Fordham University

Finding equations for a genus 5 Riemann surface with 30 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 genus 5 Riemann surface with automorphism group (30,2) in the GAP library of small groups. The quotient of this surface by its automorphism group has genus zero, and the quotient morphism is branched over three points with ramification indices (2,6,15).

We use GAP and Magma to compute equations of this Riemann surface. The main tools are the Eichler trace formula, black-box commands in GAP and Magma for obtaining matrix generators of a representation of a finite group having a specified character.

Obtaining candidate polynomials in Magma and GAP

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 autcv10.txt used below. 
 Magma V2.21-7     Tue Mar 22 2016 15:32:06 on Davids-MacBook-Pro-2 [Seed = 
3362718502]

+-------------------------------------------------------------------+
|       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"
> G:=SmallGroup(30,2);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,5,[2,6,15]);
Set seed to 0.


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


------------------------------------------------------------------
Class |   1  2    3    4    5    6   7   8     9    10    11    12
Size  |   1  5    1    1    2    2   5   5     2     2     2     2
Order |   1  2    3    3    5    5   6   6    15    15    15    15
------------------------------------------------------------------
p  =  2   1  1    4    3    6    5   4   3    12    11    10     9
p  =  3   1  2    1    1    6    5   2   2     6     6     5     5
p  =  5   1  2    4    3    1    1   8   7     4     3     4     3
------------------------------------------------------------------
X.1   +   1  1    1    1    1    1   1   1     1     1     1     1
X.2   +   1 -1    1    1    1    1  -1  -1     1     1     1     1
X.3   0   1  1 -1-J    J    1    1-1-J   J  -1-J     J  -1-J     J
X.4   0   1  1    J -1-J    1    1   J-1-J     J  -1-J     J  -1-J
X.5   0   1 -1 -1-J    J    1    1 1+J  -J  -1-J     J  -1-J     J
X.6   0   1 -1    J -1-J    1    1  -J 1+J     J  -1-J     J  -1-J
X.7   +   2  0    2    2   Z1 Z1#2   0   0    Z1    Z1  Z1#2  Z1#2
X.8   +   2  0    2    2 Z1#2   Z1   0   0  Z1#2  Z1#2    Z1    Z1
X.9   0   2  0-2-2*J  2*J   Z1 Z1#2   0   0    Z2 Z2#11  Z2#7  Z2#2
X.10  0   2  0-2-2*J  2*J Z1#2   Z1   0   0  Z2#7  Z2#2    Z2 Z2#11
X.11  0   2  0  2*J-2-2*J Z1#2   Z1   0   0  Z2#2  Z2#7 Z2#11    Z2
X.12  0   2  0  2*J-2-2*J   Z1 Z1#2   0   0 Z2#11    Z2  Z2#2  Z2#7


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

# denotes algebraic conjugation, that is,
#k indicates replacing the root of unity w by w^k

J = RootOfUnity(3)

Z1     = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | 0, 0, 1, 1 ]

Z2     = (CyclotomicField(15: Sparse := true)) ! [ RationalField() | 0, 0, 0, 0,
-1, -1, -1, -1 ]


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

[2]     Order 2       Length 5      
        Rep G.1

[3]     Order 3       Length 1      
        Rep G.2^2

[4]     Order 3       Length 1      
        Rep G.2

[5]     Order 5       Length 2      
        Rep G.3^2

[6]     Order 5       Length 2      
        Rep G.3

[7]     Order 6       Length 5      
        Rep G.1 * G.2^2

[8]     Order 6       Length 5      
        Rep G.1 * G.2

[9]     Order 15      Length 2      
        Rep G.2^2 * G.3^2

[10]    Order 15      Length 2      
        Rep G.2 * G.3^2

[11]    Order 15      Length 2      
        Rep G.2^2 * G.3

[12]    Order 15      Length 2      
        Rep G.2 * G.3


Surface kernel generators:  [ G.1, G.1 * G.2 * G.3, G.2^2 * G.3^4 ]
Is hyperelliptic?  false
Is cyclic trigonal?  true
Multiplicities of irreducibles in relevant G-modules:
I_1      =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1      =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1 ]
I_2      =[ 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
S_2      =[ 0, 0, 2, 1, 0, 0, 1, 2, 1, 1, 0, 1 ]
H^0(C,2K)=[ 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1 ]
I_3      =[ 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1 ]
S_3      =[ 0, 2, 1, 1, 2, 1, 3, 3, 2, 1, 3, 2 ]
H^0(C,3K)=[ 0, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 1 ]
I2timesS1=[ 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 3, 1 ]
Is clearly not generated by quadrics? true
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 30 and degree 8
[
    [-1 0 0 0 0]
    [0 -z^7 + z^3 + z^2 z^7 - z^5 - z^4 - z^3 - z^2 + z + 1 0 0]
    [0 -z^5 + 1 z^7 - z^3 - z^2 0 0]
    [0 0 0 0 z^5 - 1]
    [0 0 0 -z^5 0],

