Manin involutions for elliptic pencils and discrete integrable systems
MMANIN INVOLUTIONS FOR ELLIPTIC PENCILSAND DISCRETE INTEGRABLE SYSTEMS
MATTEO PETRERA, YURI B. SURIS, KANGNING WEI AND REN ´E ZANDERInstitut f ¨ur Mathematik, MA 7-1Technische Universit¨at Berlin, Str. des 17. Juni 136, 10623 Berlin, GermanyA
BSTRACT . We contribute to the algebraic-geometric study of discrete integrable systemsgenerated by planar birational maps:(a) we find geometric description of Manin involutions for elliptic pencils consisting ofcurves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index1), and(b) we characterize special geometry of base points ensuring that certain compositionsof Manin involutions are integrable maps of low degree (quadratic Cremona maps). Inparticular, we identify some integrable Kahan discretizations as compositions of Manininvolutions for elliptic pencils of higher degree.
1. I
NTRODUCTION
Intimate relation of the theory of integrable systems to algebraic geometry is well ap-preciated in these days. In the present paper, we address this relation for a very basicclass of integrable systems, namely for discrete integrable systems generated by birationalmaps of CP with a rational integral of motion and an invariant measure with a rationaldensity (whereas the emphasis is put on the integral of motion). In such a system, orbitsare confined to invariant curves (level sets of the integral), and on each invariant curvethe map induces an automorphism.For general reasons, invariant curves must have genus zero or one, since only in thesecases the induced automorphisms on the invariant curves can be of infinite order (non-periodic). Our main object of interest will be rational elliptic surfaces (i.e., surfaces bira-tionally equivalent to a plane, admitting a fibration by elliptic curves). A classification ofpencils of elliptic curves in a plane was given by Bertini, a modern proof of this result isdue to Dolgachev [8]. It says that any such pencil is birationally equivalent to a Halphenpencil of index m ∈ N , in which a generic curve is of degree 3 m and has multiplicity m ateach of nine base points.Planar maps preserving pencils of elliptic curves appeared over and over again in thetheory of discrete integrable systems. Probably, the most prominent example is givenby QRT maps [9, 22, 24], which preserve pencils of biquadratic curves. Further examplesare given by (autonomous versions of) discrete Painlev´e equations [11, 23], as well as theso called HKY maps which preserve pencils of curves of higher degrees [15]. Recently,further examples appeared in the context of the so called Kahan discretization [14], [17,21], [5–7]. A sort of a classification of such maps, based on the Dolgachev’s classificationof rational elliptic surfaces, was given in [3] and sounds almost tautologically: a birational E-mail: [email protected], [email protected], [email protected],[email protected] . a r X i v : . [ n li n . S I] A ug MATTEO PETRERA, YURI B. SURIS, KANGNING WEI AND REN ´E ZANDER map preserving an Halphen pencil (of index m ) either preserves each fiber or interchangesthe fibers in a nontrivial way.In the present paper, we are occupied with a construction of integrable maps preserv-ing a pencil of elliptic curves, based only on the pencil itself. The basic idea is to com-pose two (non-commuting) birational involutions preserving the pencil. This constructionis almost obvious for QRT maps, where one can always use the horizontal and the ver-tical switches as suitable involutions (the horizontal switch assigns to any point of a bi-quadratic curve the second intersection point of the curve with the horizontal line throughthe original point; the definition of the vertical switch is analogous). In [12, 13] this con-struction was extended to a (projectively equivalent) case of maps preserving a pencil ofquartic curves with two double points, and it was observed that the corresponding in-volutions are closely related to the so called Manin involutions , which were introducedin [16] for pencils of cubic curves. Recall that, for a pencil of cubic curves with nine basepoints p , . . . , p , the Manin involution centered at p assigns to each point p the thirdintersection point of the line ( p p ) with the curve of the pencil passing through p . In [18],it was shown that Kahan discretizations of canonical Hamiltonian systems with cubicHamiltonian can be characterized as compositions of Manin involutions in the case of aspecial geometry of the base points of the cubic pencil, see [19] for a related result. Wealso mention that compositions of Manin involutions for cubic pencils appeared in thetheory of discrete integrable systems already in [10, 26].Birational involutions of the plane are classical and well studied objects in algebraicgeometry. Their classification was given by Bertini [2] and says that every non-trivial bi-rational involution of P is birationally conjugate to exactly one of the following: a deJonqui`eres involution of degree d > d witha unique singular point p which is an ordinary multiple point of multiplicity d − p for a pencil of cubic curves with nine base points p , . . . , p is a de Jonqui`eres involution with center p defined by a hyperelliptic curve ofdegree d = p and eight Weierstrass points p , . . . , p (we are indebted to I. Dolgachev for explaining this).In the present paper, our goal is to use Manin involutions for elliptic pencils to con-struct integrable dynamical systems. We start by finding a geometric formulation ofManin involutions for elliptic pencils consisting of curves of higher degree. This is achievedvia a birational conjugation from the canonical construction for cubic pencils. It seems,however, that the geometric construction directly in terms of the original pencils is notavailable in the literature. The most non-trivial contribution consists in finding the con-ditions under which the involutions and their compositions are of low degree, thus pro-ducing simple and attractive examples of integrable birational maps.2. E LLIPTIC PENCILS AND CUBIC PENCILS
Throughout the paper, we work over the field C . We consider pencils of curves in P ,i.e., families of curves P = { C λ } parametrized by λ ∈ P , C λ = (cid:110) [ x : x : x ] ∈ P : F ( x , x , x ) + λ G ( x , x , x ) = (cid:111) . ANIN INVOLUTIONS FOR ELLIPTIC PENCILS AND DISCRETE INTEGRABLE SYSTEMS 3
Here F , G are two homogeneous polynomials of degree d . The points of the set B = (cid:110) [ x : x : x ] ∈ P : F ( x , x , x ) = G ( x , x , x ) = (cid:111) are called base points of the pencil P . As usual, they are counted with multiplicities. Wewill assume that the multiplicities of each base point on both curves F = G = type of the pencil is then ( d ; ( n ) ( n ) ( n ) . . . ) where d is the degree of the curves of the pencil, n the number of simple base points, n the number of double base points, n the number of triple base points and so on. Thepencil itself will be denoted by P (cid:0) d ; p m , p m , . . . p m N N (cid:1) ,which refers to the degree d and the list of base points p i with their respective multiplici-ties m i , so that N = n + n + n + . . .. Multiplicities m i = d = ∑ k n k k . (1)Through any point [ x : x : x ] ∈ P \ B , there passes a unique curve C λ of the pencil,with λ = − F ( x , x , x ) / G ( x , x , x ) .Our main interest is in the elliptic pencils , for which generic curves of the pencil are ofgenus g =
1. According to the degree-genus formula, the genus of irreducible curves ofthe pencil is given by: g = ( d − )( d − ) − ∑ k n k k ( k − ) =
1. (2)We remark that by virtue of (1), the latter equation is equivalent to3 d = ∑ k n k k , (3)where the right-hand side is the total number of base points (counted with multiplicities).Examples:(1) A pencil of the type (
3; 9 ) of cubic curves with nine simple base points.(2) A pencil of the type (
4; 8 ) of curves of degree 4 with eight simple and two doublepoints. By an automorphism of P , we can send the double points to infinity (say,to [ ] and [ ] ), then in the affine coordinates ( x / x , x / x ) , we get apencil of biquadratic curves. Such pencils are pretty well studied and have plentyof applications in the theory of discrete integrable systems [9, 22].(3) A pencil of the type (
6; 6 ) of curves of degree 6 with six simple points, threedouble points and two triple points.(4) A pencil of the type (
6; 9 ) of curves of degree 6 with nine double points, i.e., aHalphen pencil of index 2. Remark.
