A note on the stratification by automorphisms of smooth plane curves of genus 6

0.23 MB·

by  Eslam Badr

A NOTE ON THE STRATIFICATION BY AUTOMORPHISMS OF SMOOTH PLANE
CURVES OF GENUS 6
ESLAMBADRANDELISALORENZOGARC´IA
Abstract. Inthisnote,wegiveaso-calledrepresentativeclassificationforthestratabyautomorphismgroup
of smooth k-planecurves of genus 6, wherek isafixed separable closureof afield k of characteristic p=0 or
p>13. Westartwithaclassificationalreadyobtainedin[3]andwemimicthetechniques in[10]and[11].
Interestingly, inthe waytoget these familiesforthe differentstrata, wefindtwo remarkablephenomenons
that did not appear before. One is the existence of a non 0-dimensional final stratum of plane curves. At a
7 firstsightitmaysoundodd,butwewillseethat thisisanormalsituationforhigher degrees andwewillgive
1
aexplanationforit.
0
Weexplicitlydescriberepresentativefamiliesforallstrata,exceptforthestratumwithautomorphismgroup
2
Z/5Z. Herewefindtheseconddifferencewiththelowergenuscaseswheretheprevioustechniquesdonotfully

n work. Fortunately, we are still able to prove the existence of such family by applying a version of Lu¨roth’s
a
theorem indimension2.
J

1
1. Introduction
2

Letkbeafieldofcharacteristicp 0,andfixaseparablealgebraicclosurekofk. Byasmoothk-planecurve
] ≥
T C of genus g, we mean a smooth projective curve C that admits a non-singular plane model F = 0 P2
{ C } ⊆ k
N over k of degree d, in this case, the genus g equals 1(d 1)(d 2).
2 − −
.
h It might happen for some genus g that no smooth k-plane curves exist. The first genus for which there exist
t smooth k-plane curves are: 0,1,3,6,... The curves of genus 0 are isomorphic to the projective line, and the
a
m curves of genus 1 are elliptic ones, which are quite well understood. For genus 3, we get plane quartic curves,
[ and different arithmetics properties have been investigated by many people around. We mention, for example,

aclassificationuptoisomorphismwithgoodpropertiesthatcanbefoundin[10,11],orthestudyoftheirtwists
1
v in [11,12].
5
For genus 6, the dimension of the (coarse)moduli space of smooth curves of genus g =6 over k is equal
6 M6
0 to 3g−3 = 15. The stratum MP6l of smooth k-plane curves of genus 6 has dimension equal to 21−9 = 12,
6 since there are 21 monic monomial of degree 5 in 3 variables and all the isomorphisms are given by projective
0
matricesofsize3 3. Inparticular,thisdimensionislargerthanthedimensionofthehyperellipticlocus,which
1. ×
is 2g 1=11. Aclassification,upto k-isomorphism,to smoothplane curvesofgenus6,andthe determination
0 −
7 of the full automorphism groups are given by the first author, et al. in [3].
1 The aim of this note is to refine the classification made in [3], by providing better families for the strata in
:
v Pl. By better families, we mean that the set of normalforms with parameters describing the different strata
M6
Xi of MP6l give a bijectionwith the points ineachstratumandthe parametersaredefined overthe field ofmoduli
r oftherepresentingpoint. Inparticular,welookforcomplete andrepresentativefamiliesoverk,whicharegood
a
substitutes to the notion of universal families of (coarse) moduli spaces, especially when the spaces have no
extra structures. These concepts have been introduced by R. Lercier, et al. in [10, 2].
§
We find a representative family for each of the strata in [3] by using the techniques in [10,11] except for a
special one. For this family, we do not write downthe representativefamily, but we provethat it exists and we
give a recipe to construct it. We do it by applying a version of Lu¨roth’s theorem in dimension 2.
Anotherinterestingphenomenonappearingistheexistenceofafinalstratumofplanecurveswhosedimension
is not zero. By a final stratum we mean a stratum not containing any other proper stratum. One could expect
2010 Mathematics Subject Classification. 11G30,14D22,14H50,37P45.
Key words and phrases. Modulispace,Planecurves,Representative families.
E.BadrissupportedbyMTM2013-40680-P.
1
2 E.BADRANDE.LORENZO
thatbyaddingrestrictionsinthe parametersofafamilydefining astratumwithagivenautomorphismsgroup,
one could get bigger automorphism groups until obtaining a zero-dimensional stratum. This happens for all
the families except for one. For this family eachrestrictionin the parametersprovidinga biggerautomorphism
group yields a singular curve. We find an explanation for this fact: this family can be embedded in a family of
curvesofgenus6withthesameautomorphismgroupforwhichwecancarryoutthepreviousoperationwithout
getting singular curves, the key point is they are not plane curves anymore. Moreover,we prove that this may
happen in general for higher genera.
Finally,wementionthattheobtainedrepresentativeclassificationforsmoothquinticcurvesbyautomorphism
groups is the first step for the computation of the twists of these curves. This will be done by the first author
in his thesis [1].
Acknowledgments. The authors would like to thank Francesc Bars Cortina for his fruitful comments and
suggestions for the preparation of this note.