    [   -z^5       0       0       0       0]
    [      0 z^5 - 1       0       0       0]
    [      0       0 z^5 - 1       0       0]
    [      0       0       0    -z^5       0]
    [      0       0       0       0    -z^5],

    [1 0 0 0 0]
    [0 0 -z^5 0 0]
    [0 -z^5 + 1 z^7 - z^3 - z^2 0 0]
    [0 0 0 -z^7 + z^3 + z^2 z^5 + z^4 - z - 1]
    [0 0 0 -z^7 + z^5 + z^4 + z^3 + z^2 - z - 1 -1]
]
Matrix Surface Kernel Generators:
[
    [-1 0 0 0 0]
    [0 -z^7 + z^3 + z^2 z^7 - z^5 - z^4 - z^3 - z^2 + z + 1 0 0]
    [0 -z^5 + 1 z^7 - z^3 - z^2 0 0]
    [0 0 0 0 z^5 - 1]
    [0 0 0 -z^5 0],

    [z^5 0 0 0 0]
    [0 -z^5 - z^4 + z + 1 -1 0 0]
    [0 z^7 - z^5 - z^4 - z^3 - z^2 + z + 1 z^5 + z^4 - z - 1 0 0]
    [0 0 0 -z^7 + z^5 + z^4 + z^3 + z^2 - z - 1 -1]
    [0 0 0 z^5 + z^4 - z - 1 z^7 - z^5 - z^4 - z^3 - z^2 + z + 1],

    [z^5 - 1 0 0 0 0]
    [0 -z^7 + z^5 + z^4 + z^3 + z^2 - z - 1 -z^5 + 1 0 0]
    [0 1 0 0 0]
    [0 0 0 -z^5 + 1 -z^7 + z^5 + z^4 + z^3 + z^2 - z - 1]
    [0 0 0 -z^7 + z^3 + z^2 z^5 + z^4 - z - 1]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 3
Dimension 2
Multiplicity 2
[
    x_0^2,
    x_3^2 + (-z^7 + z^5 + z^4 + z^3 + z^2 - z - 1)*x_3*x_4 + (z^5 - 1)*x_4^2
]
I2 contains a 2-dimensional subspace of CharacterRow 8
Dimension 4
Multiplicity 2
[
    x_0*x_1,
    x_0*x_2,
    x_1*x_3 - x_2*x_4,
    x_1*x_4 - z^5*x_2*x_3 + (-z^5 - z^4 + z + 1)*x_2*x_4
]
Finding cubics:
I3 contains a 1-dimensional subspace of CharacterRow 2
Dimension 2
Multiplicity 2
[
    x_0^3,
    x_0*x_3^2 + (-z^7 + z^5 + z^4 + z^3 + z^2 - z - 1)*x_0*x_3*x_4 + (z^5 - 
        1)*x_0*x_4^2
]
I3 contains a 1-dimensional subspace of CharacterRow 3
Dimension 1
Multiplicity 1
[
    x_1^2*x_3 + (-2*z^7 + 2*z^4 + 2*z^3 + 2*z^2 - 2*z - 2)*x_1^2*x_4 + (-4*z^4 +
        4*z)*x_1*x_2*x_3 + 2*x_1*x_2*x_4 - z^5*x_2^2*x_3 + (-z^5 + z^4 - z + 
        1)*x_2^2*x_4
]
I3 contains a 1-dimensional subspace of CharacterRow 5
Dimension 2
Multiplicity 2
[
    x_0*x_1^2 + (-z^5 - z^4 + z + 1)*x_0*x_1*x_2 - z^5*x_0*x_2^2,
    x_1^2*x_3 - 2*x_1*x_2*x_4 + z^5*x_2^2*x_3 + (z^5 + z^4 - z - 1)*x_2^2*x_4
]
I3 contains a 2-dimensional subspace of CharacterRow 7
Dimension 6
Multiplicity 3
[
    x_0^2*x_3,
    x_0^2*x_4,
    x_1^3 + 3*z^5*x_1*x_2^2 + (-z^7 + z^3 + z^2)*x_2^3,
    x_1^2*x_2 + (-z^5 - z^4 + z + 1)*x_1*x_2^2 + 1/3*(z^7 - z^5 - z^4 - z^3 - 
        z^2 + z + 1)*x_2^3,
    x_3^3 + (-z^5 - z^4 + z + 1)*x_3*x_4^2 + (-z^7 + z^3 + z^2)*x_4^3,
    x_3^2*x_4 + (-z^7 + z^5 + z^4 + z^3 + z^2 - z - 1)*x_3*x_4^2 + (z^5 - 
        1)*x_4^3
]
I3 contains a 2-dimensional subspace of CharacterRow 8
Dimension 6
Multiplicity 3
[
    x_0*x_3^2 + (-z^5 + 1)*x_0*x_4^2,
    x_0*x_3*x_4 + 1/2*(-z^7 + z^5 + z^4 + z^3 + z^2 - z - 1)*x_0*x_4^2,
    x_1^3 + (-z^7 + z^5 + z^4 + z^3 + z^2 - z - 1)*x_1*x_2^2 + (-z^7 + z^3 + 
        z^2)*x_2^3,
    x_1^2*x_2 + (-z^5 - z^4 + z + 1)*x_1*x_2^2 - z^5*x_2^3,
    x_3^3 + (-3*z^5 + 3)*x_3*x_4^2 + (-z^7 + z^3 + z^2)*x_4^3,
    x_3^2*x_4 + (-z^7 + z^5 + z^4 + z^3 + z^2 - z - 1)*x_3*x_4^2 + 1/3*(z^5 + 
        z^4 - z - 1)*x_4^3
]
I3 contains a 2-dimensional subspace of CharacterRow 9
Dimension 4
Multiplicity 2
[
    x_0*x_1^2 + z^5*x_0*x_2^2,
    x_0*x_1*x_2 + 1/2*(-z^5 - z^4 + z + 1)*x_0*x_2^2,
    x_1^2*x_3 + (-z^5 - z^4 + z + 1)*x_1*x_2*x_3 - z^5*x_2^2*x_3,
    x_1^2*x_4 + (-z^5 - z^4 + z + 1)*x_1*x_2*x_4 - z^5*x_2^2*x_4
]
I3 contains a 4-dimensional subspace of CharacterRow 11
Dimension 6
Multiplicity 3
[
    x_0^2*x_1,
    x_0^2*x_2,
    x_0*x_1*x_3 - x_0*x_2*x_4,
    x_0*x_1*x_4 - z^5*x_0*x_2*x_3 + (-z^5 - z^4 + z + 1)*x_0*x_2*x_4,
    x_1*x_3^2 + (-z^7 + z^5 + z^4 + z^3 + z^2 - z - 1)*x_1*x_3*x_4 + (z^5 - 
        1)*x_1*x_4^2,
    x_2*x_3^2 + (-z^7 + z^5 + z^4 + z^3 + z^2 - z - 1)*x_2*x_3*x_4 + (z^5 - 
        1)*x_2*x_4^2
]
I3 contains a 2-dimensional subspace of CharacterRow 12
Dimension 4
Multiplicity 2
[
    x_0*x_1*x_3 + (-z^5 - z^4 + z + 1)*x_0*x_2*x_3 + x_0*x_2*x_4,
    x_0*x_1*x_4 + z^5*x_0*x_2*x_3,
    x_1*x_3^2 + (-z^5 + 1)*x_1*x_4^2 - 2*x_2*x_3*x_4 + (z^7 - z^5 - z^4 - z^3 - 
        z^2 + z + 1)*x_2*x_4^2,
    x_1*x_3*x_4 + 1/2*(-z^7 + z^5 + z^4 + z^3 + z^2 - z - 1)*x_1*x_4^2 - 
        1/2*z^5*x_2*x_3^2 + (-z^5 - z^4 + z + 1)*x_2*x_3*x_4 + 1/2*(-z^7 + z^3 +
        z^2)*x_2*x_4^2
]

