Fordham University

A pencil of genus 6 Riemann surfaces with automorphism group (24,12)

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 1-parameter family of genus 6 Riemann surfaces with automorphism group \( (24,12)\) in the GAP library of small groups, which is the symmetric group \(S_4\). The quotient of any member of this family by its automorphism group has genus zero, and the quotient morphism is branched over four points with ramification indices (2,2,2,4).

We use Magma to compute equations of one member of this family, and give a conjectural description of this family.

Obtaining candidate polynomials in Magma

We use some Magma code originally developed by David Swinarski during a visit to the University of Sydney in June/July 2011.Here is the file autcv10e.txt used below. 
Magma V2.21-7     Mon Mar 28 2016 15:07:40 on Davids-MacBook-Pro-2 [Seed = 
2503623460]

+-------------------------------------------------------------------+
|       This copy of Magma has been made available through a        |
|                   generous initiative of the                      |
|                                                                   |
|                         Simons Foundation                         |
|                                                                   |
| covering U.S. Colleges, Universities, Nonprofit Research entities,|
|               and their students, faculty, and staff              |
+-------------------------------------------------------------------+

Type ? for help.  Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> MatrixGens,MatrixSKG,Q,C:=RunExample(SmallGroup(24,12),6,[2,2,3,4]);
Set seed to 0.


Character Table of Group G
--------------------------


-----------------------
Class |   1  2  3  4  5
Size  |   1  3  6  8  6
Order |   1  2  2  3  4
-----------------------
p  =  2   1  1  1  4  2
p  =  3   1  2  3  1  5
-----------------------
X.1   +   1  1  1  1  1
X.2   +   1  1 -1  1 -1
X.3   +   2  2  0 -1  0
X.4   +   3 -1 -1  0  1
X.5   +   3 -1  1  0 -1



Conjugacy Classes of group G
----------------------------
[1]     Order 1       Length 1      
        Rep Id(G)

[2]     Order 2       Length 3      
        Rep G.3

[3]     Order 2       Length 6      
        Rep G.1

[4]     Order 3       Length 8      
        Rep G.2

[5]     Order 4       Length 6      
        Rep G.1 * G.3


Surface kernel generators:  [ G.3, G.1 * G.3 * G.4, G.2^2 * G.3, G.1 * G.2^2 * 
G.3 * G.4 ]
Is hyperelliptic?  false
Is cyclic trigonal?  false
Multiplicities of irreducibles in relevant G-modules:
I_1      =[ 0, 0, 0, 0, 0 ]
S_1      =[ 0, 0, 0, 1, 1 ]
H^0(C,K) =[ 0, 0, 0, 1, 1 ]
I_2      =[ 1, 0, 1, 0, 1 ]
S_2      =[ 2, 1, 3, 1, 3 ]
H^0(C,2K)=[ 1, 1, 2, 1, 2 ]
I_3      =[ 1, 1, 1, 4, 5 ]
S_3      =[ 2, 2, 2, 8, 8 ]
H^0(C,3K)=[ 1, 1, 1, 4, 3 ]
I2timesS1=[ 1, 1, 2, 5, 5 ]
Is clearly not generated by quadrics? false
Plane quintic obstruction?  false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 24 and degree 8
[
    [-1 -1  0  0  0  0]
    [ 0  1  0  0  0  0]
    [ 0  0 -1  0  0  0]
    [ 0  0  0 -1  0  0]
    [ 0  0  0  0  1  0]
    [ 0  0  0  1  0  1],

    [ 0  1  0  0  0  0]
    [-1 -1  0  0  0  0]
    [-1  0  1  0  0  0]
    [ 0  0  0 -1  0 -1]
    [ 0  0  0  0  1  1]
    [ 0  0  0  1  0  0],

    [-1 -1  1  0  0  0]
    [ 0  0 -1  0  0  0]
    [ 0 -1  0  0  0  0]
    [ 0  0  0  0  1  0]
    [ 0  0  0  1  0  0]
    [ 0  0  0 -1 -1 -1],

