A new class of integrable maps of the plane: Manin transformations with involution curves
AA new class of integrable maps of the plane: Manin transformationswith involution curves
Peter H. van der KampDepartment of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia.Email: [email protected] 22, 2020
Abstract
For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curveof the pencil in exactly one non base-point of the pencil. Involution curves can be used to construct integrablemaps of the plane which leave invariant a cubic pencil.
Keywords:
Integrable map of the plane, Manin transformation, Involution, Invariant, Pencil of cubic curves.
One clear definition of an integrable map is the notion of Liouville integrability, which requires the existence ofsufficiently many invariant functions in involution with each other [3, 12]. One way to obtain such maps is byreduction from integrable lattice equations [5, 11]. For planar maps, integrability is equivalent to the preservationof a pencil of curves and measure-preservation [8]. A rather large (18-parameter) family of integrable maps of theplane was obtained by Quispel, Roberts and Thompson (QRT) [7]. These maps preserve a pencil of biquadraticcurves, and have been studied in the context of algebraic geometry of elliptic surfaces in [2, 9].Consider a linear pencil of cubic curves in the ( u, v )-plane of the form P ( C ) := F ( u, v ) − CG ( u, v ) = 0 . (1)Such a pencil has (at most) 3 base-points, which are the solutions of F = G = 0 (Bezout’s theorem). A straightline through a (non-singular) base-point p will intersect each curve in the pencil in 2 other points. Thus one candefine a map which interchanges these 2 points. Such a map, denoted ι p , is coined a Manin [4] involution in [2,Section 4.2] and a p -switch in [10]. The composition of two Manin involutions τ p,q = ι q ◦ ι p is an integrable map ofthe plane, as it leaves invariant a pencil of curves and is measure preserving [10, Proposition 8].Similarly, one can construct involutions that leave invariant pencils of curves of degree N = 2, or N = 4 [10].For N = 2, the point p , which is called the involution point of ι p , can be chosen to be any point but one of thebase-points of the pencil. For N = 4, one requires the pencil to have two base-points which are double points ofboth F = 0 and G = 0, and these points are taken as involution points. In [10] it was shown that this constructiondoes not generalise to maps which preserve a pencil of degree > N = 4 pencil, namely a biquadratic pencil. Such a pencil has 2 double base-points, at ( ∞ , , ∞ ). The corresponding involutions are the horizontal switch ι , and the vertical switch ι , and the QRTmap is the composition τ = ι ◦ ι [2, page viii].In [6] a new type of Manin involutions was introduced. These involutions switch the 2 points that are notbase-points in the intersection of a curve of the pencil with a specified curve of degree >
1. For example, consider apencil P of degree N = 4 with 2 double base-points and 8 simple ones. Taking 4 base-points of P , including the 2double base-points, one constructs the unique pencil of degree N = 2 which has those points as simple base-points.Then each curve of this pencil intersects each curve of P in precisely two other points. Thus an involution can bedefined that switches these 2 points. 1 a r X i v : . [ n li n . S I] S e p n this letter, we describe another type of Manin involutions. Recently, a planar map γ [1, Equation (1.1)]was obtained by taking an open boundary reduction from the Q δ = 0) quad-equation, and it was shown to be acomposition of two involutions, γ = ι q ◦ ι p , where one involution point, e.g. p , is a simple base point of a singularcubic pencil, and the other involution point, q = q ( C ), depends on the particular curve in the pencil. We describethe special geometry of these so-called involution curves, which lead to novel classes of maps that leave invariantpencils of cubic curves, and are measure-preserving. The map γ found in [1] is given by γ ( u, v ) = ( u + v ) ( au + b ( au + v + 1) v ) b ( u + ( ub + v + 1) v ) (( a + b ) uv + a ( bu + v ) v + a ( u + v ) ) (cid:18) u ( au + ( abu + bv + a ) v ) a ( u + ( au + v + 1) v ) , v (cid:19) . (2)It leaves invariant a singular pencil of cubic curves (of genus 0), of the form (1) with F ( u, v ) = v (1 + u + v ) + au ( ub + uv + v ) , G ( u, v ) = uv. (3)This pencil has 5 base-points, in homogeneous coordinates: b = (0 : 0 : 1) , b = (0 : − , b = (1 : 0 : 0) , b = (1 : − , b = (1 : − a : 0) , (4)of which the first one is a double point. The simple base-points yield the following b -switches: ι b ( u, v ) = (cid:32) u (cid:0) abu v + abuv + abuv + buv + bv + au + buv + 2 bv + vb (cid:1) bv ( v + 1) ( v + u + 1) ( au + v + 1) , au ( au + v + 1) ( v + u + 1) vb (cid:33) ,ι b ( u, v ) = (cid:18) bv ( v + 1) ua ( vb + 1) , v (cid:19) ,ι b ( u, v ) = (cid:18) vb ( v + u + 1) ( u + v ) buv + bv + au + vb , au ( u + v ) buv + bv + au + vb (cid:19) ,ι b ( u, v ) = (cid:18) vb ( au + v + 1) ( au + v ) a ( abuv + bv + vb + u ) , u ( au + v ) abuv + bv + vb + u (cid:19) . One can check that each of the compositions γ ◦ ι b i and ι b i ◦ γ is a involution. In fact, they are p -switches, where p depends not only on a, b but also on C , the parameter of the pencil P ( C ). We define γ ◦ ι b i = ι h i and ι b i ◦ γ = ι k i .The involution point of a p -switch δ can be calculated as follows. Starting with x = ( u, v ), determine x = δ ( x ) , x = γ ( x ) , x = δ ( x ) . The involution point p of δ = ι p is obtained as the intersection of the lines x x and x x .Following this procedure, for each map ι h i and ι k i , the variables u, v can be eliminated. Explicit expressions forthe involution points h i and k i in terms of a, b, C are provided in Appendix A. The points h i , k i are in the intersectionof the pencil P ( C ) with curves that we denote by H i , K i . These curves can be obtained as follows. For an involutionpoint p ( a, b, C ) eliminate (using a Groebner basis) the variable C from the set of equations { u = p , v = p } . Weobtain H := uvab + v b + au + av = 0 H := auv + v + u + v = 0 H := vbu a + au + (cid:0) a + b (cid:1) uv + 2 auv + av + av = 0 H := uvb + v + u + v = 0 K := a bu v + a u + (cid:0) ba + ab (cid:1) uv + 2 uvab + bv a + b v = 0 K := u vab + a u + (cid:0) ab + b (cid:1) uv + 2 uvab + b v + b v = 0 K := au + vb = 0 K := a bu v + a u + (cid:0) ba + b (cid:1) uv + 2 uvab + b v + b v = 0 . b b b b b H t H H t H t K t K K K t t )at the base-points of the pencil.Each of the above curves intersects the pencil P ( C ), with F, G given by (3) in some (or all) of the base-points of thepencil. The type of intersection at the base-points of the pencil is given in table 1. Note that all curves in P ( C ) aretangent to each other at base point b . The intersection number of the quadratic curve H with the pencil P ( C )equals 2 · K with the pencil P ( C ) equals 3 · H i , K i , apart from the intersection at the base-points,there is only one other simple intersection between the curve and the pencil. This makes it possible to define, foreach curve in the pencil, the involution point to be this unique (non-base point) intersection point. In the sequel, we refer to a curve which consists of involution points as an involution curve.
Definition 1.
A curve Q is an involution curve for a cubic pencil P ( C ) if the sum of the intersection numbersat the base-points of P ( C ) is · degree ( Q ) -1. Let Q be an involution curve for a cubic pencil P ( C ) whose set of base-points is denoted B . An involution ι Q can be constructed as follows. For any given p (cid:54)∈ B let C p be the value of C such that p is on the curve R = P ( C p ).We take q to be the unique non base-point intersection of Q and R . There are four cases to consider. If p = q is aflex point of R then ι Q ( p ) = p . If p = q is a not a flex point of R then ι Q ( p ) is the the unique intersection r (cid:54) = p ofthe tangent line to R at p and R . If p (cid:54) = q and the line pq is tangent to R at p then ι Q ( p ) = p . Finally, if p (cid:54) = q intersects R in a third point r , then ι Q ( p ) = r .In Figure 1 we illustrate the action of a q -switch on a cubic curve R by connecting points p ∈ R to their images ι q ( p ) using straight lines through the involution point q ∈ R . Three of the straight lines in Figure 1 are tangent tothe red curve R , and for two points p we have ι q ( p ) = p .Figure 1: The action of a q -switch on a cubic curve.3 .1 Involution curves for singular cubic pencils The pencil in the previous section, apart from being singular, has a special extra feature, namely that all curvesare tangent in one point. For such pencils the following is clear.
Proposition 2.