From this output, we see that this Riemann surface is cyclic trigonal. Thus, its canonical ideal is not generated by quadrics.

The first isotypical subspace of \(S_2\) corresponding to the character \( \chi_{3}\). This yields a polynomial of the form 


c_1*(x_0^2) + c_2*(x_3^2 + (-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_3*x_4 + (z_30^5 - 1)*x_4^2)
Assume that \(c_1,c_2\) are both nonzero. Then after scaling \(x_0\) and dividing, we may assume that \(c_1=c_2=1\).

The second isotypical subspace corresponds to the character \(\chi_4\). The surface kernel generators are block diagonal with blocks of sizes \(1 \times 1\), \(2 \times 2\), and \(2 \times \). Thus we use the first two polynomials shown here to generate one copy of \(V_{4}\) and use the FindParallelBases command to find a second ordered basis such that the group G acts by the same matrices on both 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*x_3 + 1/2*(-z^7 + z^5 + z^4 + z^3 + z^2 - z - 1)*x_1*x_4 + 1/2*(-z^5 - z^4 + z + 1)*x_2*x_3
    + 1/2*(-z^7 + z^3 + z^2 - 1)*x_2*x_4]
[1/2*(z^7 - z^5 - z^4 - z^3 - z^2 + z + 1)*x_1*x_3 + (-z^5 + 1)*x_1*x_4 - x_2*x_3 + 1/2*(z^7 - 
    z^5 - z^4 - z^3 - z^2 + z + 1)*x_2*x_4]
This yields polynomials of the form 

c_3*(x_0*x_1)+c_4*(x_1*x_3 + 1/2*(-z^7 + z^5 + z^4 + z^3 + z^2 - z -
1)*x_1*x_4 + 1/2*(-z^5 - z^4 + z + 1)*x_2*x_3 + 1/2*(-z^7 + z^3 + z^2
- 1)*x_2*x_4)
c_3*(x_0*x_2) + c_4*(1/2*(z^7 - z^5 - z^4 - z^3 - z^2 + z + 1)*x_1*x_3 + (-z^5 + 1)*x_1*x_4 -     x_2*x_3 + 1/2*(z^7 - z^5 - z^4 - z^3 - z^2 + z + 1)*x_2*x_4)
Assume that \(c_3\) is nonzero. After dividing by \(c_3\) , we may assume that \(c_3=1\). By comparing the characters for \(I_3\) and \(I_2 \otimes S_1\) we see that the cubic generators needed to define this trigonal Riemann surface come from the isotypical subspace of \(S_3\) with character 8. Again, we find three ordered bases such that the action of G is given by the same matrices with respect to all three ordered bases: 

