A 1-parameter family of genus 6 Riemann surfaces with automorphism group (24,8)

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 surface 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 (2,2,3,4). Moreover, this family contains the cyclic trigonal Riemann surface with automorphism group (72,15).

We use Magma to compute equations of these Riemann surfaces.

Obtaining candidate polynomials in `Magma`

Previously we found equations for the Riemann surface with automorphism group (72,15). Its cyclic trigonal equation was \( y^3 = (x^4-2 \sqrt{-3} x^2+1) (x^4+ 2 \sqrt{-3} x^2+1)^2\). We also found matrix generators for the action of this automorphism group on the holomorphic differentials.

We begin with the conjecture that the cyclic trigonal equation for this family is \( y^3 = (x^4-tx^2+1) (x^4+ t x^2+1)^2\). We find the subgroup of the group (72,15) that preserves this equation and check that it is the desired group.

First, we write the canonical ideal associated to this trigonal equation for a generic value of \(t\), and load the matrix generators from the group (72,15):

Magma V2.21-7     Fri May  6 2016 21:20:33 on Davids-MacBook-Pro-2 [Seed = 
3250312281]

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Type ? for help.  Type -D to quit.
> K<z_36>:=CyclotomicField(36);
> z_12:=z_36^3;
> z_9:=z_36^4;
> z_6:=z_36^6;
> z_3:=z_36^12;
> t:=17+z_36^5;
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> X:=Scheme(P5,[
> x_0*x_2-x_1^2,
> x_0*x_4-x_1*x_3,
> x_0*x_5-x_2*x_3,
> x_1*x_4-x_0*x_5,
> x_1*x_5-x_2*x_4,
> x_3*x_5-x_4^2,
> x_0^3+t*x_0^2*x_2+x_0*x_2^2+x_3^3-t*x_3^2*x_5+x_3*x_5^2,
> x_0^2*x_1+t*x_0*x_1*x_2+x_1*x_2^2+x_3^2*x_4-t*x_3*x_4*x_5+x_4*x_5^2,
> x_0^2*x_2+t*x_0*x_2^2+x_2^3+x_3^2*x_5-t*x_3*x_5^2+x_5^3
> ]);
> A:=Matrix([
> [0,  0,  0,  -z_9/2,  z_36^13/2,  z_9/2],
> [0,  0,  0,  -z_36^13,  0,  -z_36^13],
> [0,  0,  0,  z_9/2,  z_36^13/2,  -z_9/2],
> [z_36^14/2,  z_36^5/2,  -z_36^14/2,  0,  0,  0],
> [-z_36^5,  0,  -z_36^5,  0,  0,  0],
> [-z_36^14/2,  z_36^5/2,  z_36^14/2,  0,  0,  0]
> ]);
> B:=Matrix([
> [0,  0,  0,  z_12^5,  0,  0],
> [0,  0,  0,  0,  z_3^2,  0],
> [0,  0,  0,  0,  0,  -z_12^5],
> [z_12,  0,  0,  0,  0,  0],
> [0,  z_3,  0,  0,  0,  0],
> [0,  0,  -z_12,  0,  0,  0]
> ]);
> GL6K:=GeneralLinearGroup(6,K);
> G:=sub<GL6K | A,B>;

Next, we compute the subgroup of \(G\) that preserves \(X\):


> nm:=Inverse(NumberingMap(G));
> L:=[];
> for i:=1 to 72 do
for>   if Automorphism(P5,nm(i))(X) eq X then
for|if>     L:=Append(L,nm(i));
for|if>   end if;
for> end for;
> #L;
24
> H:=sub<GL6K | L>; 
> IdentifyGroup(H);
<24, 8>

We search \(H\) to find a set of surface kernel generators:


> load "autcv10e.txt";
> ASKG:=AllSurfaceKernelGenerators(H,[2,2,3,4]);
> ASKG[1];
[
    [0 0 0 0 0 z_36^9 - z_36^3]
    [0 0 0 0 z_36^6 0]
    [0 0 0 -z_36^9 + z_36^3 0 0]
    [0 0 z_36^3 0 0 0]
    [0 -z_36^6 + 1 0 0 0 0]
    [-z_36^3 0 0 0 0 0],

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

    [z_36^6 - 1          0          0          0          0          0]
    [         0 z_36^6 - 1          0          0          0          0]
    [         0          0 z_36^6 - 1          0          0          0]
    [         0          0          0    -z_36^6          0          0]
    [         0          0          0          0    -z_36^6          0]
    [         0          0          0          0          0    -z_36^6],

    [0 0 0 -z_36^3 0 0]
    [0 0 0 0 z_36^6 - 1 0]
    [0 0 0 0 0 z_36^3]
    [-z_36^9 + z_36^3 0 0 0 0 0]
    [0 -z_36^6 0 0 0 0]
    [0 0 z_36^9 - z_36^3 0 0 0]
]

Checking the equations in `Magma`

We check that for two values of \(t\), we obtain a 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 did not check that two members in this family 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 \(t=1\):

> K<z_12>:=CyclotomicField(12);
> z_6:=z_12^2;
> z_3:=z_12^4;
> t:=1;
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> X:=Scheme(P5,[
> x_0*x_2-x_1^2,
> x_0*x_4-x_1*x_3,
> x_0*x_5-x_2*x_3,
> x_1*x_4-x_0*x_5,
> x_1*x_5-x_2*x_4,
> x_3*x_5-x_4^2,
> x_0^3+t*x_0^2*x_2+x_0*x_2^2+x_3^3-t*x_3^2*x_5+x_3*x_5^2,
> x_0^2*x_1+t*x_0*x_1*x_2+x_1*x_2^2+x_3^2*x_4-t*x_3*x_4*x_5+x_4*x_5^2,
> x_0^2*x_2+t*x_0*x_2^2+x_2^3+x_3^2*x_5-t*x_3*x_5^2+x_5^3
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> A:=Matrix([
> [0, 0, 0, 0, 0, z_12^5],
> [0, 0, 0, 0, z_6, 0],
> [0, 0, 0, z_12^-1, 0, 0],
> [0, 0, z_12, 0, 0, 0],
> [0, -z_6+1, 0, 0, 0, 0],
> [-z_12, 0, 0, 0, 0, 0]
> ]);
> B:=Matrix([
> [0, 0, 1, 0, 0, 0],
> [0, -1, 0, 0, 0, 0],
> [1, 0, 0, 0, 0, 0],
> [0, 0, 0, 0, 0, 1],
> [0, 0, 0, 0, -1, 0],
> [0, 0, 0, 1, 0, 0]
> ]);
> C:=Matrix([
> [z_3, 0, 0, 0, 0, 0],
> [0, z_3, 0, 0, 0, 0],
> [0, 0, z_3, 0, 0, 0],
> [0, 0, 0, z_3^2, 0, 0],
> [0, 0, 0, 0, z_3^2, 0],
> [0, 0, 0, 0, 0, z_3^2]
> ]);
> Order(A);
2
> Order(B);
2
> Order(C);
3
> Order( (A*B*C)^(-1));
4
> GL6K:=GeneralLinearGroup(6,K);
> IdentifyGroup(sub<GL6K | A,B,C>);
<24, 8>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-z_12*x_5
(-z_12^2 + 1)*x_4
z_12*x_3
(-z_12^3 + z_12)*x_2
z_12^2*x_1
(z_12^3 - z_12)*x_0
and inverse
-z_12*x_5
(-z_12^2 + 1)*x_4
z_12*x_3
(-z_12^3 + z_12)*x_2
z_12^2*x_1
(z_12^3 - z_12)*x_0
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
x_2
-x_1
x_0
x_5
-x_4
x_3
and inverse
x_2
-x_1
x_0
x_5
-x_4
x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
(z_12^2 - 1)*x_0
(z_12^2 - 1)*x_1
(z_12^2 - 1)*x_2
-z_12^2*x_3
-z_12^2*x_4
-z_12^2*x_5
and inverse
-z_12^2*x_0
-z_12^2*x_1
-z_12^2*x_2
(z_12^2 - 1)*x_3
(z_12^2 - 1)*x_4
(z_12^2 - 1)*x_5

