Finding equations of a 1-parameter family of genus 7 Riemann surfaces with 48 automorphisms

Magaard, Shaska, Shpectorov, and Völklein give tables of smooth Riemann surfaces of genus \( 3 \leq g \leq 10\) with automorphism groups \(G\) satisfying \( \# G > 4(g-1)\). Their list is based on a computer search by Thomas Breuer.

They list a 1-parameter family of genus 7 Riemann surfaces with automorphism group (48,48) in the GAP library of small groups. The quotient surface of any member of this family 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 developed by David Swinarski during a visit to the University of Sydney in June/July 2011. Here is the file autcv10c.txt used below.

Magma V2.21-10    Sat Mar 12 2016 14:51:50 on ace-math01 [Seed = 1291741551]
Type ? for help.  Type -D to quit.
> load "autcv10c.txt";
Loading "autcv10c.txt"
> G:=SmallGroup(48,48);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,7,[2,2,2,4]);
Set seed to 0.


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


--------------------------------------
Class |   1  2  3  4  5  6  7  8  9 10
Size  |   1  1  3  3  6  6  8  6  6  8
Order |   1  2  2  2  2  2  3  4  4  6
--------------------------------------
p  =  2   1  1  1  1  1  1  7  3  3  7
p  =  3   1  2  3  4  5  6  1  8  9  2
--------------------------------------
X.1   +   1  1  1  1  1  1  1  1  1  1
X.2   +   1 -1  1 -1  1 -1  1 -1  1 -1
X.3   +   1  1  1  1 -1 -1  1 -1 -1  1
X.4   +   1 -1  1 -1 -1  1  1  1 -1 -1
X.5   +   2 -2  2 -2  0  0 -1  0  0  1
X.6   +   2  2  2  2  0  0 -1  0  0 -1
X.7   +   3  3 -1 -1  1  1  0 -1 -1  0
X.8   +   3 -3 -1  1 -1  1  0 -1  1  0
X.9   +   3 -3 -1  1  1 -1  0  1 -1  0
X.10  +   3  3 -1 -1 -1 -1  0  1  1  0



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

[2]     Order 2       Length 1      
        Rep G.2

[3]     Order 2       Length 3      
        Rep G.4

[4]     Order 2       Length 3      
        Rep G.2 * G.4

[5]     Order 2       Length 6      
        Rep G.1

[6]     Order 2       Length 6      
        Rep G.1 * G.2

[7]     Order 3       Length 8      
        Rep G.3

[8]     Order 4       Length 6      
        Rep G.1 * G.2 * G.4

[9]     Order 4       Length 6      
        Rep G.1 * G.4

[10]    Order 6       Length 8      
        Rep G.2 * G.3


Surface kernel generators:  [ G.1 * G.2 * G.3, G.1 * G.3 * G.5, G.1 * G.2 * 
G.3^2, G.1 * G.3^2 * G.5 ]
Is hyperelliptic?  false
Is cyclic trigonal?  false
Multiplicities of irreducibles in relevant G-modules:
I_1      =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1      =[ 0, 0, 1, 0, 0, 0, 0, 0, 1, 1 ]
H^0(C,K) =[ 0, 0, 1, 0, 0, 0, 0, 0, 1, 1 ]
I_2      =[ 2, 0, 0, 0, 0, 1, 1, 1, 0, 0 ]
S_2      =[ 3, 0, 0, 1, 1, 2, 3, 2, 1, 0 ]
H^0(C,2K)=[ 1, 0, 0, 1, 1, 1, 2, 1, 1, 0 ]
I_3      =[ 0, 2, 4, 0, 1, 2, 2, 2, 5, 5 ]
S_3      =[ 0, 3, 5, 0, 2, 3, 3, 4, 7, 8 ]
H^0(C,3K)=[ 0, 1, 1, 0, 1, 1, 1, 2, 2, 3 ]
I2timesS1=[ 0, 2, 4, 0, 2, 3, 3, 3, 6, 6 ]
Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 48 and degree 16
[
    [-1  0  0  0  0  0  0]
    [ 0  1  1  1  0  0  0]
    [ 0  0  0 -1  0  0  0]
    [ 0  0 -1  0  0  0  0]
    [ 0  0  0  0 -1  0  1]
    [ 0  0  0  0  0 -1 -1]
    [ 0  0  0  0  0  0  1],