> FindParallelBases(MatrixG,[C[5][1],C[5][2]],[C[5][3],C[5][4]]);
[x_1^3 + (-2*z^5 + 2)*x_1^2*x_2 + (z^7 - z^5 - z^4 - z^3 - z^2 + z + 1)*x_1*x_2^2 + (-z^7 + z^3 
    + z^2 - 2)*x_2^3]
[1/2*(z^5 - z^4 + z - 1)*x_1^3 + 1/2*(-2*z^7 - z^5 + 2*z^4 + 2*z^3 + 2*z^2 - 2*z - 2)*x_1^2*x_2 
    + 1/2*(2*z^7 - 2*z^3 - 2*z^2 + 3)*x_1*x_2^2 + 1/2*(z^5 - z^4 + z - 1)*x_2^3]
> FindParallelBases(MatrixG,[C[5][1],C[5][2]],[C[5][5],C[5][6]]);
[x_3^3 + (3*z^7 + 3*z^5 - 3*z^4 - 3*z^3 - 3*z^2 + 3*z + 3)*x_3^2*x_4 + (-3*z^5 + 3*z^4 - 3*z + 
    3)*x_3*x_4^2 + x_4^3]
[(z^4 - z)*x_3^3 - 3/2*x_3^2*x_4 + 1/2*(-3*z^7 - 3*z^5 + 3*z^4 + 3*z^3 + 3*z^2 - 3*z - 
    3)*x_3*x_4^2 + 1/2*(z^5 - z^4 + z - 1)*x_4^3]

This yields the polynomials 

c_5*(x_0*x_3^2 + (-z_30^5 + 1)*x_0*x_4^2)+c_6*(x_1^3 + (-2*z_30^5 + 2)*x_1^2*x_2 + (z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_1*x_2^2 + (-z_30^7 + z_30^3 + z_30^2 - 2)*x_2^3)+c_7*(x_3^3 + (3*z_30^7 + 3*z_30^5 - 3*z_30^4 - 3*z_30^3 - 3*z_30^2 + 3*z_30 + 3)*x_3^2*x_4 + (-3*z_30^5 +3*z_30^4 - 3*z_30 + 3)*x_3*x_4^2 + x_4^3),
c_5*(x_0*x_3*x_4 + 1/2*(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_0*x_4^2)+c_6*(1/2*(z_30^5 - z_30^4 + z_30 - 1)*x_1^3 + 1/2*(-2*z_30^7 - z_30^5 + 2*z_30^4 + 2*z_30^3 + 2*z_30^2 - 2*z_30- 2)*x_1^2*x_2 + 1/2*(2*z_30^7 - 2*z_30^3 - 2*z_30^2 + 3)*x_1*x_2^2 + 1/2*(z_30^5 - z_30^4 + z_30 - 1)*x_2^3)+c_7*((z_30^4 - z_30)*x_3^3 - 3/2*x_3^2*x_4 + 1/2*(-3*z_30^7 - 3*z_30^5 + 3*z_30^4 + 3*z_30^3 + 3*z_30^2 -3*z_30 - 3)*x_3*x_4^2 + 1/2*(z_30^5 - z_30^4 + z_30 - 1)*x_4^3)
Assume that \(c_5\) and \(c_6\) are nonzero. Then after scaling \(x_1,x_2\) and dividing, we may assume that \(c_5=c_6=1\).

For generic values of \(c_4,c_7\), the intersection of these 5 polynomials in \(\mathbb{P}^4\) is zero-dimensional. Here is an example showing this: 