2. Preliminaries
2.1. Complete and representative families. It is well known that the (coarse) moduli spaces are
g
M
algebraicvarietieswhosegeometricpointsgivea classificationofisomorphismclassesofsmoothcurvesofgenus
g. The existence of universal families for a moduli space helps to recover the information on its points and
allows to write down the attached objects to a point of this space. However,universal families do not exist for
the moduli space .
g
M
R. Lercier, et al in [10, 2] introduced three good substitutes for the notion of universal family in our case:
§
complete, finite and representative families.
We recall the basic definitions of such families over a field k of characteristic p=0 or p>2g+1. For more
details, we refer to [10, 2]:
§
Definition 2.1. Let S be a scheme overk. A curveofgenus g 2 overS is a morphismofschemes C S that
≥ →
is proper and smooth with geometrically irreducible fibers of dimension 1 and genus g.
Definition 2.2. [complete, finite, representative family] Let C be a curve over a scheme S, and assume that
eachgeometricfiberofCcorrespondstoapointofafixedstratumS M . WethengetamorphismfC :S S
g
⊆ →
overk. ThefamilyC Siscomplete (resp. representative)forthestratumS iffC issurjective(resp. bijective)
→
on F-points for every algebraic extension F/k. If the family is complete and all the fibers of fC are finite and
with bounded cardinality, we say that the family is finite. In particular, if a family is finite the dimension of
the family is equal to the dimension of the scheme S.
The family C S is geometrically complete (representative) if it is complete (representative) after extending
→
the scalers to k.
We also remark the following:
Remark 2.3. If a family defined over k is geometrically representative, then it is representative and complete
(see [10, Lemma 2.2]). Moreover, in this case the field of moduli is a field of definition.
2.2. Smooth plane curves of genus 6: the strata Pl(G). Consider the set Pl of all isomorphism
M6 Mg
classes of smooth curves of , admitting a non-singular plane model over k. For simplicity, we call such
g
M g
curves smooth k-plane curves. For a finite group G, we denote by Pl(G) the set of all elements [C] of Pl
Mg Mg
]
suchthatG is isomorphicto asubgroupofthe full automorphismgroupAut(C). We use the notation Pl(G)
Mg
for the subset of Pl(G) whose elements satisfies G=Aut(C).
Mg ∼
]
ThefinitegroupsGforwhich Pl(G)isnon-emptyaredeterminedin[3]. Moreover,geometricallycomplete
M6
families that depend on a fixed injective representation ̺: Aut(C)֒ PGL (k) are given.
3
→
Theorem 2.4 (Badr-Bars). Let k be a fixed separable closure of a field k of characteristic p with p = 0 or
p > 13. The following table gives the complete list of automorphism groups of non-singular plane curves of
degree 5 over k, along with geometrically complete families over k for the associated strata. We denote by L
i,B
a homogeneous polynomial of degree i in the variables X,Y,Z B .
{ }\{ }
A NOTE ON THE STRATIFICATION BY AUTOMORPHISMS OF SMOOTH PLANE CURVES OF GENUS 6 3
Case G ̺(G) F̺(G)(X,Y,Z)
1 GAP(150,5)1 [ζ5X:Y :Z],[X:ζ5Y :Z] X5+Y5+Z5
[X:Z:Y],[Y :Z:X]
2 GAP(39,1) [X:ζ13Y :ζ1130Z],[Y :Z:X] X4Y +Y4Z+Z4X
3 GAP(30,1) [X:ζ15Y :ζ1151Z],[X:Z:Y] X5+Y4Z+YZ4
4 Z/20Z [X:ζ4 Y :ζ5 Z] X5+Y5+XZ4
20 20
5 Z/16Z [X:ζ16Y :ζ1162Z] X5+Y4Z+XZ4
6 Z/10Z [X:ζ120Y :ζ150Z] X5+Y5+XZ4+β2,0X3Z2
7 D10 [X:ζ5Y :ζ52Z],[X:Z:Y] X5+Y5+Z5+β3,1XY2Z2+β4,3X3YZ
8 Z/8Z [X:ζ8Y :ζ84Z] X5+Y4Z+XZ4+β2,0X3Z2
9 S3 [X:ζ3Y :ζ32Z] X5+Y4Z+YZ4+β2,1X3YZ+β3,3X2(cid:0)Z3+Y3(cid:1)+
[X:Z:Y] +β4,2XY2Z2
10 Z/5Z [X:Y :ζ5Z] Z5+L5,Z
11 Z/4Z [X:ζ4Y :ζ42Z] X5+X(cid:0)Z4+Y4(cid:1)+β2,0X3Z2+β3,2X2Y2Z+β5,2Y2Z3
12 Z/4Z [X:Y :ζ4Z] Z4L1,Z+L5,Z
13 Z/3Z [X:ζ3Y :ζ32Z] X5+Y4Z+YZ4+β2,1X3YZ+
+X2(cid:0)β3,0Z3+β3,3Y3(cid:1)+β4,2XY2Z2
14 Z/2Z [X:Y :ζ2Z] Z4L1,Z+Z2L3,Z+L5,Z
15 {1} [X:Y :Z] L5(X,Y,Z)
Table1: Geometricallycompletefamilies overk
The algebraic restrictions on the coefficients, so that each family is smooth, geometrically irreducible, and
has no larger automorphism group are not given.
Remark 2.5. The geometrically complete families over k given in Theorem 2.4 are not necessarily represen-
tative or even complete. For example, the smooth Q-plane curve C : X5+Y5+ 1XZ4+X3Z2 = 0 has two
2
representatives in the above list, which are X5+Y5+XZ4 √2X3Z2 =0 and none of them is defined over Q.
