Connectivity of the branch locus of moduli space of rational maps

Abstract

Milnor proved that the moduli space Md{\rm M}_{d} of rational maps of degree
d≥2d \geq 2 has a complex orbifold structure of dimension 2(d−1)2(d-1). Let us
denote by Sd{\mathcal S}_{d} the singular locus of Md{\rm M}_{d} and by
Bd{\mathcal B}_{d} the branch locus, that is, the equivalence classes of
rational maps with non-trivial holomorphic automorphisms. Milnor observed that
we may identify M2{\rm M}_2 with C2{\mathbb C}^2 and, within that
identification, that B2{\mathcal B}_{2} is a cubic curve; so B2{\mathcal B}_{2}
is connected and S2=∅{\mathcal S}_{2}=\emptyset. If d≥3d \geq 3, then Sd=Bd{\mathcal
S}_{d}={\mathcal B}_{d}. We use simple arguments to prove the connectivity of
it.

Full text

arXiv:1502.03001v2 [math.DS] 13 Feb 2015
CONNECTIVITY OF THE BRANCH LOCUS OF MODULI SPACE OF RATIONAL
MAPS
RUBEN A. HIDALGO AND SA ´
UL QUISPE
Abstract. Moduli space Mdof rational maps of degree d≥2 has a orbifold structure, with a
singular locus Sd. Inside Mdthere is also its branch locus Bd, consisting of those equivalence
classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may
identify M2with C2and, within that identification, that B2is a cubic curve; so B2is connected and
S2=∅. If d≥3, then Bd=Sd. In this paper we use simple arguments to prove the connectivity of
Bd.
1. Introduction
The space Ratdof complex rational maps of degree d≥2 can be identified with a Zariski open
set of the (2d+1)-dimensional complex projective space P2d+1
C; this is the complement of the
algebraic variety defined by the resultant of two polynomials of degree at most d. The group of
M¨obius transformations PSL2(C) acts on Ratdby conjugation; φ, ψ ∈Ratdare said to be equivalent
if there is some T∈PSL2(C) so that ψ=T◦φ◦T−1.
The PSL2(C)-stabilizer of a rational map φ∈Ratd, denoted as Aut(φ), is called the group of
holomorphic automorphisms of φ. It is well known that Aut(φ) is finite.
The quotient space Md=Ratd/PSL2(C) is the moduli space of rational maps of degree d.
Silverman [7] obtained that Mdcarries the structure of an affine geometric quotient and Milnor
[6] proved that it also carries the structure of a complex orbifold of dimension 2(d−1) (Milnor
also obtained that M2C2). Levy [4] noted that Mdis a rational variety (he also observed that the
order of Aut(φ) is bounded above by a constant depending on d).
Let us denote by π: Ratd→Mdthe holomorphic branch cover obtained by the above conju-
gation action, that is, its deck group is PSL2(C). This map fails to be a covering exactly at those
rational maps with non-trivial group of holomorphic automorphisms. Let us denote by Bd⊂Md
the locus of branch values of π.
Using Milnor’s identification M2=C2, the locus B2corresponds to the cubic y2=x3; the cuspid
corresponds to the class of a rational map φ(z)=1/z2with Aut(φ)D3(dihedral group of order
6) and all other points in the cubic corresponds to those classes of rational maps with the cyclic
group Z2as group of holomorphic automorphisms. In this way, B2is connected. In this paper we
prove the connectivity of Bdfor every d≥2.
Theorem 1. The branch locus Bdis connected.
2000 Mathematics Subject Classification. 37P45, 37F10, 37P05.
Key words and phrases. Rational maps, Automorphisms, Moduli spaces, Branch locus.
Partially supported by Project Fondecyt 1150003. 1

