Monodromy of rational curves on toric surfaces

For an ample line bundle $\mathcal{L}$ on a complete toric surface $X$, we consider the subset $V_{\mathcal{L}} \subset \vert \mathcal{L} \vert$ of irreducible nodal rational curves contained in the smooth locus of $X$. We study the monodromy map from the fundamental group of $V_{\mathcal{L}}$ to the permutation group on the set of nodes of a reference curve $C \in V_{\mathcal{L}}$. We show that the monodromy has to preserve a certain obstruction map $\varPsi_{X}$ from the set of nodes of $C$ to a finite set depending solely on $X$. Provided that $\mathcal{L}$ is sufficiently big (in a sense we precise below), we show that the image of the monodromy is exactly the group of deck transformations of the map $\varPsi_{X}$. Along the way, we provide a handy tool to compute the image of the monodromy for any pair $(X, \mathcal{L})$. Eventually, we provide some examples of pairs $(X, \mathcal{L})$ with small $\mathcal{L}$ and for which the image of the monodromy is strictly smaller than expected.


Introduction
For an ample line bundle L on a complete toric surface X , we consider the variety V L ⊂ |L | of irreducible nodal rational curves contained in the smooth locus of X . For a general curve C ∈ V L , any loop in V L based at C induces a permutation on the set of nodes of the curve C . This action is recorded by the monodromy map µ L : π 1 V L ,C → Aut nodes of C .
The map µ L plays an important role in different contexts. First, the image of µ L can be thought of as a first approximation of the fundamental group π 1 V L ,C . From this perspective, the study of the map µ L is in the line of the work [DL81] on fundamental groups of complement to discriminant varieties, see [CL18a], [CL18b] and [Sal17] for recent results. The study of the map µ L contributes also to the Galois theory of enumerative problems in algebraic geometry. We refer to [Har79] and [Vak06] for the foundation and [SW15], [Est19] and [EL18] for recent developpements. Eventually, the works [Har86], [Tyo07] and [Tyo14] illustrate how the Severi Problem on toric surfaces can be decomposed into the study of the map µ L and a separate problem in deformation theory.
Apart from the above references, the author was unable to find contribution to the study of the map µ L in the literature. In particular, the image of µ L seems not to be known whenever X is neither CP 2 nor a Hirzebruch surface.
In order to determine im(µ L ), we investigate possible obstructions to permute the nodes of a curve C ∈ V L . One obstruction comes from the existence of a canonical covering ρ :X → X of X by an other toric surfaceX . IfC ⊂X is a nodal rational curve such that ρ(C ) = C , a node p of C comes from a pair p 1 , p 2 of (possibly identical) points inC mapping to p. The pair p 1 p −1 2 , p 2 p −1 1 is a subset of the kernel of the group homomorphismX ⊃ (C * ) 2 → (C * ) 2 ⊂ X induced by ρ. In Section 4, we encoded the decoration p → p 1 p −1 2 , p 2 p −1 1 of the nodes of C by the obstruction map Ψ X and show that the image of µ L is necessarily a subgroup of the group of deck transformations of Ψ X .
In the present paper, we show that the map Ψ X is the only general obstruction. For any constant ≥ 1, we denote by C ≥ (X ) the set of ample line bundles L on X satisfying deg L |C ≥ for any curve C ⊂ X . We prove the following. Theorem 1 is a consequence of Theorems 4 and 5 proven in Section 6. We provide explicit constants (X ) in Proposition 6.14. However, we do not address the question of minimality of those constants in this paper and the reader interested in a particular pair (X , L ) not falling under the hypotheses of Proposition 6.14 is referred to Theorem 4.
The obstruction map Ψ X already appeared implicitly in [Tyo14, §4.1]. Recall that a singular point of X is necessarily a toric fixed point p ∈ X . In such case, the point p is a cyclic quotient singularities C 2 (Z/m p Z) for a unique integer m p ≥ 2. It turns out that the cardinal of the target of Ψ X is gcd({m p })/2 +1 where the gcd is taken over all toric fixed points p ∈ X (take m p = 1 for smooth p). In particular, the map Ψ X does not provide any obstruction when X is smooth.

Conjecture 2. For any smooth complete toric surface X , the image of the monodromy map µ L is the full permutation group on the nodes of C .
The latter is motivated by computations using the present methods and other methods involving tropical geometry. With the above conjecture, we wish to emphasize that there is a substantial room for improvement in Theorem 1. The latter theorem and the nature of the map Ψ X imply that the surjectivity (or non-surjectivity) of the map µ L is an intrinsic property of the toric surface X , for sufficiently big L at least. In Section 7, we show that the general situation is slightly more subtle. We provide several examples of pairs (X , L ) for which Ψ X is trivial and µ L is however not surjective. For a more intrinsic definition, consider the lattice N dual to M , that is, the lattice of oneparameter subgroups of X • . In dimension 2, a fan F ⊂ N R := N ⊗ R is a collection of half-rays supported by primitive vectors in N submitted to some extra properties. Call the latter collection of vectors the support of the fan F . The toric surface X is obtained by gluing affine charts coming from the cones of the fan F , see [Ful93,§1.4]. Practically, the support of the fan F giving rise to X is the collection of inner normals to the edges of the polygon ∆ ⊂ M R . The notion of inner or outer normal depends on the choice of an orientation on N that we fix once and for all.
Through the text, we denote by n the number of rays of the fan F ⊂ N R defining the toric surface X and fix once and for all a counter-clockwise cyclical indexation of these rays in Z/nZ. For any j ∈ Z/nZ, we denote by n j ∈ N the primitive integer vector supporting the j t h ray of F . The group action of X • onto itself extends to the whole X . To each ray of F corresponds the closure of an X • -orbit of dimension 1 in X that we refer to as a toric divisor of X . For any j ∈ Z/nZ, we denote by D j CP 1 the orbit corresponding to the j t h ray of F . The classes of the divisors D j , j ∈ Z/nZ generate the Picard group Pic(X ) H 2 (X , Z), see [Ful93,§3.4]. In particular, any divisor class [D] ∈ Pic(X ) is determined by its intersections multiplicities with the D j , j ∈ Z/nZ. On the toric surface X , a line bundle L is ample if and only if it is very ample if and only if j := deg L |D j > 0 for any j ∈ Z/nZ, see [CLS11, Theorems 6.3.13 and 6.1.14]. As mentioned above, the integers j , j ∈ Z/nZ, determine the line bundle L . Moreover, they satisfy j ∈Z/nZ j · n j = 0 (1) Conversely, every collection j j ∈Z/nZ satisfying the above equation is given by j = deg L |D j for some line bundle L on X , see [CLS11, Proposition 6.4.1]. This gives an explicit description of the set of ample line bundles C (X ) on X as formal sums over the rays of F . For ≥ 1, we define An equivalent description of C (X ) can be given in terms of lattice polygons in M R . The polygon ∆ ⊂ M R representing a line bundle L ∼ = j j ∈Z/nZ is given as the region bounded by the concatenation of the vectors 1 · v 1 , ..., n · v n where v j ∈ M is the negative normal to n j ∈ N . By equation (2), the concatenation closes up and ∆ is a well defined lattice polygon whose edges are indexed by Z/nZ. Denote by ∆ j the j t h edge of ∆ and by ∆ j , j +1 the vertex ∆ j ∩ ∆ j +1 . Note that, by construction, the vertex ∆ n ∩∆ 1 is the origin 0 ∈ M . It follows that line bundles in C (X ) are in bijective correspondence with lattice polygons ∆ in M R whose (inner) normal fan is exactly F and such that ∆ n ∩ ∆ 1 = 0. We will denote by L ∆ ∈ C (X ) the line bundle corresponding to ∆. For a point C ∈ |L |, we abusively denote by C the corresponding curve C ⊂ X . We denote by V L ⊂ |L | the space of irreducible nodal rational curves contained in the smooth locus of X . For simplicity, we also denote V ∆ := V L ∆ . According to [Tyo07,Proposition 4.1], the set V L is non-empty and irreducible (recall here that L is ample). Moreover, each curve C ∈ |L | can be parametrized explicitly as follows, see [Tyo07,Section 4]. Choose coordinates X • (C * ) 2 and let n j = (α j , β j ) in the induced coordinates on M . If L ∼ = j j ∈Z/nZ , then any irreducible rational curve C ∈ |L | admits a parametrization of the form where z 0 , w 0 ∈ C * and a j ,l ∈ CP 1 = C ∪ {∞}. In the above formula, any factor t − a j ,l with a j ,l = ∞ is to be replaced with the constant factor 1. Such a representation is unique up to the action of PGL 2 (C) on the parameter t . As a consequence, the variety V L has dimension |∂∆ ∩ M | − 1. Recall also that any curve C ∈ V L has exactly | int Z (∆)| nodes, see [Kho78]. Let us now define the monodromy map µ L . Below, we use Aut(E ) to denote the group of permutations on a finite set E and Aut( f ) to denote the group of deck transformations of a map f between finite sets. For a reference curve C ∈ V L , we define the monodromy map as follows. Let γ := C θ θ∈[0,1] ⊂ V L be a loop based at C , that is C = C 0 = C 1 . For any trivialization such that Φ(0, _ ) = id, the permutation Φ(1, _ ) on the set of nodes of C only depends on the class [γ] ∈ π 1 V L ,C . Then, we define µ L [γ] := Φ(1, _ ) −1 .