We do allow infinitely near base points , at which the curves of the pencil have tosatisfy certain conditions tangency up to certain order. In the formulations of our generalresults about geometry of Manin involutions, we silently assume that the geometry ofbase points is generic, in particular, that there are no incidental collinearities. However,
MATTEO PETRERA, YURI B. SURIS, KANGNING WEI AND REN ´E ZANDER all our main examples are non-generic with a plenty of incidental collinearities, since it isexactly this feature that allows for a substantial drop of degree of the resulting birationalmaps. We hope this will not lead to any confusions.3. M
ANIN INVOLUTIONS
For cubic curves, one has a simple geometric interpretation of the addition law. Cor-respondingly, there is a simple geometric construction of certain birational involutions of P induced by pencils of cubic curves, cf. [16, p. 1376], [25, p. 35]. These were dubbed Manin involutions in [9, Sect. 4.2].
Definition 1. (Manin involutions for cubic pencils) Consider a nonsingular cubic curve C in P , and a point p ∈ C. The
Manin involution on C with respect to p is the map I C , p : C → C defined as follows: for a generic p (cid:54) = p , theimage I C , p ( p ) is the unique third intersection point of C with the line ( p p ) ; for p = p , the line ( p p ) should be interpreted as the tangent line to C at p . Consider a pencil P = { C λ } of cubic curves in P with at least one nonsingular member.Let p be a base point of the pencil. The Manin involution I P , p : P (cid:57)(cid:57)(cid:75) P is a birational mapdefined as follows. For any p ∈ P which is not a base point, I P , p ( p ) = I C λ , p ( p ) , where C λ isthe unique curve of the pencil through the point p. For elliptic pencils of degree higher than 3, the geometric construction of Manin in-volutions seems to be unknown. The only exception are the vertical and the horizontalswitches in biquadratic pencils, of which the famous QRT maps are composed [9, 22].They can be immediately translated to a construction of generalized Manin involutions forquartic pencils with two double points, with respect to the both double points [13]. Thedefinition of the generalized Manin involution I C , p for a quartic curve C and a doublepoint p ∈ C , resp. of the generalized Manin involution I P , p for a quartic pencil withtwo double points, one of them being p , literally coincides with Definition 1. This is jus-tified by the fact that any line through a double point p ∈ C still intersects the quarticcurve C at two further points.The main goal of this paper is to elaborate on the geometric definition of Manin invo-lutions in arbitrary elliptic pencils birationally equivalent to a Halphen pencil of index 1,i.e., to a cubic pencil.To find a birational equivalence, one can resolve the multiple base points by means ofsuitable birational transformations. Often, the simplest way of doing this is by a sequenceof quadratic Cremona transformations . Recall that a generic quadratic Cremona transforma-tion φ : P (cid:57)(cid:57)(cid:75) P has three distinct fundamental points I ( φ ) = { p , p , p } which areblown up to three lines ( q q ) , ( q q ) , ( q q ) , respectively. The three lines ( p p ) , ( p p ) , ( p p ) are blown down to the points q , q , q , respectively, which build the indetermi-nacy set of the inverse map: I ( φ − ) = { q , q , q } . A practical way to construct such amap consists in finding homogeneous polynomials φ ( x , x , x ) of degree 2 vanishing atthe fundamental points p , p , p . Geometrically, we are speaking about the set of conicsin P through p , p , p . The space of solutions of this linear system is two-dimensional: αφ + βφ + γφ , where φ , φ , φ are homogeneous polynomials of ( x , x , x ) of degree2. The map φ : [ x : x : x ] (cid:55)→ [ u : u : u ] = [ φ ( x , x , x ) : φ ( x , x , x ) : φ ( x , x , x )] (4) ANIN INVOLUTIONS FOR ELLIPTIC PENCILS AND DISCRETE INTEGRABLE SYSTEMS 5 is the sought after birational map P (cid:57)(cid:57)(cid:75) P . A different choice of a basis φ , φ , φ of thenet corresponds to a linear projective transformation of the target plane P .Note that the pre-image of a generic line au + bu + cu = P is the conic a φ + b φ + c φ = p , p , p ) in the source plane P . Itfollows that for any regular point p of φ , the pencil of lines P ( q ) through q = φ ( p ) in P corresponds to the pencil of conics P ( p , p , p , p ) in P .4. E XAMPLE : A QUARTIC PENCIL WITH TWO DOUBLE BASE POINTS
Geometry of the base points.
Consider an elliptic pencil in P of the type (
4; 8 ) , E = P ( p , . . . , p , p , p ) .Thus, E consists of quartic curves with 8 simple base points p , . . . , p and two doublebase points p , p . The position of the ten base points is not arbitrary: for a genericconfiguration of ten points, there exists just one curve of degree 4 through these points,having the prescribed two of them as double points. On the other hand, for a genericconfiguration of nine points, there is a one-parameter family (a pencil) of curves of de-gree 4 through these points, having the prescribed two of them as double points (nineincidence conditions plus four second order conditions, altogether 13 linear conditions,while a generic curve of degree 4 has 14 non-homogeneous coefficients). Counting theintersection numbers, we see that all curves of the pencil pass through a further simplepoint (indeed, seven simple points and two double points contribute 7 × + × = Proposition 1.
In a generic pencil P ( p , . . . , p , p , p ) , one of the curves is reducible andconsists of the line ( p p ) and a cubic curve passing through all ten base points p , . . . , p .Proof. Fix any point p ∈ ( p p ) different from p , p , and consider the unique curve C of the pencil through p . If the line ( p p ) would not be a component of this curve,then the intersection number of C with the line ( p p ) would be at least 2 × + =
5, acontradiction. Thus, the curve C is reducible and contains the line ( p p ) as one of thecomponents. Another component is a cubic curve through p , . . . , p (with p , p beingsimple points on the cubic). (cid:3) Remark 1.
If the reducible curve C happens to contain ( p p ) as a double line, then the remain-ing component is a conic through eight base points p , . . . , p . Birational reduction to a cubic pencil.
Consider a pencil E = P ( p , . . . , p , p , p ) .Let φ : P (cid:57)(cid:57)(cid:75) P be a quadratic Cremona map with the fundamental points p , p , p .Thus, φ blows down the lines ( p p ) , ( p p ) , ( p p ) to points denoted by q , q , q ,respectively, and blows up the points p , p , p to the lines ( q q ) , ( q q ) , ( q q ) . Allother base points p i , i =
2, . . . , 8, are regular points of φ , their images will be denoted by q i = φ ( p i ) . Proposition 2.