±
3. Representative families for Pl(̺(G))
M6
ThissectionisdevotedtogiverepresentativefamiliesoverkforthedifferentstratainTheorem2.4fork-plane
g
curves of genus 6. As in [10], the first step is computing the normalizers of the groups ρ(G) in Theorem 2.4.
Throughout this section, k has characteristic p=0 or p>13.
3.1. Computation of the normalizers N (k). We start with the families in Theorem 2.4, which are
̺(G)
]
geometricallycomplete overk for eachofthe strata Pl(̺(G)). Isomorphismsbetweentwocurvesinthe same
M6
family, in particular with identical automorphism group̺(G) in PGL (k), are clearly given by 3 3 projective
3
×
matrices in the normalizer N (k) over k.
̺(G)
Notations. By D(k), we mean the group of diagonal matrices in PGL (k), and by T (k), we denote its
3 X
subgroup of elements the shapes [aX :Y :Z] for some a k. Similarly, we define T (k) and T (k).
Y Z
∈
1 0
Second, through the embedding GL (k)֒ PGL (k): A , we can identify GL (k) with its image
2 3 2
→ 7→ 0 A!
in PGL (k). We denote this image by GL (k). In the same way, we define GL (k) and GL (k).
3 2,X 2,Y 2,Z
Third, the following subgroups ofPGL (k) are also considered: S˜ := [X :Z :Y],[Z :X :Y] , G := [X :
3 3 03
h i h
Z :Y],[X :ζ Y :Z] , and G := [X :Z :X], [X :Y :ζ Z] .
3 05 5
i h i
]
Theorem 3.1. Let ̺(G) be one of the automorphism groups given by Theorem 2.4, such that Pl(̺(G)) is
M6
not 0-dimensional. The normalizer N (k) of ̺(G) in PGL (k) is generated by:
̺(G) 3
N 1 (k)=PGL3(k); N̺(Z/5Z)(k)=GL2,Z(k);
• { } •
• N̺(Z/2Z)(k)=GL2,Z(k); • N̺(S3)(k)=hTX(k),G03i;
N̺(Z/3Z)(k)= D(k),S˜3 ; N̺(Z/8Z)(k)= D(k),[Z :Y :X] ;
• h i • h i
• N̺(Z/4Z)(k)=GL2,Z(k); • N̺(D10)(k)=hTX(k), G05i ;
N̺(Z/4Z)(k)= D(k),[Z :Y :X] ; N̺(Z/10Z) =D(k).
• h i •
Proof. TheTheoremisastraightforwardimplicationfromthewell-knownresultthatsaysthattwonon-singular
matricescommuteifandonlyifthereisacommonbasisinwhichbothofthemdiagonalizeoroneisamultipleof
1WeusetheGAPlibrary[6]notations forsmallfinitegroup.
4 E.BADRANDE.LORENZO
theidentity. Asanexample,weprovethecases̺(Z/3Z)and̺(S )simultaneously,andtheremainingsituations
3
are proven in the same way: if φ ∈ N̺(Z/3Z)(k), then φ−1diag(1,ζ3,ζ32)φ = diag(1,ζ3,ζ32), or diag(1,ζ32,ζ3).
Hence, φ is diagonal or a permutation of the variables, up to a re-scaling. In particular, φ is a product
of an element of D(k) and an element of S˜, which gives the situation for ̺(Z/3Z). On the other hand
3
N̺(S3)(k) ⊆ N̺(Z/3Z)(k). Furthermore, if φ ∈ N̺(Z/3Z)(k) such that φ−1[X : Z : Y]φ is of order 2 in ̺(S3) =
[X :ζ Y :ζ2Z], [X :Z :Y] , then
h 3 3 i
φ diag(a,ζr,1),[aX :ζrZ :Y] a k and 0 r 2 .
∈{ 3 3 | ∈ ≤ ≤ }
Rewriting diag(a,ζr,1) as diag(a,1,1)diag(1,ζ ,1)r, and [aX : ζrZ : Y] as diag(a,ζr,1)[X : Z : Y], gives the
3 3 3 3
conclusion for ̺(S ). (cid:3)
3
3.2. Representative families for Pl(̺(G)). We study the existence of representative families over k for
M6
the different strata, i.e families of curves C S whose points are in natural bijection with the subvarieties.
g →
R. Lercier, et al. in [10] explicitly constructed such families for smooth plane quartics in order to determine
unique representatives for the isomorphism classes of smooth plane quartics over finite fields. We also refer to
the second author’s thesis [11, Ch. 2] for such parametrization of smooth plane quartic curves over number
fields.
Theorem 3.2 (Representativefamilies). The following table shows representative families over k for each stra-
]
tumofsmoothk-planecurves ofgenus6,withnon-trivialautomorphism groupoforder =5. For Pl(̺(Z/5Z))
6 M6
a geometrically complete family is shown.