2 RUBEN A. HIDALGO AND SA ´
UL QUISPE
Let us denote by Sd⊂Mdthe singular locus of Md, that is, the set of points over which Md
fails to be a topological manifold. As already seen above, as M2C2, so S2=∅. But, if d≥3,
Sd=Bd[5], in particular, Sd,∅. Theorem 1then asserts the following.
Corollary 1. The singular locus Sdof Mdis connected.
Theorem 1states that given any two rational maps φ, ψ ∈Ratd, both with non-trivial group of
holomorphic automorphisms, there a conjugated ρ∈Ratdof ψand there is a continuous family
Θ: [0,1] →Ratdwith Θ(0) =φ,Φ(1) =ρand Aut(Θ(t)) non-trivial for every t. Even if
Aut(φ)Aut(ψ), we may not ensure that Aut(Θ(t)) stay in the same isomorphic class; this comes
from the existence of rigid rational maps [3] (in the non-cyclic situation).
Remark 1. In the 80’s Sullivan provided a dictionary between dynamic of rational maps and the
dynamic of Kleinian groups [8]. If we restrict our Klenian groups to be co-compact Fuchsian
groups of a fixed genus g ≥2, then we will be dealing with closed Riemann surfaces of genus g.
The moduli space of Riemann surfaces Mghas the structure of an orbifold. The branch locus in Mg
is the set of isomorphic classes of Riemann surfaces with non-trivial holomorphic automorphisms.
In [1]it was proved that in general the branch locus is non-connected; a great difference with the
connectivity of brach locus for rational maps.
2. Rational maps with non-trivial group of holomorphic automorphisms
It is well known that the group of holomorphic automorphisms of φ∈Ratd, where d≥2,
is a finite subgroup of PSL2(C) which are known to be (see, for instance, [2]) either the trivial
group or isomorphic to a cyclic group or the dihedral group or the alternating groups A4,A5or the
symmetric group S4. Moreover, for each possible group above there are rational maps admitting it
as group of holomorphic automorphisms [3].
If Gis either a finite cyclic group, a dihedral group, A4,A5or S4, then we denote by Bd(G)⊂
Mdthe locus of classes of rational maps φwith Aut(φ) admitting a subgroup isomorphic to G. We
say that Gis admissible for dif Bd(G),∅.
If Gis either cyclic or dihedral or A4, then there may be some elements in Bd(G) with group
of holomorphic automorphisms non-isomorphic to G(i.e., they admit more holomorphic automor-
phisms). If Gis either isomorphic to S4or A5, then Bd(G) may have isolated points [3], so it is
not connected in general.
2.1. Admissibility of the non-cyclic cases. Miasnikov-Stout-Williams observed the following
admissibility for the case of platonic groups.
Theorem 2 ([5]).Let d ≥2.
(1) A4is admissible for d if and only if d is odd.
(2) A5is admissible for d if and only if d is congruent modulo 30 to either 1,11,19,21.
(3) S4is admissible for d if and only if d is coprime to 6.

