Generators of invariant linear system on tropical curves for finite isometry group
aa r X i v : . [ m a t h . AG ] M a y Generators of invariant linear system ontropical curves for finite isometry group
Song JuAe
Abstract
For a tropical curve Γ and a finite subgroup K of the isometrygroup of Γ , we prove, extending the work by Haase, Musiker and Yu([3]), that the K -invariant part of the complete linear system associ-ated to a K -invariant effective divisor on Γ is finitely generated. Let R ( D ) denote the set consisting of rational functions corresponding tothe complete linear system | D | for an effective divisor D on a tropical curve Γ , where a tropical curve means a metric graph possibly with unboundededges. R ( D ) becomes a tropical semimodule. The projective space R ( D ) / R is naturally identified with the complete linear system | D | . Haase, Musikerand Yu showed that R ( D ) is finitely generated ([3, Theorem 6]). A tropicalsubsemimodule R ′ of R ( D ) corresponds to a linear subspace Λ of | D | . Thislinear subspace Λ is called a linear system associated to R ′ .In this paper, we recall some basic facts of tropical curves in Section 2.Then in Section 3, we observe the K -invariant set R ( D ) K of R ( D ) and provethat R ( D ) K is actually finitely generated, where K is a finite subgroup of theisometry group of Γ . Our proof is basically analogous to that of [3], but it isnot perfectly compatible, i.e. the K -invariant set S K of the generator set S of R ( D ) defined in [3, Lemma 6] is not a generator set of R ( D ) K . We find such aset corresponding to S , which we call S ( D ) K . The condition defining S ( D ) K is tangibly given from geometric information. Also, the construction of aharmonic morphism with degree | K | from Γ to the quotient tropical curve Γ ′ of Γ by K precedes. We follow Chan’s natural construction ([2]) with alittle bit of adaptation. Finally, using the harmonic morphism, we prove that1 ( D ) K is finitely generated as a tropical semimodule in our main theorem(Theorem 3.11). When D is K -invariant, we can identify R ( D ) K / R withthe K -invariant linear subsystem | D | K and then | D | K is finitely generatedby S ( D ) K / R . In this section, we briefly recall the theories of tropical curves ([5]), divi-sors on tropical curves ([5]), harmonic morphisms of tropical curves ([2], [3],[4]), and chip-firing moves on tropical curves ([3]), which we need later.
In this paper, a graph means an unweighted, finite connected nonempty multi-graph. Note that we allow the existence of loops. For a graph G , the sets ofvertices and edges are denoted by V ( G ) and E ( G ), respectively. The valence val( v ) of a vertex v of G is the number of edges emanating from v , where wecount each loop as two. A vertex v of G is a leaf end if v has valence one. A leaf edge is an edge of G having a leaf end.An edge-weighted graph ( G, l ) is the pair of a graph G and a function l : E ( G ) → R > ∪ {∞} called a length function , where l can take the value ∞ on only leaf edges. A tropical curve is the underlying ∞ -metric space of anedge-weighted graph ( G, l ). For a point x on a tropical curve Γ obtained from( G, l ), if the distances between x and all points on Γ other than x are infinity,then x is called a point at infinity , else, x is said to be a finite point . For theabove tropical curve Γ , ( G, l ) is said to be its model . There are many possiblemodels for Γ . We construct a model ( G ◦ , l ◦ ) called the canonical model of Γ as follows: when Γ is a circle, we determine V ( G ◦ ) as the set consistingof one arbitrary point on Γ , else when Γ is the ∞ -metric space obtainedfrom only one edge with length of ∞ , V ( G ◦ ) consists of the two endpointsof Γ (those are points at infinity) and an any point on Γ as the origin, else,generally, we determine V ( G ◦ ) := { x ∈ Γ | val( x ) = 2 } , where the valenceval( x ) is the number of connected components of U x \ { x } with U x beingany sufficiently small connected neighborhood of x in Γ . Since connectedcomponents of Γ \ V ( G ◦ ) consist of open intervals, whose lengths determinethe