MODULI OF CURVES
Cui Yin
A Dissertation
in
Mathematis
Presented tothe Faulties of the University ofPennsylvaniainPartial
FulllmentoftheRequirementsfortheDegreeofDotorofPhilosophy
2003
David Harbater
Supervisor of Dissertation
David Harbater
Thisthesisouldnothavebeenwrittenwithoutthehelpofmyadvisor,D.Harbater.
I would like toexpress my gratitude tohim for suggesting this problemto me, for
hispatiene andinsight,forhisonstant supervising myworkonit,forhisdetailed
omments on drafts of this thesis, and aboveall for the enthusiasm he radiatesfor
disovering new mathematis.
I would like to thank F. Pop, L. Shneps and R. Pries for very illuminating
disussions.
This is also the best time to thank the professors and the support sta at the
math department of the University of Pennsylvania. In partiular, many thanks
go to C. Chai, T. Chinburg, C. Croke, R. Donagi, M. Gerstenhaber, J. Shaneson,
S. Shatz and J. Burns.
Iwould alsoliketothank myfriends Kursat,JeandRohit fortheir friendship.
I amvery grateful to my parents and my parents-in-lawfor their help, support
and love.
THE MAPPING CLASS GROUP AND SPECIALLOCI IN MODULI OF
CURVES
Cui Yin
David Harbater,Advisor
The general problem this thesis is onerned with is that of studying the
sub-varieties of the moduli spae M
g;[n℄
orresponding to urves with extra
automor-phisms. A typial urve with marked points has no automorphisms; but some do,
depending upon the hoie of urves and position of marked points. This gives us
ertain subvarieties in the modulispae. ForRiemann surfaes, these subvarieties
are haraterized by speifying a nite group of mapping-lasses whose ation on
a urve is xed topologially. This thesis builds upon previous work by Gonz
alez-Dez, Harvey and Shneps. Gonzalez-Dez and Harvey [GH92℄ onsidered these
irreduiblesubvarietiesforgenusg 2urveswithoutmarkedpointsoverthe
om-plex numbers, where they have studiedthe oarse modulispae for surfaes with a
speied automorphismgroup. Cornalba [Cor87℄ gives a omplete lassiation of
the irreduiblesubvarietiesorrespondingtotheurveswhoseautomorphismgroup
ontains a xed yli subgroup of prime order, in the ase g 1;n = 0 over the
omplex numbers. Later Shneps [Sh02a ℄ onsidered the situationof genus 0with
In this thesis, we onsider more generalases in higher genusand inharateristi
p, withn marked points,orresponding tothemarked urves whoseautomorphism
1 Introdution 1
1.1 Bakground . . . 1
1.2 Outline of the thesis . . . 5
1.3 Preliminaries of moduliof urves . . . 8
2 Notions of Speial Loi 14 2.1 Dierential speial loi . . . 15
2.2 Algebrai speial loi . . . 22
3 Speial loi in low genus 30 3.1 Speial loiingenus 0 . . . 31
3.1.1 Genus zero overthe omplexnumbers C . . . 31
3.1.2 Examples in harateristi 5 . . . 34
3.1.3 Examples in harateristi 3 . . . 36
3.1.4 Speial loiof genus0 in harateristip . . . 36
4.1 Splittingand surjetivity over the omplex numbers . . . 46
4.2 Splittingand surjetivityonditions ingenuszero ase inall
Introdution
1.1 Bakground
The purpose of this thesis is to study speial loi in the moduli spae M
g;n [resp.
M
g;[n℄
℄parametrizingthe smoothprojetiveurves ofgenusg withn ordered [resp.
unordered℄ marked points. Over the omplex numbers, this spae is losely related
to the pure [resp. full℄ mapping lass group
g;n [resp.
g;[n℄
℄, whih is its orbifold
fundamental group. One an apply this relationship to study Galois theory,
espe-ially viathe Grothendiek-Teihmuller group d
GT ([Sh02b ℄).
Thegeneralproblemthisthesisisonernedwithisthatofstudyingthe
subvari-eties ofthe modulispaeM
g;[n℄
orrespondingtourveswith extraautomorphisms,
where a typial urve with marked points has no automorphisms; but some do,
are haraterized by speifying a nite group of mapping-lasses whose ation on
a urve is xed topologially. After onsidering this lassial ase, we will extend
these ideas to urves over more general elds,inludingthe harateristi pase.
We begin by realling the situation in the lassial ase, before outliningwhat
willbe donelater inthisthesis. LetS beanorientable topologialsurfaeof genus
g equipped with n distint ordered marked points s
1 ;:::;s
n
(we say that S is of
type (g;n)). A parametrized (ordered) marked Riemann surfae of genus g is a
Riemann surfae X of genus g with n distint ordered marked points x
1
;:::;x
n
together with a parametrization, i.e., a dieomorphism : S ! X suh that
(s
i )=x
i
foreahi. TwoparametrizedmarkedRiemannsurfaesX (withmarked
points x
1
;:::;x
n
and parameterization ) and X 0
(with marked points x 0
1
;:::;x 0
n
and parameterization 0
) are said to be isomorphi if there exists an isomorphism
: X !X 0
of Riemann surfaes with (x
i )= x
0
i
for eah i and a dieomorphism
h :S !S with h(s
i )=s
i
, for eah i, whih is isotopito the identity, via afamily
of dieomorphismsh
t
: S !S with h
t (s
i ) =s
i
, for t 2 [0;1℄ and eah i, suh that
the following diagramommutes:
S ! X h ? ? y ? ? y S 0 ! X 0
TheTeihmullerspaeT
g;n
isthesetofisomorphismlassesofparameterizedmarked
in afew smallases).
Themappinglassgroup
g;[n℄
atsontheTeihmullerspaeT
g;n
. Theunordered
modulispae M
g;[n℄
is realizedas the quotient of the Teihmuller spae T
g;n
by the
ationofthemappinglassgroup
g;[n℄
. Similarly,theorderedmodulispaeM
g;n is
the quotient of T
g;n
by the pure subgroup
g;n of
g;[n℄
. The permutationgroup S
n
ats naturally on M
g;n
by permuting the marked points on the Riemann surfaes.
For any subgroup G2S
n
, we write M
g;n
(G)=M
g;n =G.
Let'beanelementof nite orderin the fullor puremappinglass group(the
group of homotopy lasses of dieomorphisms of a surfae), then we onsider the
set ofpointsinTeihmuller spaexed by '. The imageof this set inthe quotient
modulispae M
g;n
(G)isalledthe speiallous of 'and denotedM
g;n
(G;'). The
speial lous of ' is losely related to the moduli spae M(T) of the topologial
quotientT =S='. Moreexpliitly,let['℄denotetheassoiatedpermutationofthe
marked points and letG S
n
be the subgroup generated by the disjoint yles of
['℄. Then there is anatural overing map of nite degree[Sh02a ℄
' :
f
M
g;n
(G;') !M(T) (1.1.1)
The morphism(1.1.1) orresponds to a grouphomomorphism
Norm
g;n (G)
(')! (T) (1.1.2)
where
g;n
(G) is the preimage of G under the anonial surjetion
g;[n℄ ! S
These two groups are the orbifold fundamental groups of M
g;n
(G;') and M(T)
respetively.
Ifthe homomorphism(1.1.2)issurjetive,wesaythat'satisesthe surjetivity
ondition, whihorrespondstotheondition thatthemorphism(1.1.1)isadegree
1overing. Ifthehomomorphism(1.1.2)issplit, wesaythat'satisesthesplitting
ondition. This additional ondition says that the natural orbifold strutures on
f
M
g;n
(G;') and M(T)are the same exept for the automorphism' at eah point.
When g =0,every niteorderelement of
g;[n℄
satisesboth the surjetivityand
splitting onditions [Sh02a℄.
This thesis builds upon previous work by Gonzalez-Dez, Harvey and Shneps.
Gonzalez-DezandHarvey[GH92℄onsideredtheseirreduiblesubvarietiesforgenus
g 2 urves without marked points over the omplex numbers, where they have
studied the oarse modulispae for surfaes with a speiedautomorphism group.