Under the map φ : a) Quartic curves of the original pencil E in P correspond to curves of a cubic pencil P ( q , . . . , q , q , q ) MATTEO PETRERA, YURI B. SURIS, KANGNING WEI AND REN ´E ZANDER with nine base points in P ; the point q is not a base point of the latter pencil. b) For i =
2, . . . , 8 , the pencil of lines P ( q i ) in P corresponds to the pencil of conics P ( p i , p , p , p ) in P . c) The pencils of lines P ( q ) , P ( q ) in P correspond to the pencils of lines P ( p ) , P ( p ) in P .Proof. a) The total image of a quartic curve C ∈ E is a curve of degree 8. Since C passesthrough p , its total image contains the line ( q q ) . Since C passes through p and p with multiplicity 2, its total image contains the lines ( q q ) and ( q q ) withmultiplicity 2. Dividing by the linear defining polynomials of all these lines, wesee that the proper image of C is a curve of degree 8 − =
3. This curve hasto pass through all points q i , i =
2, . . . , 8. The curve C of degree 4 has no otherintersections with the line ( p p ) different from two double points p and p ,therefore its proper image does not pass through q . On the other hand, the curve C of degree 4 has one additional intersection point with each of the lines ( p p ) and ( p p ) , different from the simple point p and the double point p , resp. p .Therefore, its proper image passes through q , resp. q , with multiplicity 1.b) This follows from the fact that p i , i =
2, . . . , 8, are regular points of φ .c) Consider the total pre-image of a line through q . It is a conic through p , p , p whose defining polynomial vanishes on the line ( p p ) . Thus, the conic is re-ducible and contains that line. Dividing by the defining polynomial of this line (ofdegree 1), we see that the proper pre-image is a line which must pass through p .Similarly, the proper pre-image of a line through q is a line through p . (cid:3) Let S be the elliptic surface obtained from P by blowing up the ten base points p i , i =
1, . . . , 10. Let E i be the exceptional divisor classes of the blow-ups. The Picard groupof S is Pic ( S ) = Z D (cid:76) Z E (cid:76) . . . (cid:76) Z E . The class of a generic curve of the pencil E is4 D − E − E − E − E − E − E − E − E − E − E . (5)The quadratic Cremona map of Proposition 2 corresponds to the following change ofbasis of the Picard group: D (cid:48) = D − E − E − E , E (cid:48) = D − E − E , E (cid:48) = D − E − E , E (cid:48) = D − E − E , (6)and E (cid:48) i = E i for i =
2, . . . , 8.One can check that E (cid:48) is a redundant class, in the sense that the class (5) of a generalcurve of the pencil is expressed through E (cid:48) , . . . , E (cid:48) only:4 D − E − . . . − E − E − E = D (cid:48) − E (cid:48) − E (cid:48) − . . . − E (cid:48) . (7) ANIN INVOLUTIONS FOR ELLIPTIC PENCILS AND DISCRETE INTEGRABLE SYSTEMS 7
This corresponds to the fact that q is not a base point of the φ -image of the pencil E .Note that E (cid:48) = D − E − E is the class of (the proper transform of) the line ( p p ) in P . Blowing down E (cid:48) on S , we obtain the surface S (cid:48) which is a minimal elliptic surface(blow-up of P at nine points), whose anti-canonical divisor class coincides with (7).Statement b) of Proposition 2 translates to relations D (cid:48) − E (cid:48) i = D − E − E − E − E i in the Picard group (for i =
2, . . . , 8), while statement c) translates as D (cid:48) − E (cid:48) = D − E and D (cid:48) − E (cid:48) = D − E .4.3. Manin involutions.
In the new coordinates, where the pencil consists of cubic curves,Manin involutions I q i with respect to the base points q i of the pencil are defined as in Def-inition 1: for a point q which is not a base point, I q i ( q ) is the unique third intersection ofthe the line ( q i q ) with the cubic curve of the pencil passing through q . We now pull backthis construction to the original pencil in the old coordinates. Definition 2. (Manin involutions for pencils of the type (
4; 8 ) ) Consider a pencil E = P ( p , . . . , p , p , p ) . There are two kinds of Manin involutions. Involutions I ( ) i , j , i , j ∈ {
1, . . . , 8 } , defined in terms of the pencil of conics C i , j = P ( p i , p j , p , p ) . Given a point p which is not a base point of E , there is a unique conic of C i , j passing through p anda unique quartic curve of E passing through p. We set I ( ) i , j ( p ) = p (cid:48) , where p (cid:48) is the unique furtherintersection point of those two curves. This intersection is unique, since the intersection numberof the conic with the quartic is × = , while the intersections at the points p i , p j , p , p , andp count as + + + + = . Involutions I ( ) , I ( ) defined in terms of the pencils of lines P ( p ) , resp . P ( p ) . For instance, the involution I is defined as follows. Given a point p which is not a base point of E , we set I ( ) ( p ) = p (cid:48) , where p (cid:48) is the unique third intersection of the line ( p p ) and the quarticcurve of E passing through p. This intersection is unique, since p is a double point of the curve. Indeed:1) Due to point b) of Proposition 2, for any i =
2, . . . , 8, the Manin involution with re-spect to q i is conjugated to the map defined as above in terms of conics through p , p , p , and p i . Remarkably, while in the construction of the conjugating Cremonamap the roles of the simple base points p and p i are asymmetric, in the resultingmap I ( ) i the points p and p i are on an equal footing. More generally, I ( ) i , j = I ( ) j , i ,where the map on the left-hand side should be understood as conjugated to I q j under the quadratic Cremona map with the fundamental points p i , p , p , whilethe right-hand side should be understood as conjugated to I q i under the quadraticCremona map with the fundamental points p j , p , p .2) Due to point c) of Proposition 2, Manin involutions I q , I q on P are conjugatedto the maps I ( ) , I ( ) on P defined in terms of lines through p , p , respectively.Again, while the construction depends on the choice of a simple base point p , theresulting map does not depend on this choice. MATTEO PETRERA, YURI B. SURIS, KANGNING WEI AND REN ´E ZANDER
The involution I ( ) i , j has all base points of the pencil as singularities (indeterminacypoints). For instance, it blows up the point p k to the conic through p i , p j , p k , p , p . How-ever, a composition I ( ) j , k ◦ I ( ) i , j with three distinct simple base points p i , p j , p k is well defined at p k and maps it to p i .Moreover, this composition can be characterized as the unique map acting on the ellipticcurves of the pencil as the shift mapping p k to p i . In particular, this composition does notdepend on j .5. E XAMPLE : A SEXTIC PENCIL WITH THREE DOUBLE BASE POINTS AND TWO TRIPLEBASE POINTS
Birational reduction to a cubic pencil.
Consider an elliptic pencil in P of the type (
6; 6 ) , E = P ( p , . . . , p , p , p , p , p , p ) ,consisting of curves of degree 6 with six simple base points p , . . . , p , three double basepoints p , p , p , and two triple base points p , p . We reduce it to a cubic pencil in twosteps. Step 1.
Apply a quadratic Cremona map φ (cid:48) with the fundamental points p , p , p (the both triple base points and one of the double base points). Thus, φ (cid:48) blows downthe lines ( p p ) , ( p p ) , ( p p ) to the points denoted by q , q , q , respectively, andblows up the points p , p , p to the lines ( q q ) , ( q q ) , ( q q ) . All other base points p i , i =
1, . . . , 8 are regular points of φ (cid:48) and their images are denoted by q i = φ (cid:48) ( p i ) . Proposition 3.