length function l ◦ . If a model ( G, l ) of Γ has no loops, then ( G, l ) is saidto be a loopless model of Γ . For a model ( G, l ) of Γ , the loopless model for2 G, l ) is obtained by regarding all midpoints of loops of G as vertices and byadding them to the set of vertices of G . The loopless model for the canonicalmodel of a tropical curve is called the canonical loopless model .For terminology, in a tropical curve Γ , an edge of Γ means an edge ofthe underlying graph G ◦ of the canonical model ( G ◦ , l ◦ ). Let e be an edgeof Γ which is not a loop. We regard e as a closed subset of Γ , i.e., includingthe endpoints v , v of e . The relative interior of e is e ◦ = e \ { v , v } . Fora point x on Γ , a half-edge of x is a connected component of U x \ { x } withany sufficiently small connected neighborhood U x of x .For a model ( G, l ) of a tropical curve Γ , we frequently identify a vertex v (resp. an edge e ) of G with the point corresponding to v on Γ (resp. theclosed subset corresponding to e of Γ ). Let Γ be a tropical curve. An element of the free abelian group Div( Γ )generated by points on Γ is called a divisor on Γ . For a divisor D on Γ , its degree deg( D ) is defined by the sum of the coefficients over all points on Γ .We write the coefficient at x as D ( x ). A divisor D on Γ is said to be effective if D ( x ) ≥ x in Γ . If D is effective, we write simply D ≥
0. The setof points on Γ where the coefficient(s) of D is not zero is called the support of D and written as supp( D ).A rational function on Γ is a constant function of −∞ or a piecewiselinear function with integer slopes and with a finite number of pieces, takingthe value ±∞ only at points at infinity. Rat( Γ ) denotes the set of rationalfunctions on Γ . For a point x on Γ and f in Rat( Γ ) which is not constant −∞ , the sum of the outgoing slopes of f at x is denoted by ord x ( f ). If x isa point at infinity and f is infinite there, we define ord x ( f ) as the outgoingslope from any sufficiently small connected neighborhood of x . Note when Γ is a singleton, for any f in Rat( Γ ), we define ord x ( f ) := 0. This sum is 0 forall but finite number of points on Γ , and thusdiv( f ) := X x ∈ Γ ord x ( f ) · x is a divisor on Γ , which is called a principal divisor . Two divisors D and E on Γ are said to be linearly equivalent if D − E is a principal divisor.We handle the values ∞ and −∞ as follows: let f, g in Rat( Γ ) take thevalue ∞ and −∞ at a point x at infinity on Γ respectively, and y be any3oint in any sufficiently small neighborhood of x . When ord x ( f ) + ord x ( g )is negative, then ( f ⊙ g )( x ) := ∞ . When ord x ( f ) + ord x ( g ) is positive,then ( f ⊙ g )( x ) := −∞ . Remark that the constant function of −∞ on Γ dose not determine a principal divisor. For a divisor D on Γ , the completelinear system | D | is defined by the set of effective divisors on Γ being linearlyequivalent to D .The set of R with two tropical operations: a ⊕ b := max { a, b } and a ⊙ b := a + b becomes a semiring called the tropical semiring , where both a and b are in R .For a divisor D on a tropical curve, let R ( D ) be the set of rational functions f = −∞ such that D + div( f ) is effective. When deg( D ) is negative, | D | isempty, so is R ( D ). Otherwise, from the argument in Section 3 of [3], D isnot empty and consequently so is R ( D ). Hereafter, we treat only divisors ofnonnegative degree. Lemma 2.2.1 (cf. [3, Lemma 4]) . R ( D ) becomes a tropical semimoduleon R by extending above tropical operations onto functions, giving pointwisesum and product. By the definition of ord x ( f ) for a point x at infinity and f in Rat( Γ ), wecan prove Lemma 2.2.1 in the same way of [3, Lemma 4].For a tropical subsemimodule M of ( R ∪ {±∞} ) Γ (or of R Γ ), f in M iscalled an extremal of M when it implies f = g or f = g that any g and g in M satisfies f = g ⊕ g . Remark 2.2.2 ([3, Proposition 8]) . Any finitely generated tropical subsemi-module f M of R Γ is generated by the extremals of f M . With the adaptation for ±∞ , we can prove the following lemma in sameway as the above remark. Lemma 2.2.3.