Rational simple Harnack curves
In Section 4, we will see that the obstructions of the monodromy map µ L can be explicitly described when the reference curve C ∈ V L is a rational simple Harnack curve. In this section, we introduce the required knowledge about these algebraic curves.
In order to define the latter curves, we need and fix coordinates (z, w) on X • (C * ) 2 . The complex conjugation on X • (C * ) 2 extends to an anti-holomorphic involution conj on X . A curve C ∈ |L | is real if conj(C ) = C and we denote RC the fixed locus of conj |C . Recall the amoeba map Definition 3.1. A real curve C ∈ |L | is a (possibly singular) simple Harnack curve if the restriction of the amoeba map A : C ∩ (C * ) 2 → R 2 is at most 2-to-1.
The above definition is shown to be equivalent to the original definition [MR01, Definitions 2 and 3] in [MR01, Theorem 1]. In order to grasp some of the geometry of these curves, we can observe that the amoeba of a smooth simple Harnack C curve is modelled on the polygon ∆ pierced at the points int(∆) ∩ M , see for instance [Lan15, Figure 2], [CL18a, Figure 1] and [Ola17, Figure 1]. The boundary of A (C ) is exactly A (RC ), see [Mik00,Lemma 8]. In particular, the embedding of RC in RX ∆ can be recovered from A (C ), or equivalently from ∆, see for instance [Mik00,Figures 4 and 5]. Topologically, the curve C is obtained as the inflation of its amoeba. The real locus RC ⊂ C is dividing and has the maximal number of connected components, that is p a C + 1.
Singular simple Harnack curves are now obtained from smooth simple Harnack curves by contracting some of the holes of the amoeba. The latter procedure results in the contraction of some of the compact ovals of RC ⊂ (C * ) 2 to points. A local model for such contractions is given by z 2 +w 2 = ε, 0 ≤ ε < 1. In particular, the only singularities of simple Harnack curves are real isolated double points.
The existence of smooth simple Harnack curves in |L | is guaranteed by [Mik00, Corollary A4]. For singular ones, the existence is addressed in [KO06, Theorem 6], [Bru15, Theorems 2 and 10], [CL18a, Theorem 3] in various contexts. For rational simple Harnack curves, there is an explicit construction. Recall that any real rational curve C ∈ |L | admits a parametrization as in (3) with z 0 , w 0 ∈ R * and a j ,l ∈ RP 1 . Fix an orientation on RP 1 so that the collection of parameters a j ,l ∈ RP 1 inherits a cyclical ordering. As a consequence of [Bru15, Theorem 10], we have the following.
Proposition 3.2. A real rational curve in V L is a simple Harnack curve if and only if it can be parametrized as in (3) with z 0 , w 0 ∈ R * and a j ,l ∈ RP 1 such that for any j ∈ Z/nZ and 1 ≤ l ≤ j , we have a ( j −1), j −1 < a j ,l < a ( j +1),1 .
In particular, there always exist rational simple Harnack curves in the linear system |L |.
Recall that for smooth simple Harnack curves C ∈ |L ∆ |, the order map ord of [FPT00] establishes a bijective correspondence between the set of compact connected components of RC ∩ (C * ) 2 and the set of lattice points int Z (∆) := ∆ ∩ M , see [Mik00,Corollary 10]. We now aim to extend this correspondence to rational simple Harnack curves. To do so, we use the existence of deformations of rational simple Harnack curves to smooth simple Harnack curves. The existence of such deformations can be proven using the machinery of [KO06, §4.5] in general toric surfaces, see [Ola17]. As a more direct approach, take P (z, w) to be a real Laurent polynomial of Newton polygon ∆ defining C . Then, we can find a real polynomial R with Newton polygon ∆ such that, for any node ν ∈ C , we have R(ν) > 0 (respectively R(ν) < 0) if Hess P (ν) is positive definite (respectively negative definite). A small deformation of P in the direction of R gives the desired smoothing.
According to [Mik00,Lemma 11], the map ord on a smooth simple Harnack curve C can be described as follows. Let c ⊂ RC ∩ (C * ) 2 be the unique connected component joining the two consecutive toric divisors D n and D 1 . For any compact component c of RC ∩ (C * ) 2 , draw a path γ ⊂ A (C ) joining A (c) to A (c ). By the 2-to-1 property of the amoeba map, the lift A −1 (γ) is a loop in C ∩(C * ) 2 invariant by complex conjugation. There is exactly one orientation of the latter loop such that the corresponding homology class (a, b) ∈ H 1 ((C * ) 2 , Z) satisfies (−b, a) ∈ int Z (∆) (note the sign mistake in the sixth line of the proof of [Mik00, Lemma 11]). Then, we have ord(c ) = (−b, a). Definition 3.3. For a rational simple Harnack curve C ∈ V L ∆ , a node ν ∈ C , define the order map where (a, b) ∈ H 1 ((C * ) 2 , Z) is the homology class of the (carefully oriented) loop The fact that the (−b, a) ∈ int Z (∆) (with the appropriate orientation of A −1 (γ)) follows from continuity by applying the above construction to a nearby smooth simple Harnack curve. Proof. This is a direct consequence of the existence of smoothings and the fact that the order map on smooth simple Harnack curve is bijective, see [Mik00, Corollary 10].
For an ample line bundle L = L ∆ on X and a rational simple Harnack curve C ∈ V L , define the monodromy map by "composition" of the monodromy map µ L of (4) with the order map of Definition 3.3. Formally, we have µ ∆ [γ] = ord •Φ 1, _ • ord −1 where Φ is the trivialization used to define µ L in (4).

Obstructions
In this section, we investigate the obstructions to permute nodes of a reference curve C ⊂ X along loops in V L .
Let X ⊃ X • be the complete toric surface constructed from a fan F ⊂ N R , where N is the lattice of one-parameters subgroup of the algebraic torus X • . LetÑ ⊂ N be the sublattice generated by the support of the fan F . Regarded as a fan inÑ R , the fan F gives rise to a different complete toric surfaceX whenever the inclusionÑ ⊂ N is strict. If we denote byM the lattice dual toÑ and definẽ X • = Hom g r M , C * , we have the following short exact sequences The second and third sequences are obtained by applying successively the functors Hom g r _ , C * and Hom g r _ , Z to the first sequence. AsÑ has finite index in N , there exists for any n ∈ N an integer λ ∈ Z such that λn ∈Ñ . For m ∈M , we can define n, m := m λn λ ∈ Q. This construction leads to a pairing N /Ñ ×M /M → C * (n, m ) → e 2i π〈n,m 〉 inducing an isomorphism N /Ñ Hom g r M /M , C * =: G X . We wish to emphasize that a choice of coordinates on either of X • , N or M induces coordinates on the two other spaces. Note also that we have a priori non-natural isomorphismM /M N /Ñ . Coordinates on N andÑ provide such an isomorphism. According to [CLS11, Proposition 3.3.7], the group homomorphismX • → X • extends to a morphism of toric surfaces ρ :X → X realizing X as the quotient ofX by the group G X ⊂X • . The group G X acts freely onX except at the fixed point of theX • -action. It implies in particular that all the X • -fixed point in X are cyclic quotient singularities when G X is non-trivial.
Any line bundle L ∈ C (X ) with intersection sequence j j ∈Z/nZ pulls back to a line bundlẽ L ∈ C (X ) with the same intersection sequence. The restriction of the induced map ρ * : |L | → |L | to VL necessarily lands in the space of irreducible rational curves in |L |, with singularities possibly worst than nodes. However, we clearly have ρ −1 * V L ⊂ VL . In the lemma below, we denote by Q X the quotient of the group G X that identifies pairs of inverse elements. Formally, we have Q X = {g , g −1 } | g ∈ G X .
Lemma 4.1. The map ρ * : |L | → |L | induces a |G X |-to-1 map from ρ −1 * V L ⊂ VL to V L . For any curve C ∈ V L and any curveC ∈ VL such that ρ(C ) = C , any double point ν ∈ C is the image by ρ of two pointsp 1 ,p 2 ∈C in the same G X -orbit inX . If g ∈ G X is such that g ·p 1 =p 2 , the correspondence ν → p 1 ,p 2 defines a map Ψ X ,C : nodes of C → Q X ν → g , g −1 that is independent of the choice of the curveC .
In the above lemma, the two pointsp 1 andp 2 may coincide, in which case the double point ν ∈ C is the image of a double point ofC .
