A tropical analogue of the Hessian group
aa r X i v : . [ n li n . S I] A p r A tropical analogue of the Hessian group
NOBE Atsushi
Department of Mathematics, Faculty of Education, Chiba University,1-33 Yayoi-cho Inage-ku, Chiba 263-8522, Japan
Abstract
We investigate a tropical analogue of the Hessian group G , the group of linear automor-phisms acting on the Hesse pencil. Through the procedure of ultradiscretization, the grouplaw on the Hesse pencil reduces to that on the tropical Hesse pencil. We then show that thedihedral group D of degree three is the group of linear automorphisms acting on the tropicalHesse pencil. The Hessian group G ≃ Γ ⋊ SL (2 , F ) is a subgroup of P GL (3 , C ), the group of lineartransformations on the projective plane P ( C ), where Γ = ( Z / Z ) and SL (2 , F ) is the speciallinear group over the finite field F of characteristic three. The Hessian group is generated by thefollowing four linear transformations g = g = ζ
00 0 ζ g = ζ ζ ζ ζ g = ζ
00 0 ζ , where ζ denotes the primitive third root of 1. The name, “Hessian” group, comes from the factthat G is the group of linear automorphisms acting on the Hesse pencil [1, 4]. The Hesse pencilis a one-dimensional linear system of plane cubic curves in P ( C ) given by f ( x , x , x ; t , t ) := t (cid:0) x + x + x (cid:1) + t x x x = 0 , where ( x , x , x ) is the homogeneous coordinate of P ( C ) and the parameter ( t , t ) ranges over P ( C ) [1]. The curve composing the pencil is called the Hesse cubic curve (see figure 1).Figure 1: Several members of the Hessepencil.Each member of the pencil is denoted by E t ,t andthe pencil itself by { E t ,t } ( t ,t ) ∈ P ( C ) . The nine basepoints of the pencil are given as follows p = (0 , , − p = (0 , , − ζ ) p = (0 , , − ζ ) p = (1 , , − p = (1 , , − ζ ) p = (1 , , − ζ ) p = (1 , − , p = (1 , − ζ , p = (1 , − ζ , . Any smooth curve in the pencil has the nine base pointsas its inflection points, and hence they are in the Hesseconfiguration [1, 4]. We choose p as the unit of addi-tion of the points on the Hesse cubic curve.The group E t ,t [3] of three torsion points on E t ,t consists of the nine base points p , p , · · · , p . The map p (1 , p (0 ,
1) 1nduces the group isomorphism E t ,t [3] ≃ Γ, which is the normal subgroup of G generated bythe elements g and g . Therefore, the action of Γ fixes the parameter ( t , t ) of the Hesse pencil.Let α : G → P GL (2 , C ) be a map given by α ( g ) : ( t , t ) = ( x x x , x + x + x ) ( t ′ , t ′ ) = ( x ′ x ′ x ′ , x ′ + x ′ + x ′ ) , where g ∈ G and g : ( x , x , x ) ( x ′ , x ′ , x ′ ). Then we have Ker( α ) ⊃ Γ = h g , g i . Actually,Γ is a subgroup of Ker( α ) of index two.On the other hand, α ( g ) and α ( g ) act effectively on P GL (2 , C ): α ( g ) : ( t , t ) ( t ′ , t ′ ) = (3 t + t , t − t ) α ( g ) : ( t , t ) ( t ′ , t ′ ) = ( t , ζ t ) . Thus g and g induce the action on the Hesse pencil independent of its additive group structure.We can easily check the following relation α ( g ) = α ( g ) = 1 . It follows that we have α ( G ) = h α ( g ) , α ( g ) i ≃ T , where T is the tetrahedral group. Thus the group α ( G ) acts on P GL (2 , C ) as the permutationsamong the following 12 elements λ := t t , ζ λ, ζ λ, − λ λ , ζ − ζ λ λ , ζ − ζ λ λ , − ζ λ ζ λ , ζ − ζ λ ζ λ , ζ − λ ζ λ , − ζ λ ζ λ , ζ − λ ζ λ , ζ − ζ λ ζ λ . The Hesse pencil contains four singular members with multiplicity three corresponding to thefollowing ( t , t ) [1] ( t , t ) = (0 , , (1 , − , (1 , − ζ ) , (1 , − ζ ) . Denote these points by s i ( i = 1 , , ,
