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
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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
space of solutions of this linear system is two-dimensional: αφ0+βφ1+γφ
2,where
φ0,φ
1,φ
2are homogeneous polynomials of (x0,x
1,x
2)of degree 2. The map
φ:[x0:x1:x2] →[u0:u1:u2]=[φ0(x0,x
1,x
2):φ1(x0,x
1,x
2):φ2(x0,x
1,x
2)]
(4)
is the sought after birational map P2 P2. A different choice of a basis φ0,φ
1,φ
2
of the net corresponds to a linear projective transformation of the target plane P2
2.
Note that the pre-image of a generic line au0+bu1+cu2=0inthetargetplane
P2
2is the conic aφ0+bφ1+cφ2=0 (passing through p1,p
2,p
3) in the source plane
P2
1. It follows that for any regular point pof φ, the pencil of lines P(1;q) through
q=φ(p) in P2
2corresponds to the pencil of conics P(2;p, p1,p
2,p
3)in P2
1.
4 Example: A Quartic Pencil with Two Double Base Points
4.1 Geometry of the Base Points
Consider an elliptic pencil in P2of the type (4;8122),
E=P(4;p1,...,p
8,p
2
9,p
2
10).
Thus, Econsists of quartic curves with 8 simple base points p1,...,p
8and two dou-
ble base points p9,p
10. The position of the ten base points is not arbitrary: for a
generic configuration 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 generic configuration of nine points, there is a one-parameter family (a pencil)
of curves of degree 4 through these points, having the prescribed two of them as dou-
ble points (nine incidence conditions plus four second order conditions, altogether 13
linear conditions, while a generic curve of degree 4 has 14 non-homogeneous coef-
ficients). Counting the intersection numbers, we see that all curves of the pencil pass
through a further simple point (indeed, seven simple points and two double points
contribute 7 ×1+2×4=15 to the intersection number 16).
More information on the configuration of the ten base points is contained in the
following statement.
Proposition 1 In a generic pencil P(4;p1,...,p
8,p
2
9,p
2
10), one of the curves is
reducible and consists of the line (p9p10)and a cubic curve passing through all ten
base points p1,...,p
10.
Proof Fix any point p∈(p9p10)different from p9,p10, and consider the unique
curve Cof the pencil through p. If the line (p9p10)would not be a component of
this curve, then the intersection number of Cwith the line (p9p10)would be at least
2×2+1=5, a contradiction. Thus, the curve Cis reducible and contains the
line (p9p10)as one of the components. Another component is a cubic curve through
p1,...,p
10 (with p9,p10 being simple points on the cubic).
Math Phys Anal Geom (2021) 24: 6
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Remark 1 If the reducible curve Chappens to contain (p9p10)as a double line, then
the remaining component is a conic through eight base points p1,...,p
8.
4.2 Birational Reduction to a Cubic Pencil
Consider a pencil E=P(4;p1,...,p
8,p
2
9,p
2
10).Letφ:P2
1 P2
2be a quadratic
Cremona map with the fundamental points p1,p
9,p
10. Thus, φblows down the
lines (p9p10),(p1p10),(p1p9)to points denoted by q1,q
9,q
10, respectively, and
blows up the points p1,p
9,p
10 to the lines (q9q10),(q1q10),(q1q9). All other base
points pi,i=2,...,8, are regular points of φ, their images will be denoted
by qi=φ(pi).
Proposition 2 Under the map φ:
a) Quartic curves of the original pencil Ein P2
1correspond to curves of a cubic
pencil
P(3;q2,...,q
8,q
9,q
10)
with nine base points in P2
2; the point q1is not a base point of the latter pencil.
b) For i=2,...,8, the pencil of lines P(1;qi)in P2
2corresponds to the pencil of
conics
P(2;pi,p
1,p
9,p
10)
in P2
1.
c) The pencils of lines P(1;q9),P(1;q10)in P2
2correspond to the pencils of lines
P(1;p9), P(1;p10)
in P2
1.
Proof
a) The total image of a quartic curve C∈Eis a curve of degree 8. Since Cpasses
through p1, its total image contains the line (q9q10).SinceCpasses through p9
and p10 with multiplicity 2, its total image contains the lines (q1q10)and (q1q9)
with multiplicity 2. Dividing by the linear defining polynomials of all these
lines, we see that the proper image of Cis a curve of degree 8−5=3. This curve
has to pass through all points qi,i=2,...,8. The curve Cof degree 4 has no
other intersections with the line (p9p10)different from two double points p9and
p10, therefore its proper image does not pass through q1. On the other hand, the
curve Cof degree 4 has one additional intersection point with each of the lines
(p1p9)and (p1p10), different from the simple point p1and the double point p9,
resp. p10. Therefore, its proper image passes through q10,resp.q9, with
multiplicity 1.
b) This follows from the fact that pi,i=2,...,8, are regular points of φ.
Math Phys Anal Geom (2021) 24: 6 Page 7 of 26 6
c) Consider the total pre-image of a line through q9. It is a conic through
p1,p
9,p
10 whose defining polynomial vanishes on the line (p1p10). Thus, the
conic is reducible and contains that line. Dividing by the defining polynomial
of this line (of degree 1), we see that the proper pre-image is a line which must
pass through p9. Similarly, the proper pre-image of a line through q10 is a line
through p10.
Let Sbe the elliptic surface obtained from P2by blowing up the ten base points pi,
i=1,...,10. Let Eibe the exceptional divisor classes of the blow-ups. The Picard
group of Sis Pic(S) =ZDZE1...ZE10. The class of a generic curve of
the pencil Eis
4D−E1−E2−E3−E4−E5−E6−E7−E8−2E9−2E10.(5)
The quadratic Cremona map of Proposition 2 corresponds to the following change of
basis of the Picard group:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
D=2D−E1−E9−E10,
E
1=D−E9−E10,
E
9=D−E1−E10,
E
10 =D−E1−E9,
(6)
and E
i=Eifor i=2,...,8.