Proof. Let us describe the map ρ * : VL → |L | in coordinates. Choose coordinates on M andM and let (z, w) and (z,w) be the induces coordinates on X • andX • respectively. Denote by A := a b c d ∈ GL 2 (Z) the matrix of the map M →M where we use line vectors. The mapÑ → N is given by A t in the dual bases and the mapX • → X • is given by (z,w) → z aw b ,z cw d . According to (3), any curvẽ C ∈ VL can be parametrized bẽ where n j = (α j ,β j ) in the coordinates given onÑ . Note that if n j = (α j , β j ) in the coordinates given on N , then (α j , β j ) = (α j ,β j ) · A t . Then, the curve ρ(C ) is parametrized by As can be read from the above parametrization, the image curve is in |L |. After moding out the action of PGL 2 (C) on the parameter of bothφ and φ, we deduce that there are exactly |G X |-many parametrizationsφ leading to a given parametrization φ: these parametrizations correspond to the choice of pairs (z 0 ,w 0 ) such that ρ(z 0 ,w 0 ) = (z 0 , w 0 ) for a given pair (z 0 , w 0 ). This proves the first part of the statement. For the second part of the statement, the pointsp 1 ,p 2 are given respectively byφ(t 1 ) andφ(t 2 ) where t 1 , t 2 ∈ CP 1 are the two distinct points defined by φ(t 1 ) = φ(t 2 ) = p. It remains to show that g is uniquely defined and independent of the choice of the preimage curveC . The action of G X is free except at the torus-fixed point ofX so that g is uniquely defined except ifp 1 =p 2 is a fixed point. In the latter case, the point p is a fixed point in X , that is a singular point of the surface X . It implies that ν is a higher singularity than just a node in C . This leads to a contradiction and implies that g is unique. As seen above, the curvesC satisfying ρ(C ) = C differ from one another by a toric translation in G X . Hence, the element g does not depend on the choice ofC . Proof. Let γ : S 1 → V L be a loop based at C . For any θ ∈ S 1 , the map Ψ X ,C θ defined on the curve C θ := γ(θ) is continuous in θ. It follows that the permutation on the nodes of C induced by the loop γ has to preserve the fiber of the map Ψ X ,C .
In the rest of the text, we will simply denote Ψ X ,C by Ψ X . We now describe the obstruction map Ψ X in terms of the order map given at the end of Section 2. Let ∆ ⊂ M R be the lattice polygon  Proof. Below, we show that the maps Ψ X and Ψ ∆ •ord coincide under the appropriate identification Q X Q ∆ . The second part of the statement follows from Corollary 4.2.
Recall the notations introduced in Section 2 and let us fix coordinates on N such that n 1 = (1, 0). It follows that M ∆ = αZ ⊕ Z where the integer α satisfies both α = gcd {α j } 1≤ j ≤n and α = M ∆ : M . We can then identify Q ∆ 0, 1, ..., α/2 and Ψ ∆ (n, m) = d(n, αZ) where d is the Euclidean distance between closed subsets of R. At last, fix coordinates onÑ such that n 1 = (1, 0). Then, the map X • → X • is given by (z,w) → z α ,w and G X is identified with the group of α t h roots of unity in thẽ z-coordinate. As in Lemma 4.1, we consider a curveC ∈ VL parametrized bỹ such that ρ(C ) = C . If we denoteφ =: (φ 1 ,φ 2 ) the coordinate functions, then C is parametrized by φ := (φ 1 , φ 2 ) = φ α 1 ,φ 2 . Let us now describe Ψ X and Ψ ∆ • ord in coordinates. On the one hand, any double point p ∈ C parametrized by φ corresponds to an unordered pair {t , The only crucial point here is that the latter loop is the the image by φ of a path : [0, 1] → CP 1 joining t to t , see Definition 3.3. We have then that Ψ ∆ ord(p) = d (a, αZ). Now, we deduce from Cauchy's integral formula that Under the above identification Q X Q ∆ , the latter implies that the maps Ψ X and Ψ ∆ • ord coincide. This concludes the proof.
Proof. Let α := M ∆ : M with α ≥ 2. Then, there are exactly α/2 ≥ 1 classes in Q ∆ that are distinct from the class of the lattice M ∆ . In order to prove the statement, it suffices to show that there exists a lattice point in int Z (∆) that projects to any such class. Indeed, the map Ψ ∆ will have strictly more than one fiber under the assumption of the statement. As the monodromy µ ∆ has to preserve these fibers by Proposition 4.3, the map µ ∆ cannot be surjective. First, we claim that there exists a lattice triangle T ⊂ ∆ such that T ∩ ∂∆ ∩ M = vertices of T . To see this, consider a lattice triangle T ⊂ ∆ obtained as the convex hull of three consecutive lattice points on ∂∆. If T = ∆, then we are done. If ∆ = T and T does not prove the claim, the convex hull of any other triple of consecutive and non-collinear lattice points on ∂∆ will.
Take T ⊂ ∆ a lattice triangle such that T ∩ ∂∆ ∩ M = vertices of T . By construction, the vertices of T are in M ∆ . We claim that Ψ ∆ (T ∩ M ) \ vertices of T always contains Q ∆ \ 0. To see this, take coordinates on M such that (0, 0), (0, 1) and (p, αq) are the vertices of T where m, p, q ∈ Z >0 . In particular, we have M ∆ = Z ⊕ αZ and Ψ ∆ (n, m) = d m, αZ when identifying Q ∆ 0, 1, ..., α/2 . The triangle T contains exactly one lattice point on the union of the two horizontal lines of respective heights h and αq − h, for any integer 0 < h < αq. Indeed, there is exactly one lattice point at height h in the interior of the parallelogram conv T, (p, αq + 1) . Since Ψ ∆ maps this point to d h, αZ and h can assume all the integer values between 0 and αq, the claim follows.
Remark 4.5. If p j ∈ X is the toric fixed point corresponding to the vertex ∆ j , j +1 , then p j is a singular point of X if and only if m j := n j , n j +1 : N ≥ 2. In such case, the point p j is a cyclic quotient singularity of X of type C 2 (Z/m j Z), see [Ful93,Chapter 2,§2.2]. From there, this is not hard to see that M ∆ : M = gcd {m j } j ∈Z/nZ . In particular, we have |Q ∆ | = gcd({m j })/2 + 1.

Monodromy in weighted projective planes
In this section, the lattice polygon ∆ ⊂ M R is a triangle such that Z (∆ 1 ) ≥ 2. In particular, the associated toric surface X ∆ is a weighted projective plane. We define the subset V ∆,1 ⊂ V ∆ ⊂ |L ∆ | to be the space of rational curves having only one intersection point with each of the two toric divisors D 2 , D 3 ⊂ X ∆ . We require moreover that these two points are distinct from the three toric fixed points of X ∆ .
Recall that if C ∈ V ∆ is a rational simple Harnack curve, we can consider the monodromy map µ ∆ : π 1 V ∆ ,C → Aut int Z (∆) . According to Proposition 3.2, the simple Harnack curve C can be taken in V ∆,1 by simply requiring that all the parameters a 2,l (respectively a 3,l ) are equal to each other. In particular, the map µ ∆ restricts to V ∆,1 without ambiguity.
Let M ∆,1 ⊂ M be the lattice generated by ∆ 1 ∩ Z 2 and the vertex ∆ 2 ∩ ∆ 3 . Denote by Q ∆,1 the quotient of the cyclic group M M ∆,1 that identifies pairs of opposite points. Finally, define the obstruction map Ψ ∆,1 : int Z (∆) → Q ∆,1 to be the restriction of the quotient map M → Q ∆,1 . The main statement of this section is the following.
Lemma 5.1. Up to a toric translation in X ∆ , any curve C ∈ V ∆,1 admits a parametrization of the form where {a} := a 1 , ..., a ⊂ C * .
Proof. By definition of V ∆,1 , the two points C ∩ D 2 and C ∩ D 3 are distinct. Up to the action of PGL 2 (C) by pre-composition on the parametrization (3), we can assume that C ∩D 2 is parametrized by ∞ and C ∩ D 3 by 0. By definition of V ∆,1 again, none of the remaining parameters a j ,l of (3) can be equal to either 0 or ∞, otherwise the curve C would pass through a toric fixed point. Then, the parametrization (3) gives We recover the announced formula after translating by (z −1 For any integer k ∈ 1, ..., q/2 such that q = 2k and any parameter {a} := {a 1 , ..., a } ⊂ C * , define the polynomial where σ j is the elementary symmetric polynomial of degree j on variables. In the case q = 2k, the formula above leads to a polynomial whose support is contained in an affine sublattice of index 2.
