set I w := ι ( E + w ) ⊆ I ∞ w . Then I t ( γ ) ∩ I h ( γ ) = { ι ( γ ) } and ι ( γ ) ∈ I h ( γ ) is minimal for any γ ∈ E b (Γ). Example 2.1.
To illustrate the definitions consider the curve on Figure 1. Then, • the root is u and the edges are oriented upwards; • v (cid:23) v , u (cid:23) v , but v and u are incomparable; • t ( γ ) = v and h ( γ ) = v ; • E + v = { e , e , γ } , E + v = { e , e } , and E + v = { e , γ } . ILYA TYOMKIN e e e e e γ γ Figure 2.
A stable rational tropical curve with 5 marked ends. • I ∞ v = { , , , , } and I ∞ v = { , , } ; • ι ( γ ) = ι ( γ ) = 2 and ι ( e i ) = i for all i ≤ • the edge γ is essential, and γ is not; • finally, I v = { , , } , I v = { , } .2.2. The cross-ratio.
The algebraic cross-ratio.
Recall that the classical cross-ratio (or double ra-tio ) of four distinct points p , . . . , p ∈ P is defined by the formula λ ( P ; p , . . . , p ) = ( p − p )( p − p )( p − p )( p − p ) . In particular, if ( p , p , p ) = (1 , , ∞ ) then p = λ ( P ; p , . . . , p ). It is well-knownthat λ is a coordinate on the moduli space of smooth connected rational curveswith four marked points λ : M , ∼ −→ P \ { , , ∞} .2.2.2. The tropical cross-ratio.
Let (Γ; e ) be a rational tropical curve. In [Mik07],Mikhalkin defined the tropical double ratio of the two pairs { e i , e i } and { e i , e i } (or the tropical cross ratio of Γ with respect to e i , . . . , e i ) to be the signed lengthof the intersection of the oriented geodesic paths joining e i to e i and e i to e i ,where the sign is positive if and only if the orientations are compatible. Plainly,the tropical cross-ratio is stable under tropical modifications/contractions.We say that γ ∈ E b (Γ) separates e t , e j from e d , e l if and only if e t , e j belong toone of the two connected components of Γ \ { γ } , and e d , e l to another. Then thetropical cross-ratio of Γ with respect to e i , . . . , e i is given by λ tr (Γ; e i , . . . , e i ) := X γ ∈ E b (Γ) ǫ ( γ, i ) | γ | ;where(2.1) ǫ ( γ, i ) = , γ separates the ends e i , e i from e i , e i , − , γ separates the ends e i , e i from e i , e i , Example 2.2.
Consider the rational tropical curve on Figure 2. The commonpart of the geodesic paths joining e to e and e to e has length | γ | , but theorientations are opposite. Hence λ tr (Γ; e , e , e , e ) = −| γ | . Alternatively, γ separates e , e from e , e , and γ separates e from e , e , e , which also leads tothe conclusion that λ tr (Γ; e , e , e , e ) = 0 · | γ | + ( − · | γ | = −| γ | . In a similarway one checks that λ tr (Γ; e , e , e , e ) = 0 and λ tr (Γ; e , e , e , e ) = | γ | + | γ | . NUMERATION OF RATIONAL CURVES WITH CROSS-RATIO CONSTRAINTS 7
The objectives of the paper.
The setting.
We fix once and for all • a pair of dual lattices N and M ; • n , . . . , n r ∈ N such that P ri =1 n i = 0; • sublattices n i ∈ L i ⊆ N , 1 ≤ i ≤ r , such that N/L i are torsion-free; • F -points ζ i ∈ ( T N /T L i )( F ) for all 1 ≤ i ≤ r ; • F -point λ = ( λ , . . . , λ s ) ∈ T s Z ( F ); • s × J with integral entries 1 ≤ J ij ≤ r . Notation.
By abuse of notation, we write (cid:3) i , . . . , (cid:3) i when addressing a quadrupleof objects parameterized by the i -th row of matrix J ; e.g, q i , . . . , q i or e i , . . . , e i .Let Σ ⊂ N R be the fan generated by the rays ρ i := Span R + ( n i ) for 1 ≤ i ≤ r . Wedenote by X := X Σ the corresponding toric variety, and by O i the closure in X ofthe T L i -orbit corresponding to ζ i . Then O i ∩ X ρ i is given by x m = ζ i ( m ), m ∈ L i ,since n i ∈ L i , where X ρ i is the open subscheme corresponding to the ray ρ i . Finally,we set λ tr := ν ( λ ) ∈ Q s and O tr i := { m ν ( x m ( p )) | p ∈ ( T N ∩ O i )( F ) } ⊆ N Q .Notice that O tr i is the preimage of ζ tr i := ν ( ζ i ) ∈ N Q / ( L i ) Q .2.3.2. The objectives.
In this paper we compare between:
The stable morphisms f : ( C ; q ) → X, where ( C ; q ) is a smooth irreducible rationalcurve with r marked points such that Degree and tangency profile: div ( f ∗ x m ) = P ( n i , m ) q i , Toric constraint: f ( q i ) ∈ O i for all i ≤ r , Cross-ratio constraint: λ ( C ; q i , q i , q i , q i ) = λ i for all i ≤ s ;and the stable rational N Q -parameterized Q -tropical curves h : (Γ; e ) → N R with r unbounded ends for which Degree and multiplicity profile: h ( u i ) = n i for all i ≤ r , Affine constraint: h ( v i ) ∈ O tr i for all i ≤ r , Tropical cross-ratio constraint: λ tr (Γ; e i , e i , e i , e i ) = λ tr i for all i ≤ s Notation.
The sets of such morphisms and tropical curves are denoted by W and W tr respectively.The goal of the paper is to construct a natural map Tr : W → W tr and to study itsfibers. Under certain regularity conditions we prove that the fibers are non-empty,and count the number of points in each fiber.2.4. A toy example.
Let us consider the case of rational curves of class (1 ,
1) in P × P passing through two general points and having prescribed cross-ratio withrespect to the four points in the complement of the big torus.The setting is as follows: M = N = Z , r = 6, n = (cid:0) (cid:1) , n = − (cid:0) (cid:1) , n = (cid:0) (cid:1) , n = − (cid:0) (cid:1) , n = n = 0, L i = N for 1 ≤ i ≤ L i = { } for i = 5 , ζ i = 1 for1 ≤ i ≤ ζ i ∈ ( F × ) for i = 5 , s = 1, J = (1 2 3 4), λ = λ . Thus, X is thetoric surface obtained by removing the four zero-dimensional orbits from P × P , O i = X for 1 ≤ i ≤ O i = ζ i for i = 5 ,
6. We shall also assume that ζ , ζ ,and λ are general in an appropriate sense. We do not assume that n i are primitive or non-zero. ILYA TYOMKIN (0 ,
0) (11 , e ′ e ′ e ′ e ′ e e e e Figure 3.