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

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

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

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

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

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

    [-1  0  0  0  0  0  0]
    [ 0  0  0  1  0  0  0]
    [ 0  0 -1 -1  0  0  0]
    [ 0 -1  0  0  0  0  0]
    [ 0  0  0  0  0  1  0]
    [ 0  0  0  0 -1  0  0]
    [ 0  0  0  0  0  1  1]
]
Finding quadrics:
I2 contains a 2-dimensional subspace of CharacterRow 1
Dimension 3
Multiplicity 3
[
    x_0^2,
    x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2,
    x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2
]
I2 contains a 2-dimensional subspace of CharacterRow 6
Dimension 4
Multiplicity 2
[
    x_1^2 - 2*x_1*x_3 - x_2^2 + x_3^2,
    x_1*x_2 - 2*x_1*x_3 - 1/2*x_2^2 + x_2*x_3,
    x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 - 1/2*x_6^2,
    x_4*x_5 - 1/2*x_4*x_6 + 1/2*x_5*x_6 - 1/4*x_6^2
]
I2 contains a 3-dimensional subspace of CharacterRow 7
Dimension 9
Multiplicity 3
[
    x_0*x_4,
    x_0*x_5,
    x_0*x_6,
    x_1^2 - x_3^2,
    x_1*x_2 - x_2*x_3,
    x_2^2 - 2*x_2*x_3,
    x_4^2 - x_5^2,
    x_4*x_6 + 1/2*x_6^2,
    x_5*x_6 - 1/2*x_6^2
]
I2 contains a 3-dimensional subspace of CharacterRow 8
Dimension 6
Multiplicity 2
[
    x_0*x_1,
    x_0*x_2,
    x_0*x_3,
    x_1*x_5 - x_3*x_4 - x_3*x_6,
    x_1*x_6 + x_2*x_5 - x_2*x_6,
    x_2*x_4 + x_3*x_6
]

The output above shows that the ideal contains quadrics from four isotypical subspaces of \(S_2\). Note that the matrices and equations shown above are over the rational numbers.

The first isotypical subspace, which corresponds to the character \( \chi_1\) in the character table shown above, yields two polynomial of the form


c_1*(x_0^2)+c_2*(x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2)+c_3*(x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2),
c_4*(x_0^2)+c_5*(x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2)+c_6*(x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2),

Assume that \( (c_1,c_2)\) and \( (c_4,c_5)\) are linearly independent. Then after row reducing we may assume that \( (c_1,c_2) = (1,0)\) and \( (c_4,c_5) = (0,1)\). Assume that \(c_3\) is nonzero. Then after rescaling \(x_0\) we may assume that \(c_3=c_1 =1\). Assume that \(c_6\) is nonzero. Then after rescaling \(x_1,x_2,x_3\) we may assume that \(c_5=c_6 =1\).

The second isotypical subspace corresponds to the character \( \chi_6\) in the character table shown above. Note that the matrix surface kernel generators have a block diagonal form with blocks of size \(1 \times 1\), \(3 \times 3\), and \(3 \times 3\). We therefore let the first two polynomials shown here generate one copy of \(V_{6}\) and use the FindParallelBases function to find an ordered bases of the span of the last two polynomials such that the action of \(G\) is given by the same matrices relative to both ordered bases.


> GL7K:=Parent(MatrixGens[1]);
> MatrixG:=sub<GL7K | MatrixGens>;
> FindParallelBases(MatrixG,[Q[2][1],Q[2][2]],[Q[2][3],Q[2][4]]);
[x_4^2 - 2*x_4*x_5 + 2*x_4*x_6 + x_5^2 - 2*x_5*x_6]
[   -2*x_4*x_5 + x_4*x_6 - x_5*x_6 + 1/2*x_6^2]