One can check that E
1is a redundant class, in the sense that the class (5)ofa
general curve of the pencil is expressed through E
2,...,E
10 only:
4D−E1−...−E8−2E9−2E10 =3D−E
2−E
3−...−E
10.(7)
This corresponds to the fact that q1is not a base point of the φ-image of the pencil E.
Note that E
1=D−E9−E10 is the class of (the proper transform of) the line (p9p10)
in P2. Blowing down E
1on S, we obtain the surface Swhich is a minimal elliptic
surface (blow-up of P2at nine points), whose anti-canonical divisor class coincides
with (7).
Statement b) of Proposition 2 translates to relations D−E
i=2D−E1−E9−
E10 −Eiin the Picard group (for i=2,...,8), while statement c) translates as
D−E
9=D−E9and D−E
10 =D−E10.
4.3 Manin Involutions
In the new coordinates, where the pencil consists of cubic curves, Manin involutions
Iqiwith respect to the base points qiof the pencil are defined as in Definition 1: for
a point qwhich is not a base point, Iqi(q) is the unique third intersection of the the
line (qiq) with the cubic curve of the pencil passing through q. We now pull back
this construction to the original pencil in the old coordinates.
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Definition 2 (Manin involutions for pencils of the type (4;8122))
Consider a pencil E=P(4;p1,...,p
8,p
2
9,p
2
10). There are two kinds of Manin
involutions.
1) Involutions I(2)
i,j ,i, j ∈{1,...,8}, defined in terms of the pencil of conics
Ci,j =P(2;pi,p
j,p
9,p
10).
Given a point pwhich is not a base point of E, there is a unique conic of Ci,j passing
through p, and a unique quartic curve of Epassing through p.WesetI(2)
i,j (p) =p,
where pis the unique further intersection point of those two curves. This intersection
is unique, since the intersection number of the conic with the quartic is 2 ×4=8,
while the intersections at the points pi,p
j,p
9,p
10,andpcount as 1+1+2+2+1=
7.
2) Involutions I(1)
9,I(1)
10 defined in terms of the pencils of lines
P(1;p9), resp.P(1;p10).
For instance, the involution I9is defined as follows. Given a point pwhich is not a
base point of E,wesetI(1)
9(p) =p,wherepis the unique third intersection of the
line (p9p) and the quartic curve of Epassing through p. This intersection is unique,
since p9is a double point of the curve.
Indeed
1) Due to point b) of Proposition 2, for any i=2,...,8, the Manin involution
with respect to qiis conjugated to the map defined as above in terms of con-
ics through p1,p9,p10,and pi. Remarkably, while in the construction of the
conjugating Cremona map the roles of the simple base points p1and piare
asymmetric, in the resulting map I(2)
1,i the points p1and piare on an equal foot-
ing. More generally, I(2)
i,j =I(2)
j,i , where the map on the left-hand side should be
understood as conjugated to Iqjunder the quadratic Cremona map with the fun-
damental points pi,p
9,p
10, while the right-hand side should be understood as
conjugated to Iqiunder the quadratic Cremona map with the fundamental points
pj,p
9,p
10.
2) Due to point c) of Proposition 2, Manin involutions Iq9,I
q10 on P2
2are conju-
gated to the maps I(1)
9,I(1)
10 on P2
1defined in terms of lines through p9,p
10,
respectively. Again, while the construction depends on the choice of a simple
base point p1, the resulting map does not depend on this choice.
The involution I(2)
i,j has all base points of the pencil as singularities (indeterminacy
points). For instance, it blows up the point pkto the conic through pi,p
j,p
k,p
9,p
10.
However, a composition
I(2)
j,k ◦I(2)
i,j
with three distinct simple base points pi,p
j,p
kis well defined at pkand maps it
to pi. Moreover, this composition can be characterized as the unique map acting
on the elliptic curves of the pencil as the shift mapping pkto pi. In particular, this
composition does not depend on j.
Math Phys Anal Geom (2021) 24: 6 Page 9 of 26 6
5 Example: A Sextic Pencil with Three Double Base Points and Two
TripleBasePoints
5.1 Birational Reduction to a Cubic Pencil
Consider an elliptic pencil in P2of the type (6;613223),
E=P(6;p1,...,p
6,p
2
7,p
2
8,p
2
9,p
3
10,p
3
11),
consisting of curves of degree 6 with six simple base points p1,...,p
6, three double
base points p7,p
8,p
9, and two triple base points p10,p
11. We reduce it to a cubic
pencil in two steps.
Step 1. Apply a quadratic Cremona map φwith the fundamental points p9,p
10,
p11 (the both triple base points and one of the double base points). Thus, φblows
down the lines (p10p11),(p9p11),(p9p10)to the points denoted by q9,q
10,q
11,
respectively, and blows up the points p9,p
10,p
11 to the lines (q10q11),(q9q11),
(q9q10). All other base points pi,i=1,...,8 are regular points of φand their
images are denoted by qi=φ(pi).
Proposition 3 The change of variables φmaps a pencil
E=P(6;p1,...,p
6,p
2
7,p
2
8,p
2
9,p
3
10,p
3
11)
of sextic curves to a pencil
P(4;q1,...,q
6,q
10,q
11,q2
7,q2
8)
of quartic curves with eight simple base points and two double base points. The point
q9is not a base point of the latter pencil.
