European Journal of Mathematics (2021) 7:1046–1073
https://doi.org/10.1007/s40879-021-00479-4
RESEARCH ARTICLE
On the singularity structure of Kahan discretizations of a
class of quadratic vector fields
René Zander1
Received: 9 March 2020 / Accepted: 20 May 2021 / Published online: 30 June 2021
© The Author(s) 2021
Abstract
We discuss the singularity structure of Kahan discretizations of a class of quadratic
vector fields and provide a classification of the parameter values such that the corre-
sponding Kahan map is integrable, in particular, admits an invariant pencil of elliptic
curves.
Keywords Birational map ·Integrable map ·Elliptic curve ·Elliptic pencil
Mathematics Subject Classification 37J70 ·14H70 ·14E05
1 Introduction
The Kahan discretization scheme was introduced in the unpublished notes [13]as
a method applicable to any system of ordinary differential equations in Rnwith a
quadratic vector field
f(x)=Q(x)+Bx+c,x∈Rn,
where each component of Q:Rn→Rnis a quadratic form, while B∈Rn×nand
c∈Rn. Kahan’s discretization reads as
x−x
2ε=Q(x,x)+1
2B(x+x)+c,(1)
This research is supported by the DFG Collaborative Research Center TRR 109 “Discretization in
Geometry and Dynamics”.
BRené Zander
1Institut für Mathematik, MA 7-1, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin,
Germany
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On the singularity structure of Kahan discretizations… 1047
where
Q(x,x)=1
2(Q(x+x)−Q(x)−Q(x))
is the symmetric bilinear form corresponding to the quadratic form Q. Equation (1)is
linear with respect to xand therefore defines a rational map x=φε(x).SinceEq.(1)
remains invariant under the interchange x↔xwith the simultaneous sign inversion
ε→−ε, one has the reversibility property φ−1
ε(x)=φ−ε(x). In particular, the map
φεis birational.
In this paper, we consider the class of two-dimensional quadratic differential equa-
tions
˙x
˙y=1−γ1
1(x,y)1−γ2
2(x,y)1−γ3
3(x,y)J∇H(x,y), (2)
where
H(x,y)=γ1
1(x,y)γ2
2(x,y)γ3
3(x,y),
and
i(x,y)=aix+biy
are linear forms, with ai,bi∈C,J=01
−10and γ1,γ
2,γ
3∈R\{0}.
Integrability of the Kahan maps φ:C2→C2was established for several cases of
parameters (γ1,γ
2,γ
3):If(γ1,γ
2,γ
3)=(1,1,1), then (2) is a canonical Hamiltonian
system on R2with homogeneous cubic Hamiltonian. For such systems, a rational
integral for the Kahan map φwas found in [4,17]. The Kahan maps for the cases
(γ1,γ
2,γ
3)=(1,1,2)and (γ1,γ
2,γ
3)=(1,2,3)were treated in [6,17,20]. In all
three cases, the level sets of the integral for both the continuous time system and the
Kahan discretization have genus 1. If (γ1,γ
2,γ
3)=(1,1,0), then (2) is a Hamiltonian
vector field on R2with linear Poisson tensor and homogeneous quadratic Hamiltonian.
In this case, a rational integral for the Kahan map φwas found in [5]. The level sets
of the integral have genus 0.
In this paper, we study the singularity structure of the Kahan discretization as a
birational quadratic map φ:CP2→CP2. Based on general classification results by
Diller and Favre [9], we provide the following classification for the Kahan map φof
(2) depending on the values of the parameters (γ1,γ
2,γ
3):
Theorem 1.1 Let φ:CP2→CP2be the Kahan map of (2). The sequence of degrees
d(m)of iterates φmgrows exponentially, so that the map φis non-integrable, except
for the following cases:
(i) If (γ1,γ
2,γ
3)=(1,1,1), (1,1,2), (1,2,3), the sequence d(m)of degrees grows
quadratically. The map φadmits an invariant pencil of elliptic curves. The degree
of a generic curve of the pencil is 3,4,6, respectively.
