https://doi.org/10.1007/s11040-021-09376-4
Manin Involutions for Elliptic Pencils and Discrete
Integrable Systems
Matteo Petrera1·Yuri B. Suris1·Kangning Wei1·Ren´
e Zander1
Received: 28 August 2020 / Accepted: 20 January 2021 /
©The Author(s) 2021
Abstract
We contribute to the algebraic-geometric study of discrete integrable systems gener-
ated by planar birational maps: (a) we find geometric description of Manin involu-
tions for elliptic pencils consisting of curves of higher degree, birationally equiva-
lent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special
geometry of base points ensuring that certain compositions of Manin involutions are
integrable maps of low degree (quadratic Cremona maps). In particular, we iden-
tify some integrable Kahan discretizations as compositions of Manin involutions for
elliptic pencils of higher degree.
Keywords Elliptic curve ·Elliptic pencil ·Birational map ·Integrable map
Mathematics Subject Classification (2010) 37J70 ·14H70 ·14H45
This research is supported by the Deutsche Forschungsgemeinschaft (DFG) Collaborative Research
Center TRR 109 “Discretization in Geometry and Dynamics”.
Yuri B. Suris
[email protected]berlin.de
Matteo Petrera
Kangning Wei
Ren´
e Zander
1Institut f¨ur Mathematik, MA 7-1, Technische Universit¨
at Berlin, Str. des 17. Juni 136, 10623
Berlin, Germany
Math Phys Anal Geom (2021) 24: 6
Published online: 4 March 2021
1 Introduction
Intimate relation of the theory of integrable systems to algebraic geometry is well
appreciated in these days. In the present paper, we address this relation for a very
basic class of integrable systems, namely for discrete integrable systems generated
by birational maps of CP2with a rational integral of motion and an invariant measure
with a rational density (whereas the emphasis is put on the integral of motion). In
such a system, orbits are confined to invariant curves (level sets of the integral), and
on each invariant curve the map induces an automorphism.
For general reasons, invariant curves must have genus zero or one, since only in
these cases 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 birationally equivalent to a plane, admitting a fibration by elliptic curves). A
classification of pencils of elliptic curves in a plane was given by Bertini, a modern
proof of this result is due to Dolgachev [8]. It says that any such pencil is birationally
equivalent to a Halphen pencil of index m∈N, in which a generic curve is of degree
3mand has multiplicity mat each of nine base points.
Planar maps preserving pencils of elliptic curves appeared over and over again in
the theory of discrete integrable systems. Probably, the most prominent example is
givenbyQRTmaps[9,22,24], which preserve pencils of biquadratic curves. Further
examples are given by (autonomous versions of) discrete Painlev´
e equations [11,
23], as well as the so called HKY maps which preserve pencils of curves of higher
degrees [4,15]. 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 classification of rational elliptic surfaces, was given in [3]
and sounds almost tautologically: a birational map preserving an Halphen pencil (of
index m) either preserves each fiber or interchanges the fibers in a nontrivial way.
In the present paper, we are occupied with a construction of integrable maps pre-
serving a pencil of elliptic curves, based only on the pencil itself. The basic idea is
to compose two (non-commuting) birational involutions preserving the pencil. This
construction is almost obvious for QRT maps, where one can always use the hori-
zontal and the vertical switches as suitable involutions (the horizontal switch assigns
to any point of a biquadratic curve the second intersection point of the curve with
the horizontal line through the original point; the definition of the vertical switch is
analogous). In [12,13] this construction was extended to a (projectively equivalent)
case of maps preserving a pencil of quartic curves with two double points, and it was
observed that the corresponding involutions are closely related to the so called Manin
involutions, which were introduced in [16] for pencils of cubic curves. Recall that, for
a pencil of cubic curves with nine base points p1,...,p
9, the Manin involution cen-
tered at p1assigns to each point pthe third intersection point of the line (p1p) with
the curve of the pencil passing through p.In[18], it was shown that Kahan discretiza-
tions of canonical Hamiltonian systems with cubic Hamiltonian can be characterized
as compositions of Manin involutions in the case of a special geometry of the base
points of the cubic pencil, see [19] for a related result. We also mention that com-
positions of Manin involutions for cubic pencils appeared in the theory of discrete
integrable systems already in [10,26].
6Page2of26 Math Phys Anal Geom (2021) 24: 6
Birational involutions of the plane are classical and well studied objects in alge-
braic geometry. Their classification was given by Bertini [2] and says that every
non-trivial birational involution of P2is birationally conjugate to exactly one of the
following: a de Jonqui`
eres involution of degree d>2 (which fixes an irreducible
curve of degree dwith a unique singular point pwhich is an ordinary multiple point
of multiplicity d−2), a Geiser involution, or a Bertini involution. We refrain from
giving precise definitions here and refer the reader to the original source. A modern
proof of the Bertini classification was given in [1]. The place of Manin involutions
in the Bertini classification is as follows: a Manin involution centered at p1for a
pencil of cubic curves with nine base points p1,...,p
9is a de Jonqui`
eres involu-
tion with center p1defined by a hyperelliptic curve of degree d=5 and of genus 3,
with a triple point p1and eight Weierstrass points p2,...,p
9(we are indebted to I.
Dolgachev for explaining this).
In the present paper, our goal is to use Manin involutions for elliptic pencils to
construct integrable dynamical systems. We start by finding a geometric formulation
of Manin involutions for elliptic pencils consisting of curves of higher degree. This
is achieved via 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 not available in the literature. The most non-trivial contribution
consists in finding the conditions under which the involutions and their composi-
tions are of low degree, thus producing simple and attractive examples of integrable
birational maps.