This yields polynomials of the form


c_7*(x_1^2 - 2*x_1*x_3 - x_2^2 + x_3^2)+c_8*(x_4^2 - 2*x_4*x_5 + 2*x_4*x_6 + x_5^2 - 2*x_5*x_6),
c_7*(x_1*x_2 - 2*x_1*x_3 - 1/2*x_2^2 + x_2*x_3) + c_8*(-2*x_4*x_5 + x_4*x_6 - x_5*x_6 + 1/2*x_6^2),

Assume that \( c_7 \) is nonzero. Then after dividing by \(c_7\), we may assume that \(c_7 = 1\)

The third isotypical subspace corresponds to the character \( \chi_7\). Again using the block diagonal form of the matrix surface kernel generators, we let \( \operatorname{Span} ( x_1^2 - x_3^2, x_1 x_2 - x_2 x_3, x_2^2 - 2 x_2 x_3)\) be one copy of \( V_7\), and use the FindParallelBases function to find ordered bases of such that the action of \(G\) is given by the same matrices relative to all three ordered bases.


> FindParallelBases(MatrixG,[Q[3][4],Q[3][5],Q[3][6]],[Q[3][1],Q[3][2],Q[3][3]\
]);
[x_0*x_4 + x_0*x_5 + x_0*x_6]
[          x_0*x_4 + x_0*x_5]
[        2*x_0*x_4 + x_0*x_6]
> FindParallelBases(MatrixG,[Q[3][4],Q[3][5],Q[3][6]],[Q[3][7],Q[3][8],Q[3][9]\
]);
[x_4^2 + 2*x_4*x_6 - x_5^2 + x_6^2]
[        x_4*x_6 - x_5*x_6 + x_6^2]
[                2*x_4*x_6 + x_6^2]

This yields candidate polynomials of the form


c_9*(x_0*x_4 + x_0*x_5 + x_0*x_6)+c_10*(x_1^2 - x_3^2)+c_11*(x_4^2 + 2*x_4*x_6 - x_5^2 + x_6^2),
c_9*(x_0*x_4 + x_0*x_5)+c_10*(x_1*x_2 - x_2*x_3)+c_11*(x_4*x_6 - x_5*x_6 + x_6^2),
c_9*(2*x_0*x_4 + x_0*x_6)+c_10*(x_2^2 - 2*x_2*x_3)+c_11*(2*x_4*x_6 + x_6^2),

Assume that \(c_9\) is nonzero. Then after dividing by \(c_9\), we may assume that \(c_9 = 1\).

The fourth isotypical subspace corresponds to the character \( \chi_8\) in the character table shown above. Again using the block diagonal form of the matrix surface kernel generators, we let \( \operatorname{Span} \langle x_1 x_5 - x_3 x_4 - x_3 x_6, x_1 x_6 + x_2 x_5 - x_2 x_6, x_2 x_4 + x_3 x_6 \rangle\) be one copy of \( V_8\), and use the FindParallelBases function to find an ordered bases of the span of the first three polynomials such that the action of \(G\) is given by the same matrices relative to all three ordered bases.


> FindParallelBases(MatrixG,[Q[4][4],Q[4][5],Q[4][6]],[Q[4][1],Q[4][2],Q[4][3]\
]);
[  x_0*x_1 + x_0*x_3]
[2*x_0*x_1 - x_0*x_2]
[x_0*x_2 - 2*x_0*x_3]

This yields the candidate polynomials


c_12*(x_0*x_1 + x_0*x_3)+c_13*(x_1*x_5 - x_3*x_4 - x_3*x_6),
c_12*(2*x_0*x_1 - x_0*x_2)+c_13*(x_1*x_6 + x_2*x_5 - x_2*x_6),
c_12*(x_0*x_2 - 2*x_0*x_3)+c_13*(x_2*x_4 + x_3*x_6)

Assume that \( c_{12} \) is nonzero. Then after dividing by \(c_{12}\) , we may assume that \( c_{12} =1\).