The toy example: the two curves in W tr .The space W thus consists of the stable maps f : ( C ; q , . . . , q ) → P × P forwhich q = f ∗ ( { }× P ), q = f ∗ ( {∞}× P ), q = f ∗ ( P ×{ } ), q = f ∗ ( P ×{∞} ), λ ( C ; q , . . . , q ) = λ , and f ( q i ) = O i ∈ ( F × ) for i = 5 , W explicitly. Indeed, pick the coordinate t on C suchthat t ( q ) = λ, t ( q ) = 1 , t ( q ) = 0 , t ( q ) = ∞ . Since f is of class (1 ,
1) and q , . . . , q are the pullbacks of the boundary divisor, f is given by an equation ofthe form t (cid:16) c t − λt − , c t (cid:17) , for some c , c ∈ F . Thus, the affine constraint is givenby (cid:16) c t ( q i ) − λt ( q i ) − , c t ( q i ) (cid:17) = O i for i = 5 ,
6. After eliminating t ( q ) and t ( q ) oneobtains one linear and one quadratic equation in c , c . Thus, for a general choiceof λ, O , O , the space W consists of two F -points, and hence there exist preciselytwo rational curves of class (1 ,
1) in P × P satisfying the constraints.Let us now describe the set W tr : it consists of the parameterized rational tropicalcurves with six unbounded ends e , . . . e , such that the slopes of e , . . . , e are (cid:0) (cid:1) , − (cid:0) (cid:1) , (cid:0) (cid:1) , − (cid:0) (cid:1) respectively, the ends e i are contracted to the points ν ( O i ) ∈ R for i = 5 ,
6, and λ tr (Γ; e , e , e , e ) = ν ( λ ). If ν ( λ ) > ν ( O ) , ν ( O ) ∈ R are tropically general, e.g. ν ( O ) = (0 , , ν ( O ) = (11 , , ν ( λ ) = 5, then one cancheck that there exist precisely two parameterized rational tropical curves in theset W tr as in Figure 3. It follows from the Correspondence theorem (Theorem 5.1),that the map Tr : W → W tr is bijective in this particular example.3. Tropicalization
We employ the canonical tropicalization of [Tyo12] to construct the tropicaliza-tion map Tr : W → W tr : Let f : ( C ; q ) → X be an element of W . Fix a completediscretely valued field of definition K of ( C ; q ), and a uniformizer π . Since ( C ; q )is rational, its stable model is defined over the ring of integers K o ⊂ K . Wedefine the underlying graph of the tropicalization Γ to be the dual graph of thestable reduction of ( C ; q ). The vertices corresponding to the components of C arecalled finite , and those that correspond to the marked points are infinite . Theedges corresponding to nodes of the reduction are called bounded , and the edgescorresponding to marked points are called unbounded (or ends ). The length of anend is set to be ∞ , and the length of the edge corresponding to a node p of thereduction is defined to be ν ( η ) if the total space of the stable model is given by NUMERATION OF RATIONAL CURVES WITH CROSS-RATIO CONSTRAINTS 9 q = λq = 0 q = 1 q = ∞ P K ; y ∞ ∞ L ; yE ; λy u u u u | γ | = ν ( λ ) yz = λν ( λ ) > γe e e e Figure 4.
The stable model of a rational curve with four markedpoints for which ν ( λ ( C ; q , . . . , q )) > xy = η ∈ K o in a (´etale) neighborhood of p (Figure 4). The function h is definedas follows: If a finite vertex w corresponds to a component C w of the reductionthen h ( w )( m ) := ν ( π ) · ord C w ( f ∗ ( x m )) = e − K ord C w ( f ∗ ( x m )). If u i corresponds tothe marked point q i then h ( u i )( m ) := ord q i ( f ∗ ( x m )). The parametrization h mapsany finite vertex w to h ( w ), all bounded edges to the straight intervals joining theimages of the attached vertices, and the ends e i to the rays h ( v i ) + Q + h ( u i ).The curve h : (Γ; e ) → N R is a rational N Q -parameterized Q -tropical curve by[Tyo12, Lemma 2.23]. Plainly, the degree and the multiplicity profile constraints aresatisfied by the construction. To see that the affine constraint is satisfied, recall that O i ∩ X ρ i is given by the equations x m = ζ i ( m ) for all m ∈ L i , and ( n i , m ) = 0 for all m ∈ L i since n i ∈ L i . Thus, h ( v i )( m ) = ord C vi ( f ∗ ( x m )) = ord q i ( f ∗ ( x m )) = ζ i ( m )for all m ∈ L i . Hence h ( v i ) ∈ O tr i . Finally, the tropical cross-ratio constraint issatisfied by the following: Lemma 3.1.
Let ( C ; q , . . . , q ) be a smooth rational curve with four marked pointsover the field K , and (Γ; e , . . . , e ) its tropicalization. Then λ tr = ν ( λ ) .Proof. Choose the coordinate y : C ∼ −→ P such that ( q , q , q , q ) = ( λ, , , ∞ ) , and consider the trivial model P K o over the ring of integers K o ⊂ K . Assume, first,that the reduction of q is different from 0, 1, and ∞ . Then ν ( λ ) = ν (1 − λ ) = 0,and P K o is the stable model of ( C ; q , . . . , q ). Hence λ tr = 0 = ν ( λ ).Case 1: ν ( λ ) >
0. To construct the stable model, consider the blow up B of P K o with respect to the ideal generated by λ and y . Then the reduction of B consistsof the strict transform L of P k and of the exceptional divisor E ≃ P k . The two components intersect at one point, q and q specialize to distinct points of E , q and q to distinct points of L , and none of them to E ∩ L . Hence B is the stablemodel of ( C ; q , . . . , q ), see Figure 4. Furthermore, the stable model is given by yz = λ near the node of the reduction, and hence the length of the unique boundededge of Γ is ν ( λ ). Thus, λ tr = ν ( λ ).Case 2: ν (1 − λ ) >
0. In this case the stable model is the blow up of P K o withrespect to the ideal (1 − λ, − y ). Moreover, q and q specialize to distinct pointsof the exceptional divisor, and q and q to distinct points of the strict transform of P k . The length of the unique bounded edge of Γ is ν (1 − λ ). Thus, λ tr = 0 = ν ( λ ).Case 3: ν ( λ ) <
0. This time the stable model is the blow up of P K o with respectto ( λ − , /y ). Furthermore, q and q specialize to distinct points of the exceptionaldivisor, and q and q to distinct points of the strict transform of P k . The lengthof the unique bounded edge of Γ is − ν ( λ ). Thus, λ tr = ν ( λ ). (cid:3) Realization