Any finitely generated tropical subsemimodule M of R ( D ) ⊂ ( R ∪ {±∞} ) Γ is generated by the extremals of M . Let
Γ, Γ ′ be tropical curves, respectively, and ϕ : Γ → Γ ′ be a continuousmap. The map ϕ is called a morphism if there exist a model ( G, l ) of Γ and4 model ( G ′ , l ′ ) of Γ ′ such that the image of the set of vertices of G by ϕ isa subset of the set of vertices of G ′ , the inverse image of the relative interiorof any edge of G ′ by ϕ is the union of the relative interiors of a finite numberof edges of G and the restriction of ϕ to any edge e of G is a dilation bysome non-negative integer factor deg e ( ϕ ). Note that the dilation factor on e with deg e ( ϕ ) = 0 represents the ratio of the distance of the images of anytwo points x and y except points at infinity on e to that of original x and y . If an edge e is mapped to a vertex of G ′ by ϕ , then deg e ( ϕ ) = 0. Themorphism ϕ is said to be finite if deg e ( ϕ ) > e of G . For anyhalf-edge h of any point on Γ , we define deg h ( ϕ ) as deg e ( ϕ ), where e is theedge of G containing h .Let Γ ′ be not a singleton and x a point on Γ . The morphism ϕ is harmonicat x if the number deg x ( ϕ ) := X h h ′ deg h ( ϕ )is independent of the choice of half-edge h ′ emanating from ϕ ( x ), where h isa connected component of the inverse image of h ′ by ϕ . The morphism ϕ is harmonic if it is harmonic at all points on Γ . One can check that if ϕ is afinite harmonic morphism, then the numberdeg( ϕ ) := X x x ′ deg x ( ϕ )is independent of the choice of a point x ′ on Γ ′ , and is said the degree of ϕ ,where x is an element of the inverse image of x ′ by ϕ . If Γ ′ is a singletonand Γ is not a singleton, for any point x on Γ , we define deg x ( ϕ ) as zeroso that we regard ϕ as a harmonic morphism of degree zero. If both Γ and Γ ′ are singletons, we regard ϕ as a harmonic morphism which can have anynumber of degree.Let ϕ : Γ → Γ ′ be a harmonic morphism between tropical curves. For f in Rat( Γ ), the push-forward of f is the function ϕ ∗ f : Γ ′ → R ∪ {±∞} defined by ϕ ∗ f ( x ′ ) := X x ∈ Γϕ ( x )= x ′ deg x ( ϕ ) · f ( x ) . The pull-back of f ′ in Rat( Γ ′ ) is the function ϕ ∗ f ′ : Γ → R ∪{±∞} defined by ϕ ∗ f ′ := f ′ ◦ ϕ . We define the push-forward on divisors ϕ ∗ : Div( Γ ) → Div( Γ ′ )5y ϕ ∗ ( D ) := X x ∈ Γ D ( x ) · ϕ ( x ) . One can check that deg( ϕ ∗ ( D )) = deg( D ) and ϕ ∗ (div( f )) = div( ϕ ∗ f ) forany divisor D on Γ and any f in Rat( Γ ) (cf. [1, Proposition 4.2]). In [3], Haase, Musiker and Yu used the term subgraph of a tropical curveas a compact subset of the tropical curve with a finite number of connectedcomponents and defined the chip firing move
CF( f Γ , l ) by a subgraph f Γ of a tropical curve e Γ and a positive real number l as the rational functionCF( f Γ , l )( x ) := − min( l, dist( x, f Γ )), where dist( x, f Γ ) is the infimum of thelengths of the shortest path to arbitrary points on f Γ from x . They provedthat every rational function on a tropical curve is an (ordinary) sum of chipfiring moves (plus a constant) ([3, Lemma 2]) with the concept of a weightedchip firing move . This is a rational function on a tropical curve having twodisjoint proper subgraphs f Γ and f Γ such that the complement of the unionof f Γ and f Γ in e Γ consists only of open line segments and such that therational function is constant on f Γ and f Γ and linear (smooth) with integerslopes on the complement. A weighted chip firing move is an (ordinary) sumof chip firing moves (plus a constant) ([3, Lemma 1]).With unbounded edges, their definition of chip firing moves needs a littlecorrection. Let Γ be a subgraph of a tropical curve Γ which does not haveany connected components consisting only of points at infinity and l a positivereal number or infinity. The chip firing move by Γ and l is defined as therational function CF( Γ , l )( x ) := − min( l, dist( x, Γ )). Lemma 2.4.1.