Proof The total image of a curve C∈Eis a curve of degree 12. Since Cpasses
through p9,p
10,p
11 with the multiplicities 2,3,3, its total image contains the lines
(q10q11),(q9q11),(q9q10)with the same multiplicities. Dividing by the linear defin-
ing polynomials of all these lines, we see that the proper image of Cis a curve of
degree 12 −8=4. This curve passes through all points qi,i=1,...,8 (for i=7,8
with multiplicity 2). The curve Cof degree 6 has no other intersections with the line
(p10p11)different from two triple points p10 and p11, therefore its proper image does
not pass through q9. On the other hand, the curve Cof degree 6 has one additional
intersection point with each of the lines (p9p10)and (p9p11), different from the dou-
ble point p9and the triple point p10, respectively p11. Therefore, its proper image
passes through q11,resp.q10, with multiplicity 1.
Step 2. Apply a quadratic Cremona map φ with the fundamental points q7,q
8(the
both double base points), and one of the simple base points. As we know from
Proposition 2, the image of the pencil P(4;q1,...,q
6,q
10,q
11,q2
7,q2
8)under φ
is a pencil of cubic curves with nine base points. The nature of the composition
φ ◦φdepends on the choice of the simple base point qidesignated as the third
fundamental point of φ, and is different in the cases i=1,...,6andi=10,11.
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It turns out that the first option contains all the possibilities for the different sorts
of Manin involutions, therefore we restrict our attention to this case, taking, for
definiteness, i=6. Thus, let φ have three fundamental points q6,q
7,q
8.Itblows
down the lines (q6q7),(q6q8),(q7q8)to points r8,r
7,r
6, respectively, and blows
up the points q6,q
7,q
8to the lines (r7r8),(r6r8),(r6r7). All other base points
qi,i=1,...,5,10,11 are regular points of φ, their images will be denoted by
ri=φ(qi). As follows from Propositions 3, 2, we have:
Proposition 4 The change of coordinates φ=φ ◦φ:P2
1 P2
2maps a pencil
E=P(6;p1,...,p
6,p
2
7,p
2
8,p
2
9,p
3
10,p
3
11)
of sextic curves in P2
1to a pencil
P(3;r1,...,r
5,r
7,r
8,r
10,r
11)
of cubic curves with nine base points in P2
2. The points r6and r9are not base points
of this cubic pencil.
Properties of the birational change of coordinates φ=φ ◦φon P2are eas-
ily obtained. It is a Cremona map of degree 4 which blows down the lines (p9p10),
(p9p11),(p10p11)to the points r11,r10,r9, respectively, and blows down the con-
ics C(p6,p
7,p
9,p
10,p
11),C(p6,p
8,p
9,p
10,p
11),C(p7,p
8,p
9,p
10,p
11)to the
points r8,r7,r6, respectively. Moreover, φblows up the points p9,p10,p11 to the
lines (r10r11),(r9r11),(r9r10), respectively, and the points p6,p7,p8to the con-
ics C(r7,r
8,r
9,r
10,r
11),C(r6,r
8,r
9,r
10,r
11),C(r6,r
7,r
9,r
10,r
11), respectively.
Points pi,i=1,...,5, are regular points of φ, their images are ri=φ(pi).The
pre-image of a generic line in P2is a quartic curve passing through p6,...,p
11 (the
points p10 and p11 being of multiplicity 2). In particular, for any regular point p,the
pencil of lines P(1;r) through r=φ(p) in P2
2corresponds to the pencil
P(4;p, p6,p
7,p
8,p
2
9,p
2
10,p
2
11)
of quartic curves in P2
1.
Proposition 5 The change of coordinates φ=φ ◦φ:P2
1 P2
2has the following
properties:
a) For i=1,...,5, the pencil of lines P(1;ri)in P2
2corresponds to the pencil
P(4;pi,p
6,p
7,p
8,p
2
9,p
2
10,p
2
11)
of quartic curves in P2
1.
b) For i=10,11, the proper pre-images of lines of the pencil P(1;ri)in P2
2are
cubics of the respective pencil
P(3;p6,p
7,p
8,p
9,p
2
10,p
11), P(3;p6,p
7,p
8,p
9,p
10,p
2
11)
in P2
1.
Math Phys Anal Geom (2021) 24: 6 Page 11 of 26 6
c) For i=7,8, the proper pre-images of lines of the pencil P(1;ri)in P2
2are
conics of the respective pencil
P(2;p7,p
9,p
10,p
11), P(2;p8,p
9,p
10,p
11)
in P2
1.
Proof
a) This follows from the fact that pi,i=1,...,5 are regular points of φ.
b) Consider the total pre-image of a line through r10. It is a quartic curve passing
through p6,...,p
11,havingp10,p
11 as double points. Its defining polynomial
vanishes on the line (p9p11), which blows down to r10. Thus, the quartic is
reducible and contains that line. Dividing by the defining polynomial of the line,
we see that the proper pre-image is a cubic passing through p6,p
7,p
8,p
10,p
11,
with p10 being a double point.
c) Consider the total pre-image of a line through r7. It is a quartic curve passing
through p6,...,p
11,havingp10,p
11 as double points. Its defining polynomial
vanishes on the conic C(p6,p
8,p
9,p
10,p
11), which blows down to r7. Thus,
the quartic is reducible and contains that conic. Dividing by the defining poly-
nomial of the conic, we see that the proper pre-image is a conic passing through
p7,p
9,p
10,p
11.