Let P ( C ) be a pencil of cubic curves, which has 1 double base point b and simple base-points b , b , b , b such that all curves are tangent at b . Let Q be a curve such that either • degree( Q )=2, b is on Q , Q is tangent at b and contains one other point b i with i ∈ { , , } , or • degree( Q )=3, b is a double point on Q , Q is tangent at b and contains two distinct points b i with i ∈ { , , } .Then Q is an involution curve. Example
Consider the pencil P ( C ) with F = 3 u + 6 u v + 9 uv + 12 v − u − uv + 24 v ,G = (2 v + u )(35 u − uv + 26 v − u + 52 v ) . (5)It has a double base point at (0,0), simple base-points at (2,-1), (1,1), (0,-2), and (331971/549181, 394350/549181).All curves have tangent line T = 20 u + 43 v + 3 = 0 at (2,1). The pencil Q ( D ) := 38 u + 41 uv − v + 9 u − D ( u + 2 v )(2 u + v − P ( C ), which have simple base-points at (0,0), (1,1), (2,-1), and tangent line T at (2,-1). An involution is defined for any curve in Q ( D ). We have illustrated the action of ι (0 , − ◦ ι Q ( − / ontwo different curves in P ( C ) in Figure 2.Figure 2: The action of ι (0 , − ◦ ι Q ( − / on P ( − /
10) (left) and P (13 /
60) (right).The action of ι (0 , − ◦ ι Q (25 / and ι Q ( − / ◦ ι Q (25 / on the same curves in P ( C ) are illustrated in Figures 3and 4. In these, and subsequent figures, the involution points are indicated by red dots.Figure 3: The action of ι (0 , − ◦ ι Q (25 / on P ( − /
10) and P (13 / ι Q ( − / ◦ ι Q (25 / on P ( − /
10) and P (13 / Proposition 3.
Let P ( C ) be a pencil of cubic curves, which has 1 double base point b and simple base-points b , b , b , b , b and possibly some tangency at some of the base-points (in which case there are less of them). Let Q be curve such that either • degree( Q )=1, b is a simple point on Q , or • degree( Q )=1, 2 simple base point of P ( C ) are simple points on Q , or • degree( Q )=2, 4 base-points of P ( C ) including b are simple points on Q , or • degree( Q )=3, b is a double point on Q , 4 other base-points of P ( C ) are simple points on Q .Then Q is an involution curve. Example
The pencil P ( C ) with (5) admits the involution curves: L := 3 x − y − , and Z := 5509 x + 3032 y − x − xy + 6064 y . The line L intersects P ( C ) in the points (0 , − , (cid:18) C − C − , − C − C − (cid:19) . The singular cubic Z is 1 curve in a 3-parameter family of curves which have b as double point and intersect P ( C )in base-points b i , for all i = 0 , . . . ,
4. The remaining intersection point for Z is (cid:18) C − C + 13156506 C − C − C + 31299114 C − , − C − C + 8301321 C − C − C + 31299114 C − (cid:19) . The action of the composition of ι Z and ι L on selected curves of P ( C ) are given in Figure 5.Figure 5: The action of ι Z ◦ ι L on P ( − /
10) and P (13 / .2 Involution curves for non-singular cubic pencils It is possible to construct involution curves for non-singular cubic pencils.
Proposition 4.
Let P ( C ) be a non-singular pencil of cubic curves, with at least 7 simple base-points b , . . . , b .Let Q be a curve of degree 3 such that b is a double point on Q and b , . . . , b are simple points on Q . Then Q isan involution curve. Example
We construct a pencil of cubic curves with finite base-points(0 , , (2 , − , (1 , , (0 , − , (5 , , (3 , , ( − , , ( − , . This fixes a pencil P ( C ) with F = 135 u v − uv + 1436 v + 1173 u + 2737 uv − v − u − vG = 391 u − u v + 673 uv − v − uv + 1539 v − u + 6186 v, (6)which has 9-th base point (126933249 / , / ,
0) and simple points (2 , − , , − , , − ,
3) in commonwith P ( C ): Q := 597 u − u v − uv + 104 v − u − uv + 208 v = 0 . The unique non base-point intersection between Q and P ( C ) is given by (cid:18) −
598 5174263414217 C + 26456591132843 C + 44024545626872 C + 239475428206082293729901271491 C + 7965008023759238 C + 8157021862775051 C + 2203322144658228 , − C + 7577888329154480 C + 11109497069357891 C + 52123323670479602293729901271491 C + 7965008023759238 C + 8157021862775051 C + 2203322144658228 (cid:19) . The action of ι (0 , ◦ ι Q on two curves of the pencil is illustrated in Figure 6.Figure 6: The action of ι (0 , ◦ ι Q on the two curves of the pencil P ( C ) given by (6) which contain the points(97 / , −