In this section we show that the image of the tropicalization map contains all regular curves, and describe the fiber of the tropicalization map over such curves.Let us start by reminding the notion of regularity.Let h : (Γ; e ) → N R be an element of W tr . For γ ∈ E b (Γ) (resp. γ ∈ E ∞ (Γ)) set n γ := h ( h ( γ )) − h ( t ( γ )) | γ | (resp. n γ := h ( u ), where u is the infinite vertex attached to γ ). Then n γ ∈ N by the definition of parameterized tropical curves, and its integrallength is called the multiplicity of γ . Caution! In [Tyo12], n γ denotes the primitiveintegral vector in the same direction. Consider now the following complex(4.1) θ : M w ∈ V f (Γ) N ⊕ M γ ∈ E b (Γ) Z → M γ ∈ E b (Γ) N ⊕ r M i =1 ( N/L i ) ⊕ s M i =1 Z given by1 γ n γ + s X i =1 ǫ ( γ, i ) and a w X γ ǫ ( γ, w ) a w + r X i =1 δ ( w, v i )( a w + L i ) , where ǫ ( γ, i ) is given by (2.1), and ǫ ( γ, w ) = , w = t ( γ ) , − , w = h ( γ ) , δ ( w, v i ) = (cid:26) , w = v i , G , let θ G be the map in the complex (4.1) ⊗ Z G . We denote E G (Γ , h ; O tr , λ tr ) := Ker( θ G ) and E G (Γ , h ; O tr , λ tr ) := Coker( θ G ) . If G = Z we omit G in the notation of E • G . Definition 4.1 (cf. [Tyo12, Definitions 2.45, 2.55]) . We say that (Γ , h ; O tr , λ tr ) is G -regular if E G (Γ , h ; O tr , λ tr ) = 0, and G -superabundant otherwise. Remark . (i) Plainly, G -regularity is independent of the orientation of Γ.(ii) By the structure theorem of finitely generated abelian groups, Q -regularityis equivalent to E (Γ , h ; O tr , λ tr ) being a torsion group, since the latter is finitelygenerated, and tensor product preserves cokernels. Similarly, k -regularity is equiv-alent to E (Γ , h ; O tr , λ tr ) being a torsion group of order prime to the characteristic NUMERATION OF RATIONAL CURVES WITH CROSS-RATIO CONSTRAINTS 11 of k . Hence (Γ , h ; O tr , λ tr ) is k -regular if and only if it is Q -regular and the orderof E (Γ , h ; O tr , λ tr ) is not divisible by the characteristic of k .(iii) If (Γ , h ; O tr , λ tr ) is Q -regular then E k × (Γ , h ; O tr , λ tr ) = 0 since k is alge-braically closed. Moreover, the morphism of algebraic tori over Spec( k ) corre-sponding to (4.1) is surjective, the set of k -points of its kernel is E k × (Γ , h ; O tr , λ tr ),and the kernel is reduced if and only if the order of E (Γ , h ; O tr , λ tr ) is not divisibleby the characteristic. To see the latter, choose bases in (4.1) such that the mapis given by a matrix of the form (0 | D ), where D = diag( d i ) is diagonal. Thendet( D ) = 0 by Q -regularity, E k × (Γ , h ; O tr , λ tr ) ≃ ( k × ) r × Q i Ker( k × (cid:3) di −−−→ k × ), andthe kernel is scheme-theoretically isomorphic to T r Z ,k × Q i µ d i ,k , where r is the rankof E (Γ , h ; O tr , λ tr ), and µ d i ,k ⊳ T Z ,k are the groups of roots of unity of orders d i .(iv) If (Γ , h ; O tr , λ tr ) is Q -regular and E (Γ , h ; O tr , λ tr ) = 0 then, using the no-tation above, r = 0 and the scheme-theoretic length of the kernel Q i µ d i ,k ofthe morphism of algebraic tori over Spec( k ) corresponding to (4.1) is equal to Q i d i = | ⊕ i Z /d i Z | = |E (Γ , h ; O tr , λ tr ) | . Further, by (ii) above, (Γ , h ; O tr , λ tr ) is k -regular if and only if the characteristic of k does not divide Q i d i , or, equivalently |E k × (Γ , h ; O tr , λ tr ) | = Q i d i = |E (Γ , h ; O tr , λ tr ) | . Theorem 4.2 (Realization) . Let h : (Γ; e ) → N R be an element of W tr , and K ⊂ F a complete discretely valued subfield of definition of O , λ , and (Γ , h ) . Assume that λ tr i = 0 for all i , and (Γ , h ; O tr , λ tr ) is Q -regular. Then(1) h : (Γ; e ) → N R belongs to the image of Tr : W → W tr .(2) If (Γ , h ; O tr , λ tr ) is k -regular, Γ is three-valent, and E (Γ , h ; O tr , λ tr ) = 0 then the fiber of the tropicalization map Tr over h : (Γ; e ) → N R consists of exactly |E (Γ , h ; O tr , λ tr ) | morphisms f : ( C ; q ) → X , and all morphisms in the fiber aredefined over K .Remark . It is sufficient to prove the theorem under the assumption that λ tr i > i . Indeed, the general case reduces to this by replacing λ i with λ − i if λ tr i < ,
4) in the corresponding rows of matrix J . Thus, fromnow on we will always assume that λ tr i > for all i . The space of rational curves with given tropicalization.
In this sub-section we introduce explicit coordinates on the space of rational curves with r marked points tropicalizing to a given stable rational tropical curve with markedends (Γ; e ).Let ( C ; q ) be a smooth rational curve with marked points tropicalizing to (Γ; e ).For each finite vertex w ∈ V f (Γ), denote by y w the coordinate on C such that(4.2) y w ( q r ) = ∞ , y w ( q a ) = 0 , y w ( q b ) = 1 , where a and b are the minimal and the maximal indices in I w respectively. Noticethat | I w | = | E + w | = deg( w ) − ≥ e ) is stable, and hence (4.2)makes sense.Let γ ∈ E b (Γ) be a bounded edge. If a and b are as above then a = ι ( γ ) and thecoordinate y t ( γ ) − y t ( γ ) ( q a ) y t ( γ ) ( q b ) − y t ( γ ) ( q a ) satisfies (4.2) for the vertex w = h ( γ ). Hence y t ( γ ) − y t ( γ ) ( q a ) = ( y t ( γ ) ( q b ) − y t ( γ ) ( q a )) y h ( γ ) . Thus, | γ | = ν ( y t ( γ ) ( q b ) − y t ( γ ) ( q a )) by the very definition of | γ | , since the stablemodel of C is given by ( y t ( γ ) − y t ( γ ) ( q a )) y h ( γ ) = y t ( γ ) ( q b ) − y t ( γ ) ( q a ) in a neighborhoodof the node corresponding to γ . Finally, we define α C ∈ T E b (Γ) Z ( R ) and β C ∈ A E (Γ) ( R ) by setting α Cγ := y t ( γ ) ( q b ) − y t ( γ ) ( q a ) π | γ | e K and β Cγ := y t ( γ ) ( q ι ( γ ) );and for each γ ∈ E b (Γ) a linear functionΨ γ ( y ) := β Cγ + π | γ | e K α Cγ y. Let us summarize the properties of α = α C , β = β C , and y : Proposition 4.3. (1) If γ, γ ′ ∈ E + w are distinct then β γ − β γ ′ ∈ T Z ( R ) . Further-more, if γ ∈ E + w is minimal (resp. maximal) then β γ = 0 (resp. β γ = 1 ).