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Magaard, Shaska, Shpectorov, and Völklein list smooth Riemann surfaces of genus \( g \leq 10\) with automorphism groups \(G\) satisfying \( \# G > 4(g-1)\). Their list is based on a computer search by Breuer.
They list a genus 6 Riemann surface with automorphism group (120,34) in the GAP library of small groups. This is the symmetric group \(S_5\). The quotient of this surface by its automorphism group has genus zero, and the quotient morphism is branched over three points with ramification indices (2,4,6).
We use Macaulay2 and Magma to compute an equations of this surface. The main tools are the Eichler trace formula, black-box commands in Magma for obtaining matrix generators of a representation of a finite group having a specified character, and a partial computation of a flattening stratification using Gröbner bases in Macaulay2
Magma V2.21-7 Wed Mar 23 2016 17:03:33 on Davids-MacBook-Pro-2 [Seed =
1841840010]
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Type ? for help. Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(120,34);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,6,[2,4,6]);
Set seed to 0.
Character Table of Group G
--------------------------
-----------------------------
Class | 1 2 3 4 5 6 7
Size | 1 10 15 20 30 24 20
Order | 1 2 2 3 4 5 6
-----------------------------
p = 2 1 1 1 4 3 6 4
p = 3 1 2 3 1 5 6 2
p = 5 1 2 3 4 5 1 7
-----------------------------
X.1 + 1 1 1 1 1 1 1
X.2 + 1 -1 1 1 -1 1 -1
X.3 + 4 -2 0 1 0 -1 1
X.4 + 4 2 0 1 0 -1 -1
X.5 + 5 1 1 -1 -1 0 1
X.6 + 5 -1 1 -1 1 0 -1
X.7 + 6 0 -2 0 0 1 0
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 10
Rep (1, 2)
[3] Order 2 Length 15
Rep (1, 2)(3, 4)
[4] Order 3 Length 20
Rep (1, 2, 3)
[5] Order 4 Length 30
Rep (1, 2, 3, 4)
[6] Order 5 Length 24
Rep (1, 2, 3, 4, 5)
[7] Order 6 Length 20
Rep (1, 2, 3)(4, 5)
Surface kernel generators: [
(1, 5)(2, 3),
(1, 3, 5, 4),
(1, 4)(2, 3, 5)
]
Is hyperelliptic? false
Is cyclic trigonal? false
Multiplicities of irreducibles in relevant G-modules:
I_1 =[ 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 0, 1 ]
I_2 =[ 1, 0, 0, 0, 1, 0, 0 ]
S_2 =[ 1, 1, 0, 1, 2, 1, 0 ]
H^0(C,2K)=[ 0, 1, 0, 1, 1, 1, 0 ]
I_3 =[ 0, 0, 1, 1, 1, 0, 3 ]
S_3 =[ 0, 0, 2, 2, 1, 1, 5 ]
H^0(C,3K)=[ 0, 0, 1, 1, 0, 1, 2 ]