Case G Fρ(G)(X;Y;Z) Parametersrestrictions
1 GAP(150,5) X5+Y5+Z5 -
2 GAP(39,1) X4Y +Y4Z+Z4X -
3 GAP(30,1) X5+Y4Z+YZ4 -
4 Z/20Z X5+Y5+XZ4 -
5 Z/16Z X5+Y4Z+XZ4 -
6 Z/10Z X5+Y5+aXZ4+X3Z2 a=0,1/4
7 D10 X5+aX(Y55++c(ZY55)++ZX5Y)2+ZX23+YbZX3YZ, aa36==6 0,3b36=5−15
8 Z/8Z X5+Y4Z+aXZ4+X3Z2 a6=−0,1/4
9 S3 a3X5+Y4Z+YZ4+a2X3YZ+abX2(cid:0)Z3+Y3(cid:1)+cXY2Z2, 6 n.s
d2X5+Y4Z+YZ4+dX2(cid:0)Z3+Y3(cid:1)+eXY2Z2, n.s
f4X5+Y4Z+YZ4+fXY2Z2 f =0, 3125
10 Z/5Z Z5+XY(X+Y)(X+aY)(X+bY) ab(a 1)(b6 1)−(a 16b)=0,n.b
11 Z/4Z X5+X3Z2+Y2Z3+aX2Y2Z+X(bY4+cZ4), − bc=−0, c=− 76
X5+X2Y2Z+X(dY4+eZ4)+Y2Z3, 6 de=60−20
X5+f(Y2Z3+X(Y4+Z4)) f3= 6(3)3,0
12 Z/4Z X5+c(X3Y2+aX2Y3+bcXY4+c2Y5+Z4Y) 6n.s−,n4.b
X5+s(XY4+Y5+Z4Y)
X5+e(X2Y3+fXY4+eY5+Z4Y),
X5+g(X2Y3+XY4+Z4Y)
13 Z/3Z a3X5+Y4Z+YZ4+a2X3YZ+aX2(cid:0)bY3+cZ3(cid:1)+dXY2Z2,
e2X5+Y4Z+YZ4+X2(cid:0)eY3+fZ3(cid:1)+gXY2Z2, n.s,e=f
h2X5+Y4Z+YZ4+hX2Z3+sXY2Z2, n.s6
t2X5+Y4Z+YZ4+tX2Z3 t=0, 3125
14 Z/2Z Z4Y+Z2(X3+XY2+aY3)+L5,Z 6n.s,n1.0b24
Table2: Completefamiliesoverk
The families that are modified respect to the ones in Table 1 are highlighted. The automorphism groups
remain the same one than in Table 1.
Theparametersrestrictionscomefromavoidingsingularequationsandlargerautomorphismgroups. Weuse
the abbreviations “n.s” and “n.b” for non-singularity and no bigger automorphismgroup, when it is tedious to
write down the restrictions.
Proof. (Cases with G ≇ Z/5Z) Clearly, the zero dimensional strata families are representative over k, since
each represents a single point in the (coarse) moduli space . For the rest of cases, except for the case with
6
M
G Z/5Z, we will use the same techniques used in [10] and [11]. We give a detailed example with the case
≃
G D .
10
≃
A NOTE ON THE STRATIFICATION BY AUTOMORPHISMS OF SMOOTH PLANE CURVES OF GENUS 6 5
We start with the more general family than in Theorem 2.4: aX5+b(Y5+Z5)+cX3YZ +dXY2Z2 = 0.
We know a,b=0 to avoidgetting the singular points (1:0:0) and(0:1:0). So, after rescalingandrenaming
6
variables, we obtain the geometrically complete family: X5+Y5+Z5+aX3YZ+bXY2Z2 =0.
If b = 0, then a = 0 to avoid having a bigger automorphism group, so again, after rescaling and renaming
6
variables, we can work with the family: X5+c(Y5+Z5)+X3YZ = 0. Now, it is easy to check that all the
matricesin N (see Theorem3.1)carryingequationsinthis family intoequationsagaininthe family, leave
ρ(D10)
the equation invariant. In other words, a curve with parameter a in this family is only isomorphic to itself,
which implies that the component in the family is representative over k.
If b = 0, we work with the family X5+a(Y5+Z5)+bX3YZ +XY2Z2 = 0. Again any matrix in N
6 ρ(D10)
leaving the family invariant, fixes each equation.
So, the family given by these two components is representative over k for the stratum with D . (cid:3)
10
Before proving the last part if the Theorem, we need some previous results.
Lemma 3.3. The family : Z5+XY(X +Y)(X +aY)(X +bY) = 0 is a geometrically complete family
(a,b)
C
over k for the stratum of smooth k-plane curves of genus 6, with automorphism group isomorphic to Z/5Z. In
particular, the associated scheme has dimension 2.
Proof. The family Z5+L = 0 is a geometrically complete family over k for the stratum, by Theorem 2.4.
5,Z
Moreover, L should factored in k[X,Y] into pairwise distinct linear factors, otherwise, it will be singular.
5,Z
Now, up to k-isomorphism, we change X and Y, separately, to make one of the factors equals to X = 0 and
another to Y = 0. Second, re-scale X, Y and Z simultaneously to get the factor (X +Y) in the factorization
of L . Now, we can write the family as : Z5 +XY(X +Y)(X +aY)(X +bY) = 0. This family is
5,Z (a,b)
C
]
geometrically complete over k for Pl(̺(Z/5Z)), and finite (we justify this next), so the dimension is 2.
M6
Isomorphisms from the curve to another curve in this family come from transformations
(a,b)
C
α β αt+β
: t ,
γ δ! 7→ γt+δ
sending the set 0,1, ,a,b to a set 0,1, ,c,d . The set of such transformations is a group and it is
{ ∞ } { ∞ } T
isomorphic to S . Moreover,it is generated by
5
a(b 1) 1 a
τ (a,b)=(a, − ), τ (a,b)=( , ), τ (a,b)=(b,a).