A weighted chip firing move on a tropical curve is a linearcombination of chip firing moves having integer coefficients (plus a constant).Sketch of proof.
We use the same notations as in their proof. All we haveto do is to show the construction for the case with l = ∞ . Especially,it is sufficient to check the case that Γ consists only of points at infinity.Supposing that Γ has only one point gives only two situations. Firstly, Γ contains a finite point. Then f can be written as ± s · CF( Γ , ∞ ) plus aconstant, where s is the slope of f on the complement. Secondly, Γ consistsonly of one point at infinity. Taking a finite point x , then f can be written6s ± s · (CF( f − ([ f ( x ) , ∞ ]) , ∞ ) − CF( { x } , ∞ )) plus a constant with same s as the first situation. Suppose that Γ has plural points. Γ must contain atleast one finite point. Let x i be the intersection of Γ and the closure of L i .Note that Γ = { x , · · · , x k } , where k is no less than two. With the slope s i of f on e i := L i ⊔ { x i } , f is P ki =1 ( ± s i · CF( Γ \ e i , ∞ )) plus a constant.The next lemma is proven in the same way of [3, Lemma 2] and showsthe appropriateness of this definition. Lemma 2.4.2.
Every rational function on a tropical curve is a linear com-bination of chip firing moves having integer coefficients (plus a constant).
A point on Γ with valence two is said to be a smooth point. We sometimesrefer to an effective divisor D on Γ as a chip configuration . We say that asubgraph Γ of Γ can fire on D if for each boundary point of Γ there areat least as many chips as the number of edges pointing out of Γ . A set ofpoints on a tropical curve Γ is said to be cut set of Γ if the complement ofthat set in Γ is disconnected. R ( D ) K In this section, for an effective divisor D on a tropical curve and a finitesubgroup K of the isometry group of the tropical curve, we find a generatorset of the K -invariant set R ( D ) K of R ( D ) and then, show that R ( D ) K isfinitely generated as a tropical semimodule. When D is K -invariant, R ( D ) / R is identified with the K -invariant linear system | D | K , so | D | K is finitelygenerated by the generators of R ( D ) K modulo tropical scaling. Remark 3.1 ([3, Lemma 6]) . Let e Γ be a tropical curve, e D be a divisor on e Γ and S be the set of rational functions f in R ( e D ) such that the supportof e D + div( f ) does not contain any cut set of e Γ consisting only of smoothpoints. Then (1) S contains all the extremals of R ( e D ) , (2) S is finite modulo tropical scaling, and (3) S generates R ( e D ) as a tropical semimodule. R ( e D ) is a subset of R e Γ ,the proof is applied even in the case that R ( e D ) is a subset of ( R ∪ {±∞} ) e Γ with preparations in Section 2. Also, the above remark throws the relationbetween S and e D into relief, hence hereafter we write S for e D as S ( e D ).Note that a tropical subsemimodule of R ( e D ) is not always finitely generated.Consider the tropical subsemimodule of R ([0]) corresponding to | [0] | \ { [0] } on a tropical curve [0 , Γ be a tropical curve, D an effective divisor on Γ and K a subgroupof the isometry group of Γ . One can expect the relation between R ( D ) and S ( D ) to be analogous to that of their K -invariant counterparts R ( D ) K and S ( D ) K , but in vain. Indeed, the next example objects. Example 3.2.