For technical reason, we define instead the polynomial with full support where ε( j ) = ( j − 1)/2 if p is even and ε( j ) = j /2 otherwise. For any k ∈ 1, ..., q/2 , let us denote by J k ⊂ {0, ..., } the support of the polynomial P k,{a} . Define also the multiplier λ k ∈ C whose coordinates λ k, j are given by Proposition 5.2. Let C ∈ V ∆,1 parametrized by φ := φ {a} as in (6). Then, we have the following bijective correspondence Assume moreover that C is a simple Harnack curve, then we have im P(k, _ ) = ord •Ψ ∆,1 −1 (k). In Corollary 5.3. The restriction of µ ∆ to π 1 V ∆,1 ,C maps to the group of deck transformations of Ψ ∆,1 .
Proof. The proof is similar to the proof of Corollary 4.2.
Proof of Proposition 5.2. By assumption, the curve C has only nodes as singularities. Any node ν ∈ C corresponds to an unordered pair {t , t } ⊂ C * such that φ(t ) = φ(t ). By (6), the latter equality is equivalent to In particular, there exists a unique k ∈ 1, ... q/2 such that t t , t t = e 2i πk/q , e −2i πk/q . If q = 2k, there exists a unique t ∈ C * such that φ(t ) = φ t e 2i πk/q = ν. Moreover, the parameter t where the penultimate equivalence is obtained after multiplying both sides by 1/2i e i πkp q . Conversely, any root t of P k,{a} is such that φ(t ) = φ t e 2i πk/q is a node of C . The case q = 2k is similar, except that the multiplication by e 2i πk/q = −1 is an involution so that we cannot distinguish t and t as above. This is why we adapted the definition of P k,{a} in (8).
So far, we showed that the map P is surjective and that each restriction P(k, _ ) is injective. If there exists a node ν ∈ C in the image of P(k, _ ) and P(k , _ ) for k = k , then ν admits strictly more than two preimages by φ. This is a contradiction with the fact that ν is a node. We deduce that P is a bijection.
Assume now that C is a rational simple Harnack curve. Consider a node ν ∈ C corresponding to a root t of the polynomial P k,{a} . In order to prove that Ψ ∆,1 ord(ν) = k, we proceed as in the proof of Proposition 4.3. On the one hand, we have ord(p) = (−b, a) where (a, b) ∈ H 1 (C * ) 2 , Z is the homology class of a certain loop in C ∩ (C * ) 2 . The only crucial point here is that the latter loop is the the image by φ of a path : [0, 1] → CP 1 invariant by complex conjugation and joining t to t e 2i πk/q . We have then that Ψ ∆,1 ord(ν) = d (a, qZ). On the other hand, denoting φ := (φ 1 , φ 2 ), we deduce from Cauchy's integral formula that This concludes the proof.
Remark 5.4. For a rational simple Harnack curve C ∈ V ∆,1 , we can actually show that all the roots of P k,{a} belong to the punctured line e −i πkp/q R * ⊂ C * and that their distribution on the connected components of the punctured line corresponds to the distribution of the points of Ψ −1 ∆,1 (k) ⊂ int Z (∆) on the heights k and q − k, see Figure 1.
Figure 1: The correspondence P for ∆ := conv (0, 0), (5, 0), (7, 6) and C ∈ V ∆,1 the simple Harnack curve parametrized by φ {1} (t ) = t 6 , (t − 1) 5 /t 7 . On the left, we represented the parameter t ∈ CP 1 mapped to nodes of C by φ {1} . Each node corresponds to a pair of complex-conjugated points. The roots of P k,{1} are distributed on the punctured line e −i πk7/6 R * for k ∈ 1, 2, 3 . In the center, we represented the lattice polygon ∆ together with its interior points. Finally, we represented the set Q ∆,1 on the right. The coloring of the points in CP 1 and int Z (∆) is preserved under the maps ord • φ {1} and Ψ ∆,1 .
Under the correspondence P of Proposition 5.2, it is sufficient, in order to prove Theorem 3, to show that we can permute the roots of the polynomials P k,{a} independently on k while moving in the space of parameters {a}. To that aim, we study the discriminants of the polynomials P k,{a} .
Definition 5.5. For any k ∈ 1, ..., q/2 , define the discriminant D k ⊂ Sym C * as the set of parameters {a} := {a 1 , ..., a } for which the polynomial P k,{a} has a multiple root in C * .
Recall from [GKZ08, Ch.4, §2.D, Proposition 2.7] that the map from the root-space to the coefficient- is an isomorphism onto its image C * × C −1 . For a fixed non-empty support J ⊂ 0, ..., , consider the projection onto the space of univariate polynomials with support contained in J .
where 1 J is the indicator function of J ⊂ 0, ..., . Let D J denote the J -discriminant in the space of univariate polynomials with support J . The J -discriminant is empty if |J | ≤ 2 and it is a reduced, irreducible algebraic hypersurface otherwise, see [GKZ08, Ch.9, §1.A]. For any k ∈ 0, 1, ..., q/2 , the discriminant D k ⊂ Sym C * is given by where λ k refers abusively to the coordinates-wise multiplication by the multiplier λ k . As the map R • P J k • λ k is essentially a linear projection, it follows that the discriminant D k ⊂ Sym C * is a reduced, irreducible algebraic hypersurface.
Proof. In the case q = 2k, the support J k consists of consecutive elements and |Ψ −1 ∆,1 (k)| = |J k |. The result follows in this case. Assume now that q = 2k. An element j ∈ 0, ..., is in the complement of J k if and only if In particular, the support J k contains consecutive elements except when = 2 and J k = {1}. In the latter case, we have |Ψ −1 ∆,1 (k)| = 1. This proves the statement in the case q = 2k.
Recall that the J -discriminant D J on the space of polynomials j ∈J c j · t j with support J is nonempty if and only if there exists such a polynomial with a double root at t = 1. The latter is equivalent to the coefficients c j satisfying j c j = j j · c j = 0. A general polynomial in D J is of the form j ∈J c j · c j · t j where c ∈ C * and j ∈J c j · t j is singular at t = 1. Using this description, one recovers that D J is empty if |J | ≤ 2 and that it is a reduced, irreducible algebraic hypersurface otherwise. Also, one checks easily that the only vector λ ∈ C * J such that λ · D J = D J is the vector (1, ..., 1).
Proof of Proposition 5.6. According to the description (9), the discriminant D k ⊂ Sym C * is empty if and only if the discriminant D J k is empty, if and only if |J k | ≤ 2. There are two cases: either J k = j , j +1 and then |Ψ −1 ∆,1 (k)| = 1 by Proposition 5.2, or J k does not generate Z in which case |Ψ −1 ∆,1 (k)| ≤ 1 by Lemma 5.7. This proves the first part of the statements.
Let us now show that if D k and D k are non-empty for k = k , then D k and D k are distinct. Again, we will distinguish two cases: either the supports J k and J k coincide or they do not. Assume first that J k = J k . By the description (9), the equality D k = D k is equivalent to the J k -discriminant D J k being invariant by translation by λ := λ k λ k ∈ C * J k . By Proposition 5.2, the roots of the polynomial P k,{a} are distinct from the root of P k ,{a} for general {a}. It implies that the multipliers λ k and λ k are linearly independent and then that λ = (1, ..., 1). By the above discussion, the discriminant D k is not invariant by λ. It follows that D k and D k are distinct.
Assume at last that J k = J k and take j ∈ J k \ J k . For any point c ∈ R D J k , the set R D k contains the line c + C { j } (we had the j t h coordinate line), as j ∉ J k . However, one sees easily from the above description of the J -discriminants that R D k does not contain such a line since j ∈ J k . It follows that D k and D k are distinct and the statement is proven.
Proof of Theorem 3. Fix {a} ∈ Sym C * \ k D k , and consider the monodromy map for any k ∈ 1, ..., q/2 such that D k is non-empty (there are no roots to permute otherwise, by Propositions 5.2 and 5.6). By [Est19, Theorem 1.3], we know that the map µ k is surjective provided that the support J k of P k,{a} generates Z as an affine lattice. According to Lemma 5.7, we are in this situation and the map µ k is therefore surjective. We claim that the product map of the µ k µ : is also surjective. Indeed, for a generic line L ⊂ Sym C * passing through {a}, the latter inclusion induces an isomorphism π 1 L \∪ k D k , {a} −→ π 1 Sym C * \∪ k D k , {a} , see [HT73, Théorème (0.2.1)]. As the hypersurfaces D k are pairwise distinct by Proposition 5.6, we can take L so that L∩D k ∩D k = ∅ for any pair k = k . For such a line L, we clearly have a surjection π 1 L\∪ k D k , {a} k π 1 L \ D k , {a} where each factor is induced by the inclusion map. Now, the surjectivity of each of the maps µ k imply that the restriction of µ to π 1 L \ ∪ k D k , {a} is surjective as well.
According to the correspondence of Proposition 5.2, the surjectivity of µ implies that µ ∆ π 1 V ∆,1 ,C contains the subgroup Aut Ψ ∆,1 . By Corollary 5.3, the latter containment is an equality and the result follows.