1 2 3
b a b b
−
The latest does not properly define a transformation of the curve in the family since switching the parameters
a,b does not change the equation. The first two satisfy the relations τ2 =τ3 =(τ τ )5 =1 generating a group
1 2 1 2
isomorphic to A .
5
The family defined in this way C S is finite and the fibers of fC :S S have cardinality 2 120.
→ →
(cid:3)
The family is defined over k(a,b). We are ideally looking for a family (with two parameters since we
(a,b)
C
know the dimension is two) defined over L=k(a,b) . Hence, we look for the Galois descent from k(a,b) to L.
T
This the idea behind the ad-hoc method used in [10], see the proof of Theorem 3.3.
The following asserts that the analogue of Lu¨roth’s theorem [7, Chapter IV, 2.5.5] holds in dimension 2.
Theorem 3.4. [7, Chapter V, Theorem 6.2 and Remark 6.2.1] Let L/K be a subfield extension of a purely
transcendental extension K(a,b)/K, where K is an algebraically closed field. If K(a,b) is a finite separable
extension of L, then L is also a pure transcendental extension of K.
We can now prove:
]
Claim. Thereexistsarepresentativefamilyoverk,forthestratum Pl(̺(Z/5Z))ofsmoothk-planecurves
M6
of genus 6, and the proof of Theorem 3.2 is finished.
2Another way of checking the cardinality is starting with a generic curve Z5+Q5i=0(X+αiY) and counting the (5·4·3)·2
waysofchoosingthe3rootsgoingto∞,0,1andgettingtheparametersaandb
6 E.BADRANDE.LORENZO
Proof. Consider the family , which is geometrically complete over k (see Lemma 3.3). It is defined over
(a,b)
C
k(a,b), or to be more precise, it is already defined over the subextension k(c,d) where c = a+b and d = ab.
We want to descend it to the invariant subfield L under the action of the symmetric group S on k(a,b), or
5
equivalently of A on k(c,d). In this way, we will get a representative family over k: if such a family exists, it
5
is purely transcendental of dimension 2 by Theorem 3.4, and hence given by 2 parameters.
On the other hand, Weil’s cocycle criterion [14] states that if there exists a family of isomorphisms
φ : σ ,
{ σ C(a,b) →C(a,b)}σ Gal(L/L)
∈
satisfying the cocycle condition φσ1σ2 = φσ1σ1φσ2, then there exist a descend of the curve C(a,b) over L. That
is an isomorphism φ : of smooth curves, satisfying φ σφ 1 = φ and where is defined over
(a,b) − σ
C → C ◦ C
L=k(α,β) .
T
Apriori,thecurve doesnotneedtobeaplanecurve,eveniftheisomorphismsφ are(projective)matrices
σ
C
and areplanecurves,seeProposition2.19in[4]. Butinthiscasewecanconcludethattheisomorphismφis
(a,b)
C
alsodefinedbymatrixsinceallmatricesφ thatwewilltakecanbeseeingnotonlyasmatricesinPGL (k(a,b)),
σ 3
but as matrices in GL (k(a,b)), since all of them will have the shape
2
0
∗ ∗
0 ,
 
∗ ∗
0 0 1
 
 
and then Hilbert’s Theorem 90 implies the existence of a matrix φ satisfying φ σφ 1 = φ . Another proof
− σ
◦
of this fact is [4, Theorem 2.6], that says that a K-plane curve defined over K has a K-plane model when the
degree is coprime to 3, so even if is not given by φ by a plane model over L, we can find one.
C
The Galois group Gal(k(a,b)/L) = S and generated by τ , τ and τ . We define φ = [λ( X +Y) :
5 1 2 3 τ1 −
λ( 1X +Y) : Z] where λ = 5 (a 1) 2(a b) 1a3, φ = [√5bX : √5bY : Z] and φ = [X : Y : Z] and
−a − − − − τ2 b τ3
the family φ by extending with the cocycle condition. This gives a well-defined family of
{ σ}σ Gal(k(a,b)/L) p
isomorphisms sati∈sfying the Weil cocycle condition. Hence, the mentioned descend exists. (cid:3)
Indeed, wecanconstructaexplicitmatrixφ bytakinga sufficiently generalmatrixM,andmakingHilbert’s
Theorem 90 explicit:
φ= φ τM,
τ
τ GalX(k(a,b)/L)
∈
gives an isomorphism satisfying3 φ =φ σφ 1.
σ −
◦
3.3. Detecting representatives in the list. Given a non-singular degree 5 plane curve with known non-
trivial automorphism group, we can find its representative in the classification in Theorem 3.2 by using the
remarks at the end of chapter 2 in the second author’s Thesis [11].
4. Stratification of Pl by automorphism groups
M6
The diagram in next page shows how looks like the stratification by automorphism groups of non-singular
plane quintic curves. The dimensions are computed by looking at the dimension of the schemes S in the
representative families in Theorem 3.2.
We can see in Figure 1 a phenomenon that does not happen for degree 4. We define a final stratum by
automorphism groups to be a stratum that does not properly contain any other stratum. There is a final
stratum in Pl of dimension greater than zero. This may sound odd since we could expect that by adding
M6
conditions in the parameters we would get bigger automorphisms groups. However, we will see that this is a
normal situation for degree higher than 4.
3It is a straightforward computation to check that φσσφ = φ and the meaning of a sufficiently general matrix M is that the
matrixφconstructed inthatwayisinvertible.