    [-1  0  0  0  0  0]
    [ 0  0  1  0  0  0]
    [ 0  1  0  0  0  0]
    [ 0  0  0  0 -1  0]
    [ 0  0  0 -1  0  0]
    [ 0  0  0  0  0 -1]
]
Matrix Surface Kernel Generators:
[
    [-1 -1  1  0  0  0]
    [ 0  0 -1  0  0  0]
    [ 0 -1  0  0  0  0]
    [ 0  0  0  0  1  0]
    [ 0  0  0  1  0  0]
    [ 0  0  0 -1 -1 -1],

    [-1  0  1  0  0  0]
    [ 0 -1  0  0  0  0]
    [ 0  0  1  0  0  0]
    [ 0  0  0  1  0  0]
    [ 0  0  0  0 -1  0]
    [ 0  0  0  0  1  1],

    [ 1  1  0  0  0  0]
    [-1 -1  1  0  0  0]
    [ 1  0  0  0  0  0]
    [ 0  0  0 -1 -1 -1]
    [ 0  0  0  0  0 -1]
    [ 0  0  0  1  0  1],

    [ 0 -1  0  0  0  0]
    [ 1  1 -1  0  0  0]
    [ 1  0  0  0  0  0]
    [ 0  0  0 -1 -1 -1]
    [ 0  0  0  0  0  1]
    [ 0  0  0  1  0  0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 2
Multiplicity 2
[
    x_0^2 - x_0*x_1 + x_0*x_2 + x_1^2 + x_2^2,
    x_3^2 - x_3*x_5 + x_4^2 - x_4*x_5 + x_5^2
]
I2 contains a 2-dimensional subspace of CharacterRow 3
Dimension 6
Multiplicity 3
[
    x_0^2 - x_1^2 - 2*x_1*x_2 - x_2^2,
    x_0*x_1 - x_0*x_2 - 1/2*x_1^2 + x_1*x_2 - 1/2*x_2^2,
    x_0*x_3 + x_0*x_4 - x_1*x_3 - x_1*x_4 + x_1*x_5 + x_2*x_3 + x_2*x_4 - 
        x_2*x_5,
    x_0*x_5 - x_1*x_3 + x_1*x_4 - x_2*x_3 + x_2*x_4,
    x_3^2 - x_3*x_5 + x_4^2 - x_4*x_5 - 1/2*x_5^2,
    x_3*x_4 - 1/2*x_3*x_5 - 1/2*x_4*x_5 + 1/4*x_5^2
]
I2 contains a 3-dimensional subspace of CharacterRow 5
Dimension 9
Multiplicity 3
[
    x_0^2 + 2*x_0*x_2,
    x_0*x_1 + x_0*x_2,
    x_0*x_3 - x_1*x_5,
    x_0*x_4 + x_2*x_5,
    x_1^2 - x_2^2,
    x_1*x_4 + x_2*x_3,
    x_3^2 - x_4^2,
    x_3*x_5 - 1/2*x_5^2,
    x_4*x_5 - 1/2*x_5^2
]

The output above shows that the ideal contains quadrics from three isotypical subspaces of \(S_2\).

The first isotypical subspace, which corresponds to the character \( \chi_1\), yields a polynomial of the form

\[ \begin{array}{c} c_1(x_0^2 - x_0x_1 + x_0x_2 + x_1^2 + x_2^2)+c_2(x_3^2 - x_3x_5 + x_4^2 - x_4x_5 + x_5^2) \end{array} \]

Assume that \(c_1\) is nonzero. After dividing by \(c_1\), we may assume that \(c_1=1\).

The second isotypical subspace corresponds to the character \( \chi_{3}\) in the character table shown above. Note that the matrices generating the action have a block form with blocks of size \(3 \times 3\) and \(3 \times 3\). We therefore let the first two polynomials shown generate one copy of \(V_{3}\) and use the FindParallelBases function to find two more ordered bases so that the action of \(G\) is given by the same matrices relative to these two ordered bases. 