We collect all the polynomials listed above and the assumptions we made about the coefficients \(c_i\) to obtain the following candidate polynomials: \[ \begin{array}{l} x_0^2 + x_4^2 + x_4 x_6 + x_5^2 - x_5 x_6 + x_6^2, \\ x_1^2 - x_1 x_2 + x_2^2 - x_2 x_3 + x_3^2+x_4^2 + x_4 x_6 + x_5^2 - x_5 x_6 + x_6^2, \\ x_1^2 - 2 x_1 x_3 - x_2^2 + x_3^2+c_8 (x_4^2 - 2 x_4 x_5 + 2 x_4 x_6 + x_5^2 - 2 x_5 x_6), \\ x_1 x_2 - 2 x_1 x_3 - 1/2 x_2^2 + x_2 x_3 + c_8 (-2 x_4 x_5 + x_4 x_6 - x_5 x_6 + 1/2 x_6^2), \\ x_0 x_4 + x_0 x_5 + x_0 x_6+c_{10} (x_1^2 - x_3^2)+c_{11} (x_4^2 + 2 x_4 x_6 - x_5^2 + x_6^2), \\ x_0 x_4 + x_0 x_5+c_{10} (x_1 x_2 - x_2 x_3)+c_{11} (x_4 x_6 - x_5 x_6 + x_6^2), \\ 2 x_0 x_4 + x_0 x_6+c_{10} (x_2^2 - 2 x_2 x_3)+c_{11} (2 x_4 x_6 + x_6^2), \\ x_0 x_1 + x_0 x_3+c_{13} (x_1 x_5 - x_3 x_4 - x_3 x_6), \\ 2 x_0 x_1 - x_0 x_2+c_{13} (x_1 x_6 + x_2 x_5 - x_2 x_6), \\ x_0 x_2 - 2 x_0 x_3+c_{13} (x_2 x_4 + x_3 x_6) \end{array} \] For generic values of \(c_8,c_{10},c_{11},c_{13}\), the intersection of these 10 quadrics in \(\mathbb{P}^6\) is empty. Here is an example showing this:

> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(RationalField(),6);
> c_8:=1;
> c_10:=1;
> c_11:=0;
> c_13:=1;
> X:=Scheme(P6,[
> x_0^2 + x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2,
> x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2+x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 \
+ x_6^2,
> x_1^2 - 2*x_1*x_3 - x_2^2 + x_3^2+c_8*(x_4^2 - 2*x_4*x_5 + 2*x_4*x_6 + x_5^2\
 - 2*x_5*x_6),
> x_1*x_2 - 2*x_1*x_3 - 1/2*x_2^2 + x_2*x_3 + c_8*(-2*x_4*x_5 + x_4*x_6 - x_5*\
x_6 + 1/2*x_6^2),
> x_0*x_4 + x_0*x_5 + x_0*x_6+c_10*(x_1^2 - x_3^2)+c_11*(x_4^2 + 2*x_4*x_6 - x\
_5^2 + x_6^2),
> x_0*x_4 + x_0*x_5+c_10*(x_1*x_2 - x_2*x_3)+c_11*(x_4*x_6 - x_5*x_6 + x_6^2), 
> 2*x_0*x_4 + x_0*x_6+c_10*(x_2^2 - 2*x_2*x_3)+c_11*(2*x_4*x_6 + x_6^2),
> x_0*x_1 + x_0*x_3+c_13*(x_1*x_5 - x_3*x_4 - x_3*x_6),
> 2*x_0*x_1 - x_0*x_2+c_13*(x_1*x_6 + x_2*x_5 - x_2*x_6),
> x_0*x_2 - 2*x_0*x_3+c_13*(x_2*x_4 + x_3*x_6)
> ]);
> 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_8,c_10,c_11,c_13,Degrees=>{0,0,0,0}]

o1 = S

o1 : PolynomialRing

i2 : T=S[x_0..x_6];