A NOTE ON THE STRATIFICATION BY AUTOMORPHISMS OF SMOOTH PLANE CURVES OF GENUS 6 7
Z/16Z Z/20Z GAP(150,5) GAP(30,1) GAP(39,1) dim0
Z/OOOO8✌Z✌✌✌✌✌✌✌✌✌✌✌✌Z✌Z✌//FF1OOOO50XX✵XX✵ZZ✵✵⑥✵✵✵✵⑥✵⑥✵✵⑥✵⑥✵✵⑥⑥⑥⑥⑥⑥>>D1OOOO0 cc●●●●●●●●●●●●●●●●●●●●●● OO OO ddiimm12
Z/4Z gg◆◆◆◆◆◆◆◆◆◆◆✵✵◆✵✵◆✵✵◆✵✵◆✵✵✵✵◆✵✵✵✵◆✵✵✵✵✵✵Z✵✵✵✵/OO2Z⑦✐⑦⑦✐⑦✐⑦✐⑦⑦✐⑦✐⑦✐⑦⑦✐Z??✐/4✐Z✐✐✐✐✐✐✐✐✐S✐3✐✐hh❘✐❘❘✐❘✐❘✐❘✐❘✐❘✐❘✐❘44 Z/3Z dddiiimmm347
1 dim12
{ }
Figure 1. Stratification by automorphisms group
4.1. A canonical justification. The family for the stratum with ρ(G)= [X :iY : Z] has generic compo-
h − i
nent : X5+X3Z2+Y2Z3+aX2Y2Z +bXY4+cXZ4 = 0. This stratum has dimension 3 and it is a
a,b,c
C
final stratum: no restrictionof the parameters give a largerautomorphism group. Let us work for a while with
the renormalizedfamily : X5+AX3Z2+BX2Y2Z+CXY4+XZ4+Y2Z3 =0. Now,it is easy to see
A,B,C
S
that making A=B =C =0 we geta largerautomorphismgroup. For instance, we get the new automorphism
[X :ζ Y :iZ]. However,the plane curve defined by this equation is singular.
8
We willseeanexplanationofthisfinalstratumnothavingdimensionzero. We willregardthefamily
A,B,C
S
in Pl inside a family in that is not final. When we add restrictions here, we get extra symmetries and
M6 M6
the curve is not plane anymore.
Letusstartbycomputingthefamily ofcanonicalmodelsof . Wedefinethefunctionsx=X/Z
A,B,C , ,
K SABC
and y =Y/Z.
4 4
div(x) = 2(0:0:1)+2(0:1:0) (is√4 C :1:0)=2P +2Q R
i
− − −
s=1 i=1
X X
4 4 4
div(y) = (0:0:1) (0:1:0) (is√4 C :1:0)+(1:0: A √A2 4)=P Q R + T .
− − − ± − ±2 − − − i i
s=1 q i=1 i=1
X X X
In order to compute div(dx), we work with the affine form F(x,y,1)=x5+Ax3+Bx2y2+Cxy4+x+y2.
The differential dx is an uniformizer for all points except for P and the T ’s because the tangent space to the
i
curve at these points have equation x α for some α k(A,B,C) (we have used d(x α) = dx). Then, for
− ∈ −
those points, we have to work with the expression
y(2Bx2+4Cxy2+2)
dx= dy.
−5x4+3Ax2+2Bxy2+Cy4+1
We finally get
4 4
div(dx)=P +Q+ R + T ,
i i
i=1 i=1
X X
and a basis of regular differentials is given by
dx xdx
ω = , ω = , ω =dx,
0 1 2
y y
x2dx
ω = , ω =xdx, ω =ydx.
3 4 5
y
Proposition 4.1. The ideal of the canonical model of in P5[ω ,ω ,ω ,ω ,ω ,ω ] is generated by the
A,B,C 0 1 2 3 4 5
S
polynomials
ω ω =ω2, ω ω =ω ω , ω ω =ω2, ω2 =ω ω , ω ω =ω ω , ω ω =ω ω ,
0 3 1 0 4 1 2 0 5 2 4 3 5 1 5 2 4 1 4 2 3
8 E.BADRANDE.LORENZO
ω ω2+Aω3+Bω ω2+Cω ω ω +ω2ω +ω2ω =0,
1 3 1 0 4 2 4 5 0 1 0 5
ω3+Aω ω2+Bω ω ω +Cω ω2+ω2ω +ω ω ω =0,
3 0 3 1 3 5 3 5 0 3 0 1 5
ω ω2+Aω ω ω +Bω ω ω +Cω ω2+ω2ω +ω ω ω =0.
4 3 0 3 4 2 3 5 4 5 0 4 0 2 5
We denote it by .
A,B,C
K
Proof. If ω =0, then the des-homogenizationof this ideal with respect to ω gives
0 0
6
ω5+Aω3+Bω2ω2+Cω ω4+ω4+ω2,
1 1 1 2 1 2 1 2
and we recover the affine curve for Z = 1. If ω = 0, then ω = ω = 0, ω ω = ω2, ω (ω2+ω2) = 0
SA,B,C 0 1 2 3 4 5 3 3 5
and we recover the points at infinity for : Q, R ’s.