The change of variables φ (cid:48) maps a pencil E = P ( p , . . . , p , p , p , p , p , p ) of sextic curves to a pencil P ( q , . . . , q , q , q , q , q ) of quartic curves with eight simple basepoints and two double base points. The point q is not a base point of the latter pencil.Proof. The total image of a curve C ∈ E is a curve of degree 12. Since C passes through p , p , p with the multiplicities 2,3,3, its total image contains the lines ( q q ) , ( q q ) , ( q q ) with the same multiplicities. Dividing by the linear defining polynomials of allthese lines, we see that the proper image of C is a curve of degree 12 − =
4. This curvepasses through all points q i , i =
1, . . . , 8 (for i =
7, 8 with multiplicity 2). The curve C ofdegree 6 has no other intersections with the line ( p p ) different from two triple points p and p , therefore its proper image does not pass through q . On the other hand, thecurve C of degree 6 has one additional intersection point with each of the lines ( p p ) and ( p p ) , different from the double point p and the triple point p , respectively p .Therefore, its proper image passes through q , resp. q , with multiplicity 1. (cid:3) Step 2.
Apply a quadratic Cremona map φ (cid:48)(cid:48) with the fundamental points q , q (the bothdouble base points), and one of the simple base points. As we know from Proposition 2,the image of the pencil P ( q , . . . , q , q , q , q , q ) under φ (cid:48)(cid:48) is a pencil of cubic curveswith nine base points. The nature of the composition φ (cid:48)(cid:48) ◦ φ (cid:48) depends on the choice ofthe simple base point q i designated as the third fundamental point of φ (cid:48)(cid:48) , and is differentin the cases i =
1, . . . , 6 and i =
10, 11. It turns out that the first option contains all thepossibilities for the different sorts of Manin involutions, therefore we restrict our attentionto this case, taking, for definiteness, i =
6. Thus, let φ (cid:48)(cid:48) have three fundamental points ANIN INVOLUTIONS FOR ELLIPTIC PENCILS AND DISCRETE INTEGRABLE SYSTEMS 9 q , q , q . It blows down the lines ( q q ) , ( q q ) , ( q q ) to points r , r , r , respectively, andblows up the points q , q , q to the lines ( r r ) , ( r r ) , ( r r ) . All other base points q i , i =
1, . . . , 5, 10, 11 are regular points of φ , their images will be denoted by r i = φ ( q i ) . Asfollows from Propositions 3, 2, we have: Proposition 4.
The change of coordinates φ = φ (cid:48)(cid:48) ◦ φ (cid:48) : P (cid:57)(cid:57)(cid:75) P maps a pencil E = P ( p , . . . , p , p , p , p , p , p ) of sextic curves in P to a pencil P ( r , . . . , r , r , r , r , r ) of cubic curves with nine base points in P . The points r and r are not base points of this cubicpencil. Properties of the birational change of coordinates φ = φ (cid:48)(cid:48) ◦ φ (cid:48) on P are easily obtained.It is a Cremona map of degree 4 which blows down the lines ( p p ) , ( p p ) , ( p p ) to the points r , r , r , respectively, and blows down the conics C ( p , p , p , p , p ) , C ( p , p , p , p , p ) , C ( p , p , p , p , p ) to the points r , r , r , respectively. Moreover, φ blows up the points p , p , p to the lines ( r r ) , ( r r ) , ( r r ) , respectively, and thepoints p , p , p to the conics C ( r , r , r , r , r ) , C ( r , r , r , r , r ) , C ( r , r , r , r , r ) ,respectively. Points p i , i =
1, . . . , 5, are regular points of φ , their images are r i = φ ( p i ) .The pre-image of a generic line in P is a quartic curve passing through p , . . . , p (thepoints p and p being of multiplicity 2). In particular, for any regular point p , the pencilof lines P ( r ) through r = φ ( p ) in P corresponds to the pencil P ( p , p , p , p , p , p , p ) of quartic curves in P . Proposition 5.
The change of coordinates φ = φ (cid:48)(cid:48) ◦ φ (cid:48) : P (cid:57)(cid:57)(cid:75) P has the following properties: a) For i =
1, . . . , 5 , the pencil of lines P ( r i ) in P corresponds to the pencil P ( p i , p , p , p , p , p , p ) of quartic curves in P . b) For i =
10, 11 , the proper pre-images of lines of the pencil P ( r i ) in P are cubics of therespective pencil P ( p , p , p , p , p , p ) , P ( p , p , p , p , p , p ) in P . c) For i =
7, 8 , the proper pre-images of lines of the pencil P ( r i ) in P are conics of therespective pencil P ( p , p , p , p ) , P ( p , p , p , p ) in P .Proof. a) This follows from the fact that p i , i =
1, . . . , 5 are regular points of φ . b) Consider the total pre-image of a line through r . It is a quartic curve passingthrough p , . . . , p , having p , p as double points. Its defining polynomial van-ishes on the line ( p p ) , which blows down to r . Thus, the quartic is reducibleand contains that line. Dividing by the defining polynomial of the line, we see thatthe proper pre-image is a cubic passing through p , p , p , p , p , with p beinga double point.c) Consider the total pre-image of a line through r . It is a quartic curve passingthrough p , . . . , p , having p , p as double points. Its defining polynomial van-ishes on the conic C ( p , p , p , p , p ) , which blows down to r . Thus, the quarticis reducible and contains that conic. Dividing by the defining polynomial of theconic, we see that the proper pre-image is a conic passing through p , p , p , p . (cid:3) Let S be the elliptic surface obtained from P by blowing up the eleven base points p i , i =
1, . . . , 11. Let E i be exceptional divisor classes of the blow ups. The Picard group of S is Pic ( S ) = Z D (cid:76) Z E (cid:76) . . . (cid:76) Z E . The class of a generic curve of the pencil is6 D − E − E − E − E − E − E − E − E − E − E − E . (8)The quadratic Cremona map φ (cid:48) corresponds to the following change of basis of Pic ( S ) : D (cid:48) = D − E − E − E , E (cid:48) = D − E − E , E (cid:48) = D − E − E , E (cid:48) = D − E − E , (9)and E (cid:48) i = E i for i =
1, . . . , 8. The Cremona map φ (cid:48)(cid:48) corresponds to the following changeof basis of the Picard group: D (cid:48)(cid:48) = D (cid:48) − E (cid:48) − E (cid:48) − E (cid:48) , E (cid:48)(cid:48) = D (cid:48) − E (cid:48) − E (cid:48) , E (cid:48)(cid:48) = D (cid:48) − E (cid:48) − E (cid:48) , E (cid:48)(cid:48) = D (cid:48) − E (cid:48) − E (cid:48) , (10)and E (cid:48)(cid:48) i = E (cid:48) i for i =
1, . . . , 5 and i =
9, 10, 11. Composing (9), (10), we easily compute: D (cid:48)(cid:48) = D − E − E − E − E − E − E , E (cid:48)(cid:48) = D − E − E − E − E − E , E (cid:48)(cid:48) = D − E − E − E − E − E , E (cid:48)(cid:48) = D − E − E − E − E − E , E (cid:48)(cid:48) = D − E − E , E (cid:48)(cid:48) = D − E − E , E (cid:48)(cid:48) = D − E − E , (11)and E (cid:48)(cid:48) i = E i for i =
1, . . . , 5. One can check that the classes E (cid:48)(cid:48) = D − E − E − E − E − E , E (cid:48)(cid:48) = D − E − E are redundant, in the sense that the class (8) of a general curve of the pencil E is expressedthrough E (cid:48)(cid:48) i , i (cid:54) =
6, 9:6 D − E − . . . − E − E − E − E − E − E = D (cid:48)(cid:48) − E (cid:48)(cid:48) − . . . − E (cid:48)(cid:48) − E (cid:48)(cid:48) − E (cid:48)(cid:48) − E (cid:48)(cid:48) − E (cid:48)(cid:48) . (12)This reflects the fact that r , r are not base points of the resulting cubic pencil. The redun-dant classes are the classes (of the proper transforms) of the conic C ( p , p , p , p , p ) ,resp. of the line ( p p ) in P . The surface S (cid:48) obtained by blowing down E (cid:48)(cid:48) and E (cid:48)(cid:48) on S , is a minimal elliptic surface, whose anti-canonical divisor class coincides with (12).Generic fibers of S (cid:48) are exactly the lifts of generic curves of the initial sextic pencil E .Note that statements of Proposition 5 translate as the following relations in Pic ( S ) : D (cid:48)(cid:48) − E (cid:48)(cid:48) i = D − E i − E − E − E − E − E − E , i =
1, . . . , 5, D (cid:48)(cid:48) − E (cid:48)(cid:48) = D − E − E − E − E − E − E , D (cid:48)(cid:48) − E (cid:48)(cid:48) = D − E − E − E − E .5.2. Manin involutions.
