The Group Structure on a Smooth Tropical Cubic

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1 The Group Structure on a Smooth Tropica Cubic Ethan Lake Apri 20, 2015 Abstract Just as in in cassica agebraic geometry, it is possibe to define a group aw on a smooth tropica cubic curve. In this note, we prove that the resuting group is isomorphic to the circe group S 1. 1 Preiminaries Throughout, we et T (R { },, ) be the tropica semiring, where the binary operators are defined as x y min{x, y} and x y x + y. In this ring, 0 is the mutipicative identity and is the additive identity. Note that in genera, additive inverses do not exist. The overa structure of our argument is based on [2]. Definition Let A = (a 1,..., a n ) be a finite set of integers. A tropica poynomia in n variabes x 1,..., x n with support A is an expression of the form: f = a A α a x a x an n (1) where α a T. We wi occasionay write the set (a 1,..., a n ) in vector from as a. Definition A tropica hyper surface Z(f) defined by the tropica poynomia f is the set of points in T n where f is non-inear. Z(f) is caed a tropica curve if n = 2. Additionay, we define tropica projective n-space just as in cassica agebraic geometry by setting TP n R n+1 /, where is a inear equivaence reation defined by x y x = λ y for some λ T. Athough working in projective space has some benefits, we wi focus our attention on the affine picture, since the cubics we consider wi turn out to be we-behaved at. The convex hu of A is known as the Newton poytope of f, which we denote by f. We can make a attice subdivision of f, denoted f, by forming the dua graph to f when f is considered as a tropica poynomia in TP 2. There is a natura bijection between edges of Z(f) and the edges of f, with each unbounded ray of Z(f) corresponding to an edge of f. The attice ength of the corresponding edge of f gives the mutipicity of the associated unbounded edge of Z(f). 1

2 Definition Let Γ d be the triange on the Z 2 attice with vertices at (0, 0), (0, d), (d, 0). If f fits inside Γ d but not inside Γ d 1, then C is said to have degree d. C is said to have fu support if f = Γ d. Lemma 1.1 (The Baance Condition). Let V be a vertex of a tropica curve, with E 1,..., E n the edges emanating from v. Further, et w 1,..., w n be the weights associated to each edge, with v i the primitive direction vector pointing aong E i from V. Then wi v i = 0 (2) The notion of mutipicity can aso be naturay extended to the tropica setting. Let V be a 3-vaent vertex of Z(f), with two primitive integer direction vectors v, u and associated edge weights w v, w u. The mutipicity of the vertex V is defined to be mut V = w v w u Det( v, u) (3) Note that we ony need to specify two of the primitive direction vectors, since the third can be determined from the Baance Condition. Definition A tropica curve is said to be smooth if every vertex of its associated dua graph f is 3-vaent with mutipicity 1. Definition The genus of a smooth tropica curve is the number of vertices in the interior of f whose vaence is greater than Intersections of tropica curves Many resuts from cassica agebraic geometry concerning the intersection of two curves continue to hod in the tropica setting. Theorem 1.2 (Tropica Bezout s Theorem). Let C and D be tropica curves of degree c, d. If at east one of C or D is fuy supported, then their stabe intersection consists of cd points, counting mutipicities. Intuitivey, we can see that the number of stabe intersection points of C and D must be invariant under transation of either curve in the affine pane. As such, we can position C and D so that their intersection points ie on the unbounded rays aong one given coordinate axis. It is then easy to see that they intersect in cd points by noting that the ength of any given side of the dua graph is the same for each curve. Theorem 1.3 (Tropica Bernstein s Theorem). Let C and D be any two curves intersecting transversey, with dua graphs f and g. The number of intersection points of C and D (with mutipicity) is the mixed area of f and g, that is, the number of intersection points is equa to Area( f g ) Area( f ) Area( g ), (4) where denotes the Minkowski sum. For detais, see [2, 1]. Note that we have been working in the affine picture here, and at first gance it may seem that we ve negected possibe intersection points at. However, we imit our attention to the case when at east one of f and g is fuy supported, and so ceary (f g) R 2. 2

3 Figure 1: The skeeton of a tropica amoeba, identified with a cubic curve. The curve is smooth since each vertex is 3-vaent. It has genus 1, and has 6 tentaces. 2 Divisors on smooth tropica curves Definition Let C be a smooth tropica curve in R 2. A divisor D on C is a finite forma sum D = m p p, where the sum runs over points of C. The free abeian group generated by the points on C is caed the divisor group of C, and is denoted Div(C). The sum m P is caed the degree of D. Furthermore, we define a tropica rationa function as a function of the form h = f g, where f and g are tropica poynomias with equa dua graphs. Definition Let f be a tropica poynomia. Define div(f) Div(C) as the sum div(f) = m p (5) p C Z(f) where m p is the intersection mutipicity at each point. A divisor D is caed a principa divisor if a tropica rationa function h such that D = div(h). Note that any principa divisor D has degree zero, as in the cassica setting. This foows from the tropica version of Bernstein s Theorem, since two functions with equa Newton poygons wi have identica intersection mutipicities with C. Definition Two divisors D 1 and D 2 are said to be ineary equivaent (written D 1 D 2 ) if the divisor D 1 D 2 is principa. Note that the eements of degree 0 in Div(C) form a subgroup, which we sha denote by Div 0 (C). As in the cassica case, it is easy to check that inear equivaence defines an equivaence reation on the subgroup Div 0 (C). 3