(2) If γ ∈ E b (Γ) then y t ( γ ) = Ψ γ ( y h ( γ ) ) . In particular, for any i , ( y t ( γ ) − y t ( γ ) ( q i )) = π | γ | e K α γ ( y h ( γ ) − y h ( γ ) ( q i )) . (3) Let w, w ′ ∈ V f (Γ) be finite vertices, w = w , w , . . . , w d +1 = w ′ the verticesalong the geodesic path [ w, w ′ ] , and γ , . . . , γ d the corresponding edges. Then y w = Ψ ǫ γ ◦ · · · ◦ Ψ ǫ d γ d ( y w ′ ) , where ǫ i = 1 if γ i is directed along the path [ w, w ′ ] and ǫ i = − otherwise . Inparticular, for any i , ( y w − y w ( q i )) = π e K P di =1 ǫ i | γ i | d Y i =1 α ǫ i γ i ! ( y w ′ − y w ′ ( q i )) . (4) Let ≤ i < r , w ∈ V f (Γ) , w = w , w , . . . , w d +1 = v i the vertices along thegeodesic path [ w, v i ] , and γ , . . . , γ d the corresponding edges. Then y w ( q i ) = Ψ ǫ γ ◦ · · · ◦ Ψ ǫ d γ d ( β e i ) , where ǫ i are as in (3). In particular, if w = h ( e r ) = v r then (4.3) y v r ( q i ) = β γ + π | γ | e K α γ (cid:16) . . . ( β γ d + π | γ d | e K α γ d β e i ) (cid:17) . (5) ν ( y w ( q i )) ≥ if and only if u i ≻ w , for all w ∈ V f (Γ) and ≤ i ≤ r .(6) ν ( y t ( γ ) ( q i ) − β γ ) = 0 for all γ ∈ E b (Γ) and i ∈ I ∞ t ( γ ) \ I ∞ h ( γ ) .Proof. (1) and (2) follow from the definition of α and β , (3) follows from (2) byinduction, (4) follows from (3) and the definition of β , (5) follows from (4), and (6)follows from (5) and (1). (cid:3) Set v := v r , and let a, b ∈ I v be the minimal and the maximal indices. Then thecollection { y v ( q i ) } , 1 ≤ i < r , i = a, b , is a system of coordinates on the modulispace of all rational curves with r marked points. One can check inductively onthe partially ordered sets of edges and vertices that α C and β C can be expressedexplicitly in terms of { y v ( q i ) } for curves tropicalizing to (Γ; e ). Vice versa, (4.3)expresses { y v ( q i ) } in terms of α C and β C .Thus, ( α Cγ , β Cγ ′ ) γ ′ ∈ E es (Γ) γ ∈ E b (Γ) is a system of coordinates on the space of rational curveswith r marked points tropicalizing to (Γ; e ), and C ( α C , β C ) is an open immer-sion of this space into T Z ( R ) | E b (Γ) | + ov (Γ) . Moreover, it is fairly easy to describe theimage of the immersion. Indeed, by Proposition 4.3 (1), if γ < · · · < γ d are theedges in E + w , β γ := 0 , β γ d := 1 then β γ i − β γ j ∈ T Z ( R ) for all i = j . If the latter is Notice that since Γ is a tree, there exists j such that ǫ i = 1 if and only if i ≥ j . NUMERATION OF RATIONAL CURVES WITH CROSS-RATIO CONSTRAINTS 13 satisfied for all w ∈ V f (Γ) then one defines C := P , q r := ∞ , and q i to be given bythe right-hand side of (4.3) for each 1 ≤ i < r . It is now straight-forward to verifythat ( C ; q ) tropicalizes to (Γ; e ) and ( α C , β C ) = ( α , β ). We leave the details tothe reader.4.2. The space of morphisms with given tropicalization.
Let us fix a param-eterized stable rational tropical curve with marked ends h : (Γ; e ) → N R . In thissubsection we give an explicit description of the space of morphisms f : ( C ; q ) → X tropicalizing to it, for which ( C ; q ) is a smooth connected rational curve withmarked points.Fix ( C ; q ) as above, and let α = α C , β = β C be its coordinates. Let γ ∈ E b (Γ).Define φ γ ∈ T N ( R ) by setting φ γ ( m ) := Q i ∈ I ∞ t ( γ ) \ I ∞ h ( γ ) (cid:0) y t ( γ ) ( q i ) − β γ (cid:1) ( n i ,m ) Q i/ ∈ I ∞ t ( γ ) (cid:16) y t ( γ ) ( q i ) y t ( γ ) ( q i ) − β γ (cid:17) ( n i ,m ) . Notice that φ γ is well defined since φ γ ( m ) ∈ R × for all m by Proposition 4.3 (5-6).Notice also that φ γ is a function of ( α , β ) by Proposition 4.3 (4).Let f : ( C ; q ) → X be a morphism tropicalizing to h : (Γ; e ) → N R . For each m ∈ M and w ∈ V f (Γ), let us express the rational function f ∗ ( x m ) in terms of thecoordinate y w . Since x m ∈ K ( X ) × , f ∗ ( x m ) ∈ (cid:0) F [ y w , ( y w − y w ( q r +1 )) − , . . . , ( y w − y w ( q r − )) − ] (cid:1) × . Moreover,(4.4) f ∗ ( x m ) = π e K h ( w )( m ) χ w ( m ) Y i ∈ I ∞ w ( y w − y w ( q i )) ( n i ,m ) Y i/ ∈ I ∞ w (cid:18) y w y w ( q i ) − (cid:19) ( n i ,m ) since the boundary multiplicity profile of ( C, f ) is { n i } . Plainly χ w ∈ T N ( F ).Moreover, χ w ∈ T N ( R ) ⊂ T N ( F ) by the definition of h ( w ). Lemma 4.4.
For any γ ∈ E b (Γ) the following holds (4.5) φ γ · χ t ( γ ) χ h ( γ ) · α n γ γ = 1 . Proof.
By (4.4) and Proposition 4.3 (2), the following holds for all m ∈ M : π e K h ( t ( γ ))( m ) π e K h ( h ( γ ))( m ) · χ t ( γ ) ( m ) χ h ( γ ) ( m ) · Q i/ ∈ I ∞ h ( γ ) y h ( γ ) ( q i ) ( n i ,m ) Q i/ ∈ I ∞ t ( γ ) y t ( γ ) ( q i ) ( n i ,m ) · (cid:16) π | γ | e K α γ (cid:17) P i ( n i ,m ) = 1 . Since P ri =1 n i = 0 and h ( h ( γ )) − h ( t ( γ )) = | γ | n γ , it remains to show that(4.6) φ γ ( m ) = ( π | γ | e K α γ ) − ( n γ ,m ) Q i/ ∈ I ∞ h ( γ ) y h ( γ ) ( q i ) ( n i ,m ) Q i/ ∈ I ∞ t ( γ ) y t ( γ ) ( q i ) ( n i ,m ) . After summing up the balancing conditions over the finite vertices w ⊁ t ( γ ), oneobtains the following X i/ ∈ I ∞ h ( γ ) n i = − n γ . Thus, the numerator of (4.6) is equal to Q i/ ∈ I ∞ h ( γ ) (cid:0) y t ( γ ) ( q i ) − β γ (cid:1) ( n i ,m ) by Proposi-tion 4.3 (2), which implies the lemma. (cid:3) Vice versa, if we are given a collection χ w ∈ T N ( R ) for all w ∈ V f (Γ) such that(4.5) holds for all γ ∈ E b (Γ) then the maps given by (4.4) are compatible, and hencesuch datum defines a morphism f : ( C ; q ) → X tropicalizing to h : (Γ; e ) → N R .We conclude that the space of such morphisms is given by equations (4.5) in thetrivial T N ( R ) V f (Γ) -bundle over the space of curves ( C ; q ) tropicalizing to (Γ; e ).4.3. The equations of the constraints.