Cornalba[Cor87℄givesaompletelassiationoftheirreduiblesubvarieties
orre-spondingtotheurves whoseautomorphismgroupontainsaxedyli subgroup
of prime order, inthe ase g 1;n =0over the omplex numbers. Later Shneps
[Sh02a℄ onsidered the situation of genus 0 with n marked points, and genus 1
with n = 1or 2 marked points, orresponding to the urves having a yli group
in its automorphism group, over the omplex numbers. In this thesis, we onsider
more general ases in higher genus and in harateristi p, with n marked points,
1.2 Outline of the thesis
This thesis ontinues the study of speial loi of urves disussed above, and also
arriesoverthisstudyofmoregeneralelds,inludingthoseofharateristip. The
diÆulty in generalizing the denition of \speial loi" to harateristi p is that
we need to give an equivalent purely algebrai denition of speial loi. Shneps'
originaldenition(Denition2.1.9)ofspeialloiinvolvesthemappinglassgroup,
whih is the group of homotopy lasses of orientation preserving dieomorphisms
xing [resp. permuting℄in marked pointsof S ([Sh02a ℄); so doesn't make sense in
harateristip. Chapter 2 fouses on this problem.
Setion 2.1 provides an equivalent denition of (dierential) speial loi over
the omplex numbers C without mentioning Teihmuller spae and mapping lass
group. Thisdenitioniseasiertousetoprovetheequivalenebetweenthedenition
of speial loi (in the original sense) and the denition of algebrai speial loi.
Setion 2.2gives twodenitions of algebraispeial loi,whihare equivalentwith
the denition of speial loi over the omplex numbers C. One is the \family
denition"(Denition2.2.2)whihisproved equivalenttotheoriginaldenitionby
using Hurwitz families[CH85℄. Anotheris the \permutationdenition" (Corollary
2.2.7) inthe asethat n isbigenough, wheren isthe numberof themarkedpoints
Chapter3desribesthe speialloiingenuszeroinallharateristisexpliitly,
anddesribesthepossiblepermutationsinS
n
suhthatthespeialloiisnotempty
in genus1.
If S is a sphere with n marked points, then a nite-order element of the
map-ping lass group
g;[n℄
is the lass of a dieomorphism whih is simply a rotation
around an axis (Setion 3.1.1). In setion 3.1.1, we reall some known results of
Shneps ([Sh02a ℄) whih desribe the speial loi in genus zero over the omplex
numbers expliitly. Setions 3.1.2 and 3.1.3 give some expliit examples of speial
loi in harateristi 5 and 3 by onsidering the points having non-trivial speial
automorphism group in the ordered modulispae to determine the speial loi in
the unordered modulispae. Over the omplex numbers C, a nite-order element
of the mappinglass group
g;[n℄
is the lass of adieomorphism whih is simplya
rotation aroundan axis, while inharateristi p, every nite-order automorphism
ofP 1
K
withmarked pointsistheonjugaylassofarotationaroundanaxis(i.e.,by
multiplyingrootsofunity)ortheonjugaylassofatranslation(i.e.,byaddingan
element inK) (Proposition 3.1.8). Then we give a omplete desription of speial
loiof genus0 in harateristip in Setion3.1.4.
Thereisalsoageneralizationoftheresultsingenus0tohighergenus. In
parti-ular, Proposition3.2.1desribesthepossiblepermutations['℄suhthat thespeial
lous M
1;[n℄
Chapter 4 onsiders the surjetivity and splitting onditions in all
harateris-tis in genus 0 by rst generalizing the denitions of the surjetivity and splitting
onditions to harateristi p.
The permutation group S
n
ats naturally on M
g;n
by permuting the marked
points on the Riemann surfaes. For any subgroup G 2 S
n
, we write M
g;n (G) =
M
g;n
=G. OverC, let'beanelement ofniteorderinthefullmappinglassgroup
and let T = S='. We onsider the set of points in Teihmuller spae xed by '.
The image of this set in the quotient moduli spae M
g;n
(G) is the speial lous
of ' in M
g;n
(G) and denoted M
g;n
(G;'). Shneps gave splitting and surjetivity
onditions suh that eah omponentof f
M
g;n
(G;') isas lose toM(T) aspossible
([Sh02a℄).
To generalize the splitting and surjetivity onditions to harateristi p, we
need to givean equivalent denition whih an be applied inharateristip. The
original denition of splitting and surjetivity onditions involvesthe fundamental
group of moduli spaes of urves, while the fundamental group of moduli spaes
of urves in harateristi p is very ompliated. Setion 4.1 gives the geometri
meaningofthesurjetivityondition,whihavoidsthefundamentalgroupofmoduli
spaes of urves ( Proposition 4.1.5). It also gives an equivalent desription of the
splitting ondition by nite group theory using Proposition4.1.9.
from harateristi 0to arbitraryharateristi.
1.3 Preliminaries of moduli of urves
First I would like to give a rough idea and basi examples about modulispaes of
urves.
WeletM
g
denotethesetofisomorphismlassesofsmooth,omplete,onneted
urves of genus g. The set M
g
an be given the struture of analgebrai variety.
Intuitively, we would liketo speify the algebraistruture onM
g
by requiring
it to be a universal parameter variety for families of urves of genus g. In other
words, we would like to require that there is a at family X ! M
g
of urves of
genus g suh that for any other at family X ! T of urves of genus g, there is a
unique morphism T !M
g
suh that X is the pull-bak of X. In this ase we all
X!M
g
a universal family and say that M
g
is ane moduli spae.
Unfortunately, this an not be done in general (exept for g = 0, when M
g is
a point). Nevertheless, for g 1, there is at least a variety M
g
whih has the
following properties:
(1) the set oflosed points of M
g
is inone-to-one orrespondene with the set of
isomorphism lasses of urves of genus g;
g t
lass of urves determined by the pointh(t)2M
g .
This is lassial for g = 1 (where M
g
is the j-line), and was shown by Mumford
[Mum65, Th5.11℄ for g 2. We say that M
g
, satisfying (1) and (2), is a oarse
moduli spae for urves of genus g. Moreover, Deligne and Mumford [DM69℄ have
shown that M
g
for g 2 is an irreduible quasi-projetive variety of dimension
3g 3 over any xed algebraiallylosed eld. WilliamFulton [Ful69℄ alsoproved
theirreduibilityofthemoduliofurvesofgenusg overanyeldwithharateristi
p>g+1.
The allied spae M
g;n
is very important for our study. This parametrizes the
isomorphism lasses of objets (C;x
1
;:::;x
n
), where C is a urve of genus g, and
x
1
;:::;x
n
are distint ordered pointsof C. Thus, M
g =M
g;0 .
Letus begin tolook atthe simplest ases:
(I) M 0;n = [P 1
(0;1;1)℄ n 3
fx
i =x
j
forsome i6=jg
In fat, if we have n distint points x
1
;:::;x
n 2P
1
, a unique automorphism
of P 1
takes x
1
to 0, x
2
to 1, and x
3
to 1. The remaining n 3 points are
arbitrary exept for being distint and not equalto 0;1or 1.
(II) M 1;0 =M 1;1 =A 1 j
(the aÆne linewith oordinate j)
Beause urves of genus1 with one xed point P
0
are groups,their
automor-phisms at transitively; hene M
1;0 = M
1;1
by y 2
=x(x 1)(x ) where 6=0;1. Equivalently,C
isthedoubleoverof
P 1
ramiedat0;1;1;. Oneproveseasily(orthe readeran referto[Har97,
pages 317{320℄ that C
1 = C 2
if and onlyif there is an automorphismof P 1
arrying 0;1;1;
1
(unorderedset) to0;1;1;
2
. This happens if and onlyif
2 = 1 ;1 1 ; 1 1 ; 1 1 1 ; 1 1 1 or 1 1 1 :
[E.g., note the map
(x;y)7 !(1 x;y)
arries C
toC
1
; and the map
(x;y)7 !(1=x;y=x 2 ) arries C toC 1= .℄
One must ook up an expression in invariant under these substitutions and
no more. It isustomary touse:
j =j()=256 ( 2 +1) 3 2 ( 1) 2
(It is readily heked that this j is invariant under these 6 substitutions and sine
6=max(degree ofnumerator anddenominator),noother's givethe samej. The
and the aÆneline A 1
by taking C toj() if C
= C
. Properties (1)and (2)above
an then be heked.
Now we give the moduliproblema preisestatement.