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1048 R. Zander
(ii) If (γ1,γ
2,γ
3)=(1,1,0)or (γ1,γ
2,γ
3)=(α, 1,−1),α∈R\Z∪{0},the
sequence of degrees d(m)grows linearly. The map φadmits an invariant pencil
of rational curves.
(iii) If (γ1,γ
2,γ
3)=(n,1,−1),n∈N, the sequence of degrees d(m)is bounded.
Here, (γ1,γ
2,γ
3)are fixed up to permutation and multiplication by λ∈R\{0}.
Some of the integrable cases are discussed in further detail in Sects. 4,5,6,7and 8.
2 Preliminary results
2.1 Birational maps of surfaces
Definition 2.1 Let φbe a birational map of a smooth projective surface X.Thedynam-
ical degree of the map φis defined as
λ1=lim
m→∞(φm)∗1/m,
where (φm)∗denote the induced pullback maps on the Picard group Pic(X).
Diller and Favre provide the following classification for birational maps with λ1=1:
Theorem 2.2 (Diller, Favre [9, Theorem 0.2]) Let φ:X→X be a birational map of
a smooth projective surface with λ1=1. Up to birational conjugacy, exactly one of
the following holds:
(i) The sequence (φm)∗is bounded, and φmis an automorphism isotopic to the
identity for some m.
(ii) The sequence (φm)∗grows linearly, and φpreserves a rational fibration. In
this case, φcannot be conjugated to an automorphism.
(iii) The sequence (φm)∗grows quadratically, and φis an automorphism preserv-
ing an elliptic fibration.
One says that φ:X→Xis analytically stable (AS) if (φ∗)m=(φm)∗on Pic(X).
This relates the dynamical degree λ1to the spectral radius of the induced pullback
φ∗:Pic(X)→Pic(X). Equivalently, analytic stability is characterized by the condi-
tion that there is no curve V⊂Xsuch that φn(V)∈I(φ) for some integer n⩾0,
where I(φ) is the indeterminacy set of φ(see [9, Theorem 1.14]). Therefore, the notion
of analytic stability is closely related to singularity confinement (see [15]). Indeed, a
singularity confinement pattern for a map φ:X→Xinvolves a curve V⊂Xsuch
that φ(V)=Pis a point (so that P∈I(φ−1)) and φn−1(P)∈I(φ), so that φn(P)
is a curve again for some positive integer n∈N. Such a singularity confinement
pattern can be resolved by blowing up the orbit of P. Upon resolving all singularity
confinement patterns, one lifts φto an AS map
φ:X→X.
Diller and Favre showed that for any birational map φ:X→Xof a smooth
projective surface we can construct by a finite number of successive blow-ups a surface
Xsuch that φlifts to an analytically stable birational map
φ:X→X(see [9,
Theorem 0.1]).
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On the singularity structure of Kahan discretizations… 1049
2.2 Birational quadratic maps of CP2
As shown, e.g., in [7], every quadratic birational map φ:CP2→CP2can be repre-
sented as φ=A1◦qi◦A2, where A1,A2are linear projective transformations of CP2
and qiis one of the three standard quadratic involutions:
q1:[x,y,z]→[yz,xz,xy],
q2:[x,y,z]→[xz,yz,x2],
q3:[x,y,z]→[x2
,xy,y2+xz].
In these three cases, the indeterminacy set I(φ) consists of three, respectively two, one
(distinct) singularities. The last two cases correspond to a coalescence of singularities.
Therefore, the first case is the generic one.
In the present work, we only consider the first case: φ=A1◦q1◦A2. In this case,
I(φ) ={B(1)
+,B(2)
+,B(3)
+}consists of three distinct points. Let L(1)
−denote the line
through B(2)
+,B(3)
+, and similarly for permutations of the indices 1,2,3. (We have,
e.g., B(1)
+=L(2)
−∩L(3)
−.) These lines are exceptional in the sense that they are blown
down by φto points: φ(L(i)
−)=B(i)
−. The inverse map is also quadratic with set of
indeterminacy points I(φ−1)={B(1)
−,B(2)
−,B(3)
−}.