2 Elliptic Pencils and Cubic Pencils
Throughout the paper, we work over the field C. We consider pencils of curves in P2,
i.e., families of curves P={Cλ}parametrized by λ∈P1,
Cλ=[x0:x1:x2]∈P2:F(x
0,x
1,x
2)+λG(x0,x
1,x
2)=0.
Here F,Gare two homogeneous polynomials of degree d. The points of the set
B=[x0:x1:x2]∈P2:F(x
0,x
1,x
2)=G(x0,x
1,x
2)=0
are called base points of the pencil P. As usual, they are counted with multiplicities.
We will assume that the multiplicities of each base point on both curves F=0and
G=0 (and then on all curves of the pencil) are the same. The type of the pencil is
then
(d;(n1)1(n2)2(n3)3...)
where dis the degree of the curves of the pencil, n1the number of simple base points,
n2the number of double base points, n3the number of triple base points and so on.
The pencil itself will be denoted by
Pd;pm1
1,p
m2
2,...p
mN
N,
Math Phys Anal Geom (2021) 24: 6 Page 3 of 26 6
which refers to the degree dand the list of base points piwith their respective mul-
tiplicities mi,sothatN=n1+n2+n3+.... Multiplicities mi=1 are usually
omitted.
Counting the intersection numbers, we get:
d2=
k
nkk2.(1)
Through any point [x0:x1:x2]∈P2\B, there passes a unique curve Cλof the
pencil, with λ=−F(x
0,x
1,x
2)/G(x0,x
1,x
2).
Our main interest is in the elliptic pencils, for which generic curves of the pencil
are of genus g=1. According to the degree-genus formula, the genus of irreducible
curves of the pencil is given by:
g=(d −1)(d −2)
2−
k
nk
k(k −1)
2=1. (2)
We remark that by virtue of (1), the latter equation is equivalent to
3d=
k
nkk, (3)
where the right-hand side is the total number of base points (counted with
multiplicities).
Examples
1. A pencil of the type (3;91)of cubic curves with nine simple base points.
2. A pencil of the type (4;8122)of curves of degree 4 with eight simple and two
double points. By an automorphism of P2, we can send the double points to
infinity (say, to [0:1:0]and [0:0:1]), then in the affine coordinates
(x1/x0,x
2/x0), we get a pencil of biquadratic curves. Such pencils are pretty
well studied and have plenty of applications in the theory of discrete integrable
systems [9,22].
3. Apencilofthetype(6;613223)of curves of degree 6 with six simple points,
three double points and two triple points.
4. A pencil of the type (6;92)of curves of degree 6 with nine double points, i.e., a
Halphen pencil of index 2.
Remark We do allow infinitely near base points, at which the curves of the pencil
have to satisfy certain conditions of tangency up to a certain order. In the formula-
tions of our general results about geometry of Manin involutions, we silently assume
that the geometry of base points is generic, in particular, that there are no inciden-
tal collinearities. However, all our main examples are non-generic with a plenty of
incidental collinearities, since it is exactly this feature that allows for a substantial
Math Phys Anal Geom (2021) 24: 6
6Page4of26
drop of degree of the resulting birational maps. We hope this will not lead to any
confusions.
3 Manin Involutions
For cubic curves, one has a simple geometric interpretation of the addition law. Corre-
spondingly, there is a simple geometric construction of certain birational involutions
of P2induced 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)
1) Consider a nonsingular cubic curve Cin P2, and a point p0∈C.TheManin
involution on Cwith respect to p0is the map IC,p0:C→Cdefined as follows: for
a generic p= p0,theimageIC,p0(p) is the unique third intersection point of Cwith
the line (p0p);forp=p0, the line (p0p) should be interpreted as the tangent line
to Cat p0.
2) Consider a pencil P={Cλ}of cubic curves in P2with at least one nonsingular
member. Let p0be a base point of the pencil. The Manin involution IP,p0:P2 P2
is a birational map defined as follows. For any p∈P2which is not a base point,
IP,p0(p) =ICλ,p0(p),whereCλis the unique curve of the pencil through the
point p.
For elliptic pencils of degree higher than 3, the geometric construction of Manin
involutions seems to be unknown. The only exception are the vertical and the hori-
zontal switches 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 for quartic pencils with two double points, with respect to the both dou-
ble points [13]. The definition of the generalized Manin involution IC,p0for a quartic
curve Cand a double point p0∈C, resp. of the generalized Manin involution IP,p0
for a quartic pencil with two double points, one of them being p0, literally coincides
with Definition 1. This is justified by the fact that any line through a double point
p0∈Cstill intersects the quartic curve Cat two further points.
The main goal of this paper is to elaborate on the geometric definition of Manin
involutions 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
of suitable birational transformations. Often, the simplest way of doing this is by
a sequence of quadratic Cremona transformations. Recall that a generic quadratic
Cremona transformation φ:P2
1 P2
2has three distinct fundamental points
I(φ) ={p1,p
2,p
3}whichareblownuptothreelines(q2q3),(q3q1),(q1q2), respec-
tively. The three lines (p2p3),(p3p1),(p1p2)are blown down to the points q1,q2,
q3, respectively, which build the indeterminacy set of the inverse map: I(φ−1)=
{q1,q
2,q
3}. A practical way to construct such a map consists in finding homogeneous
polynomials φ(x0,x
1,x
2)of degree 2 vanishing at the fundamental points p1,p
2,p
3.
Geometrically, we are speaking about the set of conics in P2
1through p1,p
2,p
3.The
Math Phys Anal Geom (2021) 24: 6 Page 5 of 26 6
Loading more pages...