First of all, we speify a lass of objets together with a notion of a family
of these objets over a sheme B. Roughly speaking, suh a family onsists of a
olletion of objets X
b
, for eah b. Seond, we hoose a equivalene relation
on the set S(B) of all suh families over eah B. We build a funtor F from the
ategory of shemes to that of sets by the rule
F(B)=S(B)=
and allF themodulifuntor ofourmoduliproblem. LetMbeashemesuhthat
there are naturalmaps
M
(B):F(B)!Hom(B;M)
given by
M
(B)(X)(b)=[X
b
℄, where b2B and [X
b
℄ denotes the equivalene lass
of theobjetX
b
and Hom(B;M)denotes theset ofmorphismsfromB toM. More
preisely, these maps
M
(B) determine anatural transformation
M
:F !Hom( ;M):
Denition 1.3.1. [HM98℄IfF isrepresentable by M(i.e., F Hom( ;M)),then
M
funtor F to the funtor Hom
M
= Hom( ;M) are a oarse moduli spae for the
funtor F if
1) The map
M
(pt) : F(pt) ! Hom(pt;M) is bijetive, where pt denotes any
generi point.
2) Given another sheme M 0
and a natural transformation
M 0
from F !
Hom
M 0
, there is a unique morphism : M ! M 0
suh that the assoiated
natural transformation :Hom
M
!Hom
M 0
satises
M 0
=Æ
M .
Remark 1.3.3. (1) It is easy to hek that if M is a ne modulispae, then itis
a oarse modulispae, but not vie versa.
(2) It an be heked that Denition 1.3.2 oinides with the denition we gave
atthe beginningofthis setionfor theoarse modulispae ofurvesof genus
g.
The reason that M
g
isjust a oarsemodulispae,and not ane modulispae,
is that some (speial) urves of a given genus have more automorphismsthan the
general urve of that genus. Of ourse this does not for g = 0, where the moduli
spaeisjustapoint. Butingenus1therearetwourveswithextraautomorphisms,
beyond translation and inversion. Deleting those, one obtains a ne modulispae.
In genus 2, most urves have no automorphisms other than the identity, but
g;n
ordered markedpoints,andfortheorrespondingmodulispaeM
g;[n℄
ofurveswith
unorderedmarkedpoints. (Inthemarkedsituation,thisphenomenonaneven
hap-pen in genus 0.) These speial loiof (marked) urves with extra automorphisms,
Notions of Speial Loi
The fous of this hapter is to onsider several notions of \speial lous". We
begin with the lassial ase of urves over the omplex numbers, whereShneps's
originaldenition(Denition2.1.9)ofspeialloiinvolvesthe mappinglassgroup
whih is the group of homotopy lasses of orientation preserving dieomorphisms
xing [resp. permuting℄in marked pointsof S ([Sh02a ℄), so doesn't make sense in
harateristip. Then wegeneralize this toharateristi p
Setion 2.1 provides an equivalent denition of dierential speial loi over
the omplex numbers C without mentioning Teihmuller spae and mapping lass
group. Thisdenitioniseasiertobeusedtoprovetheequivalenebetweenthe
def-inition of speial loi (in the original sense) and the denition of algebrai speial
loi. Setion2.2givestwo denitions ofalgebraispeialloi,whih are equivalent
using Hurwitz families [CH85℄. Another is the permutation denition (Corollary
2.2.7) provided that n is big enough, say n > n
0
(g), where n is the number of
marked points on the urves. Let K be an algebraially losed eld. For eah g,
there exists a number n
0
(g) suh that every nontrivial automorphism of genus g
urve over K has at most n
0
(g) xed points (Proposition 2.2.4). The permutation
denition iseasier towork with if we have enoughmarked points.
2.1 Dierential speial loi
In this thesis, we adopt some denitions from [Sh02a ℄. But sometimes we use
slightly dierent \terminology". We say dierential speial lous for the original
term speial lous.
We x S one and for all to be an orientable topologial surfae of genus g
equipped withn distintordered marked pointss
1 ;:::;s
n
. Wesay thatS isof type
(g;n).
Throughout this setion, we only work over the omplex numbers C. In order
to give the denition of speial lous, we needto givethe followingdenitions.
Denition 2.1.1. An ordered [resp. unordered℄marked Riemannsurfae isa
Rie-mannsurfaeXofgenusgtogetherwithnordered[resp.unordered℄distintmarked
pointsx
1
;:::;x
surfae of genus g onsistsof the following data:
(1) a n ordered marked Riemannsurfae (X;x
1
;:::;x
n
) of genus g;
(2) aparameterization,i.e., adieomorphism:S !X suh that(s
i )=x
i for
1in.
We say X isof type (g;n)
Denition 2.1.3. ([Sh02a, x2.1℄) Two parameterized marked Riemann surfaes
X (with marked points x
1
;:::;x
n
and parameterization ) and X 0
(with marked
pointsx 0
1 ;:::;x
0
n
and parameterization 0
) are saidto beisomorphi if there exists
an isomorphism : X ! X 0
of Riemann surfaes with (x
i ) = x
0
i
for 1 i n
and adieomorphism h:S !S with h(s
i )=s
i
, for1in,whihis isotopito
the identity, suhthat the following diagramommutes:
S ! X h ? ? y ? ? y S 0 ! X 0
Remark 2.1.4. In above denition, when we say a dieomorphism h : S ! S is
isotopi to the identity, we mean that it is isotopi to the identity via a family of
dieomorphisms h
t
:S !S with h
t (s
i )=s
i
, for t2[0;1℄ and eahi.
Remark 2.1.5. The Teihm uller spae T
g;n
is the set of isomorphism lasses of
pa-rameterized marked Riemannsurfaes of type (g;n). In fat, it is wellknown that
theTeihmullerspaeformsasimplyonnetedomplexanalytispaeofdimension
(a) We denethe full mappinglass group
g;[n℄
by setting
g;[n℄
=Di +
([S℄)=Di 0
(S);
where Di +
([S℄ denotes the group of orientation-preserving dieomorphisms
of S whih xes fs
1 ;:::;s
n
g as a set, and Di 0
(S) is the subgroup of those
whih are isotopito the identity.
(b) Wedenethe puremapping lassgroup (orpure subgroup of thefull mapping
lass group)
g;n
,by setting
g;n
=Di +
(S)=Di 0
(S);
where Di +
(S) is the subgroup of Di +
([S℄) onsisting of dieomorphisms
whih xeah marked points
i .
Remark 2.1.7. For the denition of mappinglass group, Shneps ([Sh02a℄), Hain
and Looijenga ([HL97℄) use dieomorphisms of a ompat orientable surfae of
genusg, whileGonzalez-Dez,Harveyand Malahlan ([GH92℄,[MH75℄) use
home-omorphismsof aompatorientablesurfae ofgenusg. Butthesedenitions ofthe
mapping lass group are equivalent beause every homeomorphism of a ompat
orientable surfae S of genus g an be approximated by a dieomorphismof S up
to homotopy (f. ([Hir76, Chapter 5, Lemma 1.5℄). So we an use all the results
g;[n℄ g;n
The ation is the following: if 2
g;[n℄ , let
0
denote a lifting of to a
dieo-morphism of S; then 0
maps the marked Riemann surfae (;X) to (Æ 0
;X)
([Sh02a, x2.1℄). Theunordered moduli spae M
g;[n℄
, parameterizing smoothurves
of genus g together witha unordered set of n-distint marked points, isrealized as
the quotientofthe Teihmullerspae T
g;n
by the ationofthe mappinglassgroup
g;[n℄
. Similarly, the ordered moduli spae M
g;n
, parameterizing smooth urves of
genusg together with anordered set ofn-distint marked points,isthe quotientof
T
g;n
by the pure subgroup
g;n of
g;[n℄ .
Denition 2.1.9. (f.[Sh02a, x2.1℄)If'isanelementofniteorderinthe fullor
pure mappinglass group, then we onsider the set of pointsin Teihmuller spae
xedby'. TheimageofthissetinthequotientmodulispaeM
g;n orM
g;[n℄
isalled
the dierential speial lous of '. Wedenote itby M
'
(S) or M
'
[S℄ respetively.
Now we give a notation by the following denition without mentioning T
e-ihmuller spae and mappinglass group.