Suppose that the map admits ssingularity confinement patterns, 0 ⩽s⩽3.
That means there are positive integers n1,...,ns∈Nand (σ1,...,σ
s)such that
φni−1(B(i)
−)=B(σi)
+for i=1,...,s. We assume that the niare taken to be mini-
mal and, for simplicity, we also assume that φk(B(i)
−)= φl(B(j)
−)for any k,l⩾0
and i= j. As shown by Bedford and Kim [2], one can resolve the singularity con-
finement patterns by blowing up the finite sequences B(i)
−,φ(B(i)
−), . . . , φni−1(B(i)
−).
Those sequences are also called singular orbits. In this paper, we only encounter the
situation that the orbits of different B(i)
−are disjoint. As shown in [2], one can adjust
the procedure to the more general situation.
On the blow-up surface X, the lifted map
φ:X→Xis AS, and is an automorphism
if and only if s=3. The s-tuples (n1,...,ns),(σ1,...,σ
s)are called orbit data
associated to φ. We say that the map φrealizes the orbit data (n1,...,ns),(σ1,...,σ
s).
Let H∈Pic(X)be the pullback of the divisor class of a generic line in CP2.Let
Ei,n∈Pic(X),fori⩽sand 0 ⩽n⩽ni−1, be the divisor class of the exceptional
divisor associated to the blow-up of the point φn(B(i)
−). Then Hand Ei,ngive a basis
for Pic(X), i.e.,
Pic(X)=ZH
3
i=1
ni−1
n=0
ZEi,n
that is orthogonal with respect to the intersection product, (·,·):Pic(X)×Pic(X)→
Z, and is normalized by (H,H)=1 and (Ei,n,Ei,n)=−1. The rank of the Picard
group is ni+1.
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1050 R. Zander
The induced pullback
φ∗:Pic(X)→Pic(X)is determined by (see Bedford, Kim
[2], and Diller [8])
H→ 2H−
j⩽s
Ej,nj−1,
Ei,0→ H−
j⩽s
σj=i
Ej,nj−1,i⩽s,(3)
Ei,n→ Ei,n−1,i⩽s,1⩽n⩽ni−1.
The induced pushforward
φ∗:Pic(X)→Pic(X)is determined by
H→ 2H−
j⩽s
Ej,nj−1,
Ei,0→ H−
j⩽s
σj=i
Ej,nj−1,i⩽s,(4)
Ei,n→ Ei,n−1,i⩽s,1⩽n⩽ni−1.
The maps
φ∗
,
φ∗are adjoint with respect to the intersection product (see [9, Proposition
1.1]), i.e., (
φ∗A,B)=(A,
φ∗B)for all A,B∈Pic(X).
Bedford and Kim computed the characteristic polynomial χ(λ) =det(
φ∗−λid)
explicitly for any given orbit data (see [2, Theorem 3.3]).
Let C(m)=(
φ∗)m(H)∈Pic(X)be the class of the mth iterate of a generic line.
Set
d(m)=(C(m), H), (5)
so that d(m)is the algebraic degree of the mth iterate of the map φ. Set
μi(m+j)=(C(m), Ei,j), i⩽s,0⩽j⩽ni−1.(6)
The expression on the right-hand side indeed depends on iand m+jonly: using that
the maps
φ∗,
φ∗are adjoint with respect to the intersection product and the relations
(4), we find
(C(m), Ei,j)=(C(m),
φ∗Ei,j−1)=(
φ∗C(m), Ei,j−1)=(C(m+1), Ei,j−1).
In particular, μi(m)=(C(m), Ei,0)can be interpreted as the multiplicity of B(i)
−on
the mth iterate of a generic line.
The sequence of degrees d(m)of iterates of the map φsatisfies a system of linear
recurrence relations.
Theorem 2.3 (Recurrence relations) Let φbe a birational map of CP2with three dis-
tinct indeterminacy points, and with associated orbit data (n1,...,ns),(σ1,...,σ
s).
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