Let Sbe the elliptic surface obtained from P2by blowing up the eleven base points
pi,i=1,...,11. Let Eibe exceptional divisor classes of the blow ups. The Picard
group of Sis Pic(S) =ZDZE1...ZE11. The class of a generic curve of
the pencil is
6D−E1−E2−E3−E4−E5−E6−2E7−2E8−2E9−3E10 −3E11.(8)
The quadratic Cremona map φcorresponds to the following change of basis of
Pic(S):
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
D=2D−E9−E10 −E11,
E
9=D−E10 −E11,
E
10 =D−E9−E11,
E
11 =D−E9−E10,
(9)
and E
i=Eifor i=1,...,8. The Cremona map φ corresponds to the following
change of basis of the Picard group:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
D =2D−E
6−E
7−E
8,
E
6=D−E
7−E
8,
E
7=D−E
6−E
8,
E
8=D−E
6−E
7,
(10)
Math Phys Anal Geom (2021) 24: 6
6Page12of26
and E
i=E
ifor i=1,...,5andi=9,10,11. Composing (9), (10), we easily
compute:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
D =4D−E6−E7−E8−2E9−2E10 −2E11,
E
6=2D−E7−E8−E9−E10 −E11,
E
7=2D−E6−E8−E9−E10 −E11,
E
8=2D−E6−E7−E9−E10 −E11,
E
9=D−E10 −E11,
E
10 =D−E9−E11,
E
11 =D−E9−E10,
(11)
and E
i=Eifor i=1,...,5. One can check that the classes
E
6=2D−E7−E8−E9−E10 −E11,E
9=D−E10 −E11
are redundant, in the sense that the class (8) of a general curve of the pencil Eis
expressed through E
i,i= 6,9:
6D−E1−...−E6−2E7−2E8−2E9−3E10 −3E11
=3D −E
1−...−E
5−E
7−E
8−E
10 −E
11. (12)
This reflects the fact that r6,r9are not base points of the resulting cubic pen-
cil. The redundant classes are the classes (of the proper transforms) of the conic
C(p7,p
8,p
9,p
10,p
11), resp. of the line (p10p11)in P2
1. The surface Sobtained by
blowing down E
6and E
9on S, is a minimal elliptic surface, whose anti-canonical
divisor class coincides with (12). Generic fibers of Sare 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 −E
i=4D−Ei−E6−E7−E8−2E9−2E10 −2E11,i=1,...,5,
D −E
10 =3D−E6−E7−E8−E9−2E10 −E11,
D −E
7=2D−E7−E9−E10 −E11.
5.2 Manin Involutions
We pull back the standard construction of Manin involutions for the cubic pencil in
P2
2by means of the map φto the original pencil in P2
1.
Definition 3 (Manin involutions for pencils of the type (6;613223))
Consider a pencil E=P(6;p1,...,p
6,p
2
7,p
2
8,p
2
9,p
3
10,p
3
11). There are three
kinds of Manin involutions.
1) Involutions I(4)
i,j,k,i, j ∈{1,...,6},k∈{7,8,9}. E.g., I(4)
i,j,9is defined in terms
of quartic curves of the pencil
Qi,j,9=P(4;pi,p
j,p
7,p
8,p
2
9,p
2
10,p
2
11).
Math Phys Anal Geom (2021) 24: 6 Page 13 of 26 6
Given a point pwhich is not a base point of E, there is a unique quartic curve of
Qi,j,9through p, and a unique sextic curve of Ethrough p.WesetI(4)
i,j,9(p) =p,
where pis the unique further intersection point of these two curves. This intersection
is unique, since the intersection number of the quartic with the sextic is 4 ×6=
24, while the intersections at the points pi,p
j,p
7,p
8,p
9,p
10,p
11,andpcount as
1+1+2+2+4+6+6+1=23. Involutions I(4)
i,j,k with k=7,8 are defined
similarly.
2) Involutions I(3)
i,k ,i∈{1,...,6},k∈{10,11}. E.g., I(3)
i,10 is defined in terms of
cubic curves of the pencil
Ki,10 =P(3;pi,p
7,p
8,p
9,p
2
10,p
11).
Given a point pwhich is not a base point of E, there is a unique cubic curve of Ki,10
through p, and a unique sextic curve of Ethrough p.WesetI(3)
i,10(p) =p,where
pis the unique further intersection point of these two curves. This intersection is
unique, since the intersection number of the cubic with the sextic is 3×6=18, while
the intersections at the points pi,p
7,p
8,p
9,p
10,p
11,andpcount as 1 +2+2+
2+6+3+1=17. Involutions I(3)
i,11 are defined similarly.
3) Involutions I(2)
i,j ,i, j ∈{7,8,9}, defined in terms of conics of the pencil
Ci,j =P(2;pi,p
j,p
10,p
11).
Given a point pwhich is not a base point of E, there is a unique conic of Ci,j through
p, and a unique sextic curve of Ethrough p.WesetI(2)
i,j (p) =p,wherepis the
unique further intersection point of these two curves. This intersection is unique,
since the intersection number of the conic with the sextic is 2 ×6=12, while the
intersections at the points pi,p
j,p
10,p
11,andpcount as 2 +2+3+3+1=11.
6 Quadratic Manin Maps for Special Cubic Pencils
In this section, we consider pencils of cubic curves,
E=P(3;p1,...,p
9).
Generically, a Manin involution for a cubic pencil is a birational map of degree 5 for
which all base points of the pencil are singularities (indeterminacy points). Indeed,
consider IE,pj. For any base point pi= pj, all curves Cλof the pencil pass through
pi,p
j, and have one further intersection point with the line (pipj). As a result, IE,pj
blows up any base point pi(i = j) to the line (pipj). For the same reason IE,pi
blows down this line to pj. Thus:
Proposition 6 For a cubic pencil, and for any two distinct base points piand pj,
the Manin transformation IE,pi◦IE,pjis regular at piand maps it to pj.
For a similar reason, some base points become regular points of Manin involutions
if there are collinearities among them:
Math Phys Anal Geom (2021) 24: 6
6Page14of26
Fig. 1 Pascal configuration of base points of a cubic pencil
Proposition 7 For a cubic pencil, if three distinct base points pi,p
j,p
kare
collinear, then IE,piis regular at pjand at pkand interchanges these two points.
We will say that the nine points Ai,Bi,Ci,i=1,2,3, form a Pascal
configuration, if the six points A1,A
2A3,C
1,C
2,C
3lie on a conic, and
B1=(A2C3)∩(A3C2), B2=(A3C1)∩(A1C3), B3=(A1C2)∩(A2C1).