6 Monodromy in general toric surfaces

Patchworking monodromy
In this section, we show how the inclusion of a triangle T ⊂ ∆ implies the existence of a subgroup of im(µ ∆ ) nearly isomorphic to the group Ψ T,1 of Theorem 3. Our strategy relies mainly on the use of Viro's Patchworking, see for instance [Vir08] and reference therein.
Note that the lattice polygon T := conv(w) ⊂ ∆ is a triangle. Note also that we did not specify the behaviour of the elements of G on ∂T ∩ int Z (∆). Note at last that when ∆ = conv(w), we have Ψ w = Ψ ∆,1 (for the appropriate indexation) and G w = Aut Ψ w . The main result of this section is the following.

Theorem 4. For any wedge w in ∆, the image of µ ∆ contains a w-group.
There are three cases to consider depending whether T := conv(w) has exactly 1, 2 or 3 of its edges on ∂∆. In the latter case, the polygon ∆ is itself a triangle and Theorem 4 is a consequence of Theorem 3. In the rest of this section, we restrict to the first case. The second case requires no extra arguments, simply different notations.
Let us assume that T has exactly 1 edge on ∂∆. Denote and the remaining edges of T so that , and are ordered counter-clockwise on ∂T . We fix coordinates on M such that the edge is the segment joining (0, 0) to ( , 0) and such that the vertex ∩ has coordinates (p, q) for some p, q ∈ Z >0 . The polygon T induces the subdivision ∆ := ∆ ∪ T ∪ ∆ into lattice polygons satisfying ∆ ∩T = and ∆ ∩T = . The latter subdivision is given as the domain of linearity of the piecewise linear convex function ν : ∆ → R defined by For the positive integer m := 1 + max ν(a, b) | (a, b) ∈ ∆ , define in turn the lattice polytope By construction, the projection onto ∆ identifies the non-vertical facets of ∆ ν with ∆ , T , ∆ and ∆. Therefore, the corresponding toric divisors of the toric 3-fold X ∆ ν identify with X ∆ , X T , X ∆ and X ∆ respectively. Denote X := X ∆ ∩ X T and X := X ∆ ∩ X T the torus-orbit of dimension 1 in X ∆ ν .
The coordinates on ∆ × R ⊃ ∆ ν induce coordinates (x, y, z) ∈ (C * ) 3 on the torus of X ∆ ν . It follows from [Ful93, §3.3] that the z-coordinate realizes a linear equivalence in ∆ ν between the divisors X ∆ (at z = ∞) and X ∆ ∪ X T ∪ X ∆ (at z = 0). In particular, the closure of the 1-parameter subgroup x = a, y = b ⊂ X ∆ ν intersects both X ∆ and X T transversally at one point, for any (a, b) ∈ (C * ) 2 . Therefore, we can define a vertical projection π along the latter subgroups, both upward and downward, landing in the respective tori of X ∆ and X T . We denote both projections by π and equip the tori of X ∆ and X T with coordinates (x, y) such that π(x, y, z) = (x, y) both upward and downward. Note that π induces an isomorphism from any horizontal slice {z = c} ⊂ X ∆ ν to X ∆ and an other isomorphism from (x, y, c) ∈ (C * ) 3 to (C * ) 2 ⊂ X T .
Similarly, we have that any interior point of the divisor X ∆ (respectively X ∆ ) is the limit point of a subgroup of the form x = az q , y = bz −p respectively x = az −q , y = bz p− . As above, we equip the tori of X ∆ and X ∆ with coordinates (x, y) such that the map π : (C * ) 3 → (C * ) 2 ⊂ X ∆ given by π (x, y, z) = (xz −q , y z p ) is the projection along the subgroups x = az q , y = bz −p and the map π : (C * ) 3 → (C * ) 2 ⊂ X ∆ given by π (x, y, z) = (xz q , y z −p ) is the projection along the subgroups x = az −q , y = bz p− . Again the maps π and π induce isomorphisms from (x, y, c) ∈ (C * ) 3 to (C * ) 2 ⊂ X ∆ and (C * ) 2 ⊂ X ∆ respectively.
We now give a counterpart to Viro's patchworking polynomials in terms of parametrization of rational curves. In the formula (10) below, we relabel the parameters a j ,l and the corresponding exponents (α j , β j ) of the parametrization (3). Denote by J , J T and J the set of primitive integer vectors in ∂∆ contained in ∂∆ , ∂T and ∂∆ respectively. For any j ∈ J ∪ J T ∪ J , the vector (α j , β j ) is the primitive inner normal to the edge of ∆ containing j . Note in particular that (α j , β j ) = (0, 1) for any j ∈ J T . At last, we denote by V T,w ⊂ V T the subset of curves intersecting the toric orbits corresponding to and only once. This is the analogue of V ∆,1 of Section 5. Lemma 6.2. Let {a} := a j j ∈J ∪J T ∪J ⊂ C * such that a j j ∈J , a j j ∈J T and a j j ∈J are mutually disjoint. For any z ∈ C * , define the parametrization φ z,{a} from CP 1 to X ∆ ν by and define C z := im φ z,{a} . Then, the rational curve C z converges to the curve C 0 ⊂ X ∆ ∪ X T ∪ X ∆ with irreducible components C ⊂ X ∆ , C T ⊂ X T , C ⊂ X ∆ parametrized respectively by In particular, the curve C 0 intersects the divisor X respectively X at the single point p := C ∩ C T (respectively p := C ∩ C T ). For generic parameters a j , the irreducible components of C 0 are nodal curves and the curve C T is an element of the subspace V T,w ⊂ V T .
Proof. As z tends to 0, the rational curve C z converges towards the divisor X ∆ ∪X T ∪X ∆ . Hence, the limiting curve C 0 consists of rational components C ⊂ X ∆ , C T ⊂ X T and C ⊂ X ∆ . Let us compute respective parametrizations φ {a} , φ T {a} and φ {a} . The curves C , C T and C are the respective Hausdorff limit of π •φ z,{a} CP 1 , π•φ z,{a} CP 1 and π •φ z,{a} CP 1 when z tends to 0. For φ T {a} , we have that lim z→0 π φ z,{a} (t ) = t q , t −p j ∈J T t −a j . It follows that φ T {a} is as announced in the statement. For the map φ {a} , we need to make the change of variable t = zt within the limit lim z→0 π φ z,{a} (t ) . The computation goes as follows It follows that φ {a} is as announced above. We obtain the parametrization φ {a} similarly, using the change of variable t = zt . For the second part of the statement, recall that we have a j ∈ C * for any j . Under this assumption, we read from the parametrization φ {a} that ∞ ∈ CP 1 is the only point mapping to X , from φ {a} that 0 ∈ CP 1 is the only point mapping to X and finally that φ T {a} is as in (6). In particular, the curve C T intersects both X and X at a single point. Denote p := C ∩ X and p := C ∩ X . As the curve C 0 is connected, we have p = C ∩ C T and p = C ∩ C T . Finally, it is also clear form the parametrizations φ {a} , φ T {a} and φ {a} and from the general form (3) that generic parameters a j lead to generic rational curves C , C T and C submitted to the above tangency conditions with X and X . In particular, the curves C , C T and C have only nodes as singularities.
Let U be the space of parameters z, {a j } 1≤ j ≤n involved in (10). For z, to be the curve parametrized as in (10). By extension, define as in Lemma 6.2. When the parameters {a} are real and cyclically ordered as in Proposition 3.2, the curve π Φ(z, {a}) ⊂ X ∆ is a rational simple Harnack curve for all 0 < z ≤ 1. It follows from the same proposition that the three irreducible components of Φ 0, {a} are simple Harnack curves in their respective ambient toric surfaces X ∆ , X T and X ∆ . Then, the rational curve Φ 0, {a} := C ∪C T ∪C admits an order map b) If α ∈ (respectively ), then lim z→0 p z,α ∈ X (respectively X ).
Before tackling the proof, recall that the maps π, π and π induce isomorphisms between the tori of X ∆ , X ∆ X T and X ∆ and that the induced isomorphisms between the respective first homology groups read as the identity in the coordinates systems chosen above.
Proof. Fix α ∈ int Z (∆) \ ( ∪ ). Recall from Definition 3.3 that ord 0 (p 0,α ) is given by the homology class of a loop ρ 0 contained in one of the irreducible components of Φ 0, {a} , that this loop passes through p 0,α and is invariant by complex conjugation. We can continuously deform this loop to a loop ρ z ⊂ Φ z, {a} (0 < z < 1) invariant by complex conjugation passing through the double point p z,β for some β ∈ int Z (∆). According to our choices of coordinates systems, we have that ρ 0 and π(ρ z ) have the same homology class. It follows that ord π(p z,β ) = α and that β = α. The statement a) is proven.
From part a), we know that the point p z,α for α ∈ ∪ has to converge to one of the remaining singular points of Φ 0, {a} , namely p and p (see Lemma 6.2). Again, looking at homology classes of appropriate loops passing through p z,α , we deduce that p z,α converges to p if and only if α ∈ .