Denition2.1.10. (Dierentialequivalene)LetX(withmarkedpointsx
1 ;:::;x
n )
and X 0
(with marked points x 0
1 ;:::;x
0
n
) be two ordered marked Riemann surfaes
with genus g in the ordered moduli spae M
g;n
. Let be a nite order
automor-phism ofX andlet 0
beanite orderautomorphismofX 0
, whihx eahmarked
point. Then and 0
are said to be dierentially equivalent if there exists a
dif-feomorphism : X ! X 0
with (x
i ) = x
0
i
X ! X 0 ? ? y ? ? y 0
X ! X
0
:
Similarly, Let X (with marked points x
1
;:::;x
n
) and X 0
(with marked points
x 0
1
;:::;x 0
n
) be two unordered marked Riemann surfaes with genus g in the
un-ordered modulispae M
g;[n℄
. Let be anite order automorphismof X and let 0
be a nite order automorphism of X 0
, whih x the marked points as a set. Then
and 0
are said to be dierentially equivalent if there exists a dieomorphism
:X !X 0
whih mapsthe set of marked pointsof X tothe set of marked points
of X 0
suh that the followingdiagram ommutes:
X ! X
0 ? ? y ? ? y 0
X ! X
0
Proposition 2.1.11. LetX (withmarkedpoints x
1 ;:::;x
n
) bean unordered [resp.
ordered℄ marked Riemann surfae in M
g;[n℄
[resp. M
g;n
℄ and be a nite order
automorphism of X. Pik a parameterization of X, then (X;) is a point in
Teihm uller spae. Let : S ! S be the dieomorphism indued by . Let ' be
the equivalent lass of in the full [resp. pure℄ mapping lass group.
(a) Then (X;) is xed by '.
(b) Let X 0
(with markedpoints x 0
1
;:::;x 0
n
) be anunordered [resp. ordered℄marked
Riemann surfae in M
g;[n℄ (M
g;n
). Then there exists a parameterized marked
Riemann surfae (X 0
; 0
) (where 0
spae whih is also xed by ' if and only if there exists an automorphism
of X 0
whih is dierentially equivalent to .
Proof. Weonlygiveaproofforunorderedmodulispae. Forordered modulispae,
the proof isbasially the same.
(a) By assumption of the proposition, we have the following ommutative
dia-gram: S ! X ? ? y ? ? y S ! X
whih isequivalent tothe following ommutativediagram:
S ! X id ? ? y ? ? y
S ! X
whereid:S !S istheidentity. So(X;) isxed by'inTeihmullerspae.
(b) NowsupposethatthereexistsaparameterizedmarkedRiemannsurfae(X 0
; 0
)
whih is also xed by '. Then there exists an automorphism 0
: X 0
! X 0
and a dieomorphism h : S ! S with h(s
i ) = s
i
for 1 i n whih is
isotopi tothe identity suh that the followingdiagram ommutes:
S 0 ! X 0 h ? ? y ? ? y 0 S 0 h 1 ! X 0
sine h 1
S 0 ! X 0 ? ? y ? ? y 0 S 0 ! X 0 Let = 0 1
;then we get the following ommutative diagram:
X ! X
0 ? ? y ? ? y 0
X ! X
0
So and 0
are dierentially equivalent.
Conversely,supposethereexistsanautomorphism 0
ofX 0
whihisdierentially
equivalent to . Then there exists a dieomorphism : X !X 0
whih maps the
set of marked points to the set of marked points suh that the following diagram
ommutes:
X ! X
0 ? ? y ? ? y 0
X ! X
0
Then we have the following ommutative diagram:
S
! X ! X
0 ? ? y ? ? y ? ? y 0 S
! X ! X
0
Let 0
= ; then the following diagramommutes:
S 0 ! X 0 ? ? y ? ? y 0 S 0 ! X 0
This isequivalent tothe followingommutativediagram:
where id:S !S is the identity. So(X ;)is xed by '.
Corollary 2.1.12. Let X (with marked points x
1 ;:::;x
n
) be an unordered [resp.
ordered℄markedRiemannsurfae inM
g;[n℄
[resp.M
g;n
℄and beaniteorder
auto-morphismof X. Thedierentialspeiallousof isthe setof points onthe moduli
spae M
g;n
[resp. M
g;[n℄
℄ whih have an automorphism dierentially equivalent to
.
Proof. By Denition 2.1.9, onsider the set of pointsin Teihmullerspae xed by
, then the dierential speial lous of is the image of this set in the quotient
modulispae. ByProposition2.1.11,letX 0
(withmarked pointsx 0
1
;:::;x 0
n
)beany
unordered[resp.ordered℄markedRiemannsurfaeinM
g;[n℄ (M
g;n
),thenthereexists
a parameterizedmarked Riemannsurfae (X 0
; 0
)(where 0
is aparameterization)
in Teihmuller spae whih is also xed by if and only if there exists an
auto-morphism 0
ofX 0
whihis dierentiallyequivalentto . Therefore thedierential
speiallous of is the set of points onthe modulispaeM
g;n
[resp. M
g;[n℄
℄ whih
have anautomorphismdierentiallyequivalent to .
2.2 Algebrai speial loi
Let K be an algebraiallylosed eld. Unlike the previous setion, where we only
worked over the omplex numbers C, now we generalize the denition of \speial
Remark 2.2.1. Let beaniteorder automorphismof X. Let beageneripoint
in X and orrespond to the urve A with an automorphism . And X ontains
the point orrespondingto X. Then there is aninjetive map
:Aut(A)!Aut(X):
We say that speializes if ()= .
Moreover, let 0
beanite orderautomorphismof X 0
andX ontains thepoint
orresponding to X 0
. Then there is aninjetive map
0
:Aut(A)!Aut(X 0
):
We say that speializesboth and 0
if ()= and 0
()= 0
Denition 2.2.2. Let X (with marked points x
1
;:::;x
n
) and X 0
(with marked
points x 0
1
;:::;x 0
n
) be two unordered [resp. ordered℄ marked urves with genus g in
M
g;[n℄
[resp.M
g;n ℄.
(a) A nite order automorphism of X and a nite order automorphism 0
of
X 0
, whih x the marked points as a set, are said to be algebraially
equiva-lent if there exists an irreduible subvariety X of M
g;[n℄
[resp. M
g;n
℄ and an
automorphism of the urve A whih orresponds to the generi point in
X , suh that speializes to both and 0
, where X ontains the points
orresponding to X and X 0
g;[n℄
[resp. M
g;n
℄ whih have an automorphism algebraially equivalent to . We
denote itby M
g;[n℄
( ) [resp. M
g;n ( )℄.
Proposition 2.2.3. Over the omplex numbers C, let X (with unordered [resp.
ordered℄ marked points x
1
;:::;x
n
) be a marked urve with genus g in M
g;[n℄ [resp.
M
g;n
℄. Let be a niteorder automorphismof X, whih xthe markedpoints asa
set. Thenthe algebrai speial lousof isas same as the dierential speial lous
of .
Proof. Weonlygiveaproofforunorderedmodulispae. Forordered modulispae,
the proof isbasially the same.
FirstweneedtoshowthatthespeiallousM
g;[n℄
( )isanirreduiblesubvariety
of M
g;[n℄
over the omplex numbers C. By [GH92, Theorem 1, page 79℄, we know
that the speial lous M
g;[n℄
( ) is an irreduible subvariety of M
g;[n℄
when n = 0,
i.e., in the ase of nomarked points. By a similar proof, we know that the speial
lous M
g;[n℄
( ) is anirreduible subvariety of M
g;[n℄ .
NowletY =X=h i,thenwehavethebranhdata oftheoveringmap X!Y.
By [CH85,Proposition 1.4℄, weknowthereis aoarsemodulispae forh i-Galois
overs of Y with desription branh data. (Note: the result in[CH85, Proposition
1.4℄ uses the group is abelian, here it is yli. And the base spae of the Hurwitz
family is P 1
in [CH85 , Proposition 1.4℄, but the proof is similar for general base
LetX beanunordered marked urve ofgenusg with marked pointsx
1
;:::;x
n
and 0
be anite automorphismof X 0
whihpreservesthe marked points asa set.
Suppose 0
is in the algebrai speial lous of , i.e., 0
and are algebraially
equivalent. ThenX and X 0
are in the same Hurwitz familyof h i-Galoisovers of
Y. So there exists a dieomorphism : X ! X 0
whih maps the set of marked
points of X to the set of marked points of X 0
suh that the following diagram
ommutes:
X ! X
0 ? ? y ? ? y 0
X ! X
0
i.e., and 0
are dierentially equivalent. Therefore 0
is inthe dierentialspeial
lous of .