By Pascal’s theorem, the points B1,B
2,B
3are collinear. See Fig. 1
Theorem 1 Let the points Ai,Bi,Ci,i=1,2,3, form a Pascal configuration.
Consider the pencil Eof cubic curves with these base points. Then the map
f=IE,A1◦IE,B1=IE,B1◦IE,C1(13)
=IE,A2◦IE,B2=IE,B2◦IE,C2(14)
=IE,A3◦IE,B3=IE,B3◦IE,C3(15)
is a birational map of degree 2, with I(f ) ={C1,C
2,C
3}and I(f −1)=
{A1,A
2,A
3}. It has the following singularity confinement patterns:
(C2C3)→A1→B1→C1→(A2A3), (16)
(C3C1)→A2→B2→C2→(A3A1), (17)
(C1C2)→A3→B3→C3→(A1A2). (18)
Proof We start with the following property of the addition law on a nonsingular cubic
curve C.LetP1,P
2,P
3,P
4∈C,then
P1−P3=P4−P2⇔P1+P2=P3+P4⇔(P1P2)∩(P3P4)∈C.
Math Phys Anal Geom (2021) 24: 6 Page 15 of 26 6
Thus, on any cubic curve C∈E, we have the following relations:
(A1B2)∩(A2B1)=C3∈C⇒A1−B1=A2−B2⇒IE,A1◦IE,B1=IE,A2◦IE,B2,
(B1C2)∩(B2C1)=A3∈C⇒B1−C1=B2−C2⇒IE,B1◦IE,C1=IE,B2◦IE,C2,
(A1C2)∩(B1B2)=B3∈C⇒A1−B1=B2−C2⇒IE,A1◦IE,B1=IE,B2◦IE,C2.
This proves the coincidence of all six representations in (13)–(15). Now it follows
from Proposition 6 that fhas only three indeterminacy points, I(f ) ={C1,C
2,C
3},
and similarly, I(f −1)={A1,A
2,A
3}. Moreover, Proposition 6 implies the relations
in the middle part of the singularity confinement patterns (16)–(18). The blow-up and
blow-down relations are shown with the help of Proposition 7 as follows: f(C
3)=
IE,A2◦IE,B2(C3)=IE,A2(A1)=(A1A2).
One can achieve a nice canonical form of the map from Theorem 1 by sending
points B1,B
2,B
3to infinity (cf. [18]).
Theorem 2 For a pencil of cubic curves with the base points building a Pascal
configuration, perform a linear projective transformation of P2sending the Pascal
line (B1,B
2,B
3)to infinity. Let (x, y) be the affine coordinates on the affine part
C2⊂P2. In these coordinates, the map f:(x, y) → (
x,
y) defined by (13)–(15)is
characterized by the following property. There exist constants a1,...,a
9∈Csuch
that fadmits a representation through two bilinear equations of motion of the form
x−x=a2x
x+a3(x
y+
xy) +a4y
y+a6(x +
x) +a7(y +
y) +a9,
y−y=−a1x
x−a2(x
y+
xy) −a3y
y−a5(x +
x) −a6(y +
y) −a8.(19)
These equations serve as the Kahan discretization [14,17]of the Hamiltonian
equations of motion
˙x=∂H/∂y =a2x2+2a3xy +a4y2+2a6x+2a7y+a9,
˙y=−∂H/∂x =−a1x2−2a2xy −a3y2−2a5x−2a6y−a8,(20)
for the Hamilton function
H(x,y) =1
3a1x3+a2x2y+a3xy2+1
3a4y3+a5x2+2a6xy+a7y2+a8x+a9y. (21)
Proof This is a result of a symbolic computation with MAPLE, presented in [18].
7 Quadratic Manin Maps for Special Pencils of the Type (4; 8122)
We describe the geometry of base points of a pencil of the type (4;8122)for which
one can find compositions of Manin involutions which are quadratic Cremona maps.
–Letp2,p
3,p
6,p
7be four points of P2in general position (no three of them
collinear).
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6Page16of26
– Consider three intersection points of three pairs of opposite sides of the complete
quadrangle with these vertices:
A=(p2p6)∩(p3p7), B =(p2p3)∩(p6p7), C =(p2p7)∩(p3p6). (22)
Consider the projective involutive automorphism σof P2fixing the point Cand
the line =(AB) (pointwise). The points of the pairs (p2,p
7)and (p3,p
6)
correspond under σ.
– Choose a point p9∈(p3p7), and define p10 ∈(p2p6)so that p9,p
10 correspond
under σ, or, in other words, so that the line (p9p10)passes through C.
–Let
C∈P(2;p2,p
3,p
6,p
7)be any conic of the pencil through the specified
four points. Define:
p1=the second intersection point of Cwith (p10p7),
p4=the second intersection point of Cwith (p9p6),
p5=the second intersection point of Cwith (p10p3),
p8=the second intersection point of Cwith (p9p2).
Recall that A, B, C are vertices of a self-polar triangle for C. In particular, the
projective involution σleaves Cinvariant. The points of the pairs (p1,p
8)and
(p4,p
5)correspond under σ.
We will call the pencil E=P(p1,...,p
8,p
2
9,p
2
10)aprojectively symmetric
quartic pencil with two double points.SeeFig.2.
Theorem 3 Let E=P(p1,...,p
8,p
2
9,p
2
10)be a projectively symmetric quartic
pencil with two double points. Then:
a) The projective involution σcan be represented as
σ=I(2)
1,8=I(2)
2,7=I(2)
3,6=I(2)
4,5. (23)
Fig. 2 Geometry of base points of a special quartic pencil P(4;p1,...,p
8,p
2
9,p
2
10)
Math Phys Anal Geom (2021) 24: 6 Page 17 of 26 6
b) The map
f=I(2)
i,k ◦I(2)
j,k (24)
=I(1)
9◦σ=σ◦I(1)
10 (25)
with (i, j) ∈{(1,2), (2,3), (3,4), (5,6), (6,7), (7,8)}and k∈{1,...,8}\{i, j},is
a birational map of degree 2, with I(f ) ={p4,p
8,p
10}and I(f −1)={p1,p
5,p
9}.