Let e Γ be a circle and let a map i : f G → f G which transferstwo edges to each other, where f G is the underlying graph of the canonicalloopless model of e Γ . For a point x on e Γ , we choose another point x on e Γ such that i ( x ) = x . For the group e K generated by i and the effective divisor e D = x + x , although S ( e D ) e K is empty, R ( e D ) e K is not empty. It means that S ( e D ) e K is not a generator set of R ( e D ) e K . Now, let us find a generator set for R ( D ) K that corresponds to S ( D ) for R ( D ). In the above situation, K acts on Γ naturally. We define V ( Γ ) asthe set of points x on Γ such that there exists a point y in any neighborhoodof x whose stabilizer is not equal to that of x . Lemma 3.3. V ( Γ ) is a finite set.Proof. We assume that Γ is not the ∞ -metric space obtained from only oneedge with length of ∞ . Let σ : Γ → Γ be an isometry. Then, for any edge e of Γ , the image of e by σ agrees completely with e or the intersection of e and the image of e by σ is contained in the set of the endpoints of e . In fact,if | e ∩ σ ( e ) | is infinite, then σ ( e ) is contained in e because e is an edge of Γ .It means that σ ( e ) = e . If | e ∩ σ ( e ) | is finite and e ∩ σ ( e ) contains a point on Γ other than endpoints of e , then that point has the valence of greater thantwo. It contradicts to the fact that e is an edge of Γ .From the above argument, for any edge e of Γ , we can roughly classifythe situations into four. First, σ is the identity map on e , i.e., σ fixes allpoints on e . Second, σ gives a mirror image of e . In this case, if Γ is a circleconsisting of e , the fixed points on e by σ are only antipodal points on theaxis of symmetry of σ , otherwise, the midpoint of e is fixed by σ , moreover8hen e is a loop, then the vertex connected to e is also fixed by σ . Third, σ acts as a proper rotation on e . This is possible only when Γ is a circle, and σ gives no fixed points on e . Finally, σ maps e onto other edge of Γ , thenonly the endpoints of e may be fixed by σ .Consequently, under the above assumption, since K is a finite set and Γ has finite vertices and edges, V ( Γ ) is a finite set.Let us suppose that Γ is the ∞ -metric space obtained from only oneedge with length of ∞ . Since K is a finite set, any σ in K is not a propertranslation of Γ . Each isometry of Γ other than translations fixes only onepoint on Γ . Thus, also in this case, V ( Γ ) is a finite set. Note that there canexists only one inversion. If there were two distinct, these two can generatea translation, leading | K | to infinity.We set ( G , l ) as the canonical loopless model of Γ . By Lemma 3.3,we obtain the model ( f G , e l ) of Γ by setting the K -orbit of the union of V ( G ) and V ( Γ ) as the set of vertices V ( f G ). Naturally, we can regard that K acts on V ( f G ) and also on E ( f G ). Thus, the sets V ( f G ′ ) and E ( f G ′ ) aredefined as the quotient sets of V ( f G ) and E ( f G ) by K , respectively. Let f G ′ be the graph obtained by setting V ( f G ′ ) as the set of vertices and E ( f G ′ ) asthe set of edges. Since f G is connected, f G ′ is also connected. We obtainthe loopless graph G ′ from f G ′ and the loopless model ( G , l ) of Γ from theinverse image of V ( G ′ ) by the map defined by K . Note that V ( G ) contains V ( f G ). Since K is a finite subgroup of the isometry group of Γ , the lengthfunction l ′ : E ( G ′ ) → R > ∪ {∞} , [ e ]
7→ | K e | · l ( e ) is well-defined, where [ e ]and K e mean the equivalence class of e and the stabilizer of e , respectively.Let Γ ′ be the tropical curve obtained from ( G ′ , l ′ ). Then, Γ ′ is the quotienttropical curve of Γ by K .For any edge e of G , by the Orbit-Stabilizer formula, | K e | is a positiveinteger. Thus, for ( G , l ) and ( G ′ , l ′ ), there exists only one morphism ϕ : Γ → Γ ′ that satisfies deg e ( ϕ ) = | K e | for any edge e of G .We obtain the following lemma as an extension of [2, Lemma 2.2]. Lemma 3.4.