Conversely, suppose that 0
is in the dierential speial lous of . Sine the
dierentialspeiallousM
g;[n℄
( ) isanirreduiblesubvarietyof M
g;[n℄
,there exists
a generi point in M
g;[n℄
( ) with an an automorphism of the urve A whih
orresponds tothe generi point, suh that speializesto both and 0
, where
M
g;[n℄
( ) ontains the points orresponding to X and X 0
. Therefore and 0
are
algebraiallyequivalent. Hene 0
is inthe algebrai speial lous of .
Proposition 2.2.4. Let K be an algebraially losed eld. For eah g, there exists
a leastinteger n
0
(g) suhthatevery nontrivial automorphismof genusg urve over
K has at most n
0
(g) xed points; i.e., if n >n
0
(g), then any automorphism whih
xes n distint points is an identity. In partiular, n
0
(g)=2g+2 for all g and for
we know n
0
(0) =2. For g = 1, there are only nitely many automorphisms xing
any onegivenpoint. Son
0
(1)exists. Forg 2,itiswell-known thatthereare only
nite many automorphisms, so n
0
(g) exists.
Now we show that n
0
(g)= 2g+2 for all g and for all harateristis. Suppose
X isa genus g smooth urve with an order m automorphismand n is the number
of xed points of . Sine the number of xed points of is less or equal to the
number of xed points of a power of , we may assume the automorphism has
prime order p. Let Y = X=hi; then X is a branhed overing spae of Y. By
the Riemann-Hurwitz Theorem, we have 2g 2 p(2g
Y
2)+(p 1)n. (In the
tamely ramied ase, 2g 2 = p(2g
Y
2)+(p 1)n and in the wildly ramied
ase, 2g 2 > p(2g
Y
2)+(p 1)n.) Suppose that n > 2g +2; then we get
2g 2 2p+(p 1)(2g+3) (sine g
Y
0). Sowe have g(4 2p)+1 p 0,
whih is a ontradition sine 4 2p 0 and 1 p < 0 (p 2). Therefore, we
have n 2g+2. If n=2g+2,then we an take g
Y
=0 and p=2, i.e., there is a
nontrivial order 2 automorphism of a hyperellipti urve of genus g suh that
has 2g+2 xed points. Hene n
0
(g) = 2g+2 is the least integer suh that every
nontrivialautomorphismofgenusgurveoverKhasatmostn
0
(g)xedpoints.
Remark 2.2.5. For n > n
0
(g), if an automorphism of a marked urve X xes n
points,then =1. So if two automorphisms and of a marked urve X indue
g;n
point is the identity. Therefore, the speial lous of a non-trivial automorphism
is empty. (Note: The ondition n>n
0
(g) is not reallya restrition for g =0sine
n
0
(0) = 3 and sine we need n 4 to get a non-trivial moduli spae in genus 0.)
But for unordered modulispaes, there an be non-empty speial loi even if n is
large ompared to g. This an be seen by taking a unionof nitely many orbits of
an nite orderautomorphism of the underlyingurve.
Thusifn >n
0
(g),oneanspeakintermsofpermutationsratherthan
automor-phisms,andtheseareeasiertoworkwith. Thismotivatesthefollowingproposition.
Let X be a marked urve with genus g in M
g;[n℄
. Let be a nite order
au-tomorphism of X, whih xes the marked points as a set.We denote [ ℄ for the
permutation indued by , fora point inthe unordered modulispae.
Proposition2.2.6. Forn>n
0
(g), letX (withunorderedmarkedpointsx
1
;:::;x
n )
and X 0
(with unorderedmarkedpoints x 0
1
;:::;x 0
n
) be two marked urves with genus
g in M
g;[n℄
. Let be a nite order automorphism of X, whih xes the marked
points as a set. Then there exists a nite order automorphism 0
(whih xes the
marked points as a set) of X 0
whih is dierentially equivalent to if and only if
there existsa niteorder automorphism 00
(whih xes themarked points as aset)
of X 0
suhthat [ 00
℄ is onjugate to [ ℄.
Proof. First suppose that there exists a nite order automorphism 0
(whih xes
the markedpointsasaset) ofX 0
exists a dieomorphism :X ! X whih maps the set of marked points of X to
the set of marked pointsof X 0
suh that the followingdiagram ommutes:
X ! X
0 ? ? y ? ? y 0
X ! X
0
Then 0
= 1
. So and 0
indueonjugate permutations. Let 00
= 0
,then
[ ℄ is onjugateto 00
.
Conversely, suppose that there exists a nite order automorphism 00
(whih
xes the marked points as a set) of X 0
suh that [ 00
℄ is onjugate to [ ℄ . Choose
parametrizations : S ! X for X and 0
: S ! X 0
. Let : S ! S be the
dieomorphism indued by and 0
: S ! S be the dieomorphism indued by
00
. Sine and 00
indueonjugate permutations,we know that and 0
indue
onjugate permutations.
Therefore, by [GP74, Corollary (of Isotopy Lemma), page 143℄ there exist a
dieomorphism h : S ! S whih is isotopi to identity and a dieomorphism
:S !S suhthat = 1 0
h, i.e., the following diagramommutes:
S h ! S h ? ? y ? ? y 0 S h ! S
whih isequivalent tothe following ommutativediagram:
S h ! S 0 ! X 0 h ? ? y ? ? y 0 ? ? y 00 S h ! S 0 ! X 0
So the followingdiagramommutes:
S 0 h ! X 0 h ? ? y ? ? y 00 S 0 h ! X 0 Therefore (X 0 ; 0
h)isxedby theequivalenelass of . ByProposition2.1.11,
we knowthereexists anautomorphism 0
of X 0
whihisdierentiallyequivalentto
.
Corollary 2.2.7. For n >n
0
(g), let X (with unordered marked points x
1
;:::;x
n )
be a marked urve with genus g in M
g;[n℄
and be a nite order automorphism of
X whih xes fx
1 ;:::;x
n
g as a set. The algebrai speial lous of is the set of
pointsonthemodulispaewhihhave anautomorphismwhoseinduedpermutation
of the marked points is onjugate to [ ℄.
Speial loi in low genus
This hapteronsiders speial loiingenus0 and genus 1,in allharateristis. In
genus 0 we give an expliit desription of the speial loi. In genus 1 we desribe
the possible permutationsinS
n
suh that the speial loiis not empty ingenus 1.
If S is a sphere with n marked points, then a nite-order element of the
map-ping lass group
g;[n℄
is the lass of a dieomorphism whih is simply a rotation
around an axis (Setion 3.1.1). In setion 3.1.1, we reall some known results of
Shneps ([Sh02a ℄) whih desribe the speial loi in genus zero over the omplex
numbers expliitly. Setions 3.1.2 and 3.1.3 give some expliit examples of speial
loi in harateristi 5 and 3 by onsidering the points having non-trivial speial
automorphism group in the ordered modulispae to determine the speial loi in
the unordered modulispae. Over the omplex numbers C, a nite-order element
of the mappinglass group
g;[n℄
ofP 1
K
withmarked pointsistheonjugaylassofarotationaroundanaxis(i.e.,by
multiplyingrootsofunity)ortheonjugaylassofatranslation(i.e.,byaddingan
element inK) (Proposition 3.1.8). Then we give a omplete desription of speial
loiof genus0 in harateristip in Setion3.1.4.
There is also a generalization of the results in genus 0 to higher genus. In
partiular, Proposition 3.2.1 desribes the possible permutations ['℄ suh that the
speialloiM
1;[n℄
(')isnotempty,wheretheproofisgivenbyusingthelassiation
of automorphisms ofellipti urves and Riemann-Hurwitzformula.
3.1 Speial loi in genus 0
3.1.1 Genus zero over the omplex numbers C
For the genus zero ase, the pure mapping lass group
0;n
ats on Teihmuller
spae T
g;n
freely [Sh02a , x2.1℄. So there are no speial loi in the ordered moduli
spae M
0;n
. A permutation of the ordered marked points an be realized as an
automorphism of the marked Riemann surfae [Sh02a , x3.1.1.℄. Suh points are
not orbifold pointsonthe ordered modulispae, butthey are preimages oforbifold
pointsontheunorderedmodulispaeM
0;[n℄
,sinethe havelessthann!preimages
under the ationof S
n
. The pointshaving non-trivialspeial automorphismgroup
determine where the speial loiwilllie onin the unordered modulispae M
and the speialloiinthe genuszero modulispaesfor arbitraryn. IfS is asphere
with n marked points,then anite-order element ofthe mappinglass group
g;[n℄
is the lass of a dieomorphism whih is simply a rotation around an axis. In
fat, For n 5, all nite order element in
0;[n℄
are rotations follows from [MH75,
Corollary p508℄ and [Sh02b , x4.1℄. For n = 3,
0;[n℄
= 1, For n = 4, there are
four onjugay lasses of nite order elements, whih indues dierent onjugate
permutations [Sh02b , Proof of Corollary2 inx3℄, we an see that eah onjugate
lass omesfrom arotation.