> K<z_30>:=CyclotomicField(30);
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> c_1:=1;
> c_2:=1;
> c_3:=1;
> c_4 := 1;
> c_5:=1;
> c_6:=1;
> c_7:=1;
> X:=Scheme(P4,[
> c_1*(x_0^2) + c_2*(x_3^2 + (-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_3*x_4 + (z_30^5 - 1)*x_4^2),
> c_3*(x_0*x_1)+c_4*(x_1*x_3 + 1/2*(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 -1)*x_1*x_4 + 1/2*(-z_30^5 - z_30^4 + z_30 + 1)*\
x_2*x_3 + 1/2*(-z_30^7 + z_30^3 + z_30^2- 1)*x_2*x_4),
> c_3*(x_0*x_2) + c_4*(1/2*(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_1*x_3 + (-z_30^5 + 1)*x_1*x_4 -     x_2*x_3 + 1/2*\
(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_2*x_4),
> c_5*(x_0*x_3^2 + (-z_30^5 + 1)*x_0*x_4^2)+c_6*(x_1^3 + (-2*z_30^5 + 2)*x_1^2*x_2 + (z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30\
 + 1)*x_1*x_2^2 + (-z_30^7 + z_30^3 + z_30^2 - 2)*x_2^3)+c_7*(x_3^3 + (3*z_30^7 + 3*z_30^5 - 3*z_30^4 - 3*z_30^3 - 3*z_30^2 + 3*z_30 + \
3)*x_3^2*x_4 + (-3*z_30^5 +3*z_30^4 - 3*z_30 + 3)*x_3*x_4^2 + x_4^3),
> c_5*(x_0*x_3*x_4 + 1/2*(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_0*x_4^2)+c_6*(1/2*(z_30^5 - z_30^4 + z_30 - 1)*x_1^\
3 + 1/2*(-2*z_30^7 - z_30^5 + 2*z_30^4 + 2*z_30^3 + 2*z_30^2 - 2*z_30- 2)*x_1^2*x_2 + 1/2*(2*z_30^7 - 2*z_30^3 - 2*z_30^2 + 3)*x_1*x_2^\
2 + 1/2*(z_30^5 - z_30^4 + z_30 - 1)*x_2^3)+c_7*((z_30^4 - z_30)*x_3^3 - 3/2*x_3^2*x_4 + 1/2*(-3*z_30^7 - 3*z_30^5 + 3*z_30^4 + 3*z_30^\
3 + 3*z_30^2 -3*z_30 - 3)*x_3*x_4^2 + 1/2*(z_30^5 - z_30^4 + z_30 - 1)*x_4^3)
> ]);
> Dimension(X);
0
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 have already determined that this genus 5 Riemann surface is trigonal, and therefore, the quadrics in its canonical ideal define a surface. Thus, we begin by finding a value of \(c_4\) so that the quadrics will yield a surface. We compute the leading coefficients of 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 : loadPackage("Cyclotomic")

o1 = Cyclotomic

o1 : Package

i2 : K=cyclotomicField(30);

i3 : z_30=K_0;

i4 : S=K[c_4,Degrees=>{0}];

i5 : c_1=1;

i6 : c_2=1;

i7 : c_3=1;

i8 : T=S[x_0..x_4];

i9 : I=ideal({
     c_1*(x_0^2) + c_2*(x_3^2 + (-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_3*x_4 + (z_30^5 - 1)*x_4^2),
     c_3*(x_0*x_1)+c_4*(x_1*x_3 + 1/2*(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 -1)*x_1*x_4 + 1/2*(-z_30^5 - z_30^4 + z_30 + 1)*x_2*x_3 + 1/2*(-z_30^7 + z_30^3 + z_30^2- 1)*x_2*x_4),
     c_3*(x_0*x_2) + c_4*(1/2*(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_1*x_3 + (-z_30^5 + 1)*x_1*x_4 -     x_2*x_3 + 1/2*(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_2*x_4)});

o9 : Ideal of T

i10 : L=flatten entries gens gb(I);

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

i12 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
1
c_4^2-(4/5)*ww_30^7+(4/5)*ww_30^3+(4/5)*ww_30^2+8/5
c_4
c_4^3+(-(4/5)*ww_30^7+(4/5)*ww_30^3+(4/5)*ww_30^2+8/5)*c_4
This suggests setting \( c_4^2-(4/5) z_{30}^7+(4/5) z_{30}^3+(4/5) z_{30}^2+8/5 =0\). We use Magma to compute this square root: 

> K<z_30>:=CyclotomicField(30);
> R<c_4>:=PolynomialRing(K);
> ww_30:=z_30;                                                       
> Factorization(c_4^2-(4/5)*ww_30^7+(4/5)*ww_30^3+(4/5)*ww_30^2+8/5);
[
    <c_4 + 1/5*(-6*z_30^7 - 4*z_30^6 - 2*z_30^3 + 6*z_30^2 - 2), 1>,
    <c_4 + 1/5*(6*z_30^7 + 4*z_30^6 + 2*z_30^3 - 6*z_30^2 + 2), 1>
]
Thus, we obtain \( c_4 = -1/5 (-6 z_{30}^7 - 4 z_{30}^6 - 2 z_{30} ^3 +6 z_{30}^2 - 2)\). We check that this yields a surface: 

> c_1:=1;
> c_2:=1;
> c_3:=1;
> c_4 := -1/5*(-6*z_30^7 - 4*z_30^6 - 2*z_30^3 + 6*z_30^2 - 2);
> Y:=Scheme(P4,[
> c_1*(x_0^2) + c_2*(x_3^2 + (-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_3*x_4 + (z_30^5 - 1)*x_4^2),
> c_3*(x_0*x_1)+c_4*(x_1*x_3 + 1/2*(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 -1)*x_1*x_4 + 1/2*(-z_30^5 - z_\
30^4 + z_30 + 1)*x_2*x_3 + 1/2*(-z_30^7 + z_30^3 + z_30^2- 1)*x_2*x_4),
> c_3*(x_0*x_2) + c_4*(1/2*(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_1*x_3 + (-z_30^5 + 1)*x_1*x_4 -  \
   x_2*x_3 + 1/2*(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_2*x_4)
> ]);
> Dimension(Y);
2
> IsSingular(Y);
false
For generic values of \(c_7\), the ideal defined above (with this value of \(c_4\)) is still zero-dimensional. Therefore, we next find a value of \(c_7\) so that the polynomials above define a curve. 