If both Γ and Γ ′ are not singletons, then ϕ is a finite harmonicmorphism of degree | K | .Proof. Clearly, ϕ is finite. Now we check that ϕ is harmonic and its degreeis | K | . Since K is a finite subgroup of the isometry group of Γ , for any point x on Γ and any half-edge h ′ of ϕ ( x ), each connected component of ϕ − ( h ′ )9as the same dilation factor deg h ( ϕ ), where h is a connected componentemanating from x . Therefore, for the edge e of G containing h and itsimage e ′ by ϕ , the following hold:deg x ( ϕ ) = X e h h ′ deg h ( ϕ ) = X e e e ′ deg e ( ϕ ) = | Ke | · | K e | = | K | . Where e h , e e and Ke denote a connected component of ϕ − ( h ′ ), that of ϕ − ( e ′ )and the orbit of e by K , respectively. Note that we use the Orbit-Stabilizerformula at the last equality. Accordingly, we get the conclusion.Note that whether Γ is a singleton or not agrees with whether Γ ′ is asingleton.Is R ( D ) K , the K -invariant set of R ( D ), identical to ϕ ∗ ( R ( ϕ ∗ ( D )))? Noris it. Example 3.5.
Assume the situation of Example 3.2. For a rational function f which decreases from ϕ ( x ) to ϕ ( x ) with slope one and is constant on othergraph, however f is an element of R ( ϕ ∗ ( e D )) , the pull-back of f by ϕ is notin R ( e D ) e K . Next, for R ( D ) K , the following holds. Lemma 3.6. R ( D ) K is a tropical semimodule.Proof. Let c be in R , f, g in R ( D ) K and σ in K . Since R ( D ) is a tropicalsemimodule by Lemma 2.2.1, c ⊙ f and f ⊕ g are in R ( D ). It is obvious that ⊙ and ◦ are associative and that ◦ is distributive over ⊕ from right, both( c ⊙ f ) ◦ σ and ( f ⊕ g ) ◦ σ are in R ( D ) K .Note that R ( D + div( f )) K = R ( D ) K ⊙ ( − f ) for any K -invariant rationalfunction f .The following lemma is an extension of [3, Lemma 5]. Lemma 3.7.
Let f be in Rat( Γ ) . Then, f is an extremal of R ( D ) K if andonly if there are not two proper K -invariant subgraphs Γ and Γ covering Γ such that each can fire on D + div( f ) .Proof. First, let us show the “if” part. Suppose that there are two such sub-graphs Γ and Γ . We can assume that each Γ i does not have any connectedcomponent consisting only of points at infinity. Each Γ i defines a chip firing10ove g i for a small positive number so that g i is zero on Γ i and they are non-positive. As Γ and Γ are K -invariant, so g and g are in R ( D + div( f )) K .Since g ⊕ g = 0 on Γ , we can write f as ( f + g ) ⊕ ( f + g ), i.e. f is notan extremal of R ( D ) K .Next, let us show the “only if” part. Suppose f = g ⊕ g for some g and g in R ( D ) K \ { f } . For i = 1 ,
2, there exists e g i in R ( D + div( f )) K suchthat g i = e g i ⊙ f . Let Γ i be the closure of the loci where e g i = 0. Then, theunion of Γ and Γ is Γ and each Γ i is proper. Since e g i is K -invariant, so is Γ i . Then, each Γ i can fire on D + div( f ).The term “a subgraph is infinite” means that the subgraph is a infiniteset. Lemma 3.8.