Recall that O j is given by x m = ζ j ( m )for all m ∈ L j . Thus, the morphism f : ( C ; q ) → X satisfies the constraint O j ifand only if f ∗ ( x m )( q j ) = ζ j ( m ) for all m ∈ L j .Set ζ Γ j ( m ) := π − e K h ( v j )( m ) ζ j ( m ). Then ζ Γ j ∈ ( T N /T L j )( R ), and, by (4.4), f : ( C ; q ) → X satisfies the constraint O j if and only if ζ Γ j ( m ) = χ v j ( m ) Y i ∈ I ∞ vj ( β e j − y v j ( q i )) ( n i ,m ) Y i/ ∈ I ∞ vj (cid:18) β e j y v j ( q i ) − (cid:19) ( n i ,m ) , for all m ∈ L j . Define ϕ j ∈ T N ( R ) by setting ϕ j ( m ) := Y i ∈ I ∞ vj ( β e j − y v j ( q i )) ( n i ,m ) Y i/ ∈ I ∞ vj (cid:18) β e j y v j ( q i ) − (cid:19) ( n i ,m ) . Notice that ϕ j is well defined since ϕ j ( m ) ∈ R × for all m by Proposition 4.3 (5-6).Notice also that ϕ j is a function of ( α , β ) by Proposition 4.3 (4). We conclude that f : ( C ; q ) → X satisfies the constraint O j if and only if ϕ j χ v j ≡ ζ Γ j mod T L j . Let us now reformulate the cross-ratio constraint: Pick an index 1 ≤ i ≤ s . Recallthat we assumed that λ tr i >
0. Let w i , w ′ i ∈ V f (Γ) be such that q i , q i specializeto different points of the component C w i and q i , q i specialize to different pointsof the component C w ′ i . Notice that an edge γ belongs to the geodesic path [ w i , w ′ i ]if and only if γ separates e i , e i from e i , e i . Let w ′′ i be the minimal vertex along[ w i , w ′ i ]. Set λ Γ i := π − e K λ tr i λ i ∈ T Z ( R ) and ψ i := ( y w i ( q i ) − y w i ( q i ))( y w ′ i ( q i ) − y w ′ i ( q i ))( y w ′′ i ( q i ) − y w ′′ i ( q i ))( y w ′′ i ( q i ) − y w ′′ i ( q i )) ∈ T Z ( R ) . Then, by Proposition 4.3 (3), λ ( C ; q i , . . . , q ir ) = λ i if and only if ψ i Y γ ⊂ [ w i ,w ′ i ] α γ = λ Γ i . As usual, ψ i can be expressed explicitly in terms of ( α , β ) by Proposition 4.3 (4).4.4. Proof of Theorem 4.2.
Let B ⊂ T E (Γ) Z ,K o be the K o -subscheme of points β satisfying assertion (1) of Proposition 4.3. Plainly, B is flat over K o and has purerelative dimension | E es (Γ) | . In particular, if Γ is trivalent then B ≃ Spec( K o ).Consider the K o -morphismΘ : T V f (Γ) N,K o × T E b (Γ) Z ,K o × B → T E b (Γ) N,K o × r Y i =1 ( T N,K o /T L i ,K o ) × s Y i =1 T Z ,K o NUMERATION OF RATIONAL CURVES WITH CROSS-RATIO CONSTRAINTS 15 that maps ( χ , α , β ) to (cid:18) φ γ ( α , β ) χ t ( γ ) χ h ( γ ) α n γ γ (cid:19) , (cid:0) ϕ j ( α , β ) χ v j (cid:1) , ψ i ( α , β ) Y γ ⊂ [ w i ,w ′ i ] α γ where w i , w ′ i ∈ V f (Γ) are such that q i , q i specialize to different points of thecomponent C w i and q i , q i specialize to different points of the component C w ′ i .Let S be the fiber of Θ over ( , ζ Γ , λ Γ ). Then W = S ( R ), and the ideal of S isgenerated by l := dim (cid:16) T E b (Γ) N,K o × Q ri =1 ( T N,K o /T L i ,K o ) × Q si =1 T Z ,K o (cid:17) − dim( K o )functions. Let f , . . . , f l be such generators.Let Θ k = Θ × Spec( K o ) Spec( k ) be the reduction of Θ. It follows from the defi-nition and Proposition 4.3 that the reductions of φ γ ( α , β ), ϕ j ( α , β ), and ψ i ( α , β )are independent of the reduction of α . Thus, Θ k is nothing but the product ofthe map B k → T E b (Γ) N,k × Q ri =1 ( T N,k /T L i ,k ) × Q si =1 T Z ,k given by the reduction of( φ ( α , β ) , ϕ ( α , β ) , ψ ( α , β )), and the homomorphism of k -algebraic groups θ k × : T V f (Γ) N,k × T E b (Γ) Z ,k → T E b (Γ) N,k × r Y i =1 ( T N,k /T L i ,k ) × s Y i =1 T Z ,k corresponding to (4.1). By Remark 1 (iii), the later morphism is surjective, andthe set of k -points of its kernel is E k × (Γ , h ; O tr , λ tr ). Set G := Ker( θ k × ). Then thefibers of Θ k are G -torsors over B k . In particular, the fibers are irreducible of puredimension | E es (Γ) | + rank( E (Γ , h ; O tr , λ tr )). Thus, by [Mat89, Theorem 23.1], Θis flat at any k -point, and hence so is the base change S → Spec( K o ).(1) Since W = S ( R ), it is sufficient to construct a quasi-section of S → Spec( K o ),which exists by Mumford’s existence theorem [Gro66, Proposition 14.5.10]. In ourcase, the construction is easy, and we include it for the convenience of the reader.Let p ∈ S k be a (general) closed point, and g , . . . , g d ∈ m p ⊂ O S ,p be such thattheir classes form a basis of the cotangent space at p to the underlying reducedsubscheme of S k . Then d = dim( S k ) = | E es (Γ) | + rank( E (Γ , h ; O tr , λ tr )). Considerthe reduced local subscheme Z in the local scheme S p defined by the functions g , . . . , g d . Then by Hauptidealsatz (e.g., [Mat89, Theorem 13.5]) the dimensionof any component of Z is at least dim (cid:16) T V f (Γ) N,K o × T E b (Γ) Z ,K o × B (cid:17) − l − d = 1. Onthe other hand, dim( Z k ) = 0. Thus, by Hauptidealsatz, Z is equi-dimensional ofdimension one, and π is not a zero-divisor in O ( Z ).Let Z ′ → Z be the normalization of a component of Z . Then O ( Z ′ ) is inte-grally closed in its field of fractions K ( Z ′ ) and contains K o . However, by [Bou72,Corollary 3 p.379 and Corollary 2 p.425], the integral closure of K o in a finiteextension of K is a discrete valuation ring, and hence a maximal proper subring.Thus, O ( Z ′ ) is a discrete valuation ring. Choose an embedding K ( Z ′ ) ֒ → F . Then O ( Z ′ ) = R ∩ K ( Z ′ ), and hence Z ′ → Z ⊂ S defines a point in S ( R ) = W as needed.(2) Since K o is complete and k is algebraically closed, Spec( K o ) admits no non-trivial local ´etale coverings by [Bou72, Corollary 2 p.425]. Thus, it is sufficient toshow that S is ´etale over K o and | S k | = |E (Γ , h ; O tr , λ tr ) | . By the assumption, Γ isthree-valent, and hence B = Spec( K o ). Furthermore | S k | = |E k × (Γ , h ; O tr , λ tr ) | = |E (Γ , h ; O tr , λ tr ) | by Remark 1 (iii)-(iv), since (Γ , h ; O tr , λ tr ) is k -regular. Finally,the relative tangent space of Θ at each point p ∈ S k is E k (Γ , h ; O tr , λ tr ) = 0. Thus, S is ´etale over Spec( K o ), and we are done. Remark . If we omit the k -regularity assumption in (2) then |E (Γ , h ; O tr , λ tr ) | can still be interpreted as the number of algebraic curves in the fiber of Tr butcounted with multiplicities. To see this, one considers the open neighborhood S ′ of S k ⊂ S that contains no irreducible components concentrated over the genericpoint of Spec( K o ). Then S ′ → Spec( K o ) is flat, and one can show that it is finite.Hence |E (Γ , h ; O tr , λ tr ) | , which is equal to the length of S k = S ′ k by Remark 1 (iv),is equal to the length of S ′ × Spec( K o ) Spec( K ). The latter is precisely the number ofcurves in the fiber of Tr counted with the following multiplicities: the multiplicityof f : ( C ; e ) → X is the length of the scheme concentrated at the class of the curveand cut out by the constraints on the moduli space of stable maps.5. Correspondence
In this section we prove the correspondence theorem under the assumption thatthe characteristic of k is big enough, and O tr and λ tr are tropically general, bywhich we mean the following: If a family of objects is parameterized by a conein a Q -affine space then an element in this family is tropically general for certainproblem if it does not belong to a finite union of proper affine subspaces determinedby the problem. Theorem 5.1 (Correspondence) . Assume that the constraints O and λ are suchthat O tr and λ tr are tropically general, and (5.1) s + r X i =1 rank( N/L i ) = r − . If the characteristic of k is big enough (or zero) then the map Tr : W → W tr is sur-jective and the size of the fiber over h : (Γ; e ) → N R is |E (Γ , h ; O tr , λ tr ) | . Moreover,all curves in the fiber are defined over any field of definition of (Γ , h ) . Definition 5.2.