A,B,C i
S
Tocheckthatitis non-singular,weneedto seeifthe rankofthe matrixofpartialderivativesofthe previous
generating functions has rank equal to dim(P5) dim( ) = 4 at every point4, that is, the tangent space
A,B,C
− K
has codimension 4. If ω =0, the partial derivatives of the first three equations plus the equation in the second
6
line produce 4 linearly independent vectors in the tangent space. If ω = 0, then ω is non-zero, and, the 3rd,
0 5
4th, and the 6th equations plus the equation in the last line provide 4 linearly independent vectors. Moreover,
this is independent of the choice of the parameters A,B,C. (cid:3)
Corollary 4.2. If we specialize the parameters to A = B = C = 0, then we get a smooth curve of genus 6,
whose full automorphism group has order multiple of 8 and contains
diag(ζ7,ζ5,ζ6,ζ3,ζ4,ζ5).
8 8 8 8 8 8
This curve does not admit a non-singular plane model over k. So the 11th stratum of plane curves of genus 6
in Theorem 3.2 that is final as a plane stratum, is indeed living inside a stratum of smooth curves of genus 6
which is not final.
Proof. We only need to check that the curve is not isomorphic to any one in the family 8th in Theorem
0,0,0
K
2.4withautomorphismgroupZ/8Z. Inordertocheckthatwejustlookatthe automorphism[X :ζ Y : Z]of
8
−
order8inthisfamilyactinginitscanonicalmodel. Wemimicthepreviouscomputationsandwegetthematrix
diag(ζ5,ζ6,ζ ,ζ7,ζ2,ζ5). the group generated by this matrix is clearly non conjugated to the group generated
8 8 8 8 8 8
by the one in the statement of the corollary. So, the curve does not have a smooth plane model. (cid:3)
0,0,0
K
We reinterpret the existence of these special kind of families of plane curves in terms of gr linear series as
d
follows: suppose that is such a family describing a stratum of plane curves, and let D be a divisor in Div( )
C C
that defines a g2 linear series for . In particular, D is of degree d, and the vector space (D) has dimension
d C L
2. For some specializations of the parameters in the canonical family , given by the canonical embedding
K
Φ : ֒ Pg 1(k), one gets more automorphisms. Hence more symmetries in the defining equations, which in
−
C →
turnsproducemoremeromorphicfunctionswithpolesboundedaboveby =Φ(D). Therefore,dim( ( ))>2,
D L D
and we do not get a smooth k-plane model anymore.
In our example the divisor D generating the gd-linear system is D = Q+ 4 R , and ( ) is generated
2 i=1 i L D
by 1,ω1,ω2. For the special choice of the parameters A = B = C = 0, we get D = 5Q and ( ) contains the
ω0 ω0 P L D
linearly independent functions 1,ω1,ω2,ω3, so the (projective) dimension of ( ) is greaterthan 2 and it does
ω0 ω0 ω0 L D
not define a gd linear system anymore. This why, for this choice of the parameters we do not get a smooth
2
plane model anymore, and it is due to the extra symmetries of the curve.
4.2. Non-zero dimensional final strata for higher degree. In this subsection we show examples of non-
zero dimensional final strata in Pl for infinitely many g’s.
Mg
To prove that they are final strata, that is, that there cannot be more automorphisms, we use similar
techniquestotheonesusedin[2,3]. Inparticular,weneedthenextTheorem,whichfollowswhenonecombines
[13, 1-10] or [5, Theorem 4.8], and the proof of [8, Theorem 2.1]):
§
4herewemeandedimensionofKA,B,C asavarietyoverk(A,B,C),thatis1sinceitisacurve
A NOTE ON THE STRATIFICATION BY AUTOMORPHISMS OF SMOOTH PLANE CURVES OF GENUS 6 9
Theorem 4.3 (Mitchell, Harui). Let G be a subgroup of automorphisms of a smooth k-plane curve C of degree
d 4, where k has characteristic p=0. Then one of the following situations holds:
≥
(1) G fixes a line and a point off this line.
(2) G fixes a triangle and neither line nor a point is leaved invariant. In this case, (C,G) is a descendant
of the the Fermat curve F : Xd+Yd+Zd =0 or the Klein curve K : XYd 1+YZd 1+ZXd 1=0.
d d − − −
(3) G is conjugate to a finite primitive subgroup of PGL (k) namely, the Klein group PSL(2,7), the icosa-
3
hedral group A , the alternating group A , the Hessian group Hess with 36,72,216 .
5 6
∗ ∗∈{ }
Remark 4.4. We follow the discussion in [2, 6] to deduce that Theorem 4.3 holds when the characteristic
§
p>2g+1, where g denotes the geometric genus of the curve C.
From now on we assume that k has characteristic p=0 or p>2g+1.
First, we prove:
Theorem 4.5. Let be the family of smooth k-plane curves of an odd degree d 7, defined in the following
C ≥
way:
if d 1 mod 4, then
• ≡
: Xd+Xd 2Y2+X(d 1)/2YZ(d 1)/2+aXYd 1+bXZd 1+cY(d+1)/2Z(d 1)/2 =0
− − − − − −
C
if d 3 mod 4, then
• ≡
: Xd+Xd 2Y2+X(d 1)/2YZ(d 1)/2+aXYd 1+bXZd 1+cX2Y(d 1)/2Z(d 1)/2 =0
− − − − − − −
C
Then Aut( )= [X : Y :ζ Z] , hence it is isomorphic to Z/(d 1)Z.
d 1
C h − − i −
Proof. We first note that η : (X : Y : Z) (X : Y : ζ Z) defines an automorphism of of order d 1,
d 1
7→ − − C −
and also ab = 0 by non-singularity. we will see that Aut( ) is not conjugate to any of the primitive groups in
6 C
PGL (k) mentioned in Theorem 4.3. In particular, it should fix a point, a line, or a triangle.