4 3 Tropica eiptic curves As in cassica agebraic geometry, a tropica eiptic curve means a smooth tropica curve of degree 3 and genus 1. In order for a cubic C to have genus 1, the interior vertex (1, 1) of C must be at east trivaent. If C has genus 1, it contains a unique cyce C. We wi often refer to C as the body of C. Each connected component of the set C \ C is caed a tentace of C. 3.1 Linear equivaence of points on C The aim of this section is to estabish the inear equivaence of two points depending on wether they are on the body of C or on one of its tentaces. Here, we wi show that the tentaces of C can be removed, moduo inear equivaence. This wi aow us to focus our study of the group aw on the body of C. Proposition 3.1. Let p and q be two points on the same tentace of C. Then p q. Proof. First, suppose that p and q ie on the same unbounded ray of C. Since C is smooth and has fu support, it wi have 3 unbounded rays in each of the directions (1, 0), (0, 1),and ( 1, 1). Without oss of generaity, we can focus on unbounded rays in the (1, 0) direction. Let 1, 2, 3 be the three unbounded rays of C in the (1, 0) direction, ordered by increasing y component. First, examine 1. Let p, q 1 with p be further away from the body of C than q. Suppose that p and q are cose to each other on the tentace. Let p be the tropica ine with vertex at p, and et q be the tropica ine passing trough q, whose center ies on the unbounded ray of p whose primitive direction vector is ( 1, 1). Suppose that this ray intersects other tentaces of C at the points r, s. Each intersection is transverse, and so we have div(f p ) = p + r + s and div(f q ) = q + r + s, as ong as the coseness condition between p and q is satisfied. So then div( p q ) = p q, and so p q. Of course, p and q may not be cose. In this case, we can choose a finite sequence of points p, m 1, m 2,..., m n, q on the ray shared by p and q so that p i p i+1 i, giving p q. For 2, foowing this procedure yieds div( p q ) = p q + r s, where r, s 1 (see figure). But since a points on 1 are ineary equivaent, r s and so p q. A simiar reasoning appies to the ine 3. Bounded segments of tentaces are handed anaogousy. If p and q do not ie on the same ray but are separated by one or more vertices V 1,..., V n aong the tentace, we can choose the V 1,..., V n as members of the sequence of the p i used to show equivaence for points on the same ray that are not cose. This process can be appied symmetricay to tentaces emanating in each other direction, which competes the proof. Our overa goa now is to show that the divisor group on C is isomorphic to S 1. Let s begin by finding an expicit homeomorphism from the body of C to S A homeomorphism C S 1 Since C has genus 1, C is homeomorphic to the circe group in a topoogica sense. We wi now expicity construct such a homeomorphism, using a construction for evauating distances between two points on the body of C. 4

5 3 s 2 r 1 p q p q Figure 2: A graphic showing how to estabish inear equivaence between two points on the same tentace. 5

6 Chose a fixed point O C. Let V 1,..., V n be the vertices of C abeed countercockwise, with E 1,..., E n the corresponding edges of C. We define the attice ength of an edge E i as being equa to L(E) E i / v i, where v is the primitive integer direction vector oriented aong E i. We et L be the cyce ength of C, defined by the sum of the attice engths of each edge: L = i L(E i). We now construct a homeomorphism Λ : C R/Z = S 1 which is inear in the Eucidean metric of each edge E i. We define the images of (O, V 1,..., V n ) recursivey, starting from the point O: Λ(O) = 0 (6) Λ(V 1 ) = L(E 1 ) Λ(V i+1 ) = Λ(V i ) + L(E i ), i {1,..., n 1} We can normaize Λ(V i ) by dividing out by L and identify the resuting structure with R/Z = [0, 1) = S 1. We can then define a dispacement function D between two points of C by: D : C C R/Z = S 1 (7) (p, q) Λ(q) Λ(p) Note that D is anti-commutative and satisfies the reation D(p, q) + D(q, r) = D(p, r). 4 The group aw Finay, we are abe to show that Div 0 (C)/ is equivaent to C, and can describe the resuting group structure induced on C. To begin, note that if p, q C then it is not aways possibe to construct a ine that intersects C staby at both p and q. If such a ine exists, we ca the tupe (p, q) a good pair. In the foowing emma, we et p 1 = (0, 1), p 2 = (1, 0), and p 3 = ( 1, 1) be the primitive integer direction vectors for a tropica ine. Lemma 4.1. Let p, q, p, q be points on C. Then p + q p + q D C (p, p ) = D C (q, q ) (8) Proof. We wi prove the emma in two parts, first for good pairs and then generaizing to any pairs. Assume that (p, q) and (p, q ) are both good pairs, and that p + q p + q. Then two tropica ines, such that C = {p, q, r} and C = {p, q, r}. Consider a homotopy L i of ines containing r that satisfies L 0 =, L 1 =. We consider the case where p, p are on the same edge, q, q are on the same edge, and is simpy a dispacement of aong one of the coordinate axes. We et v p and v q be the primitive integer direction vectors of the edges of C containing p and q respectivey. We assume that can be obtained by transating by δ units in the p 1 direction. Then the dispacements of p and q from the primed points are pp = p 2 δ p 1 p 2 v p v p = δ v p (9) 6