> GL6K:=Parent(MatrixGens[1]);
> MatrixG:=sub<GL6K | MatrixGens>;
> FindParallelBases(MatrixG,[Q[2][1],Q[2][2]],[Q[2][3],Q[2][4]]);
[x_0*x_3 + x_0*x_4 - 1/2*x_0*x_5 - 1/2*x_1*x_3 - 3/2*x_1*x_4 + x_1*x_5 + 
    3/2*x_2*x_3 + 1/2*x_2*x_4 - x_2*x_5]
[1/4*x_0*x_3 + 1/4*x_0*x_4 - 1/2*x_0*x_5 + 1/4*x_1*x_3 - 3/4*x_1*x_4 + 
    1/4*x_1*x_5 + 3/4*x_2*x_3 - 1/4*x_2*x_4 - 1/4*x_2*x_5]
> FindParallelBases(MatrixG,[Q[2][1],Q[2][2]],[Q[2][5],Q[2][6]]);
[                          x_3^2 - 2*x_3*x_4 + x_4^2 - x_5^2]
[1/2*x_3^2 + x_3*x_4 - x_3*x_5 + 1/2*x_4^2 - x_4*x_5]
Thus, this isotypical subspace yields the following polynomials. \[ \begin{array}{l} c_3(x_0^2 - x_1^2 - 2x_1x_2 - x_2^2) + c_4(x_0x_3 + x_0x_4 - 1/2x_0x_5 - 1/2x_1x_3 - 3/2x_1x_4 + x_1x_5 + 3/2x_2x_3 + 1/2x_2x_4 - x_2x_5)+c_5(x_3^2 - 2x_3x_4 + x_4^2 - x_5^2) \\ c_3(x_0x_1 - x_0x_2 - 1/2x_1^2 + x_1x_2 - 1/2x_2^2)+c_4(1/4x_0x_3 + 1/4x_0x_4 - 1/2x_0x_5 + 1/4x_1x_3 - 3/4x_1x_4 + 1/4x_1x_5 + 3/4x_2x_3 - 1/4x_2x_4 - 1/4x_2x_5)+c_5(1/2x_3^2 + x_3x_4 - x_3x_5 + 1/2x_4^2 - x_4x_5) \end{array} \] Assume that \(c_4\) is nonzero. Then after dividing by \(c_4\), we may assume that \(c_4 =1\).

We proceed analogously to obtain three polynomials from the isotypical subspace corresponding to the character \( \chi_5\). 

> FindParallelBases(MatrixG,[Q[3][1],Q[3][2],Q[3][5]],[Q[3][3],Q[3][4],Q[3][6]\
]);
[                                                    x_0*x_3 - x_1*x_5]
[1/2*x_0*x_3 - 1/2*x_0*x_4 - 1/2*x_1*x_5 - 1/2*x_2*x_5]
[            x_0*x_3 - x_0*x_4 + x_1*x_4 - x_1*x_5 + x_2*x_3 - x_2*x_5]
> FindParallelBases(MatrixG,[Q[3][1],Q[3][2],Q[3][5]],[Q[3][7],Q[3][8],Q[3][9]\
]);
[-2*x_4*x_5 + x_5^2]
[ x_3*x_5 - x_4*x_5]
[     x_3^2 - x_4^2]
\[ \begin{array}{l} c_6(x_0^2 + 2x_0x_2 +c_7(x_0x_3 - x_1x_5)) +c_8(-2x_4x_5 + x_5^2) \\ c_6(x_0x_1 + x_0x_2 +c_7(1/2x_0x_3 - 1/2x_0x_4 - 1/2x_1x_5 - 1/2x_2x_5) +c_8(x_3x_5 - x_4x_5) \\ c_6(x_1^2 - x_2^2) +c_7(x_0x_3 - x_0x_4 + x_1x_4 - x_1x_5 + x_2x_3 - x_2x_5) +c_8(x_3^2 - x_4^2) \end{array} \]

Assume that \(c_6\) and \(c_8\) are nonzero. After scaling \(x_3,x_4,x_5\) and dividing, we may assume that \(c_6 = c_8 = 1\).