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

i1 : loadPackage("Cyclotomic")

o1 = Cyclotomic

o1 : Package

i2 : K=cyclotomicField(30);

i3 : z_30=K_0;

i4 : S=K[c_7,Degrees=>{0}];

i5 : c_1=1;

i6 : c_2=1;

i7 : c_3=1;

i8 : c_4 = -1/5*(-6*z_30^7 - 4*z_30^6 - 2*z_30^3 + 6*z_30^2 - 2);

i9 : c_5=1;

i10 : c_6=1;

i11 : T=S[x_0..x_4];

i12 : I=ideal({
      c_1*(x_0^2) + c_2*(x_3^2 + (-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_3*x_4 + (z_30^5 - 1)*x_4^2),
      c_3*(x_0*x_1)+c_4*(x_1*x_3 + 1/2*(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 -1)*x_1*x_4 + 1/2*(-z_30^5 - z_30^4 + z_30 + 1)*x_2*x_3 + 1/2*(-z_30^7 + z_30^3 + z_30^2- 1)*x_2*x_4),
      c_3*(x_0*x_2) + c_4*(1/2*(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 +
      z_30 + 1)*x_1*x_3 + (-z_30^5 + 1)*x_1*x_4 -     x_2*x_3 + 1/2*(z_30^7
      - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_2*x_4),
      c_5*(x_0*x_3^2 + (-z_30^5 + 1)*x_0*x_4^2)+c_6*(x_1^3 + (-2*z_30^5 + 2)*x_1^2*x_2 + (z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_1*x_2^2 + (-z_30^7 + z_30^3 + z_30^2 - 2)*x_2^3)+c_7*(x_3^3 + (3*z_30^7 + 3*z_30^5 - 3*z_30^4 - 3*z_30^3 - 3*z_30^2 + 3*z_30 + 3)*x_3^2*x_4 + (-3*z_30^5 +3*z_30^4 - 3*z_30 + 3)*x_3*x_4^2 + x_4^3),
      c_5*(x_0*x_3*x_4 + 1/2*(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_0*x_4^2)+c_6*(1/2*(z_30^5 - z_30^4 + z_30 - 1)*x_1^3 + 1/2*(-2*z_30^7 - z_30^5 + 2*z_30^4 + 2*z_30^3 + 2*z_30^2 - 2*z_30- 2)*x_1^2*x_2 + 1/2*(2*z_30^7 - 2*z_30^3 - 2*z_30^2 + 3)*x_1*x_2^2 + 1/2*(z_30^5 - z_30^4 + z_30 - 1)*x_2^3)+c_7*((z_30^4 - z_30)*x_3^3 - 3/2*x_3^2*x_4 + 1/2*(-3*z_30^7 - 3*z_30^5 + 3*z_30^4 + 3*z_30^3 + 3*z_30^2 -3*z_30 - 3)*x_3*x_4^2 + 1/2*(z_30^5 - z_30^4 + z_30 - 1)*x_4^3)});

o12 : Ideal of T

i13 : L=flatten entries gens gb(I);

i14 : Lc=unique apply(L, i -> leadCoefficient i);

i15 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
1
c_7+(4/5)*ww_30^7+(6/5)*ww_30^6-(2/5)*ww_30^3-(4/5)*ww_30^2+3/5
c_7^2+((4/5)*ww_30^7+(6/5)*ww_30^6-(2/5)*ww_30^3-(4/5)*ww_30^2+3/5)*c_7
This suggests setting \(c_7+(4/5) ww_{30}^7+(6/5) ww_{30}^6-(2/5) ww_{30}^3-(4/5) ww_{30}^2+3/5 = 0\). In the next section we show that these values of \(c_{4}\) and \(c_{7}\) yield equations of a smooth curve with the correct automorphisms.