CONNECTIVITY OF THE BRANCH LOCUS OF MODULI SPACE OF RATIONAL MAPS 3
2.2. Admissibility in the cyclic case. In the case G=Cn, the cyclic group of order n≥2, the
admissibility will depend on d. First, let us observe that if a rational map admits Cnas a group
of holomorphic automorphisms, then we may conjugate it by a suitable M¨obius transformation so
that we may assume Cnis generated by the rotation T(z)=ωnz, where ωn=e2πi/n. In [5] it was
noticed that in this situation the rational map must have the form φ(z)=zψ(zn), where ψ∈Ratr, for
a suitable value of r. Since d≥2, we must have r≥1. The relations between rand dare provided
as follows.
Theorem 3 ([5]).Let d,n≥2be integers. The group Cnis admissible if and only if d is congruent
to either −1,0,1modulo n. Moreover, for such values, every rational map of degree d admitting
Cnas a group of holomorphic automorphisms is equivalent to one of the form φ(z)=zψ(zn), where
ψ(z)=Pr
k=0akzk
Pr
k=0bkzk∈Ratr,
satisfies that
(a) arb0,0, if d =nr +1;
(b) ar,0and b0=0, if d =nr;
(c) ar=b0=0, if d =nr −1.
Proof. Let φbe a rational map admitting a holomorphic automorphism of order n. By conjugating
it by a suitable M¨obius transformation, we may assume that such automorphism is the rotation
T(z)=ωnz.
(1) Let us write φ(z)=zρ(z). The equality T◦φ◦T−1=φis equivalent to ρ(ωnz)=ρ(z). Let
ρ(z)=U(z)
V(z)=Pl
k=0αkzk
Pl
k=0βkzk,
where either αl,0 or βl,0 and (U,V)=1.
The equality ρ(ωnz)=ρ(z) is equivalent to the existence of some λ,0 so that
ωk
nαk=λαk, ωk
nβk=λβk.
By taking k=l, we obtain that λ=ωl
n. So the above is equivalent to have, for k<l,
ωl−k
nαk=αk, ωl−k
nβk=βk.
So, if αk,0 or βk,0, then l−k≡0 mod (n). As (U,V)=1, either α0,0 or β0,0; so
l≡0 mod (n). In this way, if αk,0 or βk,0, then k≡0 mod (n). In this way, ρ(z)=ψ(zn) for
a suitable rational map ψ(z).
(2) It follows from (1) that φ(z)=zψ(zn), for ψ∈Ratrand suitable r. We next provide relations
between dand r. Let us write
ψ(z)=P(z)
Q(z)=Pr
k=0akzk
Pr
k=0bkzk,
where (P,Q)=1 and either ar,0 or br,0. In this way,
φ(z)=zP(zn)
Q(zn)=zPr
k=0akzkn
Pr
k=0bkzkn .
Let us first assume that Q(0) ,0, equivalently, ψ(0) ,∞. Then φ(0) =0 and the polinomials
zP(zn) and Q(zn) are relatively prime. If deg(P)≥deg(Q), then r=deg(P), ψ(∞),0, φ(∞)=∞
and deg(φ)=1+nr. If deg(P)<deg(Q), then r=deg(Q), ψ(∞)=0, φ(∞)=0 and deg(φ)=nr.

4 RUBEN A. HIDALGO AND SA ´
UL QUISPE
Let us now assume that Q(0) =0, equivalently, ψ(0) =∞. Let us write Q(u)=ulb
Q(u), where
l≥1 and b
Q(0) ,0; so deg(Q)=l+deg(b
Q). In this case,
φ(z)=P(zn)
zln−1b
Q(zn)
and the polinomials P(zn) (of degree ndeg(P)) and zln−1b
Q(zn) (of degree ndeg(Q)−1) are relatively
prime. If deg(P)≥deg(Q), then r=deg(P), ψ(∞),0, φ(∞)=∞and deg(φ)=nr. If deg(P)<
deg(Q), then r=deg(Q), ψ(∞)=0, φ(∞)=0 and deg(φ)=nr −1.
Summarizing all the above, we have the following situations:
(i) If φ(0) =0 and φ(∞)=∞, then ψ(0) ,∞and ψ(∞),0; in particular, d=nr +1. This
case corresponds to have arb0,0.
(ii) If φ(0) =∞=φ(∞), then ψ(0) =∞and ψ(∞),0; in which case d=nr. This case
corresponds to have ar,0 and b0=0.
(iii) If φ(0) =0=φ(∞), then ψ(0) ,∞and ψ(∞)=0; in particular, d=nr. This case
corresponds to have ar=0 and b0,0. But in this case, we may conjugate φby A(z)=1/z
(which normalizes hTi) in order to be in case (ii) above.
(iv) If φ(0) =∞and φ(∞)=0, then ψ(0) =∞and ψ(∞)=0; in particular, d=nr −1. This
case corresponds to have ar=b0=0.