Proof of Theorem 4. Let w be a wedge in ∆ and denote T := conv(w). Recall that the monodromy map µ ∆ : π 1 V ∆ ,C → Aut int Z (∆) of (5) is defined for a simple Harnack curve C ∈ V ∆ via the bijection ord : nodes of C → int Z (∆). Let {a} be real parameters so that C = π Φ(1, {a}) . Let us fix an arbitrary element τ ∈ Aut (Ψ w ) | int Z (T ) and show that there exists σ ∈ im(µ ∆ ) satisfying the properties a), b) of Definition 6.1 and such that σ | int Z (T ) = τ. As τ is arbitrary, the latter implies the statement we aim to prove.

Combinatorics
Recall Definition 6.1 and the notations therein. The main result of this section is the following.
Theorem 5. For any complete toric surface X , there exists a constant := (X ) > 0 such that for any lattice polygon ∆ ∈ C ≥ (X ), any subgroup G < Aut Ψ ∆ that contains a w-group for any wedge w in ∆ is the whole group Aut Ψ ∆ .
Our strategy to prove Theorem 5 is to apply Jordan's Theorem to the action of G < Aut Ψ ∆ on int Z (∆). Let ρ : G → Aut(Ω) be a transitive action of a finite group G on a finite set Ω. Recall that a block of the action ρ is a subset S ⊂ Ω such that for any g ∈ G, we have either g · S ⊂ S or g · S ⊂ Ω \ S. The action ρ is primitive is the only blocks are singletons and Ω itself. The theorem below is an elementary generalization of Jordan's Theorem [Isa08,Theorem 8.17]. The proof is almost identical. Theorem 6.5. Let ρ : G → Aut(Ω) be the action of a finite group G on a finite set Ω with orbits ψ 1 , ..., ψ q ⊂ Ω. Assume that the restriction of ρ to each orbit is primitive and that for each k ∈ {1, ..., q}, there exists a transposition in im(ρ) supported on the orbit ψ k . Then, we have Throughout the rest of this section, the group G < Aut Ψ ∆ is a group that contains a w-group for any wedge w in ∆.
Definition 6.6. A fundamental domain of a surjective map f : U → V between finite sets is a subset F ⊂ U such that f |F : F → V is a bijection. The representative of u ∈ U in the fundamental domain F is the unique u ∈ F such that f (u) = f (u ).

Lemma 6.7. a) If w is a wedge in ∆ of width at least 3, then we have
, j +2 and w := ∆ j +1 ∩ M ∪ ∆ j −1, j have both width at least 3 and if we denote T := conv(w) = conv(w ), then we have Proof. For a), choose coordinates M Z 2 such that the base of w is the segment joining (0, 0) to ( , 0) with ≥ 3 and such that the vertex of w has coordinates (p, q) with q > 0. In the present coordinates, the map Ψ w is given by Ψ w (n, m) = d(m, qZ) and Q w 0, ..., q/2 . If q = 1, then int Z conv(w) is empty and there is nothing to prove so let us assume that q ≥ 2. In order to prove a), it suffices to show that there exists (n, m) ∈ int conv(w) for any 1 ≤ m ≤ q/2 . The Euclidean length of the horizontal section of conv(w) at height m is at least the length of the section at height q/2 which is at least /2 ≥ 3/2 > 1. It follows that there exists a lattice point in int conv(w) on every such section. The statement a) is proven.
For b), denote by H the group on the left-hand side of the equality. Since both lattices 〈w〉 and 〈w 〉 are sublattices of finite index in 〈(∆ j ∪ ∆ j +1 ) ∩ M 〉, the map Ψ ∆ j ∪∆ j +1 factorizes through both Ψ w and Ψ w . Consequently, we have H < Aut Ψ ∆ j ∪∆ j +1 | int Z (T ) . Choose coordinates M Z 2 such that ∆ j +1 is the segment joining (0, 0) to ( , 0) and ∆ j −1, j = k · (p, q) where the integers , k, p and q satisfy , k ≥ 3, p and q are coprime and q > 0. In the present coordinates, the map Ψ ∆ j ∪∆ j +1 is given by Ψ ∆ j ∪∆ j +1 (n, m) = d(m, qZ). Since , k ≥ 3, the point (p + 1, q) is in int Z (T ) and the set is a fundamental domain of Ψ ∆ j ∪∆ j +1 . In order to prove b), it suffices to show that for any x ∈ int Z (T ), the transposition that sends x to its representative in F is in H . Fix x = (x 1 , x 2 ) ∈ int Z (T ) and assume first that the remainder in the division of x 2 by q is at least q/2 or it is 0. Then, there exists a unique x ∈ (n, m) ∈ int Z (T ) | pm + q ≥ nq such that x − x = (λ , 0) and the transposition τ sending x to x is in Aut Ψ w | int Z (T ) . There exists a unique x ∈ (n, m) ∈ int Z (T ) | 1 ≤ m ≤ q, pm + q ≥ nq such that x − x = λ · (p, q) and the transposition τ sending x to x is in Aut Ψ w | int Z (T ) . By the assumption on the the remainder of x 2 , the point x is in fact in F and the sought transposition is τ := τ τ τ ∈ H .
If the remainder in the division of x 2 by q is strictly less than q/2 and non zero, we claim that there exists a product of transpositions τ ∈ H such that the second coordinate of τ (x) ∈ int Z (T ) has a remainder greater or equal to q/2 . In this case, the sought transposition is τ τ(τ ) −1 ∈ H where τ is the transposition constructed in the above paragraph. The statement b) follows.
Let us prove the claim. To begin with, we translate x to x ∈ (n, m) ∈ int Z (T ) | pm + q ≥ nq using a transposition in Aut Ψ w | int Z (T ) . Then, consider the maximal integer λ ≥ 0 such that x := x + λ · (p, q) is in int Z (T ) and let h by the second coordinate of x . Then, the horizontal section of int Z (T ) at height kq −h contains at least a lattice point x (use the fact that k ≥ 3). The transposition sending x to x is in Aut Ψ w | int Z (T ) . The point x is such that the remainder of its second coordinate in the division by q is at least q/2 . The resulting permutation τ ∈ H sending x to x is the one we were looking for.

Definition 6.8. A pair of wedges in ∆ is consecutive if the wedges have the same base, this base is an edge ∆ j of ∆ and the vertices of the wedges are consecutive lattice points on ∂∆. A pair of wedges
Then the consecutive pair w, w is transitive if there exists a subset F ⊂ int Z conv(w) ∩ conv(w ) such that the map (n, m) → m induces a bijection between F and 1, ..., max(q, s)/2 . Consider the adjacent pair of wedges w = ∆ j ∩ M ∪ ∆ j −1, j −2 and w = ∆ j ∩ M ∪ ∆ j +1, j +2 . Fix coordinates M Z 2 such that ∆ j is the segment joining (0, 0) to ( , 0), the lattice point on ∆ j −1 adjacent to (0, 0) has coordinates (p, q) and the lattice point on ∆ j +1 adjacent to ( , 0) has coordinates (r, s) with , q, s > 0. Then the adjacent pair w, w is transitive if there exists a subset F ⊂ int Z conv(w) ∩ conv(w ) such that the map (n, m) → m induces a bijection between F and 1, ..., max(q, s) . We require moreover that there exists an extra point n, max(q, s) ∈ int Z conv(w) ∩ conv(w ) \ F . Remark 6.9. For a transitive consecutive pair of wedges w, w , the subset F contains a fundamental domain for both Ψ w and Ψ w . In the case of an adjacent transitive pair, the set F together with the extra point contains at least two disjoint fundamental domains for both Ψ w and Ψ w , see Figure 2. Lemma 6.10. Let ∆ ∈ C ≥3 (X ) be a lattice polygon such that any adjacent pair of wedges in ∆ is transitive. Then, there exists a subgroup G < G such that any element of G preserves both and In both cases, we pictured a subset F ⊂ int Z (∆) satisfying the assumptions of Definition 6.8 in orange.
In the adjacent case, the extra point is pictured in green.
the restriction of any g ∈ G to M ∆ is the identity and the group generated by the restriction of all g ∈ G to P ∆ is the group Aut Ψ ∆ |P ∆ .