4) in order. These s i ’s are permuted by α ( G ) as follows α ( g ) : s ←→ s , s ←→ s (1) α ( g ) : s −→ s −→ s −→ s ( s is fixed.) (2)Thus s i ’s can be corresponded to the vertices of the tetrahedron on which T ≃ α ( G ) acts.Moreover, let g = . Then we have g = ( g g ) = g . Therefore, we obtain G / Γ = h g , g i ≃ ˜ T , T is the binary tetrahedral group. Since ˜ T is isomorphic to SL (2 , F ), we obtain the semi-direct product decomposition G ≃ Γ ⋊ SL (2 , F ).The level-three theta functions θ ( z, τ ), θ ( z, τ ), and θ ( z, τ ) are defined by using the thetafunction ϑ ( a,b ) ( z, τ ) with characteristics: θ k ( z, τ ) := ϑ ( k − , )(3 z, τ ) = X n ∈ Z e πi ( n + k − ) τ e πi ( n + k − )( z + ) ( k = 0 , , , where z ∈ C and τ ∈ H := { τ ∈ C | Im τ > } . Fixing τ ∈ H , we abbreviate θ k ( z, τ ) and θ k (0 , τ )as θ k ( z ) and θ k for k = 0 , ,
2, respectively. We can easily see that the following holds θ = − θ θ = 0 . (3)Let L τ := ( − τ ) Z + (3 τ + 1) Z be a lattice in C . Consider a map ϕ : C → P ( C ), ϕ : z ( θ ( z ) , θ ( z ) , θ ( z )) . This induces an isomorphism from the complex torus C /L τ to the Hesse cubic curve E θ ′ , θ ′ . Italso induces the additive group structure on E θ ′ , θ ′ from C /L τ through the addition formulae forthe level-three theta functions [2]; let ( x , x , x ) and ( x ′ , x ′ , x ′ ) be points on E θ ′ , θ ′ , then theaddition ( x , x , x ) + ( x ′ , x ′ , x ′ ) of the points is given as follows( x , x , x ) + ( x ′ , x ′ , x ′ ) = ( x x x ′ − x x ′ x ′ , x x x ′ − x x ′ x ′ , x x x ′ − x x ′ x ′ ) . (4)The relation (3) implies that the unit of addition on E θ ′ , θ ′ induced by ϕ is p : ϕ : 0 ( θ , θ , θ ) = (0 , , −
1) = p . By using (4), we see that the actions of g and g on E θ ′ , θ ′ can be realized as the additionswith p and p , respectively( x , x , x ) g ( x , x , x ) = ( x , x , x ) + p ( x , x , x ) g ( x , ζ x , ζ x ) = ( x , x , x ) + p . Take the following representatives z k , z k , z k of the zeros of θ k ( z ) in C /L τ for k = 0 , , z z z z z z z z z = τ + τ + − τ τ +
13 5 τ + − τ τ +
13 4 τ + . Then these nine zeros are mapped into the nine inflection points on E θ ′ , θ ′ by ϕ , respectively: ϕ : z z z z z z z z z p p p p p p p p p . Let us tropicalize the Hesse pencil. For the defining polynomial f ( x , x , x ; t , t ) of the Hessecubic curve, we apply the procedure of tropicalization. Replacing + and × with max and +respectively, the polynomial f ( x , x , x ; t , t ) reduces to˜ f (˜ x , ˜ x , ˜ x ; ˜ t , ˜ t ) = max (cid:0) ˜ t + 3˜ x , ˜ t + 3˜ x , ˜ t + 3˜ x , ˜ t + ˜ x + ˜ x + ˜ x (cid:1) . In order to distinguish tropical variables form original ones, we ornament them with ˜.3 Y ✲ X ❅❅❅❅❅❅❅✁✁✁✁✁✁✁✁✁✁✁✁✁✟✟✟✟✟✟✟✟✟✟✟✟✟❅❅❅❅❅❅✁✁✁✁✁✁✁✁✁✁✁✟✟✟✟✟✟✟✟✟✟✟❅❅❅✁✁✁✁✁✁✟✟✟✟✟✟(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) O Figure 2: The curves drawn with solidlines are the regular members and the onewith broken line is the singular member ofthe tropical Hesse pencil.Let ( t , t ) be a point in P ,trop , the tropical pro-jective line. Then ˜ f can be regarded as a function˜ f : P ,trop → T , where P ,trop is the tropical projectiveplane and T := R ∪{−∞} is the tropical semi-field. Thetropical Hesse curve is the set of points such that thefunction ˜ f is not differentiable. We denote the tropicalHesse curve by C ˜ t , ˜ t . Upon introduction of the inho-mogeneous coordinate ( X := ˜ x − ˜ x , Y := ˜ x − ˜ x ) ∈ P ,trop and K := ˜ t − ˜ t ∈ P ,trop the tropical Hessecurve is denoted by C K and is given by the tropicalpolynomial F ( X, Y ; K ) := max (3 X, Y, , K + X + Y ) . Figure 2 shows the tropical Hesse curves. The one-dimensional linear system { C K } K ∈ P ,trop consisting ofthe tropical Hesse curves is called the tropical Hessepencil. The complement of the tentacles, i.e., the finitepart, of C K is denoted by ¯ C K . We denote the verticeswhose coordinates are ( K, K ), ( − K, , − K ) by V , V , and V , respectively.Let K and ε be positive numbers. Let us fix τ : τ = − K K + 2 πiε . Then the complex torus C /L τ converges into J ( C K ) in the limit ε → ϕ : J ( C K ) → R ⊂ P ,trop ,˜ ϕ : u (˜ c ( u ) , ˜ s ( u )) , where we define˜ c ( u ) := − K (cid:26)(cid:18)(cid:18) u − K K − (cid:19)(cid:19)(cid:27) + 9 K (cid:26)(cid:18)(cid:18) u − K K − (cid:19)(cid:19)(cid:27) ˜ s ( u ) := − K (cid:26)(cid:18)(cid:18) u − K K − (cid:19)(cid:19)(cid:27) + 9 K (cid:26)(cid:18)(cid:18) u − K K − (cid:19)(cid:19)(cid:27) , and (( u )) := u − Floor( u ). This map induces an isomorphism ¯ C K ≃ J ( C K ) [2, 3], where J ( C K ) isthe tropical Jacobian of C K : J ( C K ) := R / K Z = { u ∈ R | ≤ u < K } . Thus ˜ ϕ induces additive group structure on ¯ C K equipped with the unit of addition V = ˜ ϕ (0) form J ( C K ). The addition formula for ¯ C K is explicitly given in [3].The piecewise linear functions ˜ c ( u ) and ˜ s ( u ) are periodic with period 3 K and are the ul-tradiscretization of the elliptic functions c ( z ) := θ ( z, τ ) /θ ( z, τ ) and s ( z ) := θ ( z, τ ) /θ ( z, τ ),respectively [2, 3]. In the procedure of ultradiscretization, we assume u ∈ R and z = (1 − iξ ε ) u K , where ξ ε = 2 πε/ K , and take the limit ε →
0. In terms of the variable u , we put the limit of zeros z kj ( k, j = 0 , ,
2) of the level-three theta functions as follows u := lim ε → Kz = lim ε → Kz = lim ε → Kz = 0 u := lim ε → Kz = lim ε → Kz = lim ε → Kz = Ku := lim ε → Kz = lim ε → Kz = lim ε → Kz = 2 K, τ → − / ε → η : E θ ′ , θ ′ → ¯ C K so defined that the diagram commute C /L τ ε → −−−−→ J ( C K ) ϕ y y ˜ ϕ E θ ′ , θ ′ η −−−−→ ¯ C K . The inflection points of E θ ′ , θ ′ are mapped into the vertices of ¯ C K by η as follows η : p , p , p ϕ − z , z , z ε → −→ u ϕ V (5) η : p , p , p ϕ − z , z , z ε → −→ u ϕ V (6) η : p , p , p ϕ − z , z , z ε → −→ u ϕ V . (7)Now we investigate the tropical counterpart of the Hessian group G ≃ Γ ⋊ SL (2 , F ). At firstwe consider Γ ≃ ( Z / Z ) . Note that Γ = h g , g i and the actions of g and g on E t ,t is realizedas the additions with p and p , respectively. Moreover, the group generated by the additions with p and p is nothing but E t ,t [3], the group of three torsion points on E t ,t .The correspondence (5), (6), and (7) in terms of η tells us that the addition with p correspondsto that with V on C K , while that with p vanishes in the limit ε →
0. (Note that V is the unit ofaddition on C K .) Since the addition with p (resp. V ) is equivalent to that with 2 p (resp. 2 V ),the tropical analogue of Γ consists of the addition with V . Actually, it is the group C K [3] = h V i of three torsion points on C K , which is isomorphic to Z / Z . We denote the tropical analogue of agroup G by trop ( G ): trop (Γ) ≃ Z / Z . The addition with V is explicitly computed as follows( X, Y ) ⊎ V = ( X, Y ) ⊎ (0 , − K ) = ( Y − X, − X ) , where we denote the addition on C K by ⊎ and apply the addition