Checking it all in Magma 

> K<z_30>:=CyclotomicField(30);
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> c_1:=1;
> c_2:=1;
> c_3:=1;
> c_4 := -1/5*(-6*z_30^7 - 4*z_30^6 - 2*z_30^3 + 6*z_30^2 - 2);
> c_5:=1;
> c_6:=1;
> c_7:=-((4/5)*z_30^7+(6/5)*z_30^6-(2/5)*z_30^3-(4/5)*z_30^2+3/5);
> X:=Scheme(P4,[
> c_1*(x_0^2) + c_2*(x_3^2 + (-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_3*x_4 + (z_30^5 - 1)*x_4^2),
> c_3*(x_0*x_1)+c_4*(x_1*x_3 + 1/2*(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 -1)*x_1*x_4 + 1/2*(-z_30^5 - z_\
30^4 + z_30 + 1)*x_2*x_3 + 1/2*(-z_30^7 + z_30^3 + z_30^2- 1)*x_2*x_4),
> c_3*(x_0*x_2) + c_4*(1/2*(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_1*x_3 + (-z_30^5 + 1)*x_1*x_4 -  \
   x_2*x_3 + 1/2*(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_2*x_4),
> c_5*(x_0*x_3^2 + (-z_30^5 + 1)*x_0*x_4^2)+c_6*(x_1^3 + (-2*z_30^5 + 2)*x_1^2*x_2 + (z_30^7 - z_30^5 - z_30^4 - z_30^\
3 - z_30^2 + z_30 + 1)*x_1*x_2^2 + (-z_30^7 + z_30^3 + z_30^2 - 2)*x_2^3)+c_7*(x_3^3 + (3*z_30^7 + 3*z_30^5 - 3*z_30^4\
 - 3*z_30^3 - 3*z_30^2 + 3*z_30 + 3)*x_3^2*x_4 + (-3*z_30^5 +3*z_30^4 - 3*z_30 + 3)*x_3*x_4^2 + x_4^3),
> c_5*(x_0*x_3*x_4 + 1/2*(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_0*x_4^2)+c_6*(1/2*(z_30^5 - z_30^4\
 + z_30 - 1)*x_1^3 + 1/2*(-2*z_30^7 - z_30^5 + 2*z_30^4 + 2*z_30^3 + 2*z_30^2 - 2*z_30- 2)*x_1^2*x_2 + 1/2*(2*z_30^7 -\
 2*z_30^3 - 2*z_30^2 + 3)*x_1*x_2^2 + 1/2*(z_30^5 - z_30^4 + z_30 - 1)*x_2^3)+c_7*((z_30^4 - z_30)*x_3^3 - 3/2*x_3^2*x\
_4 + 1/2*(-3*z_30^7 - 3*z_30^5 + 3*z_30^4 + 3*z_30^3 + 3*z_30^2 -3*z_30 - 3)*x_3*x_4^2 + 1/2*(z_30^5 - z_30^4 + z_30 -\
 1)*x_4^3)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> A:=Matrix([
> [-1, 0, 0, 0, 0],
> [0, -z_30^7 + z_30^3 + z_30^2, z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1, 0, 0],
> [0, -z_30^5 + 1, z_30^7 - z_30^3 - z_30^2, 0, 0],
> [0, 0, 0, 0, z_30^5 - 1],
> [0, 0, 0, -z_30^5, 0]
> ]);
> B:=Matrix([
> [z_30^5, 0, 0, 0, 0],
> [0, -z_30^5 - z_30^4 + z_30 + 1, -1, 0, 0],
> [0, z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1, z_30^5 + z_30^4 - z_30 - 1, 0, 0],
> [0, 0, 0, -z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1, -1],
> [0, 0, 0, z_30^5 + z_30^4 - z_30 - 1, z_30^7 - z_30^5 - z_30^4 -z_30^3 - z_30^2 + z_30 + 1]
> ]);
> Order(A);
2
> Order(B);
6
> Order( (A*B)^(-1));
15
> GL5K:=GeneralLinearGroup(5,K);
> IdentifyGroup(sub<GL5K | A,B>);
<30, 2>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
(-z_30^7 + z_30^3 + z_30^2)*x_1 + (-z_30^5 + 1)*x_2
(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_1 + (z_30^7 - z_30^3 - z_30^2)*x_2
-z_30^5*x_4
(z_30^5 - 1)*x_3
and inverse
-x_0
(-z_30^7 + z_30^3 + z_30^2)*x_1 + (-z_30^5 + 1)*x_2
(z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_1 + (z_30^7 - z_30^3 - z_30^2)*x_2
-z_30^5*x_4
(z_30^5 - 1)*x_3
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
z_30^5*x_0
(-z_30^5 - z_30^4 + z_30 + 1)*x_1 + (z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_2
-x_1 + (z_30^5 + z_30^4 - z_30 - 1)*x_2
(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_3 + (z_30^5 + z_30^4 - z_30 - 1)*x_4
-x_3 + (z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_4
and inverse
(-z_30^5 + 1)*x_0
(-z_30^7 + z_30^5 + z_30^4 + z_30^3 + z_30^2 - z_30 - 1)*x_1 + (-z_30^7 + z_30^3 + z_30^2)*x_2
(-z_30^5 + 1)*x_1 + (z_30^7 - z_30^5 - z_30^4 - z_30^3 - z_30^2 + z_30 + 1)*x_2
(-z_30^5 - z_30^4 + z_30 + 1)*x_3 + (-z_30^7 + z_30^3 + z_30^2)*x_4
z_30^5*x_3 + (z_30^5 + z_30^4 - z_30 - 1)*x_4