For generic values of \(c_2,c_3,c_5,c_7\), the intersection of these quadrics is empty. Here is a calculation showing this: 

> K<z_24>:=CyclotomicField(24);
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> c_1:=1;
> c_4:=1;
> c_6:=1;
> c_8:=1;
> c_2:=2;
> c_3:=3;
> c_5:=5;
> c_7:=7;
> X:=Scheme(P5,[
> c_1*(x_0^2 - x_0*x_1 + x_0*x_2 + x_1^2 + x_2^2)+c_2*(x_3^2 - x_3*x_5 + x_4^2\
 - x_4*x_5 + x_5^2),
> c_3*(x_0^2 - x_1^2 - 2*x_1*x_2 - x_2^2)+c_4*(x_0*x_3 + x_0*x_4 - 1/2*x_0*x_5\
 - 1/2*x_1*x_3 - 3/2*x_1*x_4 + x_1*x_5 + 3/2*x_2*x_3 + 1/2*x_2*x_4 - x_2*x_5)+\
c_5*(x_3^2 - 2*x_3*x_4 + x_4^2 - x_5^2),
> c_3*(x_0*x_1 - x_0*x_2 - 1/2*x_1^2 + x_1*x_2 - 1/2*x_2^2)+c_4*(1/4*x_0*x_3 +\
 1/4*x_0*x_4 - 1/2*x_0*x_5 + 1/4*x_1*x_3 - 3/4*x_1*x_4 + 1/4*x_1*x_5 + 3/4*x_2\
*x_3 - 1/4*x_2*x_4 - 1/4*x_2*x_5)+c_5*(1/2*x_3^2 + x_3*x_4 - x_3*x_5 + 1/2*x_4\
^2 - x_4*x_5),
> c_6*(x_0^2 + 2*x_0*x_2) + c_7*(x_0*x_3 - x_1*x_5) + c_8*(-2*x_4*x_5 + x_5^2)\
,
> c_6*(x_0*x_1 + x_0*x_2) + c_7*(1/2*x_0*x_3 - 1/2*x_0*x_4 - 1/2*x_1*x_5 - 1/2\
*x_2*x_5) + c_8*(x_3*x_5 - x_4*x_5),
> c_6*(x_1^2 - x_2^2) + c_7*(x_0*x_3 - x_0*x_4 + x_1*x_4 - x_1*x_5 + x_2*x_3 -\
 x_2*x_5) + c_8*(x_3^2 - x_4^2)
> ]);
> Dimension(X);
-1
Therefore next we turn to Macaulay2 to compute part of a flattening stratification. (We switch software packages because, to the best of our knowledge, Magma will not compute Gröbner bases in a polynomial ring over a polynomial ring.)

Flattening stratification in Macaulay2

We compute the degree two and three elements in a Gröbner basis in Macaulay2 for the ideal generated by the candidate polynomials. 
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : S=QQ[c_2,c_3,c_5,c_7,Degrees=>{0,0,0,0}]

o1 = S

o1 : PolynomialRing

i2 : c_1=1;

i3 : c_4=1;

i4 : c_6=1;

i5 : c_8=1;

i6 : T=S[x_0..x_5];