We pull back the standard construction of Manin involutionsfor the cubic pencil in P by means of the map φ to the original pencil in P . Definition 3. (Manin involutions for pencils of the type (
6; 6 ) ) Consider a pencil E = P ( p , . . . , p , p , p , p , p , p ) . There are three kinds of Manininvolutions. Involutions I ( ) i , j , k , i , j ∈ {
1, . . . , 6 } , k ∈ {
7, 8, 9 } . E.g., I ( ) i , j ,9 is defined in terms of quarticcurves of the pencil Q i , j ,9 = P ( p i , p j , p , p , p , p , p ) . Given a point p which is not a base point of E , there is a unique quartic curve of Q i , j ,9 throughp and a unique sextic curve of E through p. We set I ( ) i , j ,9 ( p ) = p (cid:48) , where p (cid:48) is the uniquefurther intersection point of these two curves. This intersection is unique, since the intersec-tion number of the quartic with the sextic is × = , while the intersections at the pointsp i , p j , p , p , p , p , p , and p count as + + + + + + + = . Involutions I ( ) i , j , k with k =
7, 8 are defined similarly. Involutions I ( ) i , k , i ∈ {
1, . . . , 6 } , k ∈ {
10, 11 } . E.g., I ( ) i ,10 is defined in terms of cubic curvesof the pencil K i ,10 = P ( p i , p , p , p , p , p ) . Given a point p which is not a base point of E , there is a unique cubic curve of K i ,10 through pand a unique sextic curve of E through p. We set I ( ) i ,10 ( p ) = p (cid:48) , where p (cid:48) is the unique furtherintersection point of these two curves. This intersection is unique, since the intersection number ofthe cubic with the sextic is × = , while the intersections at the points p i , p , p , p , p , p ,and p count as + + + + + + = . Involutions I ( ) i ,11 are defined similarly. Involutions I ( ) i , j , i , j ∈ {
7, 8, 9 } , defined in terms of conics of the pencil C i , j = P ( p i , p j , p , p ) . A A A C C C B B B F IGURE
1. Pascal configuration of base points of a cubic pencil.
Given a point p which is not a base point of E , there is a unique conic of C i , j through p and a uniquesextic curve of E through p. We set I ( ) i , j ( p ) = p (cid:48) , where p (cid:48) is the unique further intersection pointof these two curves. This intersection is unique, since the intersection number of the conic withthe sextic is × = , while the intersections at the points p i , p j , p , p , and p count as + + + + = .
6. Q
UADRATIC M ANIN MAPS FOR SPECIAL CUBIC PENCILS
In this section, we consider pencils of cubic curves, E = P ( p , . . . , p ) .Generically, a Manin involution for a cubic pencil is a birational map of degree 5 for whichall base points of the pencil are singularities (indeterminacy points). Indeed, consider I E , p i . For any base point p j (cid:54) = p i , all curves C λ of the pencil pass through p i , p j , and haveone further intersection point with the line ( p i p j ) . As a result, I E , p i blows up any basepoint p j ( j (cid:54) = i ) to the line ( p i p j ) . For the same reason I E , p j blows down this line to p i .Thus: Proposition 6.
For a cubic pencil, the Manin transformation I E , p i ◦ I E , p j for any two distinctbase points p i and p j is regular at p i and maps it to p j . For a similar reason, some base points become regular points of Manin involutions ifthere are collinearities among them:
Proposition 7.
For a cubic pencil, if three distinct base points p i , p j , p k are collinear, then I E , p i isregular at p j and at p k and interchanges these two points. We will say that the nine points A i , B i , C i , i =
1, 2, 3, form a
Pascal configuration , if thesix points A , A A , C , C , C lie on a conic, and B = ( A C ) ∩ ( A C ) , B = ( A C ) ∩ ( A C ) , B = ( A C ) ∩ ( A C ) .By Pascal’s theorem, the points B , B , B are collinear. ANIN INVOLUTIONS FOR ELLIPTIC PENCILS AND DISCRETE INTEGRABLE SYSTEMS 13
Theorem 1.
Let the points A i , B i , C i , i =
1, 2, 3 , form a Pascal configuration. Consider the pencil E of cubic curves with these base points. Then the mapf = I E , A ◦ I E , B = I E , B ◦ I E , C (13) = I E , A ◦ I E , B = I E , B ◦ I E , C (14) = I E , A ◦ I E , B = I E , B ◦ I E , C (15) is a birational map of degree 2, with I ( f ) = { C , C , C } and I ( f − ) = { A , A , A } . It hasthe following singularity confinement patterns: ( C C ) → A → B → C → ( A A ) , (16) ( C C ) → A → B → C → ( A A ) , (17) ( C C ) → A → B → C → ( A A ) . (18) Proof.
We start with the following property of the addition law on a nonsingular cubiccurve C . Let P , P , P , P ∈ C , then P − P = P − P ⇔ P + P = P + P ⇔ ( P P ) ∩ ( P P ) ∈ C .Thus, on any cubic curve C ∈ E , we have the following relations: ( A B ) ∩ ( A B ) = C ∈ C ⇒ A − B = A − B ⇒ I E , A ◦ I E , B = I E , A ◦ I E , B , ( B C ) ∩ ( B C ) = A ∈ C ⇒ B − C = B − C ⇒ I E , B ◦ I E , C = I E , B ◦ I E , C , ( A C ) ∩ ( B B ) = B ∈ C ⇒ A − B = B − C ⇒ I E , A ◦ I E , B = I E , B ◦ I E , C .This proves the coincidence of all six representations in (13)–(15). Now it follows fromProposition 6 that f has only three indeterminacy points, I ( f ) = { C , C , C } , and simi-larly, I ( f − ) = { A , A , A } . Moreover, Proposition 6 implies the relations in the middlepart of the singularity confinement patterns (16)–(18). The blow-up and blow-down re-lations are shown with the help of Proposition 7 as follows: f ( C ) = I E , A ◦ I E , B ( C ) = I E , A ( A ) = ( A A ) . (cid:3) Theorem 2.