Cyclic trigonal approach

We begin by looking for a trigonal morphism. From the list of conjugacy classes above, we know that the order 3 elements belong to classes 3 and 4. We compute the number of fixed points of these group elements: 

> CCL:=Classes(G);
> SKG:=[ G.1, G.1 * G.2 * G.3, G.2^2 * G.3^4 ];
> NumberOfFixedPoints(G,SKG,CCL[3][3]);
7
> NumberOfFixedPoints(G,SKG,CCL[4][3]);
7
> IdentifyGroup(quo<G | CCL[3][3]>);
<10, 1>
> IdentifyGroup(quo<G | CCL[4][3]>);
<10, 1>
> IdentifyGroup(DihedralGroup(5));
<10, 1>
Thus, either of these classes yields a trigonal morphism. The quotient group in both cases is \(D_5\). One locus of seven points on the sphere with symmetry \(D_5\) is given by a bipyramid over a pentagon. This suggests either the trigonal equation \( y^3 = x(x^5-1) \) or \( y^3 = x^2(x^5-1) \). We compute the eigenvalues of the trigonal automorphism from class 3: 

> Eigenvalues(MatrixGens[2]^2);
{
    <-z^5, 2>,
    
}
The trigonal morphism corresponding to class three acts with eigenvalue \(\zeta_3\) on a three-dimensional subspace of the holomorphic differentials, and with eigenvalue \(\zeta_3^2\) on a two-dimensional subspace of the holomorphic differentials. Thus, in the notation of [AchterPries2007] we have \(r = 3, s=2\), so \(d_1 = 5, d_2 = 2\), and a basis of holomorphic differentials for this equation is given by \[ \{y \frac{dx}{y^2}, yx\frac{dx}{y^2}, g(x) \frac{dx}{y^2}, xg(x) \frac{dx}{y^2},x^2g(x) \frac{dx}{y^2} \} \] We have good evidence that the trigonal equation is \( y^3 = x^2(x^5-1) \). The canonical ideal associated to this trigonal equation is \[ \begin{array}{c} x_0 x_3-x_1 x_2,\\ x_0 x_4-x_1x_3,\\ x_2 x_4-x_3^2,\\ x_0^2 x_1-x_3x_4^2+x_2^3,\\ x_0 x_1^2-x_4^3+x_2^2 x_3 \end{array} \] We compute the action of the automorphisms on the differentials to verify this. The rotation of the sphere \( x \mapsto \zeta_5 x \) preserves \( y^3 = x^2(x^5-1) \) if \( y \mapsto \zeta_5^{-1} y\). Then \( \frac{dx}{y^2} \mapsto \frac{\zeta_5}{\zeta_5^{-2}} \frac{dx}{y^2} = \zeta_5^{3} \frac{dx}{y^2}.\) A matrix of this automorphism with respect to the basis \[ \{y \frac{dx}{y^2}, yx\frac{dx}{y^2}, x \frac{dx}{y^2}, x^2 \frac{dx}{y^2},x^3 \frac{dx}{y^2} \} \] is \[ \left[ \begin{array}{rrrrr} \zeta_5^2 & 0 & 0 & 0 & 0 \\ 0 & \zeta_5^3 & 0 & 0 & 0 \\ 0 & 0 & \zeta_5^4 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & \zeta_5 \\ \end{array} \right]. \]

The automorphism \( x \mapsto \frac{1}{x} \) interchanging the north and south poles preserves \( y^3 = x^2(x^5-1)\) if \(y \mapsto -y\). A matrix of this automorphism is \[ \left[ \begin{array}{rrrrr} 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{array} \right]. \] Finally, the trigonal automorphism \( y \mapsto \zeta_3 y\) acts via the matrix \[ \left[ \begin{array}{rrrrr} \zeta_3 & 0 & 0 & 0 & 0 \\ 0 & \zeta_3 & 0 & 0 & 0 \\ 0 & 0 & \zeta_3^2 & 0 & 0 \\ 0 & 0 & 0 & \zeta_3^2 & 0 \\ 0 & 0 & 0 & 0 & \zeta_3^2 \end{array} \right]. \] These three matrices generate the group (30,2). 


> K<z_15>:=CyclotomicField(15);
> z_5:=z_15^3;
> z_3:=z_15^5;
> A:=DiagonalMatrix([z_5^2,z_5^3,z_5^4,z_5^5,z_5]);
> B:=Matrix([
> [0,-1,0,0,0],
> [-1,0,0,0,0],
> [0,0,0,0,1],
> [0,0,0,1,0],
> [0,0,1,0,0]
> ]);
> C:=DiagonalMatrix([z_3,z_3,z_3^2,z_3^2,z_3^2]);
> GL5K:=GeneralLinearGroup(5,K);
> IdentifyGroup(sub<GL5K | A,B,C>);
<30, 2>
Finally, we search inside this matrix group to find surface kernel generators.