i7 : I=ideal {
     c_1*(x_0^2 - x_0*x_1 + x_0*x_2 + x_1^2 + x_2^2)+c_2*(x_3^2 - x_3*x_5 + x_4^2 - x_4*x_5 + x_5^2),
     c_3*(x_0^2 - x_1^2 - 2*x_1*x_2 - x_2^2)+c_4*(x_0*x_3 + x_0*x_4 - 1/2*x_0*x_5 - 1/2*x_1*x_3 - 3/2*x_1*x_4 + x_1*x_5 + 3/2*x_2*x_3 + 1/2*x_2*x_4 - x_2*x_5)+c_5*(x_3^2 - 2*x_3*x_4 + x_4^2 - x_5^2),
     c_3*(x_0*x_1 - x_0*x_2 - 1/2*x_1^2 + x_1*x_2 - 1/2*x_2^2)+c_4*(1/4*x_0*x_3 + 1/4*x_0*x_4 - 1/2*x_0*x_5 + 1/4*x_1*x_3 - 3/4*x_1*x_4 + 1/4*x_1*x_5 + 3/4*x_2*x_3 - 1/4*x_2*x_4 - 1/4*x_2*x_5)+c_5*(1/2*x_3^2 + x_3*x_4 - x_3*x_5 + 1/2*x_4^2 - x_4*x_5),
     c_6*(x_0^2 + 2*x_0*x_2) + c_7*(x_0*x_3 - x_1*x_5) + c_8*(-2*x_4*x_5 + x_5^2),
     c_6*(x_0*x_1 + x_0*x_2) + c_7*(1/2*x_0*x_3 - 1/2*x_0*x_4 - 1/2*x_1*x_5 - 1/2*x_2*x_5) + c_8*(x_3*x_5 - x_4*x_5),
     c_6*(x_1^2 - x_2^2) + c_7*(x_0*x_3 - x_0*x_4 + x_1*x_4 - x_1*x_5 + x_2*x_3 - x_2*x_5) + c_8*(x_3^2 - x_4^2)
     };

o7 : Ideal of T

i8 : L=flatten entries gens gb(I,DegreeLimit=>5);

i9 : Lc=unique apply(L, i -> leadCoefficient i);

i10 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
4
12*c_3
2
1
3*c_7^2-4
16*c_2*c_3+8*c_3-8*c_5+6*c_7
72

This output suggests \(c_3 = 0\). We repeat the calculation with this choice: 

Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : S=QQ[c_2,c_5,c_7,Degrees=>{0,0,0}]

o1 = S

o1 : PolynomialRing

i2 : c_1=1;

i3 : c_4=1;

i4 : c_6=1;

i5 : c_8=1;

i6 : c_3=0;

i7 : T=S[x_0..x_5];

i8 : I=ideal {
     c_1*(x_0^2 - x_0*x_1 + x_0*x_2 + x_1^2 + x_2^2)+c_2*(x_3^2 - x_3*x_5 + x_4^2 - x_4*x_5 + x_5^2),
     c_3*(x_0^2 - x_1^2 - 2*x_1*x_2 - x_2^2)+c_4*(x_0*x_3 + x_0*x_4 - 1/2*x_0*x_5 - 1/2*x_1*x_3 - 3/2*x_1*x_4 + x_1*x_5 + 3/2*x_2*x_3 + 1/2*x_2*x_4 - x_2*x_5)+c_5*(x_3^2 - 2*x_3*x_4 + x_4^2 - x_5^2),
     c_3*(x_0*x_1 - x_0*x_2 - 1/2*x_1^2 + x_1*x_2 - 1/2*x_2^2)+c_4*(1/4*x_0*x_3 + 1/4*x_0*x_4 - 1/2*x_0*x_5 + 1/4*x_1*x_3 - 3/4*x_1*x_4 + 1/4*x_1*x_5 + 3/4*x_2*x_3 - 1/4*x_2*x_4 - 1/4*x_2*x_5)+c_5*(1/2*x_3^2 + x_3*x_4 - x_3*x_5 + 1/2*x_4^2 - x_4*x_5),
     c_6*(x_0^2 + 2*x_0*x_2) + c_7*(x_0*x_3 - x_1*x_5) + c_8*(-2*x_4*x_5 + x_5^2),
     c_6*(x_0*x_1 + x_0*x_2) + c_7*(1/2*x_0*x_3 - 1/2*x_0*x_4 - 1/2*x_1*x_5 - 1/2*x_2*x_5) + c_8*(x_3*x_5 - x_4*x_5),
     c_6*(x_1^2 - x_2^2) + c_7*(x_0*x_3 - x_0*x_4 + x_1*x_4 - x_1*x_5 + x_2*x_3 - x_2*x_5) + c_8*(x_3^2 - x_4^2)
     };

o8 : Ideal of T

i9 : L=flatten entries gens gb(I,DegreeLimit=>5);

i10 : Lc=unique apply(L, i -> leadCoefficient i);

i11 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
3
4
12
6
16*c_5^2+24*c_5*c_7-36
96*c_5+24*c_7
16*c_5-12*c_7