Proof. Denote by G w < G the w-group in G. The subgroup G we are about to exhibit is a subgroup of In particular, any element of G will restrict to the identity on M ∆ since any element of the above group does. In order to prove the statement, we proceed to a "cyclic" induction on j ∈ Z/nZ. Let us start with j = 1 and choose a coordinate system on M such that ∆ 1 = (0, 0), ( , 0) and such that the lattice points on ∂∆ adjacent to (0, 0) and (0, ) and not in ∆ 1 have respective coordinates (p, q) and (r, s) with , q, s > 0. We know from Lemma 6.7b) that the group G 1,2 generated by G (∆ 1 ∩M )∪∆ 2,3 and G (∆ 2 ∩M )∪∆ n,1 consists of elements whose restriction to the subset int Z (∆)\conv(∆ 1 ∪ ∆ 2 ) is the identity and whose restriction to int Z conv(∆ 1 ∪∆ 2 ) generate Aut Ψ ∆ 1 ∪∆ 2 int Z (conv(∆ 1 ∪∆ 2 )) . In the present coordinates, the map Ψ ∆ 1 ∪∆ 2 is given by Ψ ∆ 1 ∪∆ 2 (n, m) = d(m, sZ). Similarly, the group G n,1 generated by G (∆ n ∩M )∪∆ 1,2 and G (∆ 1 ∩M )∪∆ n−1,n consists of elements whose restriction to the subset int Z (∆) \ conv(∆ n ∪ ∆ 1 ) is the identity and whose restriction to int Z conv(∆ n ∪ ∆ 1 ) generate Aut Ψ ∆ n ∪∆ 1 int Z (conv(∆ n ∪∆ 1 )) where Ψ ∆ n ∪∆ 1 (n, m) = d(m, qZ). We claim that the group generated by G n,1 and G 1,2 contains a subgroup consisting of elements whose restriction to int Z (∆) \ conv(∆ n ∪ ∆ 1 ) ∪ conv(∆ 1 ∪ ∆ 2 ) is the identity and whose restriction to Let us prove the claim. By the transitivity assumption on adjacent pairs, there exists a subset F ⊂ int Z conv(∆ n ∪ ∆ 1 ) ∩ conv(∆ 1 ∪ ∆ 2 ) identified with 1, ..., max(q, s) via the projection on the second coordinate. As an elementary application of the Euclidean algorithm, we have Aut d _ , qZ |F , Aut d _ , sZ |F = Aut d _ , gcd(q, s)Z |F .
It follows that each element of the group on the right-hand side of (11) is the restriction of an element in G whose restriction to int Z (∆) \ ∂ conv(∆ n ∪ ∆ 1 ) ∪ ∂ conv(∆ 1 ∪ ∆ 2 ) ∪ F is the identity. The set F contains a fundamental domain F of d _ , sZ that contains a fundamental domain F of d _ , gcd(q, s)Z . For any point x ∈ int Z conv(∆ n ∪ ∆ 1 ) \ ∂ conv(δ 1 ∪ ∆ 2 ), let τ be the transposition sending x to its representative x in F and τ be the transposition sending x to its representative x in F . Then the transposition ττ τ sends x to x . Since τ ∈ Aut d( _ , qZ) and τ ∈ Aut d _ , gcd(q, s)Z |F , the previous arguments imply that there exist g , g ∈ G whose restriction to the subset int Z (∆) \ ∂ conv(∆ n ∪ ∆ 1 ) ∪ ∂ conv(∆ 1 ∪ ∆ 2 ) is τ and τ respectively. Therefore, the element g · g · g ∈ G restricts to the permutation sending x to x on the latter subset. A symmetric reasoning can be applied to any x ∈ int Z conv(∆ 1 ∪ ∆ 2 ) \ ∂ conv(δ n ∪ ∆ 1 ). The claim follows.
To conclude, we carry inductively the same arguments and show for any j ∈ Z/nZ that we can generate Aut Ψ ∆ n ∪...∪∆ j | int Z (conv(∆ j −1 ∪∆ j )\∂ conv(∆ j −2 ∪∆ j −1 ) with restrictions of elements in G (one simply needs to replace the integer q in (11) by one of its divisors). The statement follows after carrying the induction twice "around" Z/nZ. Lemma 6.11. Let ∆ ∈ C ≥3 (X ) be a lattice polygon such that any pair of wedges in ∆ that is either adjacent or consecutive is also transitive. Then, for any x ∈ int Z (∆), there exists j ∈ Z/nZ and a permutation σ ∈ G such that σ(x) ∈ int Z conv(∆ j −1 ∪ ∆ j ) ∩ conv(∆ j ∪ ∆ j +1 ) and such that σ(x) is the only point of int Z conv(∆ j −1 ∪ ∆ j ) ∩ conv(∆ j ∪ ∆ j +1 ) in the support of the permutation σ. In particular, the group G acts transitively on any fiber of Ψ ∆ .
Proof. Pick any x ∈ int Z (∆). Then, there exists an index j ∈ Z/nZ and lattice point v ∈ ∂∆ such that x ∈ int Z conv(∆ j ∩ M ∪ {v}) . Indeed, there clearly exist indices j , k ∈ Z/nZ such that x ∈ int Z conv(∆ j ∪ ∆ k ) . If |k − j | = 1 or if x is not the intersection point of the diagonals of the quadrilateral conv(∆ j ∪ ∆ k ), we are done. Otherwise, take any v ∈ ∆ k ∩ M other than a vertex.
Let w := (∆ j ∩ M ) ∪ {v} such that x ∈ int Z conv(w) and denote G w < G be the w-group. Then, there are two cases: either there exists an element g ∈ G whose restriction to int Z (∆) \ ∂ conv(w) is a transposition sending x in int Z conv(∆ j −1 ∪ ∆ j ) ∩ conv(∆ j ∪ ∆ j +1 ) and we are done; or not. If not, we know by transitivity of consecutive pairs that there exists an element g ∈ G whose restriction to int Z (∆) \ ∂ conv(w) is a transposition sending x in int Z conv(w ) where w is the wedge consecutive to w, running clockwise along ∂∆. Repeat the same construction to the point g (x) inside int Z conv(w ) and so on until x lands either in int Z conv(∆ j −1 ∪ ∆ j ) ∩ conv(∆ j ∪ ∆ j +1 ) (and we are done) or in int Z conv(∆ j −1 ∪∆ j ) \conv(∆ j ∪∆ j +1 ). Now, by Lemma 6.10, there exists a element in G whose restriction to P ∆ is a transposition sending the image of x in int Z conv(∆ j −1 ∪ ∆ j ) ∩ conv(∆ j ∪ ∆ j +1 ) . The sought permutation σ can be taken as the product of the permutations used in the above algorithm and it has the desired property. Since int Z conv(∆ j −1 ∪∆ j )∩conv(∆ j ∪∆ j +1 ) is a subset of P ∆ , Lemma 6.10 implies that G acts transitively on any fiber of Ψ ∆ . Lemma 6.12. Let ∆ ∈ C ≥3 (X ) be a lattice polygon such that any pair of wedges in ∆ that is either adjacent or consecutive is also transitive. Then, the action of G on any fiber of Ψ ∆ is primitive.
In order to deduce Theorem 5 from Theorem 6.5, we need to show that the assumptions of Lemma 6.12 are satisfied for any ∆ ∈ C ≥ (X ) provided that is big enough.
Proposition 6.14. For any toric surface X and coordinates κ : M → Z 2 on its character lattice, define For any ∆ ∈ C ≥ (X ), any pair of wedges in ∆ that is either consecutive or adjacent is also transitive.
Remark 6.15. The constant (X , κ) of Proposition 6.14 is not intrinsic to the toric surface X as the norms |v j | depend on the choice of coordinates κ. However, the constant (X ) := min κ { (X , κ)} is well defined and intrinsic to X but harder to compute.
In order to prove the above proposition, we will need the following elementary fact. .
2) Assume moreover that either AB and C D are parallel or the lines (AB ) and (C D) intersect at a point P such that A ∈ P B . If θ denotes the angle AP D (set θ = 0 in the parallel case), then we have Proof. 1) Let θ be the positive angle at A in ABO. From the general formulas for solution of triangles, we compute that and AC = sin(θ B ) sin(θ ) BC from which we deduce the sought formula. Along the way, we use that cos(θ ) sin(θ ) = AB −BC ·cos(θ B ) BC ·sin(θ B ) . 2) Let B be the point on (B D) such that AB is parallel to C D. Under our assumptions, the point B lies on B D. In the trapezoid AB C D, we have AO AC = AB AB +C D . As above, we compute that AB = AB · AD · sin(θ A ) AD · sin(θ A − θ) + AB · sin(θ) from which we deduce the sought formula.
For any j ∈ Z/nZ, the index m j of the lattice generated by v j −1 and v j in the lattice M Z 2 is given by m j = det(v j −1 , v j ) = |v j −1 | · |v j | · sin(θ), where θ is the positive angle between v j and v j −1 . In the course of the proof below, we will use the following minoration We also warn that we use the same notation AB to denote the segment between two points A, B ∈ R 2 and to denote the Euclidean distance between them.
Proof of Proposition 6.14. Fix ∆ ∈ C ≥ (X ) and consider two consecutive wedges w and w in ∆ with common base ∆ j . Label the vertices of the convex quadrilateral conv(w ∪w ) by A, B,C , D ∈ Z 2 such that AB = ∆ j and such that A, B,C , D satisfy the assumptions of Lemma 6.16.2. Recall that C and D are consecutive lattice points on some edge ∆ k , by Definition 6.8. In order to show that w and w are transitive, it suffices to show that AO AC > 1 2 and that the horizontal section of ABO passing through the midpoint of AC have Euclidean length strictly greater than |v j |. Indeed, if we denote by A and B the intersection of the horizontal section with AO and BO respectively, then the trapezoid A A B B contains at least 2 lattice points in the interior of each integer horizontal section. By assumptions, the second coordinate of C is larger than the one of D. It follows that the consecutive pair w, w is adjacent.