Let ' be a nite-order element of the mapping lass group
g;[n℄
. We may
assume that'isarotation,say aroundthe axisthrough thenorth andsouth poles
(orresponding to the points 1;0). The north and south poles of S may or may
not be marked points, but they are always the onlyramiation points for '. The
permutation assoiated toa rotation 'is always of the form
1 :::
k
, where the
i
are disjointyles of lengthj suh that
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
jk =n if the north and south poles are not marked
jk =n 1 if one of the two poles ismarked
jk =n 2 if both poles are marked points
In the following theorem, we ompute the points havingspeial automorphism
assoiated to a permutation ['℄ whih is a produt of k disjoint yles of length j
the speial lous inM
0;n
isjustthe imageof theone weompute here inM
0;n+1 or
M
0;n+2
,under the morphismgiven by erasingthe extra marked points.
Theorem3.1.1. [Sh02a, Theorem3.5.1℄LetS beaRiemannsurfaeofgenuszero
withn markedorderedpoints,andlet'bearotationof order j withn=jk+2(i.e.,
the two xed points of ' are markedpoints of S). After replaing ' by a onjugate
of 'whih has thesame speial lousas ', wemay assume thatthe points of S are
numbered so thatthe permutation assoiated to ' is givenby
['℄=(1j)(j+12j)(j(k 1)+1jk)
LetG
' S
n
bethesubgroupgeneratedbytheabovedisjointj-yles
1 ;:::;
k
of['℄.
Let T be the orbifold quotient S=h'i, whih has k marked points with ramiation
index 1 and 2 marked points with ramiation index j.
(i) The set of xed points of ['℄ in the ordered moduli spae M(S)has j(Z=jZ)
j
disjoint onneted omponents C
, respetively onsisting ordered markings of
the form
(1;;:::; j 1
;a
1 ;a
1
;:::a
1
j 1
;:::;a
k 1 ;a
k 1 ;:::a
k 1
j 1
;0;1):
Here runs through the primitive j-th roots of unity. Eah omponent C
is
isomorphito aopy of (P 1
f0;1;;::: j 1
;1g) k 1
minus thej(k 1)lines
a
i =a
r
s
for r 6=i, 0sj 1, and is thus dened over Q a b
' G'
of j(Z=jZ)
j disjoint onneted omponents C
, the image of the C
. Eah C
is isomorphi to
(P 1
f0;1;1g) k 1
'M(T)'M
0;k+2 ;
where denotes themulti-diagonal of points with x
i =x
j
forsome i6=j and
is thus dened overQ; however the embeddings M(T)!C
M(S)=G
' are
dened overQ ab
.
(iii) IntheunorderedmodulispaeM
Sn
(S)=M(S)=S
n
, thespeiallousof'
on-sists of a single onneted omponent C. It is isomorphito the moduli spae
of M
G
(T) whih is the quotient of M(T) by the group G of all \admissible"
permutations, i.e., permutations of marked points having the same
ramia-tion index, and the spae M
G
(T) and the embedding M
G
(T)!C M
Sn (S)
are dened over Q.
Similarly, in harateristi p, we an think about the points having non-trivial
speial automorphism group in the ordered modulispae to determine the speial
loiin the unordered modulispae.
3.1.2 Examples in harateristi 5
Example 3.1.2. Inthease M
0;5
,rstwealulatethepointswith speial
0;5
The ationof onthe point takesitto(;;0;1;1),and then the transformation
bytheautomorphismx7! x
x
bringsitbakto(
;0;1;1; 1
). Thexedpoints
of are given by (;) with
=
and = 1
;
so isa rootof 3
2 2
+1. One rootis =1,but this is exludedin M
0;5 . The
remainingrootsare = 1 p 5 2 = 1 2
=3(sinethis isinharateristi 5);sothe only
xed pointof is(3;0;1;1;2). Infat,the point(3;0;1;1;2)isequivalenttothe
point (0;1;2;3;4) in the moduli spae M
0;5
sine (3;0;1;1;2) transforms to the
point(0;1;2;3;4)by the linear transformation x7! 3x 9
x 9
. Then intuitively,we an
see that (0;1;2;3;4)isxed by a translation .
Remark 3.1.3. In Example 3.1.2, only xes one pointinharateristi5, while it
xes two points inharateristi0 (f. Theorem 3.1.1).
Example 3.1.4. In the ase M
0;10
, rst we alulate the points with speial
auto-morphism group. Given a permutation = (1;2;3;4;5;6;7;8;9;10) and a point
(x
1
;:::;x
7
;1;0;1) in M
0;10
in standard representation (with three omponents
xedat0;1and1),thentheationof onthepointtakesitto(1;x
1 ;:::;x
7 ;1;0).
But in harateristi5, there isno transformationan bring itbak to the original
point (x
1 ;:::;x
7
;1;0;1) by the similar alulation in Example 3.1.2. So there is
no xed point of inM
xes twodisonnetedone-dimensionalomponentsinharateristi0(f.Theorem
3.1.1).
3.1.3 Examples in harateristi 3
Example 3.1.6. Inthease M
0;4
,rstwealulatethepointswith speial
automor-phism group. Consider the permutation =(123) and a point (;1;0;1) in M
0;4
instandardrepresentation(withthreeomponentsxedat0;1and 1). The ation
of onthe pointtakesitto(0;;1;1). Thenthe transformationy7!y+2brings
the point (0;2;1;1)bak to the original point (2;1;0;1). So the xed point of
in M
0;4
is(2;1;0;1).
Remark 3.1.7. In Example 3.1.6, only xes one pointinharateristi3, while it
xes two points inharateristi0 (f. Theorem 3.1.1).
3.1.4 Speial loi of genus 0 in harateristi p
First, here isa result todesribe the nite-order automorphismingenuszero
alge-brai urves with marked points.
Proposition 3.1.8. Let K be an algebraially losed eld. Every nite-order
au-tomorphism of P 1
K
with marked points is the onjugay lass of a rotation around
an axis (i.e., by multiplying roots of unity) or the onjugay lass of a translation
Proof. WeknowthatthegroupofautomorphismsofP
K
isisomorphitoPGL(2;K).
Sine K isanalgebraiallylosed eld,by Jordan anonialform,everyelementin
PGL(2;K)is either onjugateto
A= 0
B
B
1
0 1
C
C
A
or onjugateto
B = 0
B
B
1 0
0
2 1
C
C
A
where ;
1 ;
2
are non-zero elements inK.
If it is onjugate to A, then it has one xed point 1 and the orresponding
frational lineartransformationis z 7!z+ 1
,whihis justatranslation. Ifthe
au-tomorphismisofniteorder,thenthetranslationanonlyhappen inharateristi
p.
IfitisonjugatetoB,thenithastwoxedpoints0and1andtheorresponding
frational linear transformation is z 7! 1
2
z, whih is a omposition of a rotation
and a dilation. If the automorphism is nite order, then it is just a rotation; this
happen both in harateristi 0 and p. In harateristi 0, the rotation an have
any order;inharateristip,itsorderisprimetop,beause therearenoprimitive
p th
rootsof unity.
Thefollowingresultdesribesthespeialloiinharateristipintheaseg =0
1 n
over an algebraially losed eld K, and ' be a nite order automorphism of X
whih xes fx
1 ;:::;x
n
g as a set. Let ['℄ denote the permutation of marked points
indued by '. Let g 0
be the genus of X=' and n 0
be the number of marked points
omingfromthemarkedpointsofX. IfM
0;[n℄
(')isnotempty,then'isoftheform
1
k
where the
i
are disjoint yles of length j suh that jk =n or jk =n 1
or jk =n 2. Moreover:
(a) If p6jj, then M
0;[n℄
(') has the samedesription in harateristi 0 and p.
(b) If pjj and j >p, then M
0;[n℄
(') is empty.