Let A be a K -invariant subset of supp( D ) . If ϕ ( A ) is a cut setof Γ ′ and D ( x ) ≥ val( x ) − for any x in A , then there exists a K -invariantinfinite subgraph Γ of Γ which can fire on D and whose boundary points arein A .Proof. For such A , let Γ ′ , · · · , Γ ′ n be distinct connected components of Γ ′ \ ϕ ( A ) respectively. Note that n is no less than two since ϕ ( A ) is a cut setof Γ ′ . Clearly, for any i , the inverse image of the closure of Γ ′ i by ϕ is a K -invariant infinite subgraph of Γ we want.We call a point on Γ not being a vertex of G a K -ordinary point . Notethat if a subgraph of Γ has a K -ordinary point, topologically saying, it shouldhave infinite points. Lemma 3.9.
Let Γ be a K -invariant subgraph of Γ . If Γ is infinite andif the set of its boundary points ∂Γ contains at least one K -ordinary point,then ϕ ( ∂Γ ) is a cut set of Γ ′ and contains a point on Γ ′ not being a vertexof G ′ .Proof. For such Γ , obviously ϕ ( ∂Γ ) contains a point on Γ ′ not being avertex of G ′ . It is sufficient to check that ϕ ( ∂Γ ) is a cut set of Γ ′ . Let Γ be the closure of the complement set of Γ in Γ . This Γ is K -invariant andcontains a K -ordinary point. Thus, Γ is an infinite subgraph. Consequently, Γ ′ \ ϕ ( ∂Γ ) = ϕ ( Γ ∪ Γ ) \ ϕ ( ∂Γ ) = ( ϕ ( Γ ) \ ϕ ( ∂Γ )) ⊔ ( ϕ ( Γ ) \ ϕ ( ∂Γ )).Hence, ϕ ( ∂Γ ) is a cut set of Γ ′ .The next corollary follows from Lemma 3.8 and Lemma 3.9.11 orollary 3.10. For a subset of the support of ϕ ∗ ( D ) , we consider the fol-lowing condition ( ∗ ) :( ∗ ) it is a cut set of Γ ′ containing no vertices of G ′ and whose inverseimage by ϕ is a subset of the support of D . (1) For a subset A of supp( D ) whose image by ϕ satisfies ( ∗ ) , there existsa K -invariant infinite subgraph Γ of Γ which can fire on D and whoseboundary points are in A . (2) Let Γ be a K -invariant subgraph of Γ . If Γ is infinite and can fireon D and if the set of its boundary points consists only of K -ordinarypoints, then the image of the set of boundary points of Γ by ϕ satisfies ( ∗ ) . By Corollary 3.10, it is natural to define S ( D ) K as the set of f in R ( D ) K such that there exist no cut sets of Γ ′ contained in the support of ϕ ∗ ( D +div( f )), containing no vertices of G ′ and whose inverse image by ϕ is a subsetof the support of D + div( f ). In fact, this S ( D ) K is the set corresponding to S ( D ), i.e. S ( D ) K is a generator set of R ( D ) K . Theorem 3.11.