i3 : I=ideal({
     x_0^2 + x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2,
     x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2+x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2,
     x_1^2 - 2*x_1*x_3 - x_2^2 + x_3^2+c_8*(x_4^2 - 2*x_4*x_5 + 2*x_4*x_6 + x_5^2 - 2*x_5*x_6),
     x_1*x_2 - 2*x_1*x_3 - 1/2*x_2^2 + x_2*x_3 + c_8*(-2*x_4*x_5 + x_4*x_6 - x_5*x_6 + 1/2*x_6^2),
     x_0*x_4 + x_0*x_5 + x_0*x_6+c_10*(x_1^2 - x_3^2)+c_11*(x_4^2 + 2*x_4*x_6 - x_5^2 + x_6^2),
     x_0*x_4 + x_0*x_5+c_10*(x_1*x_2 - x_2*x_3)+c_11*(x_4*x_6 - x_5*x_6 + x_6^2),
     2*x_0*x_4 + x_0*x_6+c_10*(x_2^2 - 2*x_2*x_3)+c_11*(2*x_4*x_6 + x_6^2),
     x_0*x_1 + x_0*x_3+c_13*(x_1*x_5 - x_3*x_4 - x_3*x_6),
     2*x_0*x_1 - x_0*x_2+c_13*(x_1*x_6 + x_2*x_5 - x_2*x_6),
     x_0*x_2 - 2*x_0*x_3+c_13*(x_2*x_4 + x_3*x_6)
     });

o3 : Ideal of T

i4 : L=flatten entries gens gb(I,DegreeLimit=>8);

i5 : tally apply(L, i -> degree i)

o5 = Tally{{2, 0} => 10}
           {3, 0} => 43
           {4, 0} => 66
           {5, 0} => 14

o5 : Tally

i6 : L3=select(L, i -> degree i == {3,0});

i7 : L3c=unique apply(L3, i -> leadCoefficient i);

i8 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
4*c_8*c_10*c_13-4*c_11*c_13
2*c_8*c_10*c_11*c_13-c_11*c_13^2
c_8*c_10*c_13^2+2*c_10*c_13^2-3*c_11*c_13^2-6*c_11-3*c_13
4*c_11-2*c_13
6*c_13^2
4*c_11*c_13^2-2*c_13^3+8*c_11-4*c_13
8*c_8*c_10*c_11-4*c_8*c_10*c_13+16*c_10*c_11-8*c_10*c_13+12*c_11*c_13-6*c_13^2
24*c_10*c_11-12*c_10*c_13+24*c_11*c_13-12*c_13^2
48*c_8*c_11-24*c_8*c_13+24*c_11-12*c_13
4*c_10*c_13-4*c_11*c_13-12
12*c_8*c_10-12*c_11
4*c_8*c_11*c_13-4*c_11*c_13+12*c_8
16*c_8*c_10^2*c_13^2-16*c_8*c_10*c_11*c_13^2-64*c_8^2*c_10^2-8*c_10*c_13^3+8*c_11*c_13^3+16*c_8*c_10*c_13+8*c_13^2
12*c_11*c_13-6*c_13^2
4*c_8*c_10-2*c_13
12*c_10*c_13^2-6*c_13^3-36*c_13
2*c_8*c_13^3-2*c_13^3+12*c_8*c_13
8*c_10*c_13^2-8*c_11*c_13^2+32*c_8*c_10-40*c_13
8*c_8*c_11*c_13^2-32*c_8^2*c_10-8*c_11*c_13^2+32*c_8*c_10+40*c_8*c_13-16*c_13
32*c_8^2*c_10^2-32*c_8*c_10^2-40*c_8*c_10*c_13+16*c_10*c_13+24*c_11*c_13
8*c_8*c_10-4*c_13
4*c_13
6*c_13
2*c_13^2+4
16*c_10*c_11+16*c_11^2-8*c_10*c_13+8
2*c_8*c_10+4*c_10+6*c_11
16*c_8*c_11^2-16*c_11^2+24*c_11*c_13+8*c_8+16
2*c_10+2*c_13
4*c_8+2
8
12*c_11-6*c_13
12*c_13
72
144

After much trial and error, I conjecture that the desired 1-parameter family arises by setting the following leading coefficients to zero:


4*c_11-2*c_13
4*c_10*c_13-4*c_11*c_13-12
12*c_8*c_10-12*c_11

Thus we may describe the 1-parameter family as follows. Fix any value of \(c_{13}\). Then let \[ c_{11} = \frac{1}{2}c_{13}, \] \[ c_{10} = \frac{c_{13}^2+6}{2c_{13}}, \] and \[ c_8=\frac{1}{c_{13}^2+6}. \] This yields \[ \begin{array}{ll} x_0^2 + x_4^2 + x_4 x_6 + x_5^2 - x_5 x_6 + x_6^2,\\ x_1^2 - x_1 x_2 + x_2^2 - x_2 x_3 + x_3^2+x_4^2 + x_4 x_6 + x_5^2 - x_5 x_6 + x_6^2,\\ (c_{13}^2+6)(x_1^2 - 2 x_1 x_3 - x_2^2 + x_3^2)+x_4^2 - 2 x_4 x_5 + 2 x_4 x_6 + x_5^2 - 2 x_5 x_6,\\ (c_{13}^2+6)(x_1 x_2 - 2 x_1 x_3 - 1/2 x_2^2 + x_2 x_3) -2 x_4 x_5 + x_4 x_6 - x_5 x_6 + 1/2 x_6^2,\\ c_{13} (x_0 x_4 + x_0 x_5 + x_0 x_6)+\frac{1}{2}(c_{13}^2+6) (x_1^2 - x_3^2)+\frac{1}{2}c_{13}^2 (x_4^2 + 2 x_4 x_6 - x_5^2 + x_6^2),\\ c_{13} (x_0 x_4 + x_0 x_5)+\frac{1}{2}(c_{13}^2+6) (x_1 x_2 - x_2 x_3)+\frac{1}{2}c_{13}^2 (x_4 x_6 - x_5 x_6 + x_6^2),\\ c_{13} (2 x_0 x_4 + x_0 x_6)+\frac{1}{2}(c_{13}^2+6) (x_2^2 - 2 x_2 x_3)+\frac{1}{2}c_{13}^2 (2 x_4 x_6 + x_6^2),\\ x_0 x_1 + x_0 x_3+c_{13} (x_1 x_5 - x_3 x_4 - x_3 x_6),\\ 2 x_0 x_1 - x_0 x_2+c_{13} (x_1 x_6 + x_2 x_5 - x_2 x_6),\\ x_0 x_2 - 2 x_0 x_3+c_{13} (x_2 x_4 + x_3 x_6) \end{array} \] In the next section we show that at least two different values of \(c_{13}\) yield equations of a smooth curve with the correct automorphisms.

Checking the equations in `Magma`

We check that for two different values of \(c_{13}\), we obtain a a smooth curve with the correct automorphisms. This implies that a general member of the 1-parameter family is a smooth curve with the correct automorphisms. However, I do not know how to show that the two curves studied below are not isomorphic to each other; it is possible that we have described a point in \( \mathcal{M}_{g}\) rather than a curve in \( \mathcal{M}_g\). First we check the value \(c_{13}=1\):