7 and so qq = p 3 δ p 1 p 3 v q v q = δ v q D C (p, p ) = δ v p v p D C (q, q ) = δ v q v q = δ (10) Since p and q are moved in opposite directions aong C, D C (p, p ) = D C (q, q ). The reverse impication foows a simiar argument. Now we assume that (p, q) is not a good pair. We et p and q be two tropica ines trough p and q respectivey, with r p, s p, r q, s q being the other intersection points of p and q. We want to wigge p and q to the new ines p, q so that r p, s p, r q, s q are a preserved as intersection points. Of course, p and q get wigged to the new points p and q. We then have that p + q p + q, which impies that D C (p, p ) = D C (q, q ) from the first part of the proof. So a we have to do is wigge p and q far enough apart so that we obtain a good pair (p, q ). Since the emma hods for good pairs, it hods for arbitrary pairs. The next proposition wi aow us to induce a natura group structure on the body of C: Proposition 4.2. For any fixed point O C, the map = δ Γ O : C Div 0 (C)/ (11) is a bijection. q q O Proof. Injectivity is easy, since p O q O = p + O q + O = D(p, q) = 0 = p = q (12) where we have made use of the previous emma. Let D Div 0 (C). We need to show that a p C that satisfies D p O in order to prove surjectivity. Let D = p 1 q 1, where p 1, q 1 C. We choose p so that D C (p, p 1 ) = D C (O, q 1 ), and the ast emma tes us that p + q 1 p 1 + O, impying that D = p 1 q 1 p O. Now we take D = D 1 D 2, where D 1 = p p n and D 2 = q q n are two divisors of degree n. We et P 12 and q 12 be two points that satisfy p 1 + p 2 O + p 12 and q 1 + q 2 O + q 12. Then we have the foowing reation: D p p n (q q n ) (13) and so D D 1 D 2 where D 1 and D 2 are divisors of degree n 1. By proceeding inductivey, we can reduce a divisor of arbitrary degree to the case where n = 1. Definition We define ( C, O) as the group consisting of points on C with the group structure induced from Div 0 (C)/ such that the bijection Γ O is a group isomorphism. 7

8 p O q p q r Figure 3: An iustration showing how to add two points p and q on the body of a curve C. We first construct the tropica ine joining p and q (red ine), and mark its third point of intersection with C (point r). We then join O and r together by a ine (bue ine), and interpret p q as the third intersection point of this ine with C. 8

9 Theorem 4.3. Let p, q be any points of C, and et denote addition in the group ( C, O). The the point p q satisfies: Proof. Since Γ O is an isomorphism of groups, we have D(O, p q) = D(O, p) + D(O, q) (14) p q O = Γ O (p q) = p O + q O (15) So p q p + q O which impies p q + O p + q and so we use the ast emma to obtain D C (p, p q) = D C (O, q) (16) or equivaenty, D C (O, p q) = D C (O, p) + D C (O, q) (17) Remark In cassica agebraic geometry, we can describe the group action on a curve by geometric construction, where we project from a given origin aong points of an eiptic curve. We can perform a simiar construction in the tropica setting, by figuring out a way to add two points on the body of C. To see this, et p and q be two points on the body of C. Suppose first that (p, q) is a good pair. Then we construct the tropica ine passing through both points, and find the ine s third intersection point with the body of C. Ca this point r. If (r, O) is a good pair, construct the ine joining r and O. The third intersection of this ine with the body of C can be interpreted as p q. If (p, q) is not a good pair, we simpy wigge both points in opposite directions by an equa attice distance, and use the resuting points to construct the sum. Coroary 4.4. The map Λ : ( C, O) R/Z = S 1 is a isomorphism of groups. Proof. For any p we have Λ(p) = D(O, p). So then we just need to check that Λ respects addition: Λ(p q) = D(O, p q) = D(O, p) + D(O, q) = Λ(p) + Λ(q) (18) and so indeed, ( C, O) = S 1. References [1] Bernd Strumfes. Soving systems of poynomia equations. CBMS Regiona Conference Series in Mathematics, [2] M. Dehi Vigeand. The group aw on a tropica eiptic curve. ArXiv Mathematics e-prints, November