() If p=j and jk =n 2, then M
0;[n℄
(') is empty. If p=j and jk=n 1 or
jk =n, then M
0;[n℄
(') is isomorphi to quotient of fP 1
f0;1;1gg k 2
by S
k
, where denotes the multi-diagonal of points with x
i = x
j
for some
i6=j.
Proof. By Proposition 3.1.8, we know that if M
0;[n℄
(') is not empty, then ' is of
the form
1
k
where the
i
are disjoint yles of length j suh that jk = n or
jk=n 1 or jk=n 2.
(a) Ifp6jj,thenwe havethe j-throotsof unity. Sine theproofforharateristi
0(f.Shneps[Sh02a ,Theorem3.5.1℄)isonlyinvolvedthepuregrouptheory,
translation.
() If p=j,then ' isa translation.
If jk = n 2, then ' has no xed point sine a translation an not x two
pointspointwise.
If jk = n 1, then in the ordered modulispae M
0;n
, we know that ' xes
p 1 disjointonneted omponents, eah omponent is given by
C
i
=(0;i;:::;(p 1)i;a
1 ;:::;a
1
+(p 1)i;:::;a
k 1 ;:::;a
k 1
+(p 1)i;1);
where i =1;:::;p 1 and a
1 ;:::;a
k 1
are any numbers in the eld K suh
that all the marked pointsare distint. Inthe unorderedmodulispae M
0;n ,
alltheomponentsC
i
(aswellasallthoseomponentsorrespondingtoother
rotations having the same yle type as ') beome identied. So M
0;[n℄ (')
is isomorphi to one of C
i
, say C
1
, modulo its stabilizer in S
n = S
jk+1 . We
ould determineitsstabilizerby the similarproedurewith Shneps'proof in
(f. [Sh02a , Theorem 3.5.1℄) (where the proof only involved the pure group
theory). In fat, the stabilizer of C
1
is generated by two natural subgroups:
the rst, of order k!, orresponding to permuting the k disjointyles of ['℄;
theseond, oforderj k
,isgeneratedbythejylesthemselves. Afterompute
the quotientofC
1
by itsstabilizer,we get M
0;[n℄
(')is isomorphitoquotient
offP 1
f0;1;1gg k 2
by allpermutationsofmarkedpointsinX='whih
omesfromthemarked pointswiththesameramiationindexinX,i.e.,S
i j
If jk = n, then by a similar alulation with the ase jk = n 1, we get
M
0;[n℄
(') is isomorphi toquotient of fP 1
f0;1;1gg k 2
by S
k
, where
denotes the multi-diagonalof points with x
i =x
j .
Theorem 3.1.9 shows that there is no automorphism of order divisible by p
in harateristi p unless the order is exatly p. Also there is an automorphism
of order p, viz. translation, but this automorphism behaves dierently from an
automorphism of the same order in harateristi 0, as the examples in setions
3.1.2 and 3.1.3show.
3.2 Speial loi in genus 1
There is also a generalizationof Theorem 3.1.9 to higher genus. In partiular, for
g =1, we give someresults in the followingproposition.
Proposition 3.2.1. Let X be a marked urve of genus 1 with n marked points
x
1
;:::;x
n
overan algebraiallylosed eldK of harateristi 6=2;3 andlet 'bea
nite order automorphismof X. Let['℄2S
n
bea permutationof markedpoints for
n 5, where ['℄ is the orresponding permutation of marked points of '. Suppose
that M
1;[n℄
(') is not empty, and write ['℄ as a produt of disjoint yles
1
k .
i
(ii) the
i
areof the samelength j =2suhthat jk =n or n 1or n 2 or n 3
or n 4; or
(iii) the
i
are of the same length j = 3 suh that jk = n or n 1 or n 2 or
n 3; or
(iv) the
i
are of the same length j =4 suh that jk =n or n 1 or n 2; or
(v) one of the
i
is of length 2 and the others are of the same length j =4 suh
that jk =n 2 or jk =n 3 or jk =n 4; or
(vi) the
i
are of the same length j =6 suh that jk =n or jk =n 1; or
(vii) one of the
i
is of length 2 and the others are of the same length j =6 suh
that jk =n 2 or jk =n 3; or
(viii) one of the
i
is of length 3 and the others are of the same length j =6 suh
that jk =n 3 or jk =n 4; or
(ix) one
i
is of length 2, another is of length 3, and the remainder are all of the
same length j =6 suh thatjk=n 5 or jk =n 6.
Before we give a proof of Proposition 3.2.1, let us reall the lassiation of
automorphism group Aut(E) is a nite group of order dividing 24. More preisely,
the order of Aut(E) is givenby the following list:
(a) 2 if j(E)6=0;1728
(b) 4 if j(E)=1728 and har(K)6=2;3
() 6 if j(E)=0 and har(K)6=2;3
(d) 12 if j(E)=0=1728 and har(K)=3
(e) 24 if j(E)=0=1728 and har(K)=2
NowwegiveaproofforProposition3.2.1byusingtheRiemann-Hurwitzformula
and Theorem 3.2.2.
Proof. (of Proposition 3.2.1) Let X be a marked urve of genus 1 with n marked
pointsx
1 ;:::;x
n
overanalgebraiallylosedeld K of harateristi6=2;3and let
' be a nite order automorphismof X. Let['℄ 2 S
n
be a permutationof marked
pointsforn 5,where['℄istheorrespondingpermutationofmarked pointsof'.
Let g
0
is the genus of X=h'i. Then the possible order m of ' and their branhing
data (m
1 ;m
2
;:::;m
r
)are limited by the well-known Riemann-Hurwitzequation:
(2g 2)=m=(2g
0
2)+ r
X
i=1 (1
1
m
i ):
Here we only onsider the Riemann-Hurwitz formula in the tame ase sine if
1 2 r
the following onditions are satised ([Bre00, orollary9.4℄):
(i) lm(m
1 ;m
2
;:::;m
i 1 ;m
i+1
;:::;m
r
)=M for all i;
(ii) M divides m, and if g
0
=0;M =m;
(iii) r 6=1, and if g
0
=0, r3;
(iv) if M iseven, the numberof m
i
divisible by themaximumpowerof2dividing
M is even.
By Theorem 3.2.2, we know the possible numbers of m
i
are 2;3;4;6. Combine
the aboveonditions onRiemann-Hurwitz equation3.2, we get the possibleGalois
overings are:
(i) m =2, r=4,m
i
=2 for alli;
(ii) m =3, r=3,m
i
=3 for alli;
(iii) m =4, r=3,(m
1 ;m
2 ;m
3
)=(2;4;4);
(iv) m =6, r=3,(m
1 ;m
2 ;m
3
)=(2;3;6).
Sine X has n marked points, aording to the above possible Galois overings,
we an get the possible permutations ['℄ of marked points as in the list of this
Splitting and surjetivity in genus
zero
As disussed inChapter 1above, Shneps[Sh02a℄ introduedsplittingand
surje-tivityonditionsinthestudyofspeialloiformoduliofmarkedRiemannsurfaes.
She also proved that these onditions hold in the genus 0 ase (i.e., for marked
spheres). But her denitions used the fat that the spaes were dened over the
omplex numbers. This hapter onsiders the surjetivity and splitting onditions
inallharateristisingenus0byrstgeneralizingthedenitionsofthesurjetivity
and splittingonditions toharateristip.
The permutation group S
n
ats naturally on M
g;n
by permuting the marked
points on the Riemann surfaes. For any subgroup G 2 S
n
, we write M
g;n (G) =
M
g;n
The image of this set in the quotient moduli spae M
g;n
(G) is the speial lous
of ' in M
g;n
(G) and denoted M
g;n
(G;') Shneps gave splitting and surjetivity
onditions suh thateah omponentof f
M
g;n
(G;') isas lose toM(T) aspossible
([Sh02a℄).
Here we onsider splitting and surjetivity onditions in harateristi p. But
to dothat, werst nd new haraterizations of theseonditions overthe omplex
numbers whih do not rely on mapping lass groups or Teihmuller spae. We
then use those haraterizations as the denitions inharateristi p. The original
denition ofsplittingand surjetivityonditions involvesthe fundamentalgroupof
moduli spaes of urves, while the fundamental group of moduli spaes of urves
in harateristi p is very ompliated. Setion4.1 gives the geometri meaningof
the surjetivity ondition, whih avoids the fundamentalgroup of modulispaes of
urves ( Proposition 4.1.5). And also gives an equivalent desription of splitting
ondition by nite grouptheory using Proposition 4.1.9.