Magma V2.21-10    Thu Mar 17 2016 14:12:10 on ace-math01 [Seed = 413449490]
Type ? for help.  Type -D to quit.
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6&rt;:=ProjectiveSpace(RationalField(),6);
> c_13:=1;
> X:=Scheme(P6,[
> x_0^2 + x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2,
> x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2+x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 \
+ x_6^2,
> (c_13^2+6)*(x_1^2 - 2*x_1*x_3 - x_2^2 + x_3^2)+c_13^2*(x_4^2 - 2*x_4*x_5 + 2\
*x_4*x_6 + x_5^2 - 2*x_5*x_6),
> (c_13^2+6)*(x_1*x_2 - 2*x_1*x_3 - 1/2*x_2^2 + x_2*x_3)+ c_13^2*(-2*x_4*x_5 +\
 x_4*x_6 - x_5*x_6 + 1/2*x_6^2),
> c_13*(x_0*x_4 + x_0*x_5 + x_0*x_6)+((c_13^2+6)/2)*(x_1^2 - x_3^2)+(c_13^2/2)\
*(x_4^2 + 2*x_4*x_6 - x_5^2 + x_6^2),
> c_13*(x_0*x_4 + x_0*x_5)+((c_13^2+6)/2)*(x_1*x_2 - x_2*x_3)+(c_13^2/2)*(x_4*\
x_6 - x_5*x_6 + x_6^2),
> c_13*(2*x_0*x_4 + x_0*x_6)+((c_13^2+6)/2)*(x_2^2 - 2*x_2*x_3)+(c_13^2/2)*(2*\
x_4*x_6 + x_6^2),
> x_0*x_1 + x_0*x_3+c_13*(x_1*x_5 - x_3*x_4 - x_3*x_6),
> 2*x_0*x_1 - x_0*x_2+c_13*(x_1*x_6 + x_2*x_5 - x_2*x_6),
> x_0*x_2 - 2*x_0*x_3+c_13*(x_2*x_4 + x_3*x_6)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> A:=Matrix([
> [-1, 0, 0, 0, 0, 0, 0],
> [ 0, 0, 1, 1, 0, 0, 0],
> [ 0, 0, -1, 0, 0, 0, 0],
> [ 0, 1, 1, 0, 0, 0, 0],
> [ 0, 0, 0, 0, -1, 1, 1],
> [ 0, 0, 0, 0, 0, 0, 1],
> [ 0, 0, 0, 0, 0, 1, 0]
> ]);
> B:=Matrix([
> [-1, 0, 0, 0, 0, 0, 0],
> [ 0, 1, 1, 0, 0, 0, 0],
> [ 0, 0, -1, 0, 0, 0, 0],
> [ 0, 0, 1, 1, 0, 0, 0],
> [ 0, 0, 0, 0, 0, 0, -1],
> [ 0, 0, 0, 0, 1, -1, -1],
> [ 0, 0, 0, 0, -1, 0, 0]
> ]);
> C:=Matrix([
> [-1, 0, 0, 0, 0, 0, 0],
> [ 0, 1, 0, 0, 0, 0, 0],
> [ 0, -1, -1, 0, 0, 0, 0],
> [ 0, 0, 0, -1, 0, 0, 0],
> [ 0, 0, 0, 0, -1, 0, 0],
> [ 0, 0, 0, 0, 0, 1, 0],
> [ 0, 0, 0, 0, 0, -1, -1]
> ]);
> Order(A);
2
> Order(B);
2
> Order(C);
2
> GL7Q:=GeneralLinearGroup(7,RationalField());
> IdentifyGroup(sub<GL7Q | A,B,C>);
<48, 48>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_3
x_1 - x_2 + x_3
x_1
-x_4
x_4 + x_6
x_4 + x_5
and inverse
-x_0
x_3
x_1 - x_2 + x_3
x_1
-x_4
x_4 + x_6
x_4 + x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_1
x_1 - x_2 + x_3
x_3
x_5 - x_6
-x_5
-x_4 - x_5
and inverse
-x_0
x_1
x_1 - x_2 + x_3
x_3
x_5 - x_6
-x_5
-x_4 - x_5
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_1 - x_2
-x_2
-x_3
-x_4
x_5 - x_6
-x_6
and inverse
-x_0
x_1 - x_2
-x_2
-x_3
-x_4
x_5 - x_6
-x_6

Next, we check the value \(c_{13}=5+\zeta_{6}\):