The complex multiplicity of a curve h : (Γ; e ) → N R satisfying theconstraints O tr , λ tr is defined to be m C (Γ , h ; O tr , λ tr ) := |E (Γ , h ; O tr , λ tr ) | .Since the result of our enumerative problem over an algebraically closed fielddepends only on the characteristic we obtain the following: Corollary 5.3.
Under the assumption of the Correspondence theorem, the num-ber of stable complex rational curves f : ( C ; q ) → X satisfying general complexconstraints O , λ is equal to X (Γ ,h ) ∈W tr m C (Γ , h ; O tr , λ tr ) . The Correspondence theorem follows immediately from the Realization theoremusing the following:
Lemma 5.4.
Let h : (Γ; e ) → N R be an element of W tr . If the assumptions of theCorrespondence theorem hold then Γ is three-valent, (Γ , h ; O tr , λ tr ) is k -regular, and E (Γ , h ; O tr , λ tr ) = 0 .Proof. Consider the linear map(5.2) ϑ : M w ∈ V f (Γ) N ⊕ M γ ∈ E b (Γ) Z → M γ ∈ E b (Γ) N NUMERATION OF RATIONAL CURVES WITH CROSS-RATIO CONSTRAINTS 17 given by composing the map θ of (4.1) with the projection on the first summand.Then the space of parameterized tropical curves having the same combinatorialtype, slopes, and multiplicities as the curve h : (Γ; e ) → N R can be identified nat-urally with the cone in E Q (Γ , h ) := Ker( ϑ Q ) consisting of those elements whoseprojection to L γ ∈ E b (Γ) Q have no non-positive coordinates.Since Γ has no cycles L w ∈ V f (Γ) N Q → L γ ∈ E b (Γ) N Q is surjective, and hence sois (5.2). Thus,(5.3) dim (cid:0) E Q (Γ , h ) (cid:1) = 2 | V f (Γ) | − | E b (Γ) | . Recall that for any tropical curve one has(5.4) 3 | V f (Γ) | + | V ∞ (Γ) | + ov (Γ) = 2 | E b (Γ) | + 2 | E ∞ (Γ) | , where ov (Γ) = P w ∈ V f (Γ) ( val ( w ) − ov (Γ) ≥ | V f (Γ) | = | E b (Γ) | + 1 . After subtracting (5.5) from(5.4) and substituting to (5.3) one concludes:dim (cid:0) E Q (Γ , h ) (cid:1) = | E ∞ (Γ) | − ov (Γ) − r − − ov (Γ) . Consider the natural projection(5.6) ̺ = ̺ (Γ ,h ) : E (Γ , h ) → r M i =1 ( N/L i ) ⊕ s M i =1 Z . By the assumption, the curve h : (Γ; e ) → N R corresponds to the ̺ Q -preimage of ageneral point. Since, there are only finitely many combinatorial types of parame-terized stable rational tropical curves with given degree and boundary multiplicityprofile, we may assume that our constraint does not belong to the union of the spansof all non-maximal dimensional images of ̺ (Γ ′ ,h ′ ) Q . We conclude that ̺ Q is surjective,and hence r − − ov (Γ) ≥ s + P ri =1 rank( N/L i ) = r − ov (Γ) = 0,i.e., Γ is three-valent, and ̺ Q is an isomorphism. Hence |E (Γ , h ; O tr , λ tr ) | < ∞ and E (Γ , h ; O tr , λ tr ) = Ker( ̺ ) = 0. It follows now that (Γ , h ; O tr , λ tr ) is k -regular assoon as the characteristic of k does not divide the order of E (Γ , h ; O tr , λ tr ). (cid:3) Remark . Corollary 5.3 reduces our complex enumerative problem to a tropicalenumerative problem of finding an explicit description for W tr . A naive approachto the tropical problem would be the following:Start by preparing a complete list of three-valent trees with r ordered ends. Sincethe graph is a tree, either all its edges are ends or there exists a vertex whose starcontains exactly one non-end. Thus, balancing condition determines the slope andthe multiplicity of the non-end since the slopes and the multiplicities of the endsare given ( n , . . . , n r ). Proceeding by induction, one obtains a combinatorial typeof a parameterized stable rational tropical pseudo-curve h : (Γ; e ) → N R , whereby pseudo-curve we mean that the lengths of the bounded edges are allowed tobe arbitrary reals. For a given combinatorial type, one considers the complex(4.1) ⊗ Z Q . Then the E b (Γ)-components of θ − Q ( , ζ tr , λ tr ) provide all possible waysto equip Γ with edge lengths. Finally, thanks to Lemma 5.4, if θ Q is an isomorphismand all E b (Γ)-components of θ − Q ( , ζ tr , λ tr ) are strictly positive then the resultingcurve h : (Γ; e ) → N R belongs to W tr , and all curves in W tr are obtained this way. The number of three-valent trees with r ordered ends is (2 r − § W tr (Figure 3). However, we believe that there existmore efficient combinatorial ways (similar to lattice path algorithm or floor di-agrams) to exhibit the set W tr , which would provide an algorithm to compute P (Γ ,h ) ∈W tr m C (Γ , h ; O tr , λ tr ). Unfortunately, currently we do not know such analgorithm, and finding one is among our future projects. Let us conclude by men-tioning that the computation of the multiplicities m C (Γ , h ; O tr , λ tr ) is simple since m C (Γ , h ; O tr , λ tr ) = |E (Γ , h ; O tr , λ tr ) | = det( θ ) by its very definition. In particular,in our toy example both multiplicities are 1.5.1. The real case.