I2timesS1=[ 0, 0, 1, 1, 1, 1, 3 ]
Is clearly not generated by quadrics? false
Plane quintic obstruction? true
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 120 and degree 32
[
[ 1 0 -1 0 0 0]
[ 0 0 0 1 1 0]
[ 1 -1 -1 0 -1 0]
[-1 0 0 0 0 0]
[ 0 0 0 -1 0 0]
[ 0 0 1 1 1 1],
[-1 0 1 0 0 0]
[ 0 -1 0 0 0 1]
[ 0 0 1 0 0 0]
[ 1 -1 -1 0 -1 0]
[-1 1 0 -1 0 -1]
[ 0 0 0 0 0 1]
]
Matrix Surface Kernel Generators:
[
[ 0 0 0 0 0 1]
[ 0 0 0 -1 0 0]
[ 0 0 -1 0 0 0]
[ 0 -1 0 0 0 0]
[ 0 0 0 0 -1 0]
[ 1 0 0 0 0 0],
[ 0 0 1 0 0 1]
[-1 0 0 -1 0 0]
[ 0 0 1 1 1 1]
[ 0 0 -1 0 0 0]
[ 1 0 0 0 0 0]
[-1 1 0 -1 0 -1],
[ 0 0 0 0 -1 0]
[ 1 -1 0 1 0 1]
[ 0 1 0 0 0 0]
[ 0 0 0 1 1 0]
[ 0 0 -1 -1 -1 -1]
[ 0 -1 0 0 0 1]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
x_0^2 + x_0*x_2 - x_0*x_3 + 1/2*x_0*x_4 - 1/2*x_0*x_5 + x_1^2 - 1/2*x_1*x_2
+ 1/2*x_1*x_3 - x_1*x_4 + x_1*x_5 + x_2^2 - 1/2*x_2*x_3 - x_2*x_5 +
x_3^2 - x_3*x_4 + x_4^2 - 1/2*x_4*x_5 + x_5^2
]
I2 contains a 5-dimensional subspace of CharacterRow 5
Dimension 10
Multiplicity 2
[
x_0^2 - 2*x_1*x_5 - x_2^2 + 2*x_2*x_3 + x_3^2 - 2*x_3*x_4 - 2*x_3*x_5 +
2*x_4*x_5 - x_5^2,
x_0*x_1 - x_2*x_3 + x_2*x_4 + x_3*x_5 - x_4*x_5,
x_0*x_2 + x_1*x_5 + x_2^2 - 1/2*x_2*x_3 - x_2*x_5 - 1/2*x_3^2 + x_3*x_4 -
1/2*x_4^2 - 1/2*x_4*x_5 + x_5^2,
x_0*x_3 - x_1*x_5 - 1/2*x_2^2 + 1/2*x_2*x_3 + x_2*x_4 - x_3*x_5 - 1/2*x_4^2
+ 1/2*x_4*x_5,
x_0*x_4 - x_2*x_3 + x_2*x_4,
x_0*x_5 + x_2*x_3 - x_3*x_5,
x_1^2 + 2*x_1*x_5 + x_2^2 + x_2*x_3 - 2*x_2*x_4 - 2*x_2*x_5 - x_3^2 +
x_4*x_5 + x_5^2,
x_1*x_2 - x_2*x_4 + x_4*x_5,
x_1*x_3 + x_3*x_5 - x_4*x_5,
x_1*x_4 + x_1*x_5 + x_2^2 - x_2*x_4 - x_2*x_5 - x_3^2 + x_3*x_4 + x_3*x_5 -
1/2*x_4^2 + 1/2*x_5^2
]
The output above shows that the ideal contains quadrics from two isotypical subspaces of \(S_2\).
The first isotypical subspace, which corresponds to the character \( \chi_1\) in the character table above, yields the polynomial shown.
The second isotypical subspace is isomorphic to \(V_{5}^{\oplus 2}\). We saw no obvious way to split this into two smaller blocks. Therefore, we computed the action of the matrix generators on this block, and used Magma to find an isomorphism to \(V_{5}^{\oplus 2}\).
First, we compute the action on this 10-dimensional subspace by recomputing the action on all quadrics and restricting to this 10-dimensional space.