Next we check the value \(t=17+\zeta_{12}^5\):


> K<z_12>:=CyclotomicField(12);
> z_6:=z_12^2;
> z_3:=z_12^4;
> t:=17+z_12^5;
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> X:=Scheme(P5,[
> x_0*x_2-x_1^2,
> x_0*x_4-x_1*x_3,
> x_0*x_5-x_2*x_3,
> x_1*x_4-x_0*x_5,
> x_1*x_5-x_2*x_4,
> x_3*x_5-x_4^2,
> x_0^3+t*x_0^2*x_2+x_0*x_2^2+x_3^3-t*x_3^2*x_5+x_3*x_5^2,
> x_0^2*x_1+t*x_0*x_1*x_2+x_1*x_2^2+x_3^2*x_4-t*x_3*x_4*x_5+x_4*x_5^2,
> x_0^2*x_2+t*x_0*x_2^2+x_2^3+x_3^2*x_5-t*x_3*x_5^2+x_5^3
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> A:=Matrix([
> [0, 0, 0, 0, 0, z_12^5],
> [0, 0, 0, 0, z_6, 0],
> [0, 0, 0, z_12^-1, 0, 0],
> [0, 0, z_12, 0, 0, 0],
> [0, -z_6+1, 0, 0, 0, 0],
> [-z_12, 0, 0, 0, 0, 0]
> ]);
> B:=Matrix([
> [0, 0, 1, 0, 0, 0],
> [0, -1, 0, 0, 0, 0],
> [1, 0, 0, 0, 0, 0],
> [0, 0, 0, 0, 0, 1],
> [0, 0, 0, 0, -1, 0],
> [0, 0, 0, 1, 0, 0]
> ]);
> C:=Matrix([
> [z_3, 0, 0, 0, 0, 0],
> [0, z_3, 0, 0, 0, 0],
> [0, 0, z_3, 0, 0, 0],
> [0, 0, 0, z_3^2, 0, 0],
> [0, 0, 0, 0, z_3^2, 0],
> [0, 0, 0, 0, 0, z_3^2]
> ]);
> Order(A);
2
> Order(B);
2
> Order(C);
3
> Order( (A*B*C)^(-1));
4
> GL6K:=GeneralLinearGroup(6,K);
> IdentifyGroup(sub<GL6K | A,B,C>);
<24, 8>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-z_12*x_5
(-z_12^2 + 1)*x_4
z_12*x_3
(-z_12^3 + z_12)*x_2
z_12^2*x_1
(z_12^3 - z_12)*x_0
and inverse
-z_12*x_5
(-z_12^2 + 1)*x_4
z_12*x_3
(-z_12^3 + z_12)*x_2
z_12^2*x_1
(z_12^3 - z_12)*x_0
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
x_2
-x_1
x_0
x_5
-x_4
x_3
and inverse
x_2
-x_1
x_0
x_5
-x_4
x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
(z_12^2 - 1)*x_0
(z_12^2 - 1)*x_1
(z_12^2 - 1)*x_2
-z_12^2*x_3
-z_12^2*x_4
-z_12^2*x_5
and inverse
-z_12^2*x_0
-z_12^2*x_1
-z_12^2*x_2
(z_12^2 - 1)*x_3
(z_12^2 - 1)*x_4
(z_12^2 - 1)*x_5
>

A 1-parameter family of genus 6 Riemann surfaces with automorphism group (24,8)

Obtaining candidate polynomials in Magma

Checking the equations in Magma

Obtaining candidate polynomials in `Magma`

Checking the equations in `Magma`