formula [3]( X, Y ) ⊎ ( X ′ , Y ′ ) = (cid:0) max (cid:0) Y, X + X ′ + Y ′ (cid:1) − max (cid:0) X + 2 X ′ , Y + Y ′ (cid:1) , max (cid:0) X + Y + 2 Y ′ , X ′ (cid:1) − max (cid:0) X + 2 X ′ , Y + Y ′ (cid:1)(cid:1) . The group trop (Γ) can also be obtained by applying the procedure of ultradiscretization directlyto g and g . Let us consider the inhomogeneous coordinate ( x := x /x , y := x /x ) of P ( C ). Let g : ( x, y ) ( x ′ , y ′ ) and g : ( x, y ) ( x ′′ , y ′′ ). Then we have (cid:0)(cid:12)(cid:12) x ′ (cid:12)(cid:12) , (cid:12)(cid:12) y ′ (cid:12)(cid:12)(cid:1) = (cid:18) | y || x | , | x | (cid:19) (cid:0)(cid:12)(cid:12) x ′′ (cid:12)(cid:12) , (cid:12)(cid:12) y ′′ (cid:12)(cid:12)(cid:1) = ( | x | , | y | ) . Replacing | x | and | y | with e X/ε and e Y/ε respectively and taking the limit ε →
0, we obtain(
X, Y ) ˜ g ( Y − X, − X ) = ( X, Y ) ⊎ V ( X, Y ) ˜ g ( X, Y ) , where we denote the action on P ,trop induced form g and g by ˜ g and ˜ g , respectively.Next we consider α ( G ) ≃ T . Remember that each singular member E s i ( i = 1 , , ,
4) in theHesse pencil corresponds to the vertex of the tetrahedron on which T acts (see (1) and (2)). Thesingular members of the tropical Hesse pencil are C ∞ and C which are the tropicalization of E s E s i , ( i = 2 , , α ( g ), which permutes s and s with s and s respectively, must vanish; while the action of α ( g ), which fixes s and permutes s , s , and s cyclically, reduces to the action fixing both C and C ∞ . Therefore, we have trop ( α ( G )) ≃ trop ( T ) ≃ h i . Thus the tropical analogue of the Hessian group fixes each member of the tropical Hesse pencil.Furthermore, we consider the tropicalization of g = g . We ultradiscretize g directly as wellas g and g . The action of g on P ( C ) is given as( x, y ) g ( y, x )in the inhomogeneous coordinate. It follows that we have( X, Y ) ˜ g ( Y, X ) , where ( X, Y ) is the inhomogeneous coordinate of P ,trop and ˜ g is the action on P ,trop induced fromthat of g by applying the procedure of ultradiscretization. Thus we conclude that the tropicalanalogue of ˜ T ≃ h g , g i = G / Γ is the group of order two generated by ˜ g : trop (cid:16) ˜ T (cid:17) ≃ h ˜ g i ≃ (cid:28)(cid:18) (cid:19)(cid:29) ⊂ SL (2 , F ) . We then obtain the following theorem concerning the tropical analogue of G . Theorem 1
The dihedral group D of degree three, D = h ˜ g , ˜ g i ≃ ( Z / Z ) ⋊ (cid:28)(cid:18) (cid:19)(cid:29) where ˜ g , ˜ g ∈ P GL (3 , T ) satisfy ˜ g = ˜ g = (˜ g ˜ g ) = 1, is the group of linear automorphismsacting on the tropical Hesse pencil . The action of ˜ g on each curve of the pencil is realized as thereflection with respect to the line Y = X passing through the vertex V ; and the action of ˜ g oneach curve is realized as the addition with V .In this paper, we consider linear automorphisms acting on the Hesse pencil only. To investigatea tropical analogue of the Cremona group, the group of birational automorphisms acting on theHesse pencil, is a further problem. References [1] Artebani M and Dolgachev I,
Preprint arXiv:math/0611590v3 (2006)[2] Kajiwara K, Kaneko M, Nobe A and Tsuda T,
Kyushu J. Math. (2009) 315-338[3] Nobe A, Preprint , to appear in
RIMS Kokyuroku [4] Shaub H C and Schoonmaker H E,
Am. Math. Mon. (1931) 388-393 Identifying γ i ≃ ( γ i , ∈ ( Z / Z ) , we define the multiplication of ( γ , m ) , ( γ , m ) ∈ ( Z / Z ) ⋊ (cid:28)(cid:18) (cid:19)(cid:29) by( γ , m ) · ( γ , m ) = ( γ + m γ , m m ) ..