> K<z_120>:=CyclotomicField(120);
> GL6K:=GeneralLinearGroup(6,K);
> rho:=homGL6K | MatrixGens>;
> T:=CharacterTable(G);
> S<x_0,x_1,x_2,x_3,x_4,x_5>:=PolynomialRing(K,6);
> PB,P:=IsotypicalSubspace(G,rho,T,S,2,5);
CharacterRow 5
Dimension 10
Multiplicity 2
> S:=Parent(PB[1]);
> K:=CoefficientRing(S);
> d:=Degree(PB[1]);
> V:=VectorSpace(K,#Rows(P));
> IT:=sub< V | Rows(P)>;
> B:=Basis(IT);
> Symdrho:=[MySymmetrization(rho(G.i),d) : i in [1..#Generators(G)]];
> rho1:=[];
> for k:=1 to #Generators(G) do
for> L:=SequenceToList([SequenceToList(Coordinates(IT,V!Transpose(Symdrho[\
k]*Matrix([ [B[i][j]]: j in [1..Dimension(V)]])))) : i in [1..Dimension(IT)] ]\
);
for> M:=Transpose(Matrix([ [L[i][j]: j in [1..Dimension(IT)]] : i in [1..D\
imension(IT)] ]));
for> rho1[k]:=M;
for> end for;
> GLn1K:=GeneralLinearGroup(#B,K);
> rho1:=[GLn1K!rho1[i] : i in [1..#rho1]];
> rho1;
[
[0 0 0 -1/2 0 0 1 0 0 1]
[-2 0 1/2 -1/2 1 -1 1 1 0 1]
[0 0 0 0 0 0 0 0 0 1]
[-2 0 1 0 0 0 0 0 0 0]
[2 -1 -1/2 -1/2 -1 1 1 0 0 0]
[0 0 1/2 1/2 1 0 1 1 0 0]
[-1 0 0 -1/2 0 0 -1 0 0 -1/2]
[0 0 1/2 1/2 1 -1 1 1 -1 0]
[0 0 0 -1 -1 0 0 0 0 0]
[0 0 0 0 0 0 2 0 0 1],
[0 0 0 -1/2 0 0 1 0 0 1]
[0 1 -1 0 0 -1 -2 -1 0 -1]
[-2 0 1 -1 0 0 2 0 0 2]
[0 0 0 0 0 0 0 0 0 -1]
[0 0 1/2 1/2 1 0 1 1 0 1]
[0 0 -1 0 0 -1 -2 0 0 -1]
[1 0 0 1 0 0 0 0 0 -1/2]
[0 0 -1 0 0 0 -2 -1 0 -1]
[0 0 1/2 1/2 0 1 1 0 1 1]
[0 0 0 -1 0 0 0 0 0 0]
]
Next we compute a \(G\)-module homomorphism from Magma's generators for a five-dimensional irreducible represenation with this character to the 10-dimensional \(G\)-module with action generators shown above.
> rho2:=ActionGenerators(GModule(T[5]));
> rho2:=[ChangeRing(rho2[i],CoefficientRing(rho1[1])): i in [1..#rho2]];
> W1:=GModule(G,rho1);
> W2:=GModule(G,rho2);
> H:=SetToIndexedSet(Generators(AHom(W1,W2)));
> H;
{@
[ 1 1 -1 1 0]
[ 0 0 0 0 0]
[ 0 2 0 2 0]
[ 0 -2 2 -2 -2]
[ 1 0 -1 0 1]
[-1 0 -1 0 1]
[ 1 -2 2 -2 0]
[-1 0 -1 2 -1]
[ 1 -2 1 0 -1]
[ 0 2 -2 0 0],
[ 0 0 0 0 0]
[ 1 -1 0 -1 -1]
[ 0 0 0 0 0]
[ 0 0 0 0 0]
[ 0 0 1 0 0]
[ 0 0 0 0 -1]
[ 0 0 0 0 0]
[ 0 0 0 -1 0]
[ 0 1 0 0 1]
[ 0 0 0 0 0]
@}
> S<x_0,x_1,x_2,x_3,x_4,x_5>:=PolynomialRing(RationalField(),6);
> M:=Transpose(Matrix([Q[2]]));
> M:=ChangeRing(M,S);
> Transpose(ChangeRing(H[1],S))*M;