Setion4.2showsthatthesurjetivityandsplittingonditionsholdingeneralfor
genus0 (Theorem4.2.5 andTheorem 4.2.6),whihgeneralize theresult ofShneps
numbers
Throughout this setion, we only work over the omplex numbers C. Let S be a
topologial surfae of type (g;n) and let GS
n
be a subgroup. Let
G
(S) be the
preimage of G under the anonial surjetion ([S℄) ! S
n
. If ' is a nite-order
element of the full mapping lass group ([S℄), we let M
G
(S;') denote the image
in M
G
(S)of the set ofpointsinthe Teihmullerspae T(S)=T
g;n
whihare xed
by 'under the anonial ation of ([S℄) onT(S).
Let ['℄ denote the permutation assoiated to ', and letG
' S
n
be the group
generated by the disjoint yles of ['℄. Let us write M
'
(S) for the quotient spae
M
G
'
(S) = M(S)=G
'
, and M
'
(S;') for the whole of the speial lous of ' in
M
'
(S). We alsowrite
'
(S)for the group
G
'
(S),the preimage in ([S℄)of h['℄i
under the surjetion ([S℄)!S
n .
Remark 4.1.1. SineS is determined by (g;n), we alsowrite (S) [resp. ([S℄)℄ as
g;n (resp.
g;[n℄
) and M(S) [resp. M([S℄) ℄ asM
g;n
[resp. M
g;[n℄ ℄.
Let T = S=' and we assume that all branh points of this over (and their
preimage) are marked points. Let g 0
denote the genus of T and n 0
the number of
marked points;the fundamentalgroup of T is given by generators and relations as
1
(T)=ha
1 ;b
1 ;:::;a
g 0 ;b g 0; 1 ;::: n 0j g 0 Y i=1 (a i ;b i ) 1 n 0 =1i:
The group of inertia-preserving automorphisms of
1
(T), Aut
([
1
Aut
([
1
(T)℄)=f 2Aut
(
1
(T))j92S
n
suh that (
i )s
(i)
for 1 in 0
g
where s means \isonjugate to".
Let Aut
([S=T℄) denote the subgroup of Aut
([
1
(T)℄) onsisting of elements
whih preserve the subgroup
1
(S)2
1
(T). We introdue the followingnotations:
[S=T℄
=Aut
([S=T℄)=Inn(
1
(T))Out
([
1
(T)℄)= ([T℄);
and
S=T
=Aut
(S=T)=Inn(
1
(T))=
[S=T℄
\ (T) (T):
ForanysubgroupsHandKofagroupG,wedenoteNorm
H
(K)=Norm
G (K)\
G,whereNorm
G
(K)isthenormalizerofK inG. Foranyelementg 2G,wedenote
Norm
H
(g)=Norm
H (hgi).
Now we give the denition of surjetivity ondition of '.
Denition 4.1.2. ([Sh02a , x4.2℄) We say that a nite-order element ' 2 ([S℄)
satisesthesurjetivityondition if
S=T
= (T),i.e.,everyelementofAut
(
1 (T))
preserves the subgroup
1
(S);in other words, the homomorphism
Norm
' (S)
(')=h'i! (T)
is surjetive.
Before we give the geometri meaning of the surjetivity ondition, let us give
some similarresultsof Gonzalez-Dezand Harvey [GH92℄ inthease oftype(g;n),
order element of ([S℄). Assume that the quotient T =S=' is of genus g 0
with n 0
marked points,inluding allthe branh points.
(i) Denote byT(S;') thesubsetofpoints oftheTeihm ullerspaeT(S)=(T)
g;n
xed by'. Theneah omponent of T(S;') is isomorphi to T
g 0
;n 0
=T(T).
(ii) The set of elements of ([S℄) globally preserving eah omponent of T(S;')
in T(S) is exatly the subgroup Norm
([S℄) (').
(iii) Forevery GS
n
ontaining thepermutation['℄ assoiatedto', thequotient
f
M
G
(S;') =T(S;')=Norm
G (S)
(') isisomorphi tothe normalizationof the
speial lousM
G
(S;') M
G (S).
Remark 4.1.4 (on the proof). The proof an be given by following the proof of
Gonzalez-Dez and Harvey [GH92℄ in the ase without marked points. But there
are slight dierenes in the marked points ase from Gonzalez-Dez and Harvey's
results. In the ase withoutmarked points,T(S;') is isomorphito T
g 0
;n 0
=T(T),
whereT(S;')isirreduible([GH92,TheoremB℄).Whileinthemarkedpointsase,
T(S;')mightonsistseveralirreduibleomponents,eahomponentisisomorphi
to T
g 0
;n 0
= T(T). In fat, we an see that the map from T(S;') to T
g 0
;n 0
is not
bijetive in the marked points ase, sine there ould be dierent permutations of
themarkedpointsofSforanygivenfg 0
;n 0
g. (Cf.[Harv71,Theorem2 & Corollary
following proposition.
Proposition 4.1.5. (f. [Sh02a, x4.2℄) A nite-order element'2 ([S℄) satises
thesurjetivityonditionifandonlyifthemapfromeahomponentof f
M
'
(S;') to
M(T)isone-to-one,onsistingonlyin forgetting theorbifoldstruture of f
M
' (S;')
due to the ation of '.
Proof. Suppose that ' satises the surjetivity ondition, then by denition of
surjetivity, the homomorphism
Norm
' (S)
(')=h'i! (T)
issurjetive. ByProposition[Sh02a ,Proposition4.1.2.℄,thereis aanonial
inje-tive homomorphism
Norm
'(S)
(')=h'i! (T);
whose image is of niteindex. Therefore,
Norm
' (S)
(')=h'i! (T)
is an isomorphism. By Theorem 4.1.3, sine h'i xes every point of T(S;'), the
ation of Norm
'(S)
(') fators through the quotient group Norm
'(S)
(')=h'i, and
there is a anonial one-to-one orrespondene
T(S;')=Norm
'(S)
(')$T(S;')=(Norm
'(S)
ation of h'ixing eah point).
Now using Theorem 4.1.3, eah omponent of T(S;') is isomorphi to T
g 0
;n 0 =
T(T) and the normalization f
M
G
(S;') 'T(S;')=Norm
G (S)
(') , so eah
ompo-nent of
f
M
G
(S;') 'T(S;')=Norm
G (S)
(')$T(S;')=(Norm
' (S)
(')=h'i)
to
T(T)=(Norm
'(S)
(')$T(T)= (T)'M(T)
is one-to-one, onsisting only in forgetting the orbifold struture of f
M
'
(S;') due
to the ation of '.
Conversely,supposethat themapfromeahomponentof f
M
'
(S;')toM(T)is
one-to-one, onsisting only in forgetting the orbifold struture of f
M
'
(S;') due to
the ation of ', then by above proof, we an see that this map indues abijetive
homomorphism
Norm
'(S)
(')=h'i! (T):
Therefore, 'satises the surjetivity ondition.
Remark 4.1.6. This was stated without mentioning omponents in [Sh02a℄.
A-tually the geometri ondition in above propositiondiers slightlyfrom the one in
Shneps'paper[Sh02a℄,whereshenegletstomentionthatthemapfrom f
M
n = 5;j = 3, then ' indues a permutation (123), and M
'
(S;') onsists of two
disjoint points, while M(T) onsists of only one point. But in the ase without
marked points, f
M
'
(S;') is onneted, this issue doesnot rise. The reader an see
the paper [GH92℄.
The splittingondition is dened as follows.
Denition 4.1.7. ([Sh02a, x4.2℄) A nite-order element ' 2 ([S℄) satises the
splitting ondition if the surjetion
Norm
'(S)
(')!Norm
'(S)
(')=h'i'
S=T
splits; in other words, if we have a semi-diret produt
Norm
' (S)
(') 'h'ioNorm
' (S)
(')=h'i
Sometimes, the semi-diret produtin Denition 4.1.7 beomes diret produt,
so we give the followingdenition.
Denition 4.1.8. A nite-order element ' 2 ([S℄) satises the strong splitting
ondition if we havea diret produt
Norm
'(S)
(')'h'iNorm
'(S)
(')=h'i:
Atually, the splitting ondition an be desribed by nite group theory (i.e.,
without mentioning
'