In this subsection we assume that k = C , F = C (( t )) is the fieldof Laurent series, and hence F is the field of Puiseux series. Consider the naturalinvolution ς acting by coefficient-wise complex conjugation, and let F ς = R (( t )) beits fixed field. Assume that the toric and the cross-ratio constraints are definedover R (( t )). The goal of this section is to describe the subset W ς ⊆ W of ς -invariantpoints under the assumptions of the Correspondence theorem. Since the action of ς preserves the fibers of the tropicalization map Tr : W → W tr , we fix an element h : (Γ; e ) → N R of W tr , and analyze the action of ς on the corresponding fiber W Γ ,h .Recall that by Lemma 5.4, the assumptions of assertion (2) of Realization the-orem are satisfied. Let S be as in the proof of the theorem. Then the points in W Γ ,h = S ( R ) are uniquely determined by their reduction in S C . Notice that sincethe constraints are defined over F ς , S ( R ) admits a natural action of ς compati-ble with the complex conjugation on the reduction S C . Hence ς -invariant pointsof S ( R ) specialize to real points of S C , and pairs of ς -conjugate points to pairs ofcomplex-conjugate points.Let us now describe the set S C ( R ) of real points of S C : Consider the exactsequence (4.1) ⊗ Z R × . Then S C ( R ) is the preimages of the reduction ξ of ( , ζ Γ , λ Γ ).Thus, S C ( R ) = ∅ , if and only if the class of ξ in E R × (Γ , h ; O tr , λ tr ) is trivial, and inthis case | S C ( R ) | = |E R × (Γ , h ; O tr , λ tr ) | .Let us look closer at the groups E i R × (Γ , h ; O tr , λ tr ). Pick coordinates in (4.1) suchthat the map θ is given by a diagonal matrix D = diag ( d i ). Then(5.7) E i R × (Γ , h ; O tr , λ tr ) = E i R × / R + (Γ , h ; O tr , λ tr ) ≈ {± } ε , for i = 1 ,
2, where ε denotes the number of even d j -s. Denote by σ ( O , λ ) the imageof the reduction of ( ζ Γ , λ Γ ) in ( L ri =1 ( N/L i ) ⊕ L si =1 Z ) ⊗ Z ( R × / R + ). We shall call σ ( O , λ ) the sign of the real constraint ( O , λ ). Then the vanishing of the class of ξ in E R × (Γ , h ; O tr , λ tr ) depends only on Γ , O tr , λ tr and the sign of the constraint,rather than on the constraint itself. Definition 5.5.
The real multiplicity m R (Γ , h ; O tr , λ tr , σ ( O , λ )) of the parame-terized rational tropical curve h : (Γ; e ) → N R satisfying the constraints O tr , λ tr with respect to the sign σ ( O , λ ) is defined to be 0 if the image of ( , σ ( O , λ )) in E R × / R + (Γ , h ; O tr , λ tr ) is not trivial, and 2 ε otherwise, where ε denotes the numberof groups of even order in any decomposition E (Γ , h ; O tr , λ tr ) ≈ ⊕ Z /d i Z . Corollary 5.6.
Assume that O tr , λ tr are tropically general and (5.1) holds. Thenfor any choice of the sign σ ( O , λ ) there exist real constraints O R , λ R with given sign NUMERATION OF RATIONAL CURVES WITH CROSS-RATIO CONSTRAINTS 19 such that the number of real stable maps f : ( C ; q ) → X satisfying the constraints O R , λ R is equal to X (Γ ,h ) ∈W tr m R (Γ , h ; O tr , λ tr , σ ( O , λ )) . Proof.
Choose any constraint O , λ with tropicalizations O tr , λ tr and given sign suchthat all ζ i , λ i are convergent fractional power series with real coefficients . Then allcurves in W are defined over the subfield of convergent fractional power series,and there are only finitely many such curves. Thus, for real t small enough the ς -invariant curves specialize to real curves and the pairs of ς -conjugate curves spe-cialize to pairs of complex-conjugate curves satisfying the constraints O R , λ R . Sincedistinct curves specialize to distinct curves for t small enough, all complex curvessatisfying the constraints O R , λ R are obtained this way by Corollary 5.3. In partic-ular, the number of real stable maps is equal to the number of ς -invariant curves,which is equal to P (Γ ,h ) ∈W tr m R (Γ , h ; O tr , λ tr , σ ( O , λ )) by (5.7). (cid:3) Remark . (i) By its very definition, the real multiplicity m R (Γ , h ; O tr , λ tr , σ ( O , λ ))is bounded above by the complex multiplicity m C (Γ , h ; O tr , λ tr ), and the inequalityis strict in many cases, e.g., if m C (Γ , h ; O tr , λ tr ) is not a power of 2. In suchcases the complex count differs from the real count for any “tropical” position ofreal constraints, i.e., the position obtained by a small enough specialization of atropically general R (( t ))-constraint as in the proof of the corollary.(ii) If one chooses the sign to be totally positive, i.e., the unit element of the group( L ri =1 ( N/L i ) ⊕ L si =1 Z ) ⊗ Z ( R × / R + ), then the class of ξ in E R × (Γ , h ; O tr , λ tr ) istrivial for any (Γ , h ) ∈ W tr . Thus, such choices of sign give rise to the maximalpossible number of real stable maps f : ( C ; q ) → X satisfying real constraints in“tropical” position.(iii) As one expects, unlike the algebraically closed case, the answer to the realenumerative problem does depend on the position of the constraints. Indeed,it is easy to construct examples such that there exists (Γ , h ) ∈ W tr for which |E (Γ , h ; O tr , λ tr ) | is even. Since the curve Γ is rational, the complex (4.1) is quasi-isomorphic to a complex of the form N ⊕ M γ ∈ E b (Γ) Z → r M i =1 ( N/L i ) ⊕ s M i =1 Z . Thus, ( L ri =1 ( N/L i ) ⊕ L si =1 Z ) ⊗ Z R × maps surjectively onto E R × (Γ , h ; O tr , λ tr ),and we can choose the constraints O ′ , λ ′ with tropicalizations O tr , λ tr such thatthe class of ξ in E R × (Γ , h ; O tr , λ tr ) is non-trivial. Hence |W ς Γ ,h | = 0. Specializing t to a small positive real number we obtain constraints O ′ R , λ ′ R for which the numberof real stable maps f : ( C ; q ) → X satisfying the constraints O ′ R , λ ′ R is strictlysmaller than in the case of the totally positive sign.6. Afterword
More on parameterized tropical curves satisfying general constraints.Proposition 6.1.
Under the assumptions of Lemma 5.4, let w ′ , w ′′ ∈ V f (Γ) befinite vertices, γ ′ ∈ E (Γ) and edge, [ w ′ , w ′′ ] the geodesic path joining w ′ to w ′′ ,and [ w ′ , γ ′ ] the geodesic path containing γ ′ whose end points are w ′ and one of theendpoints of γ ′ . Then λ p p p p p Figure 5.
