Finding equations of a 1-parameter family of genus 5 Riemann surfaces with 24 automorphisms

Magaard, Shaska, Shpectorov, and Völklein give tables of 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 5 Riemann surfaces with automorphism group (24,8) in the GAP library of small groups. The quotient of any surface in this family by its automorphism group has genus zero, and the quotient morphism is branched over four points with ramification indices (2,2,2,6).

We use GAP and 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 written by David Swinarski during a visit to the University of Sydney in June/July 2011. Here is the file autcv10c.txt used below.

Magma V2.20-3     Sun Mar 20 2016 10:50:44 on Fordhamwinarski [Seed = 
4015456632]
Type ? for help.  Type -D to quit.
> load "autcv10c.txt";
Loading "autcv10c.txt"
> G:=SmallGroup(24,8);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,5,[2,2,2,6]);
Set seed to 0.


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


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


Explanation of Character Value Symbols
--------------------------------------

J = RootOfUnity(3)


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

[2]     Order 2       Length 1      
        Rep G.3

[3]     Order 2       Length 2      
        Rep G.2

[4]     Order 2       Length 6      
        Rep G.1

[5]     Order 3       Length 2      
        Rep G.4

[6]     Order 4       Length 6      
        Rep G.1 * G.2 * G.3

[7]     Order 6       Length 2      
        Rep G.2 * G.4^2

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

[9]     Order 6       Length 2      
        Rep G.3 * G.4


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

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

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

    [      1       0       0       0       0]
    [      0       0      -1       0       0]
    [      0       1      -1       0       0]
    [      0       0       0 z^4 - 1       0]
    [      0       0       0       0    -z^4]
]
Matrix Surface Kernel Generators:
[
    [-1  0  0  0  0]
    [ 0 -1  0  0  0]
    [ 0  0 -1  0  0]
    [ 0  0  0 -1  0]
    [ 0  0  0  0  1],

    [     -1       0       0       0       0]
    [      0       0      -1       0       0]
    [      0      -1       0       0       0]
    [      0       0       0       0 z^4 - 1]
    [      0       0       0    -z^4       0],

    [      -1        0        0        0        0]
    [       0        1        0        0        0]
    [       0        1       -1        0        0]
    [       0        0        0        0      z^4]
    [       0        0        0 -z^4 + 1        0],

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

The output above shows that the ideal contains quadrics from two isotypical subspaces of \(S_2\). Note that the power of z\(=\zeta_{24}\) in the surface kernel generators is always a multiple of 4. Therefore in the sequel we reduce these to z_6\(= \zeta_{6}\).

The first isotypical subspace, which corresponds to the character \( \chi_1\) in the character table shown above, yields a polynomial of the form \[ c_1(x_0^2) + c_2(x_1^2 + x_1 x_2 + x_2^2) \] Assume that \( c_1, c_2\) are nonzero. By scaling \(x_0\) and dividing, we may assume that \(c_1=c_2=1\).

The second isotypical subspace corresponds to the character \( \chi_{6}\) in the character table shown above. The matrix surface kernel generators are block diagonal with blocks of sizes \(1 \times 1\), \(2 \times 2\), and \(2 \times 2\). We use \(\operatorname{Span} \{ x_0 x_1, x_0 x_2\}\) as one copy of \(V_6\) and use the FindParallelBases to find two additional ordered bases so that the group \(G\) acts by the same matrices on all three ordered bases.


> GL5K:=Parent(MatrixGens[1]);
> MatrixG:=sub<GL5K | MatrixGens>;     
> FindParallelBases(MatrixG,[Q[2][1],Q[2][2]],[Q[2][3],Q[2][4]]);
[ x_1^2 - 2*x_1*x_2 - 2*x_2^2]
[-2*x_1^2 - 2*x_1*x_2 + x_2^2]
> FindParallelBases(MatrixG,[Q[2][1],Q[2][2]],[Q[2][5],Q[2][6]]);
[               x_3^2 + x_4^2]
[-z^4*x_3^2 + (z^4 - 1)*x_4^2]

This yields polynomials of the form


c_3*(x_0*x_1)+c_4*(x_1^2 - 2*x_1*x_2 - 2*x_2^2)+c_5*(x_3^2 + x_4^2),
c_3*(x_0*x_2)+c_4*(-2*x_1^2 - 2*x_1*x_2 + x_2^2)+c_5*(-z^4*x_3^2 + (z^4 - 1)*x_4^2)

Assume that \(c_3,c_5\) are nonzero. After scaling \(x_3,x_4\) and dividing, we may assume that \(c_3 = c_5=1\).