For a pencil of cubic curves with the base points building a Pascal configuration,perform a linear projective transformation of P sending the Pascal line (cid:96) ( B , B , B ) to infinity.Let ( x , y ) be the affine coordinates on the affine part C ⊂ P . In these coordinates, the mapf : ( x , y ) (cid:55)→ ( (cid:101) x , (cid:101) y ) defined by (13) – (15) is characterized by the following property. There existconstants a , . . . , a ∈ C such that f admits a representation through two bilinear equations ofmotion of the form (cid:40) (cid:101) x − x = a x (cid:101) x + a ( x (cid:101) y + (cid:101) xy ) + a y (cid:101) y + a ( x + (cid:101) x ) + a ( y + (cid:101) y ) + a , (cid:101) y − y = − a x (cid:101) x − a ( x (cid:101) y + (cid:101) xy ) − a y (cid:101) y − a ( x + (cid:101) x ) − a ( y + (cid:101) y ) − a . (19) These equations serve as the Kahan discretization [14, 17] of the Hamiltonian equations of motion (cid:40) ˙ x = ∂ H / ∂ y = a x + a xy + a y + a x + a y + a ,˙ y = − ∂ H / ∂ x = − a x − a xy − a y − a x − a y − a , (20) for the Hamilton functionH ( x , y ) = a x + a x y + a xy + a y + a x + a xy + a y + a x + a y . (21) Proof.
This is a result of a symbolic computation with MAPLE, presented in [18]. (cid:3)
C BA p p p p p p p p p p F IGURE
2. Geometry of base points of a special quartic pencil P ( p , . . . , p , p , p ) .7. Q UADRATIC M ANIN MAPS FOR SPECIAL PENCILS OF THE TYPE (
4; 8 ) We describe the geometry of base points of a pencil of the type (
4; 8 ) for which onecan find compositions of Manin involutions which are quadratic Cremona maps. • Let p , p , p , p be four points of P in general position (no three of them collinear). • Consider three intersection points of three pairs of opposite sides of the completequadrangle with these vertices: A = ( p p ) ∩ ( p p ) , B = ( p p ) ∩ ( p p ) , C = ( p p ) ∩ ( p p ) . (22)Consider the projective involutive automorphism σ of P fixing the point C andthe line (cid:96) = ( AB ) (pointwise). The points of the pairs ( p , p ) and ( p , p ) corre-spond under σ . • Choose a point p ∈ ( p p ) , and define p ∈ ( p p ) so that p , p correspondunder σ , or, in other words, so that the line ( p p ) passes through C . • Let
C ∈ P ( p , p , p , p ) be any conic of the pencil through the specified fourpoints. Define: p = the second intersection point of C with ( p p ) , p = the second intersection point of C with ( p p ) , p = the second intersection point of C with ( p p ) , p = the second intersection point of C with ( p p ) .Recall that A , B , C are vertices of a self-polar triangle for C . In particular, the pro-jective involution σ leaves C invariant. The points of the pairs ( p , p ) and ( p , p ) correspond under σ .We will call the pencil E = P ( p , . . . , p , p , p ) a projectively symmetric quartic pencilwith two double points . Theorem 3.
Let E = P ( p , . . . , p , p , p ) be a projectively symmetric quartic pencil with twodouble points. Then: ANIN INVOLUTIONS FOR ELLIPTIC PENCILS AND DISCRETE INTEGRABLE SYSTEMS 15 a) The projective involution σ can be represented as σ = I ( ) = I ( ) = I ( ) = I ( ) . (23) b) The map f = I ( ) i , k ◦ I ( ) j , k (24) = I ( ) ◦ σ = σ ◦ I ( ) (25) with ( i , j ) ∈ { (
1, 2 ) , (
2, 3 ) , (
3, 4 ) , (
5, 6 ) , (
6, 7 ) , (
7, 8 ) } and k ∈ {
1, . . . , 8 } \ { i , j } , is a birationalmap of degree 2, with I ( f ) = { p , p , p } and I ( f − ) = { p , p , p } . It has the followingsingularity confinement patterns: ( p p ) → p → p → p → p → ( p p ) , (26) ( p p ) → p → p → p → p → ( p p ) , (27) ( p p ) → p → p → ( p p ) . (28) c) We have: f = I ( ) ◦ I ( ) . (29) Proof.
We start with a geometric interpretation of the addition law on a generic curve
C ∈ E . Recall that the pencil E can be reduced to a pencil of cubic curves by means of thequadratic Cremona map φ based at p k , p , p for some k =
1, . . . , 8. Lines in the targetplane P , where the cubic pencil is considered, correspond in the source plane P of thepencil E to conics through p k , p , p . Now let p , q , r , s ∈ C , then, assuming that neither ofthe points p k , p , p is among p , q , r , s , we have: p − r = s − q ⇔ p + q = r + s ⇔ (cid:0) φ ( p ) φ ( q ) (cid:1) ∩ (cid:0) φ ( r ) φ ( s ) (cid:1) ∈ φ ( C ) .The geometry of the pencil E ensures the existence of a large number of quadruples ofbase points which, together with p , p , lie on a conic. Namely, the following sextuplesare conconical: ( p , p , p , p , p , p ) because p ↔ p , p ↔ p under σ , (30) ( p , p , p , p , p , p ) because p ↔ p , p ↔ p under σ , (31) ( p , p , p , p , p , p ) because p ↔ p , p ↔ p under σ , (32) ( p , p , p , p , p , p ) because p ↔ p , p ↔ p under σ , (33) ( p , p , p , p , p , p ) because p ↔ p , p ↔ p under σ , (34) ( p , p , p , p , p , p ) because p ↔ p , p ↔ p under σ . (35)The sextuples (30), (33) and (35) lie on reducible conics (cid:96) ( p , p , p ) ∪ (cid:96) ( p , p , p ) , (cid:96) ( p , p , p ) ∪ (cid:96) ( p , p , p ) and (cid:96) ( p , p , p ) ∪ (cid:96) ( p , p , p ) , respectively. One has, addi-tionally, two more sextuples lying on reducible conics: ( p , p , p , p , p , p ) − on a reducible conic (cid:96) ( p , p , p ) ∪ (cid:96) ( p , p , p ) , (36) ( p , p , p , p , p , p ) − on a reducible conic (cid:96) ( p , p , p ) ∪ (cid:96) ( p , p , p ) . (37) • From (31), (37), (36), (34) there follows: C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) (cid:51) p , C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) (cid:51) p , C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) (cid:51) p , C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) (cid:51) p .We explain how these relations are used, taking the first one as example. Theintersection C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) consists of p , p , p , and p . Upon the quadratic Cremona map φ based at p , p , p , this means that thelines ( q q ) and ( q q ) intersect at q , where q i = φ ( p i ) (the blow-ups of otherthree intersection points do not belong to the proper image of the conics). Thepoint q is one of the base points of the cubic pencil φ ( E ) . Thus, the four relationsabove imply I ( ) k ◦ I ( ) k = I ( ) k ◦ I ( ) k , k =
3, 4, 7, 8. (38) • From (35), (33), (37) there follows: C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) ⊃ ( p p ) , C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) ⊃ ( p p ) , C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) ⊃ ( p p ) , C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) ⊃ ( p p ) .Again, we explain how these relations are used, taking the first one as example.The intersection C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) consists of the point p and the line ( p p ) . Upon the quadratic Cremona map φ based at p , p , p , thepoint p is blown up to a line which does not belong to the proper image of theconics, while the line ( p p ) is blown down to the point q through which theproper images of the both conics pass. Thus, the lines ( q q ) and ( q q ) intersectat q , which is a base point of the pencil φ ( E ) . Summarizing, the four relationsabove imply I ( ) k ◦ I ( ) k = I ( ) k ◦ I ( ) k , k =
1, 4, 7, 8. (39) • From (31) , (32), (33), (34), there follows: C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) (cid:51) p , C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) (cid:51) p , C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) (cid:51) p , C ( p , p , p , p , p ) ∩ C ( p , p , p , p , p ) (cid:51) p .Exactly as before, these four relations imply I ( ) k ◦ I ( ) k = I ( ) k ◦ I ( ) k , k =
1, 2, 7, 8. (40) • In exactly the same way we prove that I ( ) k ◦ I ( ) k = I ( ) k ◦ I ( ) k , k =
3, 4, 5, 8. (41)and I ( ) k ◦ I ( ) k = I ( ) k ◦ I ( ) k , k =
3, 4, 5, 6. (42)This completes the proof of coincidence of all representations (24), as well as themiddle part of the singularity confinement patterns (26), (27).