A parameterized tropical curve in dimension 2 with aflattened three-valent vertex, and the corresponding dual subdivi-sion of the Newton polygon for r = 5 , d = 7 , s = 1 , p i = p i .(1) h ( w ′ ) = h ( w ′′ ) if and only if h contracts [ w ′ , w ′′ ] ; (2) if h ( w ′ ) ∈ h ( γ ′ ) then h ([ w ′ , γ ′ ]) is a straight interval; (3) the number of contracted edges attached to w ′ is either one or three.Proof. (1) The “if” part is clear. For the “only if” part, assume that h ( w ′ ) = h ( w ′′ ),and let η ∈ E Q (Γ , h ) ⊂ L w ∈ V f (Γ) N Q ⊕ L γ ∈ E b (Γ) Q be the class of the curve h : Γ → N R . Then η satisfies the equation(6.1) X γ ⊂ [ w ′ ,w ′′ ] a γ n γ = 0 . But (6.1) depends only on the slopes and the multiplicities of h : Γ → N R , and since(5.6) is an isomorphism, η ∈ E Q (Γ , h ) is general. Thus, (6.1) holds true identically on E Q (Γ , h ). On the other hand, the projection E Q (Γ , h ) → L γ ∈ E b (Γ) Q is surjectivesince so is L w ∈ V f (Γ) N Q → L γ ∈ E b (Γ) N Q . Thus, (6.1) holds true identically on L γ ∈ E b (Γ) Q , and hence n γ = 0 for all γ ∈ [ w ′ , w ′′ ], i.e., h contracts [ w ′ , w ′′ ].The proof of (2) is similar, but the analog of (6.1) is considered modulo the slopeof γ ′ . We leave the details to the reader. Finally, (3) follows from the balancingcondition since Γ is three-valent by Lemma 5.4. (cid:3) The dual subdivision.
Under the assumptions of Lemma 5.4, assume that N ≃ Z , and let ∆ be the Newton polygon of h : Γ → Q . One may hastilyconclude that the dual subdivision of ∆ consists of triangles and 2 k -gons withparallel opposite edges. However, this need not be the case since, for example,there may exist flattened three-valent vertices as the example on Figure 5 shows.Nevertheless, it is fairly easy to show using Lemma 5.4 and Proposition 6.1 that thedual subdivision of ∆ consists of 2 k -gons with k pairs of parallel equi-length edges(Minkowski sums of k intervals), triangles, trapezoids, pentagons (resp. hexagons)having two (resp. three) pairs of parallel edges.6.3. An algebraic approach to Realization theorem.
The goal of this sub-section is to give a sketch of an elementary algebraic proof of Realization theoremunder the assumption that (Γ , h ; O tr , λ tr ) is k -regular. We shall only explain theproof of assertion (1). Assertion (2) can be proved along the same lines, and we NUMERATION OF RATIONAL CURVES WITH CROSS-RATIO CONSTRAINTS 21 leave it to the reader. We start in the same way we proved the theorem, andintroduce the map Θ and the scheme S such that S ( R ) = W .The idea now is very simple: We think about K o -points ξ ∈ S ( K o ) as solutionsof the system of equations Θ( ξ ) = ( , ζ Γ , λ Γ ), and we construct a sequence ofapproximate solutions, i.e., ξ d ∈ T V f (Γ) N ( K o ) × T E b (Γ) Z ( K o ) × B ( K o ) for d ≥
0, suchthat for all d the following holds: ξ d +1 ≡ ξ d mod ( π d ) andΘ( ξ d ) ≡ ( , ζ Γ , λ Γ ) mod ( π d ) . Thus, there exists ξ := lim d →∞ ξ d ∈ T V f (Γ) N ( K o ) × T E b (Γ) Z ( K o ) × B ( K o ) since K o is complete, and Θ( ξ ) ≡ ( , ζ Γ , λ Γ ) mod ( π d ) for all d . Hence ξ ∈ S ( K o ) is thedesired point.Fix once and for all a point β ∈ B . We shall construct the sequence ξ d , whose B -component is β for all d . Denote by Θ β the restriction of Θ to ( b = β )-locus.By k -regularity, the reduction of Θ β modulo π is surjective, and hence there exists ξ satisfying the requirements. Assume by induction that we have constructed ξ , . . . , ξ d as needed, and for w ∈ V f (Γ), γ ∈ E b (Γ), let χ w : N → G d , α γ : Z → G d be homomorphisms, where G d := 1 + ( π d +1 ) ⊳ ( K ) × . We shall look for ξ d +1 ofthe form ξ d · ( χ , α , ). Plainly, any such ξ d +1 satisfies: ξ d +1 ≡ ξ d mod ( π d ). Lemma 6.2.
For any d ≥ , if α ≡ α ′ mod ( π d ) and β ≡ β ′ mod ( π d +1 ) then φ γ ( α , β ) ≡ φ γ ( α ′ , β ′ ) mod ( π d +1 ) ,ϕ j ( α , β ) ≡ ϕ j ( α ′ , β ′ ) mod ( π d +1 ) ,ψ i ( α , β ) ≡ ψ i ( α ′ , β ′ ) mod ( π d +1 ) . The proof of the lemma is straight-forward and is left to the reader. Let θ ′ bethe map in (4.1) ⊗ Z ( K o ) × . Then, by Lemma 6.2,Θ( ξ d · ( χ , α , )) ≡ Θ( ξ d ) · θ ′ ( χ , α ) mod ( π d +1 ) . Thus, it is sufficient to find ( χ , α ) such that(6.2) θ ′ ( χ , α ) ≡ (Θ( ξ d )) − ( , ζ Γ , λ Γ ) mod ( π d +1 ) . By k -regularity, the order of E (Γ , h ; O tr , λ tr ) ≈ ⊕ Z /d i Z does not divide the char-acteristic of k , and hence E G d (Γ , h ; O tr , λ tr ) ≈ ( ⊕ Z /d i Z ) ⊗ Z ( G d ) × = 1. Indeed, allwe have to check is that the d i -power maps G d → G d , are surjective for all i . But K o is complete, and the denominators of the expansion of (1 + π d +1 y ) /d i do notvanish. Thus, the maps are surjective, and hence (6.2) admits a solution as needed. ndex B , 14 E (Γ), 4 E ∞ (Γ), 4 E b (Γ), 4 E es (Γ), 5 E + w , 5 F , 4 I w , 5 I ∞ w , 5 J , 7 K oo , 4 K o , 4 L , 4 L i , 7 M , 4 M G , 4 N , 4 N G , 4 O i , 7 O tr i , 7 R , 4 T N , 4 T N,A , 4 V (Γ), 4 V ∞ (Γ), 4 V f (Γ), 4 X , 7[ w, w ′ ], 4 E iG (Γ , h ; O tr , λ tr ), 10 W , 7 W tr , 7 G , 15 G d , 21Ψ γ , 12 S , 15Σ, 7Θ, 14 Tr , 7 α Cγ , 12 α , λ , O , q , etc., 4 β Cγ , 12 χ w , 13deg( w ), 4 δ ( w, v i ), 10 ǫ ( γ, i ), 6 ǫ ( γ, w ), 10 e K , 4 ι , 5 λ ( P ; p , . . . , p ), 6 λ tr (Γ; e , . . . , e ), 6 λ i , 7 λ Γ i , 14 λ tr i , 7 F , 4ord Y , 4 φ γ , 13 π , 4 (cid:22) , 5 ψ i , 14 σ ( O , λ ), 18 θ , 10 θ G , 10 ν , 4 ϕ j , 14 ̺ , 17 ς , 18 ϑ , 16 ξ , 18 ξ d , 21 ζ Γ j , 14 ζ i , 7 ζ tr i , 7 e i , 4 k , 4 m C (Γ , h ; O tr , λ tr ), 16 m R (Γ , h ; O tr , λ tr , σ ( O , λ )), 18 n γ , 10 n i , 7 q i , . . . , q i ; e i , . . . , e i , etc., 7 u i , 4 v i , 4 y w , 11 | γ | , 4 References [Bou72] Nicolas Bourbaki,
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