We are expecting to find a pencil of curves, and indeed we have one coefficient, \(c_4\), which is allowed to vary.

Checking the equations in `Magma`

We check that for two different values of \(c_{4}\), 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 \( \mathcal{M}_{5}\) rather than a curve in \( \mathcal{M}_5\).

First we study \(c_4=1\).

> K<z_6>:=CyclotomicField(6);
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> c_4:=1;
> X:=Scheme(P4,[
> x_0^2 + x_1^2 + x_1*x_2 + x_2^2,
> x_0*x_1+c_4*(x_1^2 - 2*x_1*x_2 - 2*x_2^2)+x_3^2 + x_4^2,
> x_0*x_2+c_4*(-2*x_1^2 - 2*x_1*x_2 + x_2^2)-z_6*x_3^2 + (z_6 - 1)*x_4^2
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> A:=Matrix([
> [-1,0,0,0,0],
> [0,-1,0,0,0],
> [0,0,-1,0,0],
> [0,0,0,-1,0],
> [0,0,0,0,1]
> ]);
> B:=Matrix([
> [-1,0,0,0,0],
> [0,0,-1,0,0],
> [0,-1,0,0,0],
> [0,0,0,0,z_6-1],
> [0,0,0,-z_6,0]
> ]);
> C:=Matrix([
> [-1,0,0,0,0],
> [0,1,0,0,0],
> [0,1,-1,0,0],
> [0,0,0,0,z_6],
> [0,0,0,-z_6+1,0]
> ]);
> Order(A);
2
> Order(B);
2
> Order(C);
2
> Order( (A*B*C)^(-1));
6
> GL5K:=GeneralLinearGroup(5,K);
> IdentifyGroup(sub<GL5K | A,B,C>);
<24, 8>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2
-x_3
x_4
and inverse
-x_0
-x_1
-x_2
-x_3
x_4
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_2
-x_1
-z_6*x_4
(z_6 - 1)*x_3
and inverse
-x_0
-x_2
-x_1
-z_6*x_4
(z_6 - 1)*x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_1 + x_2
-x_2
(-z_6 + 1)*x_4
z_6*x_3
and inverse
-x_0
x_1 + x_2
-x_2
(-z_6 + 1)*x_4
z_6*x_3

Next, we study \(c_4 = 15+\zeta_{24}^7\):

> K<z_24>:=CyclotomicField(24);
> z_6:=z_24^4;
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> c_4:=15+z_24^7;
> X:=Scheme(P4,[
> x_0^2 + x_1^2 + x_1*x_2 + x_2^2,
> x_0*x_1+c_4*(x_1^2 - 2*x_1*x_2 - 2*x_2^2)+x_3^2 + x_4^2,
> x_0*x_2+c_4*(-2*x_1^2 - 2*x_1*x_2 + x_2^2)-z_6*x_3^2 + (z_6 - 1)*x_4^2
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2
-x_3
x_4
and inverse
-x_0
-x_1
-x_2
-x_3
x_4
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_2
-x_1
-z_24^4*x_4
(z_24^4 - 1)*x_3
and inverse
-x_0
-x_2
-x_1
-z_24^4*x_4
(z_24^4 - 1)*x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_1 + x_2
-x_2
(-z_24^4 + 1)*x_4
z_24^4*x_3
and inverse
-x_0
x_1 + x_2
-x_2
(-z_24^4 + 1)*x_4
z_24^4*x_3

Finding equations of a 1-parameter family of genus 5 Riemann surfaces with 24 automorphisms

Obtaining candidate polynomials in Magma

Checking the equations in Magma

Obtaining candidate polynomials in `Magma`

Checking the equations in `Magma`