In the above situation, the following hold: (1) S ( D ) K contains all the extremals of R ( D ) K , (2) S ( D ) K is finite modulo tropical scaling, and (3) S ( D ) K generates R ( D ) K as a tropical semimodule.Proof. (1) Suppose f is in the difference set of R ( D ) K from S ( D ) K , thenthere exists a cut set A ′ of Γ ′ contained in supp( ϕ ∗ ( D + div( f ))), containingno vertices of G ′ and such that ϕ − ( A ′ ) ⊂ supp( D + div( f )). By (1) ofCorollary 3.10, there exists a K -invariant infinite subgraph Γ of Γ whichcan fire on D + div( f ) and whose boundary points are in ϕ − ( A ′ ). Then, theclosure of Γ \ Γ can also fire on D + div( f ). Therefore, by Lemma 3.7, f isnot an extremal of R ( D ) K .(2) The push-forward of a rational function on Γ induces a natural map S ( D ) K / R → S ( ϕ ∗ ( D )) / R , [ f ] [ ϕ ∗ ( f )]. In fact, for any f in S ( D ) K , ϕ ∗ ( D + div( f )) = ϕ ∗ ( D ) + ϕ ∗ (div( f )) = ϕ ∗ ( D ) + div( ϕ ∗ ( f )), thus, ϕ ∗ ( f )is in R ( ϕ ∗ ( D )). From f ∈ S ( D ) K , there exist no cut sets of Γ ′ contained12n supp( ϕ ∗ ( D ) + ϕ ∗ (div( f ))), containing no vertices of G ′ and whose inverseimage by ϕ is a subset of supp( D + div( f )). This means that ϕ ∗ ( f ) is in S ( ϕ ∗ ( D )). Also, for any pair of f and f in [ f ], there exists c in R satisfying f = f + c . Since ϕ ∗ ( f ) = ϕ ∗ ( f + c ) = ϕ ∗ ( f ) + ϕ ∗ ( c ) = ϕ ∗ ( f ) + c , themap is well-defined. Now we show that the map is injective. Let [ f ] and[ g ] be distinct elements of S ( D ) K / R , thus div( f ) differs from div( g ). Sinceboth f and g are K -invariant, so their images ϕ ∗ (div( f )) and ϕ ∗ (div( g )) aredifferent, i.e. the map is injective. By Remark 3.1, we get the conclusion.(3) Suppose f ∈ R ( D ) K . Let N ( f ) be the number of distinct K -orbits inthe union of all K -invariant subsets of supp( D + div( f )) which is a cut setof Γ ′ containing no vertices of G ′ . We prove (3) by induction for N ( f ). If N ( f ) = 0, then f ∈ S ( D ) K from the definition of S ( D ) K . Assume that f ∈h S ( D ) K i for all N ( f ) ≤ k , where h S ( D ) K i means the tropical semimodulegenerated by S ( D ) K . We consider the case where N ( f ) = k + 1 and f / ∈ S ( D ) K . Let A be a subset of supp( D + div( f )) whose image by ϕ is a cutset of Γ ′ containing no vertices of G ′ . By (1) of Corollary 3.10, there existsa K -invariant subgraph Γ of Γ which can fire on D + div( f ) and whoseboundary points are in A . Let Γ be the closure of the complement of Γ in Γ . For any x ∈ ∂Γ i , we write the distance between x and its closest vertexof G as l x i . Let l i := min { l x i | x ∈ ∂Γ i } and g i := CF( Γ i , l i ). Then, for both i = 1 , f ⊙ g i is not equal to f and is in R ( D ) K since f, g i ∈ R ( D ) K and f = ( f ⊙ g ) ⊕ ( f ⊙ g ). By the definition of g i , N ( f ) > N ( f ⊙ g i ) and f ⊙ g i ∈ h S ( D ) K i , then f ∈ h S ( D ) K i .By Lemma 2.2.3 and the above theorem, we obtain the following corollary,which is an extension of [3, Corollary 9]. Corollary 3.12.
Let Γ be a tropical curve, D an effective divisor on Γ and K a finite subgroup of the isometry group of Γ . Then, the tropicalsemimodule R ( D ) K is generated by the extremals of R ( D ) K . This generatingset is minimal and unique up to tropical scalar multiplication. If D is K -invariant, R ( D ) K / R is naturally identified with the K -invariantlinear subsystem | D | K . In conclusion, the following statement holds fromTheorem 3.11. Theorem 3.13.
Let Γ be a tropical curve, D an effective divisor on Γ and K a finite subgroup of the isometry group of Γ . If D is K -invariant, then the K -invariant linear subsystem | D | K of | D | is finitely generated by S ( D ) K / R . eferences [1] Mathew Baker, Serguei Norine Harmonic Morphisms and HyperellipticGraphs , Int. Math. Res. Not. (2009)2914–2955.[2] Melody Chan, Tropical curves and metric graphs , Univ. of California,Berkeley thesis, 2012.[3] Christian Haase, Gregg Musiker, Josephine Yu
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