This output suggests \(4 c_5-3c_7 = 0\). We repeat the calculation with this choice: 

Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : S=QQ[c_2,c_7,Degrees=>{0,0}]

o1 = S

o1 : PolynomialRing

i2 : c_1=1;

i3 : c_4=1;

i4 : c_6=1;

i5 : c_8=1;

i6 : c_3=0;

i7 : c_5=3/4*c_7;

i8 : T=S[x_0..x_5];

i9 : I=ideal {
     c_1*(x_0^2 - x_0*x_1 + x_0*x_2 + x_1^2 + x_2^2)+c_2*(x_3^2 - x_3*x_5 + x_4^2 - x_4*x_5 + x_5^2),
     c_3*(x_0^2 - x_1^2 - 2*x_1*x_2 - x_2^2)+c_4*(x_0*x_3 + x_0*x_4 - 1/2*x_0*x_5 - 1/2*x_1*x_3 - 3/2*x_1*x_4 + x_1*x_5 + 3/2*x_2*x_3 + 1/2*x_2*x_4 - x_2*x_5)+c_5*(x_3^2 - 2*x_3*x_4 + x_4^2 - x_5^2),
     c_3*(x_0*x_1 - x_0*x_2 - 1/2*x_1^2 + x_1*x_2 - 1/2*x_2^2)+c_4*(1/4*x_0*x_3 + 1/4*x_0*x_4 - 1/2*x_0*x_5 + 1/4*x_1*x_3 - 3/4*x_1*x_4 + 1/4*x_1*x_5 + 3/4*x_2*x_3 - 1/4*x_2*x_4 - 1/4*x_2*x_5)+c_5*(1/2*x_3^2 + x_3*x_4 - x_3*x_5 + 1/2*x_4^2 - x_4*x_5),
     c_6*(x_0^2 + 2*x_0*x_2) + c_7*(x_0*x_3 - x_1*x_5) + c_8*(-2*x_4*x_5 + x_5^2),
     c_6*(x_0*x_1 + x_0*x_2) + c_7*(1/2*x_0*x_3 - 1/2*x_0*x_4 - 1/2*x_1*x_5 - 1/2*x_2*x_5) + c_8*(x_3*x_5 - x_4*x_5),
     c_6*(x_1^2 - x_2^2) + c_7*(x_0*x_3 - x_0*x_4 + x_1*x_4 - x_1*x_5 + x_2*x_3 - x_2*x_5) + c_8*(x_3^2 - x_4^2)
     };

o9 : Ideal of T

i10 : L=flatten entries gens gb(I,DegreeLimit=>5);

i11 : Lc=unique apply(L, i -> leadCoefficient i);

i12 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
2
8
4
12*c_7^2-16
192*c_7
48*c_7^2-64
This suggests \(3c_7^2-4 = 0,\) or \(c_7 = \frac{2}{\sqrt{3}}\). In the field \( \mathbb{Q}[\zeta_{24}]\), we have \(c_7 = \frac{2}{3}( \zeta_{24}^6 -2\zeta_{24}^2)\).