ANIN INVOLUTIONS FOR ELLIPTIC PENCILS AND DISCRETE INTEGRABLE SYSTEMS 17 • One sees immediately that I ( ) ◦ σ is a shift with respect to the addition law onthe curves of E , sending p → p → p → p , while σ ◦ I ( ) is a shift sending p → p → p → p . Therefore, these shifts must coincide with f . This proves (25)and the middle part of the singularity confinement pattern (28). • Collecting all the results, we see that I ( f ) = { p , p , p } and I ( f − ) = { p , p , p } ,so that f must be a quadratic Cremona map. • It remains to show the blow-up and blow-down relations in the singularity con-finement patterns (26)–(28). To see the blow-down relations on the left, we use therepresentation f = I ( ) ◦ σ . The involution σ is a projective automorphism andhas no singularities, so it suffices to study the blowing down patterns of I ( ) . Bydefinition of the map I ( ) , it is clear that it blows down the line ( p p ) to the point p , and blows down the line ( p p ) to the point p . Since f is a quadratic Cremonamap, the same holds true for the involution I ( ) ; there follows that I ( ) must blowup p to the line ( p p ) , which finishes the proof. For the blow-up relations on theright part of (26)–(28), we use f = σ ◦ I ( ) in a similar manner. (cid:3) We now turn to canonical forms of projectively symmetric quartic pencils with twodouble points, which can be achieved by projective automorphisms of P . The mostpopular one corresponds to the choice p = [ ] , p = [ ] , so that the quarticcurves become biquadratic ones. Denote the inhomogeneous coordinates on the affine part C ⊂ P by ( u , v ) . We can arrange p = ( b ) , p = ( b , 1 ) , p = ( a , − ) , p = ( − a ) , sothat (cid:96) = { u − v = } , C = ( p p ) ∩ ( p p ) = [ − ] , and σ is the Euclidean reflectionat the line (cid:96) , σ ( u , v ) = ( v , u ) .The pencil E of biquadratics reads α ( α + )( u + v − ) − ( α + ) uv + β ( u + v ) − β − λ ( u − )( v − ) =
0, (43)and is symmetric under σ . Involutions I ( ) and I ( ) are nothing but the standard verti-cal and horizontal QRT switches for this pencil, and the map f = I ( ) ◦ σ = σ ◦ I ( ) ofTheorem 3 is given by f : ( u , v ) (cid:55)→ ( (cid:101) u , (cid:101) v ) , (cid:101) u = v , (cid:101) v = α uv + β u − u − α v − β . (44)It is the “QRT root” of f = I ( ) ◦ I ( ) .To arrive at another canonical form of projectively symmetric quartic pencils with twodouble points, we perform a linear projective change of variables in P , given in the non-homogeneous coordinates by u = + β x + yx , v = + β x − yx . (45) Upon substitution (45) and some straightforward simplifications, we come to the follow-ing system (cf. [20]): (cid:40) (cid:101) x − x = x (cid:101) y + (cid:101) xy , (cid:101) y − y = ( − α ) − αβ ( x + (cid:101) x ) + (cid:0) − β ( + α ) (cid:1) x (cid:101) x − ( + α ) y (cid:101) y . (46)In order to give an intrinsic geometric characterization of this canonical form, we willneed the following observation. Proposition 8.
The following five intersection points are collinear: ( p p ) ∩ ( p p ) , ( p p ) ∩ ( p p ) , ( p p ) ∩ ( p p ) , ( p p ) ∩ ( p p ) , ( p p ) ∩ ( p p ) . Proof.
The triple of intersection points ( p p ) ∩ ( p p ) , ( p p ) ∩ ( p p ) , ( p p ) ∩ ( p p ) lies on the Pascal line for the hexagon ( p , p , p , p , p , p ) , while the triple of intersectionpoints ( p p ) ∩ ( p p ) , ( p p ) ∩ ( p p ) , ( p p ) ∩ ( p p ) lies on the Pascal line for the hexagon ( p , p , p , p , p , p ) . These hexagons correspondunder σ , therefore this holds true also for their Pascal lines. Moreover, the Pascal linesshare the point ( p p ) ∩ ( p p ) = C , therefore they must coincide. (cid:3) We will call the line containing the five intersection points from Proposition 8 the doublePascal line . Theorem 4.
For a projectively symmetric pencil of quartic curves with two double points, performa linear projective transformation of P sending the double Pascal line to infinity. By a subsequentaffine change of coordinates ( x , y ) on the affine part C ⊂ P , arrange that (cid:96) coincides with theaxis y = , p = ( − ) , and p = (
0, 1 ) . In these coordinates, the map f : ( x , y ) (cid:55)→ ( (cid:101) x , (cid:101) y ) defined by (24) – (25) is characterized by the following property. There exist a , . . . , a ∈ C witha + a = such that f admits a representation through two bilinear equations of motion of theform (cid:40) (cid:101) x − x = x (cid:101) y + (cid:101) xy , (cid:101) y − y = a − a ( x + (cid:101) x ) − a x (cid:101) x − a y (cid:101) y . (47) Proof.