It has the following singularity confinement patterns:
(p8p10)→p1→p2→p3→p4→(p5p9), (26)
(p4p10)→p5→p6→p7→p8→(p1p9), (27)
(p4p8)→p9→p10 →(p1p5). (28)
c) We have:
f2=I(1)
9◦I(1)
10 . (29)
Proof We start with a geometric interpretation of the addition law on a generic curve
C∈E. Recall that the pencil Ecan be reduced to a pencil of cubic curves by means
of the quadratic Cremona map φbased at pk,p
9,p
10 for some k=1,...,8. Lines
in the target plane P2
2, where the cubic pencil is considered, correspond in the source
plane P2
1of the pencil Eto conics through pk,p
9,p
10.Nowletp, q, r, s ∈C, then,
assuming that neither of the points pk,p
9,p
10 is among p, q, r, s,wehave:
p−r=s−q⇔p+q=r+s⇔φ(p)φ(q)∩φ(r)φ(s)∈φ(C).
The geometry of the pencil Eensures the existence of a large number of quadruples
of base points which, together with p9,p
10, lie on a conic. Namely, the following
sextuples are conconical:
(p1,p
2,p
7,p
8,p
9,p
10)because p1↔p8,p
2↔p7under σ, (30)
(p1,p
3,p
6,p
8,p
9,p
10)because p1↔p8,p
3↔p6under σ, (31)
(p1,p
4,p
5,p
8,p
9,p
10)because p1↔p8,p
4↔p5under σ, (32)
(p2,p
3,p
6,p
7,p
9,p
10)because p2↔p7,p
3↔p6under σ, (33)
(p2,p
4,p
5,p
7,p
9,p
10)because p2↔p7,p
4↔p5under σ, (34)
(p3,p
4,p
5,p
6,p
9,p
10)because p3↔p6,p
4↔p5under σ. (35)
The sextuples (30), (33)and(35) lie on reducible conics (p1,p
7,p
10)∪
(p2,p
8,p
9),(p2,p
6,p
10)∪(p3,p
7,p
9)and (p3,p
5,p
10)∪(p4,p
6,p
9),
respectively. One has, additionally, two more sextuples lying on reducible conics:
(p1,p
4,p
6,p
7,p
9,p
10)−on a reducible conic (p1,p
7,p
10)∪(p4,p
6,p
9), (36)
(p2,p
3,p
5,p
8,p
9,p
10)−on a reducible conic (p3,p
5,p
10)∪(p2,p
8,p
9). (37)
Math Phys Anal Geom (2021) 24: 6
6Page18of26
–From(31), (37), (36), (34) there follows:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
C(p1,p
6,p
3,p
9,p
10)∩C(p2,p
5,p
3,p
9,p
10)p8,
C(p1,p
6,p
4,p
9,p
10)∩C(p2,p
5,p
4,p
9,p
10)p7,
C(p1,p
6,p
7,p
9,p
10)∩C(p2,p
5,p
7,p
9,p
10)p4,
C(p1,p
6,p
8,p
9,p
10)∩C(p2,p
5,p
8,p
9,p
10)p3.
We explain how these relations are used, taking the first one as example. The
intersection C(p1,p
6,p
3,p
9,p
10)∩C(p2,p
5,p
3,p
9,p
10)consists of p3,p9,
p10,andp8. Upon the quadratic Cremona map φbased at p3,p
9,p
10, this means
that the lines (q1q6)and (q2q5)intersect at q8,whereqi=φ(pi)(the blow-
ups of other three intersection points do not belong to the proper image of the
conics). The point q8is one of the base points of the cubic pencil φ(E). Thus,
the four relations above imply
I(2)
1,k ◦I(2)
2,k =I(2)
5,k ◦I(2)
6,k ,k=3,4,7,8. (38)
–From(35), (33), (37) there follows:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
C(p2,p
6,p
1,p
9,p
10)∩C(p3,p
5,p
1,p
9,p
10)⊃(p1p9),
C(p2,p
6,p
4,p
9,p
10)∩C(p3,p
5,p
4,p
9,p
10)⊃(p4p9),
C(p2,p
6,p
7,p
9,p
10)∩C(p3,p
5,p
7,p
9,p
10)⊃(p7p9),
C(p2,p
6,p
8,p
9,p
10)∩C(p3,p
5,p
8,p
9,p
10)⊃(p8p9).
Again, we explain how these relations are used, taking the first one as example.
The intersection C(p2,p
6,p
1,p
9,p
10)∩C(p3,p
5,p
1,p
9,p
10)consists of the
point p10 and the line (p1p9). Upon the quadratic Cremona map φbased at
p1,p
9,p
10, the point p10 is blown up to a line which does not belong to the
proper image of the conics, while the line (p1p9)is blown down to the point
q10 through which the proper images of the both conics pass. Thus, the lines
(q2q6)and (q3q5)intersect at q10, which is a base point of the pencil φ(E).
Summarizing, the four relations above imply
I(2)
2,k ◦I(2)
3,k =I(2)
5,k ◦I(2)
6,k ,k=1,4,7,8. (39)
–From(31),(32), (33), (34), there follows:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
C(p3,p
6,p
1,p
9,p
10)∩C(p4,p
5,p
1,p
9,p
10)p8,
C(p3,p
6,p
2,p
9,p
10)∩C(p4,p
5,p
2,p
9,p
10)p7,
C(p3,p
6,p
7,p
9,p
10)∩C(p4,p
5,p
7,p
9,p
10)p2,
C(p3,p
6,p
8,p
9,p
10)∩C(p4,p
5,p
8,p
9,p
10)p1.