> K<z_6>:=CyclotomicField(6);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_13:=5+z_6;
> X:=Scheme(P6,[
> x_0^2 + x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2,
> x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2+x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 \
+ x_6^2,
> (c_13^2+6)*(x_1^2 - 2*x_1*x_3 - x_2^2 + x_3^2)+c_13^2*(x_4^2 - 2*x_4*x_5 + 2\
*x_4*x_6 + x_5^2 - 2*x_5*x_6),
> (c_13^2+6)*(x_1*x_2 - 2*x_1*x_3 - 1/2*x_2^2 + x_2*x_3)+ c_13^2*(-2*x_4*x_5 +\
 x_4*x_6 - x_5*x_6 + 1/2*x_6^2),
> c_13*(x_0*x_4 + x_0*x_5 + x_0*x_6)+((c_13^2+6)/2)*(x_1^2 - x_3^2)+(c_13^2/2)\
*(x_4^2 + 2*x_4*x_6 - x_5^2 + x_6^2),
> c_13*(x_0*x_4 + x_0*x_5)+((c_13^2+6)/2)*(x_1*x_2 - x_2*x_3)+(c_13^2/2)*(x_4*\
x_6 - x_5*x_6 + x_6^2),
> c_13*(2*x_0*x_4 + x_0*x_6)+((c_13^2+6)/2)*(x_2^2 - 2*x_2*x_3)+(c_13^2/2)*(2*\
x_4*x_6 + x_6^2),
> x_0*x_1 + x_0*x_3+c_13*(x_1*x_5 - x_3*x_4 - x_3*x_6),
> 2*x_0*x_1 - x_0*x_2+c_13*(x_1*x_6 + x_2*x_5 - x_2*x_6),
> x_0*x_2 - 2*x_0*x_3+c_13*(x_2*x_4 + x_3*x_6)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_3
x_1 - x_2 + x_3
x_1
-x_4
x_4 + x_6
x_4 + x_5
and inverse
-x_0
x_3
x_1 - x_2 + x_3
x_1
-x_4
x_4 + x_6
x_4 + x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_1
x_1 - x_2 + x_3
x_3
x_5 - x_6
-x_5
-x_4 - x_5
and inverse
-x_0
x_1
x_1 - x_2 + x_3
x_3
x_5 - x_6
-x_5
-x_4 - x_5
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_1 - x_2
-x_2
-x_3
-x_4
x_5 - x_6
-x_6
and inverse
-x_0
x_1 - x_2
-x_2
-x_3
-x_4
x_5 - x_6
-x_6

Computing the Betti table in `Macaulay2`

We use Macaulay2 to compute the Betti table of the ideal generated by these equations.

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

i1 : R=QQ[x_0..x_6];

i2 : c_13=1;

i3 : I=ideal {
     x_0^2 + x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2,
     x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2+x_4^2 + x_4*x_6 + x_5^2 - x_5*x_6 + x_6^2,
     (c_13^2+6)*(x_1^2 - 2*x_1*x_3 - x_2^2 + x_3^2)+c_13^2*(x_4^2 - 2*x_4*x_5 + 2*x_4*x_6 + x_5^2 - 2*x_5*x_6),
     (c_13^2+6)*(x_1*x_2 - 2*x_1*x_3 - 1/2*x_2^2 + x_2*x_3)+ c_13^2*(-2*x_4*x_5 + x_4*x_6 - x_5*x_6 + 1/2*x_6^2),
     c_13*(x_0*x_4 + x_0*x_5 + x_0*x_6)+((c_13^2+6)/2)*(x_1^2 - x_3^2)+(c_13^2/2)*(x_4^2 + 2*x_4*x_6 - x_5^2 + x_6^2),
     c_13*(x_0*x_4 + x_0*x_5)+((c_13^2+6)/2)*(x_1*x_2 - x_2*x_3)+(c_13^2/2)*(x_4*x_6 - x_5*x_6 + x_6^2),
     c_13*(2*x_0*x_4 + x_0*x_6)+((c_13^2+6)/2)*(x_2^2 - 2*x_2*x_3)+(c_13^2/2)*(2*x_4*x_6 + x_6^2),
     x_0*x_1 + x_0*x_3+c_13*(x_1*x_5 - x_3*x_4 - x_3*x_6),
     2*x_0*x_1 - x_0*x_2+c_13*(x_1*x_6 + x_2*x_5 - x_2*x_6),
     x_0*x_2 - 2*x_0*x_3+c_13*(x_2*x_4 + x_3*x_6)
     };

o3 : Ideal of R

i4 : betti res I

            0  1  2  3  4 5
o4 = total: 1 10 25 25 10 1
         0: 1  .  .  .  . .
         1: . 10 16  9  . .
         2: .  .  9 16 10 .
         3: .  .  .  .  . 1

o4 : BettiTally

By [Schreyer1986], this Betti table implies that this curve has a \(g_6^2\). We get the same Betti table with \(c_{13} = 5+\zeta_{6}\).

Finding equations of a 1-parameter family of genus 7 Riemann surfaces with 48 automorphisms

Obtaining candidate polynomials in Magma

Flattening stratification in Macaulay2

Checking the equations in Magma

Computing the Betti table in Macaulay2

Obtaining candidate polynomials in `Magma`

Flattening stratification in `Macaulay2`

Checking the equations in `Magma`

Computing the Betti table in `Macaulay2`