Let us now show that AO AC > 1 2 and A B > |v j |. We will proceed by deriving successive lower bounds for AO AC starting from the formula given in Lemma 6.16.2. As a preparation, let us show that AD ≥ 5·|v k |. Indeed, if we assume that A is the vertex ∆ j −1, j (the case A = ∆ j , j +1 is similar), then the distance AD is greater than the distance from the line (AB ) to the vertex ∆ j −2, j −1 . The latter distance is equal to det(v j , v j −1 ) · Z (∆ j −1 ) · |v j | −1 . Since Z (∆ j −1 ) ≥ 5 · max i ∈Z/nZ |v i | 2 , the claim follows. Recall that, in the terminology of Lemma 6.16.2, we have that sin(θ) < sin(θ A ), that C D = |v k | and that AB, AD ≥ 5|v k | 2 ≥ 5|v k |. Starting from the formula in Lemma 6.16.2, we deduce that In turn, we have that A B = AB · 1 − A A AO = AB · 1 − 1 2 · AC AO > AB · 1 − 4 5 = AB 5 ≥ ·|v j | 5 ≥ |v j |. We conclude that the pair w, w is transitive.
Let us now consider the adjacent pair of wedges w := (∆ j ∩ M ) ∪ ∆ j −1, j −2 and w := (∆ j ∩ M ) ∪ ∆ j +1, j +2 . Label the vertices of the convex quadrilateral conv(w ∪ w ) by A, B,C , D ∈ Z 2 such that AB = ∆ j and such that A, B,C , D satisfy the assumptions of Lemma 6.16.2. Define A ∈ AC and B ∈ B D such that A B is parallel to AB and A A AC = 1 Z (BC ) . Similarly to the case of consecutive pairs, it suffices to show that AO AC > 1 Z (BC ) and A B > |v j | in order to show that w and w are transitive. Starting from Lemma 6.16.1, we deduce that using inequality (12) on the numerator In turn, we have that A B = AB · 1 − A A AO = AB · 1 − 1 Z (BC ) · AC AO > AB · 1 − 3 5 = 2 5 AB ≥ 2· 5 |v j | > |v j |. We conclude that the adjacent pair w, w is transitive.
Proof of Theorem 5. Let := (X ) be the constant given in Proposition 6.14. Then, any group G that contains a w-group for any wedge w in ∆ acts primitively on each fiber of Ψ ∆ , according to Lemma 6.12 and Proposition 6.14. Moreover, we have that ≥ max(4, 3q − 2) where q is the constant of Lemma 6.13. According to the latter lemma, the group G contains a transposition with support in Ψ −1 ∆ (k) for each k ∈ Q ∆ . It follows now from Theorem 6.5 that G is the group of deck transformations Aut(Ψ ∆ ).

Examples with unexpected monodromy
Consider the lattice polygon ∆ := conv (0, 0), (1, 0), (0, k + 1), (−1, k) ⊂ M R depending on the integer k ≥ 2. For all k, we have that M ∆ = M and the obstruction map Ψ ∆ is therefore trivial. However, we will show in this section that the image of the monodromy map µ ∆ is strictly smaller than Aut(Ψ ∆ ) = Aut int Z (∆) for some values of k. In this section, we provide tools to treat all cases k ≥ 2 with focus on k ∈ {4, ..., 8}. Towards the end of the section, we provide a code in Mathematica ® [Inc] supporting our calculations.
In order to determine the image of µ ∆ , we will look for an explicit description of the nodes of a curve C ∈ V ∆ . The dual fan F ⊂ N R of the toric surface X ∆ consists in 4 rays generated by n 1 = (0, 1), n 2 = (−k − 1, −1), n 3 = (1, −1) and n 4 = (k, 1). Any rational curve C ∈ V ∆ has a single point of intersection with each of the 4 toric divisors D 1 to D 4 and the latter points are distinct for distinct divisors. For any curve C ∈ V ∆ parametrized as in (3), the action of PGL 2 (C) allows us to parametrize the points C ∩ D 2 , C ∩ D 3 and C ∩ D 4 by t = 1, ∞ and 0 respectively. After applying a translation in X • ∆ (C * ) 2 , the curve C can be parametrized by Any of the | int Z (∆)| = k nodes of C corresponds to a pair of distinct points {s, t } ⊂ CP 1 \ {0, 1, ∞, a} such that Since we are looking for a pair {s, t } of distinct points, we only consider the case t = s−a s−1 . The latter substitution in the first equation of the above system leads to The system of equations t = s−a s−1 , (s − a) k (s − 1) k+2 − s k (1 − a) k+1 = 0 has 2(k + 1) solutions. In order to get a manageable description of the k nodes of C , we have to get rid of some symmetries and superfluous solutions.
Observe that the substitution t = s−a s−1 is involutive, that is s = t −a t −1 . The 2 fixed points of this involution are the roots s ± = 1± 1 − a of the polynomial s 2 −2s +a. A straightforward computation shows that the roots s ± are also roots of (s − a) k (s − 1) k+2 − s k (1 − a) k+1 , implying that (1 − a) k · P s − a s − 1 .
Since neither 1 nor a is a root of P a (s), we deduce that the roots of P a (s) are invariant under the involution ι a (s) := s−a s−1 . Therefore, the 2k roots of P a (s) are divided into k orbits under the action of the involution ι a . Conversely, any polynomial P (s) whose set of roots is invariant under ι a satisfies (13). To see this, write P (s) = c 1≤ j ≤k s − s j s − ι a (s j ) .
Our next goal is to cancel the latter symmetry of the set of roots of P a (s). To that aim, consider the rational function f a : CP 1 → CP 1 of degree 2 given by f a (x) := s 2 − 2as + a (2 − a)s 2 − 2as + a 2 .
There are two interesting features of the function f a . First, the group of deck transformations of f a is exactly id, ι a . Second, for any polynomial R(s) of degree k, the polynomial P (s) := R f a (s) · (2 − a)s 2 − 2as + a 2 k (14) is a degree 2k polynomial satisfying the identity (13). Both features can be checked by hand. We claim that any polynomial P of degree 2k whose set of roots is invariant under ι a is of the form (14) for a unique polynomial R(s) of degree k. In particular, the polynomial P a (s) is of the form (14) for a unique polynomial R a (s). To see this, consider the linear endomorphism F : P (s) → (s − 1) 2k (1 − a) k · P s − a s − 1 on the C-vector space of polynomials of degree 2k. Observe that F is an involution. The set of fixed points inv(F ) of F is a vector space of dimension k. Indeed, we argued above that inv(F ) is the set of polynomials whose set of roots is globally invariant by ι a . Thus, the roots of such polynomials describe a k-dimensional subvariety in Sym 2k (C). Since a polynomial is determined by its roots up to projective equivalence, it follows that inv(F ) is (k +1)-dimensional. On the other hand, the linear map R(s) → R f a (s) · (2 − a)s 2 − 2as + a 2 k maps injectively the C-vector space of degree k polynomials into inv(F ). The claim follows. Summarizing, we proved the following.
Proposition 7.1. The following map is a bijective roots of R a → nodes of C s → φ a f −1 a (s) .
We deduce that D −1 (0) = 0, 1, − 9 16 . Take a = 1/2 as a base point, so that the curve C := φ 1/2 CP 1 is a simple Harnack curve. Let γ 1 , γ 2 and γ 3 be 3 loops in the a-space based at a = 1/2 and going around the discriminantal points a = 0, 1 and − 9 16 respectively. The roots of R 1/2 are all real with approximate values −0.4, 0.4, 0.5 and 6. If we label these roots from 1 to 4 in increasing order, we obtain by computer programming that the loops γ 1 , γ 2 and γ 3 induce respectively the permutations (12)(34), id and (24) on the roots of R 1/2 . The group generated by these permutations is isomorphic to the dihedral group D 8 ⊂ Aut 1, 2, 3, 4 of order 8 and is imprimitive since it admits the nontrivial blocks {1, 3} and {2, 4}. Alternatively, this group is the wreath product Z 2 Z 2 on the two blocks. We conclude that im(µ ∆ ) D 8 is a strict subgroup of Aut int Z (∆) . Further computations show that the blocks of the action of im(µ ∆ ) on int Z (∆) are (0, 1), (0, 3) and (0, 2), (0, 4) . Equivalently, two nodes of C are in the same block if and only if they are in the same quadrant of (R * ) 2 ⊂ X ∆ .
Below, we provide the code supporting the above computations. The interested reader can copypaste the code into a Mathematica notebook. Due to format incompatibilities, one needs to rewrite the arrows after WorkingPrecision and PlotRange before evaluating the code.