The explicit description provided in Theorem 3permits to obtain the connectivity of Bd(Cn) and
its dimension.
Corollary 2. If n ≥2and Cnis admissible for d, then Bd(Cn)is connected. Moreover,
dimC(Bd(Cn)) =









2(d−1)/n,d≡1 mod n
(2d−n)/n,d≡0 mod n
2(d+1−n)/n,d≡ −1 mod n
Proof. (1) By Theorem 3, the rational maps in Ratdadmitting a holomorphic automorphism of
order n≥2 are conjugated those of the form φ(z)=zψ(zn)∈Ratdfor ψ∈Ratras described in the
same theorem.
Let us denote by Ratd(n,r) the subset of Ratdformed by all those rational maps of the φ(z)=
zψ(zn), where ψsatisfies the conditions in Theorem 3.
If d=nr +1, then we may identify Ratd(n,r) with an open Zariski subset of Ratr; if d=nr, then
it is identified with an open Zariski subset of a linear hypersurface of Ratrand (iii) if d=nr −1,
then it is identified with an open Zariski subspace of a linear subspace of codimension two of Ratr.
In each case, we have that Ratd(n,r) is connected. As the projection of Ratd(n,r) to Mdis exactly
Bd(Cn), we obtain its connectivity.
(2) The dimension counting. We may see that, (i) if d=nr+1, then ψdepends on 2r+1 complex
parameters; (ii) if d=nr, then ψdepends on 2rcomplex parameters; and (iii) if d=nr −1, then
ψdepends on 2r−1 complex parameters. The normalizer in PSL2(C) of hTiis the 1-complex
dimensional group Nn=hAλ(z)=λz,B(z)=1/z:λ∈C− {0}i. If U∈Nn, then U◦φ◦U−1will
also have Tas a holomorphic automorphism. In fact,
Aλ◦φ◦A−1
λ(z)=zψ(zn/λn),
B◦φ◦B(z)=z/ψ(1/zn).

CONNECTIVITY OF THE BRANCH LOCUS OF MODULI SPACE OF RATIONAL MAPS 5
In this way, there is an action of Nnover Ratrso that the orbit of ψ(u) is given by the rational
maps ψ(u/t), where t∈C− {0}, and 1/ψ(1/u). In this way, we obtain the desired dimensions. 
3. Proof of Theorem 1
It is clear that Bdis equal to the union of all Bd(G), where Gruns over the admissible finite
groups for d.
If Gis admissible for dand pis a prime integer dividing the order of G(so that the cyclic group
Cpis a subgroup of G), then Cpis admissible for dand Bd(G)⊂ Bd(Cp). In this way, Bdis equal to
the union of all Bd(Cp), where pruns over all integer primes with Cpadmissible for d. Corollary 2
asserts that each Bd(Cp) is connected. Now, the connectivity of Bdwill be consequence of Lemma
1below.
Lemma 1. If p ≥3is a prime and Cpis admissible for d, then Bd(Cp)∩ Bd(C2),∅.
Proof. We only need to check the existence of a rational map φ∈Ratdadmitting a holomorphic
automorphism Tof order pand also a holomorphic automorphism Uof order 2.
First, let us consider those rational maps of the form φ(z)=zψ(zp), where (by Theorem 3) we
may assume to be of the form
ψ(z)=Pr
k=0akzk
Pr
k=0bkzk∈Ratr,
with
(a) arb0,0, if d=pr +1;
(b) ar,0 and b0=0, if d=pr;
(c) ar=b0=0, if d=pr −1.
In cases (a) and (c) we can find ψsatisfying the relation ψ(1/z)=1/ψ(z). This is possible by
considering bk=ar−k, for every k=0,1, ..., r. Now, we may see that φalso admits the holomorphic
automorphism U(z)=1/z. The automorphisms T(z)=ωpzand U(z)=1/zgenerate a dihedral
group of order 2p.
In case (b), we can consider ψso that ψ(−z)=ψ(z), which is posible to find if we assume that
(−1)kak=(−1)rakand (−1)kbk=(−1)rbk(which means that ak=bk=0 if kand rhave different
parity). In this case Tand V(z)=−zare holomorphic automorphisms of φ, generating the cyclic
group of order 2p.

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6 RUBEN A. HIDALGO AND SA ´
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[7] J.H. Silverman. The space of rational maps on P1.Duke Math. J. 94 (1998), 41–77.
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Departamento de Matem´atica y Estad´ıstica, Universidad de LaFrontera. Casilla 54-D, 4780000 Temuco, Chile
E-mail address:ruben.hidalgo@ufrontera.cl
E-mail address:saul.quispe@ufrontera.cl