Exactly as before, these four relations imply
I(2)
3,k ◦I(2)
4,k =I(2)
5,k ◦I(2)
6,k ,k=1,2,7,8. (40)
– In exactly the same way we prove that
I(2)
1,k ◦I(2)
2,k =I(2)
6,k ◦I(2)
7,k ,k=3,4,5,8. (41)
Math Phys Anal Geom (2021) 24: 6 Page 19 of 26 6
and
I(2)
1,k ◦I(2)
2,k =I(2)
7,k ◦I(2)
8,k ,k=3,4,5,6. (42)
This completes the proof of coincidence of all representations (24), as well as
the middle part of the singularity confinement patterns (26), (27).
– One sees immediately that I(1)
9◦σis a shift with respect to the addition law
on the curves of E, sending p1→p2→p3→p4, while σ◦I(1)
10 is a shift
sending p5→p6→p7→p8. 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 ) ={p4,p
8,p
10}and I(f −1)=
{p1,p
5,p
9},sothatfmust 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
the representation f=I(1)
9◦σ. The involution σis a projective automorphism
and has no singularities, so it suffices to study the blowing down patterns of
I(1)
9. By definition of the map I(1)
9, it is clear that it blows down the line (p1p9)
to the point p1, and blows down the line (p5p9)to the point p5.Sincefis a
quadratic Cremona map, the same holds true for the involution I(1)
9; there follows
that I(1)
9must blow up p9to the line (p1p5), which finishes the proof. For the
blow-up relations on the right part of (26)–(28), we use f=σ◦I(1)
10 in a similar
manner.
We now turn to canonical forms of projectively symmetric quartic pencils with
two double points, which can be achieved by projective automorphisms of P2.The
most popular one corresponds to the choice p9=[0:1:0],p10 =[1:0:0],sothat
the quartic curves become biquadratic ones. Denote the inhomogeneous coordinates
on the affine part C2⊂P2by (u, v). We can arrange p2=(1,β),p7=(β, 1),
p3=(β, −1),p6=(−1,β),sothat={u−v=0},C=(p2p7)∩(p3p6)=
[−1:1:0],A=(p2p6)∩(p3p7)=(β, β),andσis the Euclidean reflection at the
line ,
σ(u, v) =(v, u).
The pencil Eof biquadratics reads
α(α +1)(u2+v2−1)−(α +1)uv +β(u+v)−β2−λ(u2−1)(v2−1)=0,(43)
and is symmetric under σ. Involutions I(1)
9and I(1)
10 are nothing but the standard
vertical and horizontal QRT switches for this pencil, and the map f=I(1)
9◦σ=
σ◦I(1)
10 of Theorem 3 is given by
f:(u, v) → (
u,v), ⎧
⎨
⎩
u=v,
v=αuv +βu −1
u−αv −β.(44)
It is the “QRT root” of f2=I(1)
9◦I(1)
10 .
Math Phys Anal Geom (2021) 24: 6
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To arrive at another canonical form of projectively symmetric quartic pencils with
two double points, we perform a linear projective change of variables in P2,givenin
the inhomogeneous coordinates by
u=1+βx +y
x,v=1+βx −y
x. (45)
Upon substitution (45) and some straightforward simplifications, we come to the
following system (cf. [20]):
x−x=x
y+
xy,
y−y=(1−2α) −2αβ(x +
x) +1−β2(1+2α)x
x−(1+2α)y
y.(46)
In order to give an intrinsic geometric characterization of this canonical form, we
will need the following observation.
Proposition 8 The following five intersection points are collinear:
(p1p8)∩(p2p7), (p1p5)∩(p3p7), (p2p5)∩(p3p8), (p1p6)∩(p4p7), (p2p6)∩(p4p8).
Proof The triple of intersection points
(p1p8)∩(p2p7), (p1p5)∩(p3p7), (p2p5)∩(p3p8)
lies on the Pascal line for the hexagon (p1,p
5,p
2,p
7,p
3,p
8), while the triple of
intersection points
(p1p8)∩(p2p7), (p1p6)∩(p4p7), (p2p6)∩(p4p8)
lies on the Pascal line for the hexagon (p1,p
6,p
2,p
7,p
4,p
8). These hexagons cor-
respond under σ, therefore this holds true also for their Pascal lines. Moreover, the
Pascal lines share the point (p1p8)∩(p2p7)=C, therefore they must coincide.
We will call the line containing the five intersection points from Proposition 8 the
double Pascal line.
Theorem 4 For a projectively symmetric pencil of quartic curves with two double
points, perform a linear projective transformation of P2sending the double Pascal
line to infinity. By a subsequent affine change of coordinates (x, y) on the affine
part C2⊂P2, arrange that coincides with the axis y=0,p9=(0,−1), and
p10 =(0,1). In these coordinates, the map f:(x, y) → (
x,
y) defined by (24)–
(25)is characterized by the following property. There exist a0,...,a
3∈Cwith
a0+a3=2such that fadmits a representation through two bilinear equations of
motion of the form
x−x=x
y+
xy,
y−y=a0−a1(x +
x) −a2x
x−a3y
y.(47)
Proof A symbolic computation with MAPLE.