Thus we obtain the following conjectural description of the desired 1-parameter family. For generic values of \(c_2\), consider the scheme \[ \begin{array}{l} x_0^2 - x_0 x_1 + x_0 x_2 + x_1^2 + x_2^2+ c_2(x_3^2 - x_3 x_5 + x_4^2 - x_4 x_5 + x_5^2),\\ x_0 x_3 + x_0 x_4 - 1/2 x_0 x_5 - 1/2 x_1 x_3 - 3/2 x_1 x_4 + x_1 x_5 + 3/2 x_2 x_3 + 1/2 x_2 x_4 - x_2 x_5 + \frac{\sqrt{3}}{2}(x_3^2 - 2 x_3 x_4 + x_4^2 - x_5^2),\\ 1/4 x_0 x_3 + 1/4 x_0 x_4 - 1/2 x_0 x_5 + 1/4 x_1 x_3 - 3/4 x_1 x_4 + 1/4 x_1 x_5 + 3/4 x_2 x_3 - 1/4 x_2 x_4 - 1/4 x_2 x_5+\frac{\sqrt{3}}{2}(1/2 x_3^2 + x_3 x_4 - x_3 x_5 + 1/2 x_4^2 - x_4 x_5),\\ x_0^2 + 2 x_0 x_2+\frac{2}{\sqrt{3}} (x_0 x_3 - x_1 x_5) -2 x_4 x_5 + x_5^2,\\ x_0 x_1 + x_0 x_2+\frac{\sqrt{3}}{2}(1/2 x_0 x_3 - 1/2 x_0 x_4 - 1/2 x_1 x_5 - 1/2 x_2 x_5)+x_3 x_5 - x_4 x_5,\\ x_1^2 - x_2^2+\frac{\sqrt{3}}{2} (x_0 x_3 - x_0 x_4 + x_1 x_4 - x_1 x_5 + x_2 x_3 - x_2 x_5)+x_3^2 - x_4^2 \end{array} \] In the next section we show that at least two different values of \(c_{2}\) yield equations of a smooth curve with the correct automorphisms.

Checking the equations in Magma

We check that for two different values of \(c_{2}\), we obtain a smooth curve with the correct automorphisms. This implies that a general member of the pencil is a smooth curve with the correct automorphisms. However, I have not shown that the two curves studied below are not isomorphic to each other; it is possible that we have described a point in the moduli space \( \mathcal{M}_{6}\) rather than a curve in \( \mathcal{M}_6\).

The value \(c_2 = 1\) yields a singular curve. Therefore, we first study the value \(c_{2}=13\): 