[x_0^2 + x_0*x_4 - x_0*x_5 + x_1^2 - x_1*x_2 + x_1*x_3 + x_2*x_3 - 2*x_2*x_5 -
2*x_3*x_4 + x_4*x_5]
[x_0^2 + 2*x_0*x_2 - 2*x_0*x_3 - 2*x_1^2 - 2*x_1*x_3 + 2*x_1*x_4 + 2*x_2^2 -
2*x_2*x_3 + 2*x_3*x_4 - x_4^2]
[-x_0^2 + 2*x_0*x_3 - x_0*x_4 - x_0*x_5 + 2*x_1^2 - x_1*x_2 + x_1*x_3 -
2*x_1*x_4 + 2*x_1*x_5 + x_2*x_3 - 2*x_2*x_5 - x_3^2 - x_4*x_5 + 2*x_5^2]
[x_0^2 + 2*x_0*x_2 - 2*x_0*x_3 - 2*x_1^2 + 2*x_1*x_2 - 2*x_1*x_5 - 2*x_2*x_3 +
2*x_2*x_5 + 2*x_3^2 - x_5^2]
[-2*x_0*x_3 + x_0*x_4 + x_0*x_5 - x_1*x_2 - x_1*x_3 + 2*x_1*x_5 + x_2^2 -
x_2*x_3 + x_4^2 - x_4*x_5]
> Transpose(ChangeRing(H[2],S))*M;
[ x_0*x_1 - x_2*x_3 + x_2*x_4 + x_3*x_5 - x_4*x_5]
[ -x_0*x_1 + x_1*x_3 + x_2*x_3 - x_2*x_4]
[ x_0*x_4 - x_2*x_3 + x_2*x_4]
[ -x_0*x_1 - x_1*x_2 + x_2*x_3 - x_3*x_5]
[-x_0*x_1 - x_0*x_5 + x_1*x_3 - x_2*x_4 + x_3*x_5]
We check that the second block of polynomials yields suitable quadrics.
> K:=RationalField();
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> X:=Scheme(P5,[
> x_0^2 + x_0*x_2 - x_0*x_3 + 1/2*x_0*x_4 - 1/2*x_0*x_5 + x_1^2 - 1/2*x_1*x_2 \
+ 1/2*x_1*x_3 - x_1*x_4 + x_1*x_5 + x_2^2 - 1/2*x_2*x_3 - x_2*x_5 + x_3^2 - x_\
3*x_4 + x_4^2 - 1/2*x_4*x_5 + x_5^2,
> x_0*x_1 - x_2*x_3 + x_2*x_4 + x_3*x_5 - x_4*x_5,
> -x_0*x_1 + x_1*x_3 + x_2*x_3 - x_2*x_4,
> x_0*x_4 - x_2*x_3 + x_2*x_4,
> -x_0*x_1 - x_1*x_2 + x_2*x_3 - x_3*x_5,
> -x_0*x_1 - x_0*x_5 + x_1*x_3 - x_2*x_4 + x_3*x_5
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> A:=Matrix([
> [0, 0, 0, 0, 0, 1],
> [0, 0, 0, -1, 0, 0],
> [0, 0, -1, 0, 0, 0],
> [0, -1, 0, 0, 0, 0],
> [0, 0, 0, 0, -1, 0],
> [1, 0, 0, 0, 0, 0]
> ]);
> B:=Matrix([
> [0, 0, 1, 0, 0, 1],
> [-1, 0, 0, -1, 0, 0],
> [0, 0, 1, 1, 1, 1],
> [0, 0, -1, 0, 0, 0],
> [1, 0, 0, 0, 0, 0],
> [-1, 1, 0, -1, 0, -1]
> ]);
> Order(A);
2
> Order(B);
4
> Order( (A*B)^(-1));
6
> GL6K:=GeneralLinearGroup(6,K);
> IdentifyGroup(sub<GL6K | A,B>);
<120, 34>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
x_5
-x_3
-x_2
-x_1
-x_4
x_0
and inverse
x_5
-x_3
-x_2
-x_1
-x_4
x_0
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
-x_1 + x_4 - x_5
x_5
x_0 + x_2 - x_3
-x_1 + x_2 - x_5
x_2
x_0 + x_2 - x_5
and inverse
x_1 - x_4 + x_5
-x_1 - x_3 + x_4
x_4
x_1 - x_2 + x_5
x_0 - x_3 + x_4
x_1