Math Phys Anal Geom (2021) 24: 6 Page 21 of 26 6
8 Quadratic Manin Maps for Special Pencils of the Type (6; 613223)
In this section, we consider the following example considered in detail in [7,17,
21,27]. Let f:P2 P2be the birational map given in the non-homogeneous
coordinates x=(x, y) on the affine part C2⊂P2by two bilinear relations between
(x, y) and (
x,
y) =f(x,y):
x−x=γ1(2(x)3(
x)+2(
x)3(x))J ∇1
+γ2(1(x)3(
x)+1(
x)3(x))J ∇2
+γ3(1(x)2(
x)+1(
x)2(x))J ∇3,(48)
where (γ1,γ
2,γ
3)=(1,2,3),i(x)=aix+biyare linear forms with ai,b
i∈C,
and J=01
−10
. This map is the Kahan discretization of the quadratic flow
˙
x=γ12(x)3(x)J ∇1+γ21(x)3(x)J ∇2+γ31(x)2(x)J ∇3,(49)
which can be put as
˙
x=1
(1(x))γ1−1(2(x))γ2−1(3(x))γ3−1J∇H0(x), (50)
where
H0(x)=(1(x))γ1(2(x))γ2(3(x))γ3. (51)
Integrability of the Kahan discretization (48) was demonstrated in [7,17,21]for
(γ1,γ
2,γ
3)=(1,1,1), (1,1,2),and(1,2,3). Sections 6and 7deal with gener-
alizations of the cases (γ1,γ
2,γ
3)=(1,1,1), (1,1,2), the present one deals with
(γ1,γ
2,γ
3)=(1,2,3).
As shown in [7,17,21], the map fdefined by (48) with (γ1,γ
2,γ
3)=(1,2,3)
admits an integral of motion:
H(x)=H0(x)
L+(x)L−(x)M+(x)M−(x)Q(x),(52)
where
L±(x)=1±3d312(x),
M±(x)=1±(3d231(x)−d123(x)),
Q(x)=1−9d2
312
2(x)+16d2
122
3(x),
with
dij =aibj−ajbi.
Thus, the phase space of fis foliated by the pencil of invariant curves
Cλ=H0(x)−λL+(x)L−(x)M+(x)M−(x)Q(x)=0. (53)
The pencil has deg =6 and contains two reducible curves: C0={H0(x, y) =0},
consisting of the lines {i(x, y) =0},i=1,2,3, with multiplicities 1,2,3, and C∞,
consisting of the conic Q(x,y) =0 and the four lines L±(x, y) =0, M±(x, y) =0.
Math Phys Anal Geom (2021) 24: 6
6Page22of26
All curves Cλpass through the set of base points which is defined as C0∩C∞.See
Fig. 3. One easily computes the 11 (distinct) base points of the pencil. They are given
by:
– six base points of multiplicity 1 on the line 1=0:
p1=−b1
5d12d31
,a1
5d12d31 ,p
2=−b1
3d12d31
,a1
3d12d31 ,p
3=−b1
d12d31
,a1
d12d31 ,
p4=−p3,p
5=−p2,p
6=−p1;
– three base points of multiplicity 2 on the line 2=0:
p7=−b2
4d12d23
,a2
4d12d23 ,p
8=[b2:−a2:0],p
9=−p7;
– and two base points of multiplicity 3 on the line 3=0:
p10 =−b3
3d23d31
,a3
3d23d31 ,p
11 =−p10.
Fig. 3 The curves C0,C∞,C−0.002 of the sextic pencil (in red, blue and green, respectively) for 1(x)=
y/6, 2(x)=3x−y,3(x)=4x+y
Math Phys Anal Geom (2021) 24: 6 Page 23 of 26 6
The map fis a quadratic Cremona map with the indeterminacy points I(f ) =
{p6,p
9,p
11}and I(f −1)={p1,p
7,p
10}, and has the following singularity
confinement patterns:
(p9p11)→p1→p2→p3→p4→p5→p6→(p7p10),
(p6p11)→p7→p8→p9→(p1p10),
(p6p9)→p10 →p11 →(p1p7).
Unlike the previous two sections, we will not derive here general geometric con-
ditions for the base points of the pencil which ensure that certain compositions
of Manin involutions are quadratic Cremona maps. Rather, we will give here the
corresponding statements for the pencil (53).
Theorem 5 The map fcan be represented as compositions of the Manin involutions
in the following ways:
f=I(4)
i,k,m ◦I(4)
j,k,m =I(3)
i,n ◦I(3)
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.
9 Conclusions
The contribution of this paper is two-fold:
– Finding geometric description of Manin involutions for elliptic pencil consist-
ing of curves of higher degree, birationally equivalent to cubic pencils (Halphen
pencils of index 1).
– Characterizing special geometry of base points ensuring that certain composi-
tions of Manin involutions are integrable maps of low degree (quadratic Cremona
maps). As particular cases, we identify some integrable Kahan discretizations as
compositions of Manin involutions.
It should be mentioned that both issues can and should be studied also for
Halphen pencils of index m>1. For instance, for a Halphen pencil of index
2, P(6;p2
1,...,p
2
9), one can propose the following construction of involutions Ipi,
i=1,...,9. For any p∈P2different from the base points, consider the cubic curve
through pand p2,...,p
9. Its intersection number with the Halphen’s curve of degree
6 through pis 3 ×6=18. The intersections with eight base points p2,...,p
9and
with pcount as 8×2+1=17, so there is exactly one remaining intersection point p.
We declare p=Ip1(p). One can see that the so defined involutions Ipi:P2 P2
are Bertini involutions (of degree 17 in the generic situation). However, one can
show that the birational map of degree 3 from [3, eq. (6)] is a composition of two
such involutions, due to very special geometry of the base points, involving infintely
Math Phys Anal Geom (2021) 24: 6
6Page24of26
near ones. We hope to be able to identify in the future work further low degree inte-
grable birational maps as compositions of fundamental involutions defined by elliptic
pencils.
Funding Open Access funding enabled and organized by Projekt DEAL.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as
you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons
licence, and indicate if changes were made. The images or other third party material in this article are
included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the
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from the copyright holder. To view a copy of this licence, visit http://creativecommonshorg/licenses/by/4.0/.
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