> K<z_24>:=CyclotomicField(24);
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> c_7:=-1/3*(-2*z_24^6 + 4*z_24^2);
> c_5:=1/c_7;
> c_2:=13;
> X:=Scheme(P5,[
> x_0^2 - x_0*x_1 + x_0*x_2 + x_1^2 + x_2^2+c_2*(x_3^2 - x_3*x_5 + x_4^2 - x_4\
*x_5 + x_5^2),
> x_0*x_3 + x_0*x_4 - 1/2*x_0*x_5 - 1/2*x_1*x_3 - 3/2*x_1*x_4 + x_1*x_5 + 3/2*\
x_2*x_3 + 1/2*x_2*x_4 - x_2*x_5+c_5*(x_3^2 - 2*x_3*x_4 + x_4^2 - x_5^2),
> 1/4*x_0*x_3 + 1/4*x_0*x_4 - 1/2*x_0*x_5 + 1/4*x_1*x_3 - 3/4*x_1*x_4 + 1/4*x_\
1*x_5 + 3/4*x_2*x_3 - 1/4*x_2*x_4 - 1/4*x_2*x_5+c_5*(1/2*x_3^2 + x_3*x_4 - x_3\
*x_5 + 1/2*x_4^2 - x_4*x_5),
> x_0^2 + 2*x_0*x_2+ c_7*(x_0*x_3 - x_1*x_5) -2*x_4*x_5 + x_5^2,
> x_0*x_1 + x_0*x_2 + c_7*(1/2*x_0*x_3 - 1/2*x_0*x_4 - 1/2*x_1*x_5 - 1/2*x_2*x\
_5) + x_3*x_5 - x_4*x_5,
> x_1^2 - x_2^2 + c_7*(x_0*x_3 - x_0*x_4 + x_1*x_4 - x_1*x_5 + x_2*x_3 - x_2*x\
_5) + x_3^2 - x_4^2
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> A:=Matrix([
> [-1, -1, 1, 0, 0, 0],
> [0, 0, -1, 0, 0, 0],
> [0, -1, 0, 0, 0, 0],
> [0, 0, 0, 0, 1, 0],
> [0, 0, 0, 1, 0, 0],
> [0, 0, 0, -1, -1, -1]
> ]);
> B:=Matrix([
> [-1, 0, 1, 0, 0, 0],
> [0, -1, 0, 0, 0, 0],
> [0, 0, 1, 0, 0, 0],
> [0, 0, 0, 1, 0, 0],
> [0, 0, 0, 0, -1, 0],
> [0, 0, 0, 0, 1, 1]
> ]);
> C:=Matrix([
> [1, 1, 0, 0, 0, 0],
> [-1, -1, 1, 0, 0, 0],
> [1, 0, 0, 0, 0, 0],
> [0, 0, 0, -1, -1, -1],
> [0, 0, 0, 0, 0, -1],
> [0, 0, 0, 1, 0, 1]
> ]);
> Order(A);
2
> Order(B);
2
> Order(C);
3
> Order( (A*B*C)^(-1));
4
> GL6K:=GeneralLinearGroup(6,K);
> IdentifyGroup(sub);
<24, 12>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_0 - x_2
x_0 - x_1
x_4 - x_5
x_3 - x_5
-x_5
and inverse
-x_0
-x_0 - x_2
x_0 - x_1
x_4 - x_5
x_3 - x_5
-x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
x_0 + x_2
x_3
-x_4 + x_5
x_5
and inverse
-x_0
-x_1
x_0 + x_2
x_3
-x_4 + x_5
x_5
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
x_0 - x_1 + x_2
x_0 - x_1
x_1
-x_3 + x_5
-x_3
-x_3 - x_4 + x_5
and inverse
x_1 + x_2
x_2
x_0 - x_1
-x_4
x_3 - x_5
x_3 - x_4
Next, we check the value \(c_{2}=-19+\zeta_{24}^7\): 
> c_2:=-19+z_24^7;
> X:=Scheme(P5,[
> x_0^2 - x_0*x_1 + x_0*x_2 + x_1^2 + x_2^2+c_2*(x_3^2 - x_3*x_5 + x_4^2 - x_4\
*x_5 + x_5^2),
> x_0*x_3 + x_0*x_4 - 1/2*x_0*x_5 - 1/2*x_1*x_3 - 3/2*x_1*x_4 + x_1*x_5 + 3/2*\
x_2*x_3 + 1/2*x_2*x_4 - x_2*x_5+c_5*(x_3^2 - 2*x_3*x_4 + x_4^2 - x_5^2),
> 1/4*x_0*x_3 + 1/4*x_0*x_4 - 1/2*x_0*x_5 + 1/4*x_1*x_3 - 3/4*x_1*x_4 + 1/4*x_\
1*x_5 + 3/4*x_2*x_3 - 1/4*x_2*x_4 - 1/4*x_2*x_5+c_5*(1/2*x_3^2 + x_3*x_4 - x_3\
*x_5 + 1/2*x_4^2 - x_4*x_5),
> x_0^2 + 2*x_0*x_2+ c_7*(x_0*x_3 - x_1*x_5) -2*x_4*x_5 + x_5^2,
> x_0*x_1 + x_0*x_2 + c_7*(1/2*x_0*x_3 - 1/2*x_0*x_4 - 1/2*x_1*x_5 - 1/2*x_2*x\
_5) + x_3*x_5 - x_4*x_5,
> x_1^2 - x_2^2 + c_7*(x_0*x_3 - x_0*x_4 + x_1*x_4 - x_1*x_5 + x_2*x_3 - x_2*x\
_5) + x_3^2 - x_4^2
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_0 - x_2
x_0 - x_1
x_4 - x_5
x_3 - x_5
-x_5
and inverse
-x_0
-x_0 - x_2
x_0 - x_1
x_4 - x_5
x_3 - x_5
-x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
x_0 + x_2
x_3
-x_4 + x_5
x_5
and inverse
-x_0
-x_1
x_0 + x_2
x_3
-x_4 + x_5
x_5
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
x_0 - x_1 + x_2
x_0 - x_1
x_1
-x_3 + x_5
-x_3
-x_3 - x_4 + x_5
and inverse
x_1 + x_2
x_2
x_0 - x_1
-x_4
x_3 - x_5
x_3 - x_4