3
We just handle the situation for the Hessian groups Hess , since A (so does A ) has no elements of order
6 5
∗
d 1 6, and the Klein group PSL(2,7) (the only simple group of order 168) has no elements of order 6 or
− ≥
8 inside. Moreover, any element of Hess has order 1, 2, 3, 4, or 6. Then Hess does not appear as the full
≥ ∗ ∗
automorphism group of , except possibly for d = 7 and = 216. But, following the same discussion that we
C ∗
did for [3, Proposition 12], one easily deduces that there exists no smooth k-plane curve of degree 7, whose
automorphism group is conjugate to Hess .
216
It is well known that Aut(F ) = 6d2 and Aut(K ) = 3(d2 3d+3), see, for instance, [8, Propositions
d d
| | | | −
3.3, 3.5]. Hence, can not be a descendant of the Klein curve K or the Fermat curve F , since, d 1 does
d d
C −
not divide 3(d2 3d+3), or 6d2 (except possibly when d = 7). On the other hand, the automorphisms of
−
F : X7+Y7+Z7 =0havethe shape[X :ζaY :ζbZ], [ζbZ :ζaY :X], [X :ζbZ :ζaY], [ζaY :X :ζbZ], [ζaY :
7 7 7 7 7 7 7 7 7 7
ζbZ :X], or [ζbZ :X :ζaY]. Non of these automorphisms has has order 6, so with d=7 is not a descendant
7 7 7 C
of F , as well. Therefore, Aut( )֒ PGL (k) must fix a line and a point off this line.
7 3
C →
Inparticular,thefixedlineisoneofthereferencelinesB =0withB X,Y,Z ,andthepointisoneofthe
∈{ }
reference points (1:0:0), (0:1:0), or (0:0:1), because η Aut( ) does. Consequently, all automorphisms
∈ C
of areallofthe shapes either [X :vY +wZ :sY +tZ], [vX+wZ :Y :sX+tZ],or[vX+wY :sX+tY :Z].
C
In any case, we always get s=w=0 through the term Xd 2Y2, and all automorphisms of are diagonal,say
−
C
diag(1,λ,µ). We thenobtainλ2 =λµ(d 1)/2 =λd 1 =µd 1 =1,thatis, λ=1, µ=ζ2r orλ= 1, µ=ζ2r+1
for r 0,1,...,d 2 . Consequently, −Aut( ) =−d 1. − d−1 − d−1(cid:3)
∈{ − } | C | −
Corollary 4.6. The family given by Theorem 4.5 describes a final stratum of dimension 3.
C
Second, for even degrees d 10, we construct the family:
≥
Theorem 4.7. For any d=2m 10 with m odd, let be the family of smooth k-plane curves defined by
′
≥ C
Xd+Yd+Zd+XmZm+Xd−2Y2+a2Xd−4Y4+a3Xd−6Y6+...+am 1X2Yd−2+amXm−2Y2Zm =0.
−
Then Aut( )= [X : Y :ζ Z] , in particular, it is cyclic of order d.
′ m
C h − i
10 E.BADRANDE.LORENZO
Proof. Similarly, Aut( ) is not conjugate to any of the finite primitive subgroups of PGL (k), and also is
′ 3 ′
C C
not a descendant of the Klein curve K . So Aut( ) fixes a line and a point off this line, or is a descendant
d ′ ′
C C
of the Fermat curve F . In the first situation, we deduce that it is cyclic of order d, by using the same
d
discussionofTheorem4.5throughthe monomialsXmZm andXd 2Y2. Ontheotherhand,anyautomorphism
−
of F of order d, which is not a homology type5 is conjugate, inside Aut(F ), to either [X : Y : ζ Z],
d d m
−
or [X : ζ Z : Y]. But also both automorphisms are in different conjugacy classes in PGL (k), so, if
m 3
− C
is a descendant of the Fermat curve, through a transformation A PGL (k), then we may assume that
3
∈
A 1[X : Y : ζ Z]A= [X : Y : ζ Z]. In particular, the matrix A must be a diagonal one, say diag(1,λ,µ)
− m m
− −
such that λd = µd = µm = λ2 = 1. Therefore, λ = 1 and µ = ζa for some integer a, and thus A Aut( ).
± m ∈ C
Moreover,Aut( ) Aut(F )= [X : Y :ζ Z] . This completes the proof.
d m
C ∩ h − i
(cid:3)
Corollary 4.8. The family in Theorem 4.7 describes a final stratum of dimension at least m 3.
′
C −
Remark 4.9. Theorem 4.7 and Corollary 4.8 arestill truewhen m is even, butfor simplicity weneedtoimpose
m 8 in order to exclude the primitive groups in PGL (k).
3
≥
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•Eslam Essam EbrahimFaragBadr
DepartamentMatema`tiques,Edif. C,UniversitatAuto`nomade Barcelona,08193Bellaterra, Catalonia,Spain
E-mail address: [email protected]
Departmentof Mathematics,Faculty of Science,CairoUniversity,Giza-Egypt
E-mail address: [email protected]
•Elisa Lorenzo Garc´ıa
IRMAR-Universit´e de Rennes 1,35042Rennes Cedex,France
E-mail address: [email protected]
5i.e.,itfixesonlythreepointsintheplane