A symbolic computation with MAPLE. (cid:3)
8. Q
UADRATIC M ANIN MAPS FOR SPECIAL PENCILS OF THE TYPE (
6; 6 ) In this section, we consider the following example considered in detail in [7, 17, 21, 27].Let f : P → P be the birational map given in the non-homogeneous coordinates x =( x , y ) on the affine part C ⊂ P by two bilinear relations between ( x , y ) and ( (cid:101) x , (cid:101) y ) = f ( x , y ) : (cid:101) x − x = γ ( (cid:96) ( x ) (cid:96) ( (cid:101) x ) + (cid:96) ( (cid:101) x ) (cid:96) ( x )) J ∇ (cid:96) + γ ( (cid:96) ( x ) (cid:96) ( (cid:101) x ) + (cid:96) ( (cid:101) x ) (cid:96) ( x )) J ∇ (cid:96) + γ ( (cid:96) ( x ) (cid:96) ( (cid:101) x ) + (cid:96) ( (cid:101) x ) (cid:96) ( x )) J ∇ (cid:96) , (48) ANIN INVOLUTIONS FOR ELLIPTIC PENCILS AND DISCRETE INTEGRABLE SYSTEMS 19 where ( γ , γ , γ ) = (
1, 2, 3 ) , (cid:96) i ( x ) = a i x + b i y are linear forms with a i , b i ∈ C , and J = (cid:18) − (cid:19) . This map is the Kahan discretization of the quadratic flow˙ x = γ (cid:96) ( x ) (cid:96) ( x ) J ∇ (cid:96) + γ (cid:96) ( x ) (cid:96) ( x ) J ∇ (cid:96) + γ (cid:96) ( x ) (cid:96) ( x ) J ∇ (cid:96) , (49)which can be put as ˙ x = ( (cid:96) ( x )) γ − ( (cid:96) ( x )) γ − ( (cid:96) ( x )) γ − J ∇ H ( x ) , (50)where H ( x ) = ( (cid:96) ( x )) γ ( (cid:96) ( x )) γ ( (cid:96) ( x )) γ . (51)Integrability of the Kahan discretization (48) was demonstrated in [7,17,21] for ( γ , γ , γ ) =(
1, 1, 1 ) , (
1, 1, 2 ) , and (
1, 2, 3 ) . Sections 6 and 7 deal with generalizations of the cases ( γ , γ , γ ) = (
1, 1, 1 ) , (
1, 1, 2 ) , the present one deals with ( γ , γ , γ ) = (
1, 2, 3 ) .As shown in [7, 17, 21], the map f defined by (48) admits an integral of motion: H ( x ) = H ( x ) L + ( x ) L − ( x ) M + ( x ) M − ( x ) Q ( x ) , (52)where L ± ( x ) = ± d (cid:96) ( x ) , M ± ( x ) = ± ( d (cid:96) ( x ) − d (cid:96) ( x )) , Q ( x ) = − (cid:0) d (cid:96) ( x ) + d (cid:96) ( x ) (cid:1) ,with d ij = a i b j − a j b i .Thus, the phase space of f is foliated by the pencil of invariant curves C λ = (cid:8) H ( x ) − λ L + ( x ) L − ( x ) M + ( x ) M − ( x ) Q ( x ) = (cid:9) . (53)The pencil has deg = C = { H ( x , y ) = } , consist-ing of the lines { (cid:96) i ( x , y ) = } , i =
1, 2, 3, with multiplicities 1, 2, 3, and C ∞ , consisting ofthe conic Q ( x , y ) = L ± ( x , y ) = M ± ( x , y ) =
0. All curves C λ passthrough the set of base points which is defined as C ∩ C ∞ . One easily computes the 11(distinct) base points of the pencil. They are given by: • six base points of multiplicity 1 on the line (cid:96) = p = (cid:16) − b d d , a d d (cid:17) , p = (cid:16) − b d d , a d d (cid:17) , p = (cid:16) − b d d , a d d (cid:17) , p = − p , p = − p , p = − p ; • three base points of multiplicity 2 on the line (cid:96) = p = (cid:16) − b d d , a d d (cid:17) , p = [ b : − a : 0 ] , p = − p ; • and two base points of multiplicity 3 on the line (cid:96) = p = (cid:16) − b d d , a d d (cid:17) , p = − p . The map f is a quadratic Cremona map with the indeterminacy points I ( f ) = { p , p , p } and I ( f − ) = { p , p , p } , and has the following singularity confinement patterns: ( p p ) → p → p → p → p → p → p → ( p p ) , ( p p ) → p → p → p → ( p p ) , ( p p ) → p → p → ( p p ) . xyp p p p p p p p p p F IGURE
3. The curves C , C ∞ , C − of the sextic pencil (in red, blue andgreen, respectively) for (cid:96) ( x ) = y /6, (cid:96) ( x ) = x − y , (cid:96) ( x ) = x + y .Unlike the previous two sections, we will not derive here general geometric condi-tions for the base points of the pencil which ensure that certain compositions of Manininvolutions are quadratic Cremona maps. Rather, we will give here the correspondingstatements for the pencil (53). Theorem 5.
The map f can be represented as compositions of the Manin involutions in the fol-lowing ways: f = I ( ) i , k , m ◦ I ( ) j , k , m = I ( ) i , n ◦ I ( ) j , n for any ( i , j ) ∈ { (
1, 2 ) , (
2, 3 ) , (
3, 4 ) , (
4, 5 ) , (
5, 6 ) } , k ∈ {
1, . . . , 6 } \ { i , j } , and m ∈ {
7, 8, 9 } ,n ∈ {
10, 11 } .Proof. Symbolic computation with MAPLE. (cid:3)
9. C
ONCLUSIONS
The contribution of this paper is two-fold:
ANIN INVOLUTIONS FOR ELLIPTIC PENCILS AND DISCRETE INTEGRABLE SYSTEMS 21 • Finding geometric description of Manin involutions for elliptic pencil consisting ofcurves of higher degree, birationally equivalent to cubic pencils (Halphen pencilsof index 1). • Characterizing special geometry of base points ensuring that certain composi-tions of Manin involutions are integrable maps of low degree (quadratic Cremonamaps). As particular cases, we identify some integrable Kahan discretizations ascompositions of Manin involutions.It should be mentioned that both issues can and should be studied also for Halphenpencils of index m >
1. For instance, for a Halphen pencil of index 2, P ( p , ..., p ) , onecan propose the following construction of involutions I p i , i =
1, . . . , 9. For any p ∈ P different from the base points, consider the cubic curve through p and p , . . . , p . Itsintersection number with the Halphen’s curve of degree 6 through p is 3 × =
18. Theintersections with eight base points p , . . . , p and with p count as 8 × + =
17, sothere is exactly one remaining intersection point p (cid:48) . We declare p (cid:48) = I p ( p ) . One can seethat the so defined involutions I p i : P → P are Bertini involutions (of degree 17 in thegeneric situation). However, one can show that the birational map (6) of degree 3 from [3]is a composition of two such involutions, due to very special geometry of the base points,involving infintely near ones. We hope to be able to identify in the future work further lowdegree integrable birational maps as compositions of fundamental involutions defined byelliptic pencils. 10. A CKNOWLEDGEMENT
This research is supported by the DFG Collaborative Research Center TRR 109 “Dis-cretization in Geometry and Dynamics”.R
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